Subgroups — C₇×#25984 #13203200

Order 1792.

876 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3084	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3083	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2897	C₂	1	1	yes	maximal, characteristic
4	C₇×#2897	C₂	1	1	yes	maximal, characteristic
5	C₇×#2092	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2063	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2063	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2027	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1866	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1866	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1860	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1860	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1816	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1815	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1792	C₂	1	1	yes	maximal, characteristic
16	C₂×C₇×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
17	C₂×C₇×#131	C₂²	1	1	yes	characteristic
18	C₂×C₇×#131	C₂²	1	1	yes	characteristic
19	C₇×#544	C₂²	1	1	yes	characteristic, complemented
20	C₇×#520	C₂²	1	1	yes	characteristic
21	C₇×#520	C₂²	1	1	yes	characteristic
22	C₇×#479	—	2	1	no	—
23	C₇×#479	—	2	1	no	—
24	C₇×#478	—	2	1	no	—
25	C₇×#478	—	2	1	no	—
26	C₇×#477	—	2	1	no	—
27	C₇×#476	—	2	1	no	—
28	C₇×#475	—	2	1	no	—
29	C₇×#474	—	2	1	no	—
30	C₂×C₇×#108	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₇×#108	C₂²	1	1	yes	characteristic, complemented
32	C₇×#428	C₂²	1	1	yes	characteristic, complemented
33	C₇×#428	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₇×#107	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₇×#106	C₂²	1	1	yes	characteristic, complemented
36	C₇×#424	C₂²	1	1	yes	characteristic, complemented
37	C₇×#424	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₇×#104	C₂²	1	1	yes	characteristic
39	C₂×C₇×#104	C₂²	1	1	yes	characteristic
40	C₇×#418	—	2	1	no	—
41	C₇×#418	—	2	1	no	—
42	C₇×#418	C₂²	1	1	yes	characteristic
43	C₇×#418	C₂²	1	1	yes	characteristic
44	C₇×#417	—	2	1	no	—
45	C₇×#417	—	2	1	no	—
46	C₇×#417	C₂²	1	1	yes	characteristic
47	C₇×#417	C₂²	1	1	yes	characteristic
48	C₇×#417	C₂²	1	1	yes	characteristic
49	C₇×#417	C₂²	1	1	yes	characteristic
50	C₇×#416	—	2	1	no	—
51	C₇×#416	—	2	1	no	—
52	C₇×#416	C₂²	1	1	yes	characteristic
53	C₇×#416	C₂²	1	1	yes	characteristic
54	C₂×C₇×#102	—	2	1	no	—
55	C₂×C₇×#102	—	2	1	no	—
56	C₂×C₇×#102	C₂²	1	1	yes	characteristic
57	C₂×C₇×#102	C₂²	1	1	yes	characteristic
58	C₂×C₇×#100	C₂²	1	1	yes	characteristic
59	C₇×#390	C₂²	1	1	yes	characteristic
60	C₇×#382	C₂²	1	1	yes	characteristic, complemented
61	C₇×#343	C₂²	1	1	yes	characteristic, complemented
62	C₇×#343	C₂²	1	1	yes	characteristic, complemented
63	C₇×#343	C₂²	1	1	yes	characteristic, complemented
64	C₇×#339	C₂²	1	1	yes	characteristic, complemented
65	C₇×#339	C₂²	1	1	yes	characteristic, complemented
66	C₇×#336	C₂²	1	1	yes	characteristic, complemented
67	#25984	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₇×H(5,𝔽₂)	—	2	2	no	—
69	C₇×H(5,𝔽₂)	—	2	2	no	—
70	C₇×H(5,𝔽₂)	—	4	1	no	—
71	C₇×H(5,𝔽₂)	Q₈	1	4	yes	complemented
72	C₂×C₇×#41	—	2	1	no	—
73	C₂×C₇×#41	—	2	1	no	—
74	C₂×C₇×#41	C₂³	1	1	yes	characteristic
75	C₂×C₇×#41	C₂³	1	1	yes	characteristic
76	C₂²×C₇×D₄	—	2	1	no	—
77	C₂²×C₇×D₄	—	2	1	no	—
78	C₂²×C₇×D₄	—	4	1	no	—
79	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
80	C₇×#131	—	2	2	no	—
81	C₇×#131	—	2	2	no	—
82	C₇×#131	—	4	1	no	—
83	C₇×#131	—	4	1	no	—
84	C₂×C₇×M₁₆	—	2	1	no	—
85	C₂×C₇×M₁₆	—	2	1	no	—
86	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
87	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
88	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
89	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
90	C₂²×C₇×C₈	—	2	1	no	—
91	C₂²×C₇×C₈	—	2	1	no	—
92	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
93	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
94	C₇×C₄⋊Q₈	—	2	1	no	—
95	C₇×C₄⋊Q₈	—	2	1	no	—
96	C₇×#125	—	2	1	no	—
97	C₇×#125	—	2	1	no	—
98	C₇×C₂²⋊Q₈	—	4	1	no	—
99	C₄×C₇×Q₈	—	2	1	no	—
100	C₄×C₇×Q₈	—	2	1	no	—
101	C₄×C₇×D₄	—	2	1	no	—
102	C₄×C₇×D₄	—	2	1	no	—
103	C₄×C₇×D₄	—	4	1	no	—
104	C₄×C₇×D₄	—	4	1	no	—
105	C₇×#117	—	2	1	no	—
106	C₇×#117	—	2	1	no	—
107	C₇×#117	C₂³	1	1	yes	characteristic
108	C₇×#117	C₂³	1	1	yes	characteristic
109	C₂×C₇×C₄⋊C₄	—	2	1	no	—
110	C₂×C₇×C₄⋊C₄	—	2	1	no	—
111	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
112	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
113	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
114	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
115	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
116	C₇×#108	—	4	1	no	—
117	C₇×#108	—	4	1	no	—
118	C₇×#107	—	2	1	no	—
119	C₇×#107	—	2	1	no	—
120	C₇×#107	—	2	1	no	—
121	C₇×#107	—	2	1	no	—
122	C₇×#106	—	2	1	no	—
123	C₇×#106	—	2	1	no	—
124	C₇×#106	—	2	1	no	—
125	C₇×#106	—	2	1	no	—
126	C₇×C₄⋊C₈	—	2	1	no	—
127	C₇×C₄⋊C₈	—	2	1	no	—
128	C₇×C₄⋊C₈	—	2	1	no	—
129	C₇×C₄⋊C₈	—	2	1	no	—
130	C₇×#104	—	2	2	no	—
131	C₇×#104	—	2	2	no	—
132	C₇×#103	—	2	1	no	—
133	C₇×#103	—	2	1	no	—
134	C₇×#103	—	2	1	no	—
135	C₇×#103	—	2	1	no	—
136	C₇×#103	D₄	1	1	yes	characteristic
137	C₇×#103	D₄	1	1	yes	characteristic
138	C₇×#103	D₄	1	1	yes	characteristic
139	C₇×#103	D₄	1	1	yes	characteristic
140	C₇×#102	—	2	1	no	—
141	C₇×#102	—	2	1	no	—
142	C₇×#102	—	2	1	no	—
143	C₇×#102	—	2	1	no	—
144	C₇×#102	—	2	1	no	—
145	C₇×#102	—	2	1	no	—
146	C₇×#102	—	2	1	no	—
147	C₇×#102	—	2	1	no	—
148	C₇×#102	—	2	1	no	—
149	C₇×#102	—	2	1	no	—
150	C₇×#102	—	2	1	no	—
151	C₇×#102	—	2	1	no	—
152	C₇×#102	D₄	1	1	yes	characteristic
153	C₇×#102	D₄	1	1	yes	characteristic
154	C₇×#102	D₄	1	1	yes	characteristic
155	C₇×#102	D₄	1	1	yes	characteristic
156	C₇×#100	—	4	1	no	—
157	C₇×#95	—	4	1	no	—
158	#3084	C₂×C₇	1	1	yes	characteristic, complemented
159	#3083	C₂×C₇	1	1	yes	characteristic, complemented
160	#2897	C₂×C₇	1	1	yes	characteristic
161	#2897	C₂×C₇	1	1	yes	characteristic
162	#2092	C₂×C₇	1	1	yes	characteristic, complemented
163	#2063	C₂×C₇	1	1	yes	characteristic, complemented
164	#2063	C₂×C₇	1	1	yes	characteristic, complemented
165	#2027	C₂×C₇	1	1	yes	characteristic, complemented
166	#1866	C₂×C₇	1	1	yes	characteristic, complemented
167	#1866	C₂×C₇	1	1	yes	characteristic, complemented
168	#1860	C₂×C₇	1	1	yes	characteristic, complemented
169	#1860	C₂×C₇	1	1	yes	characteristic, complemented
170	#1816	C₂×C₇	1	1	yes	characteristic, complemented
171	#1815	C₂×C₇	1	1	yes	characteristic, complemented
172	#1792	C₂×C₇	1	1	yes	characteristic
173	C₂⁴×C₇	—	2	1	no	—
174	C₂⁴×C₇	—	4	1	no	—
175	C₇×#41	—	2	1	no	—
176	C₇×#41	—	2	1	no	—
177	C₇×#41	—	2	1	no	—
178	C₇×#41	—	2	1	no	—
179	C₇×#41	—	2	4	no	—
180	C₇×#41	—	2	4	no	—
181	C₇×#41	—	4	2	no	—
182	C₇×#41	—	4	2	no	—
183	C₇×#41	C₂×Q₈	1	2	yes	—
184	C₇×#41	C₂×Q₈	1	2	yes	—
185	C₇×#41	#41	1	2	yes	—
186	C₇×#41	#41	1	2	yes	—
187	C₂×C₇×Q₈	—	4	1	no	—
188	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
189	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
190	C₂×C₇×D₄	—	2	1	no	—
191	C₂×C₇×D₄	—	2	1	no	—
192	C₂×C₇×D₄	—	2	4	no	—
193	C₂×C₇×D₄	—	2	4	no	—
194	C₂×C₇×D₄	—	2	1	no	—
195	C₂×C₇×D₄	—	2	1	no	—
196	C₂×C₇×D₄	—	2	1	no	—
197	C₂×C₇×D₄	—	2	1	no	—
198	C₂×C₇×D₄	—	2	1	no	—
199	C₂×C₇×D₄	—	4	2	no	—
200	C₂×C₇×D₄	—	4	2	no	—
201	C₂×C₇×D₄	—	4	1	no	—
202	C₂×C₇×D₄	—	4	1	no	—
203	C₂×C₇×D₄	—	4	4	no	—
204	C₂×C₇×D₄	—	4	4	no	—
205	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
206	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
207	C₂×C₇×D₄	C₂×Q₈	1	2	yes	—
208	C₂²×C₄×C₇	—	2	1	no	—
209	C₂²×C₄×C₇	—	2	1	no	—
210	C₂²×C₄×C₇	—	2	1	no	—
211	C₂²×C₄×C₇	—	2	1	no	—
212	C₂²×C₄×C₇	—	2	1	no	—
213	C₂²×C₄×C₇	—	2	1	no	—
214	C₂²×C₄×C₇	—	4	1	no	—
215	C₂²×C₄×C₇	—	4	1	no	—
216	C₂²×C₄×C₇	—	4	1	no	—
217	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
218	C₇×M₁₆	—	2	1	no	—
219	C₇×M₁₆	—	2	1	no	—
220	C₇×M₁₆	—	2	1	no	—
221	C₇×M₁₆	—	2	1	no	—
222	C₇×M₁₆	—	2	1	no	—
223	C₇×M₁₆	—	2	1	no	—
224	C₇×M₁₆	—	2	1	no	—
225	C₇×M₁₆	—	2	1	no	—
226	C₇×M₁₆	—	4	2	no	—
227	C₇×M₁₆	—	4	2	no	—
228	C₂×C₇×C₈	—	2	1	no	—
229	C₂×C₇×C₈	—	2	1	no	—
230	C₂×C₇×C₈	—	2	1	no	—
231	C₂×C₇×C₈	—	2	1	no	—
232	C₂×C₇×C₈	—	2	1	no	—
233	C₂×C₇×C₈	—	2	1	no	—
234	C₂×C₇×C₈	—	2	1	no	—
235	C₂×C₇×C₈	—	2	1	no	—
236	C₂×C₇×C₈	—	4	2	no	—
237	C₂×C₇×C₈	—	4	2	no	—
238	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
239	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
240	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
241	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
242	C₇×C₄⋊C₄	—	2	1	no	—
243	C₇×C₄⋊C₄	—	2	1	no	—
244	C₇×C₄⋊C₄	—	2	1	no	—
245	C₇×C₄⋊C₄	—	2	1	no	—
246	C₇×C₄⋊C₄	—	2	1	no	—
247	C₇×C₄⋊C₄	—	2	1	no	—
248	C₇×C₄⋊C₄	—	4	1	no	—
249	C₇×C₄⋊C₄	—	4	1	no	—
250	C₇×C₄⋊C₄	—	4	1	no	—
251	C₇×C₄⋊C₄	—	4	1	no	—
252	C₇×C₄⋊C₄	—	4	1	no	—
253	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
254	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
255	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
256	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
257	C₇×C₂²⋊C₄	—	2	1	no	—
258	C₇×C₂²⋊C₄	—	2	1	no	—
259	C₇×C₂²⋊C₄	—	4	1	no	—
260	C₇×C₂²⋊C₄	—	4	1	no	—
261	C₇×C₂²⋊C₄	—	8	2	no	—
262	C₄²×C₇	—	2	1	no	—
263	C₄²×C₇	—	2	1	no	—
264	C₄²×C₇	—	2	1	no	—
265	C₄²×C₇	—	2	1	no	—
266	C₄²×C₇	—	4	1	no	—
267	C₄²×C₇	—	4	1	no	—
268	C₂×H(5,𝔽₂)	C₂²×C₇	1	1	yes	characteristic
269	C₂×#131	C₂²×C₇	1	1	yes	characteristic
270	C₂×#131	C₂²×C₇	1	1	yes	characteristic
271	#544	C₂²×C₇	1	1	yes	characteristic, complemented
272	#520	C₂²×C₇	1	1	yes	characteristic
273	#520	C₂²×C₇	1	1	yes	characteristic
274	#479	—	2	1	no	—
275	#479	—	2	1	no	—
276	#478	—	2	1	no	—
277	#478	—	2	1	no	—
278	#477	—	2	1	no	—
279	#476	—	2	1	no	—
280	#475	—	2	1	no	—
281	#474	—	2	1	no	—
282	C₂×#108	C₂²×C₇	1	1	yes	characteristic, complemented
283	C₂×#108	C₂²×C₇	1	1	yes	characteristic, complemented
284	#428	C₂²×C₇	1	1	yes	characteristic, complemented
285	#428	C₂²×C₇	1	1	yes	characteristic, complemented
286	C₂×#107	C₂²×C₇	1	1	yes	characteristic, complemented
287	C₂×#106	C₂²×C₇	1	1	yes	characteristic, complemented
288	#424	C₂²×C₇	1	1	yes	characteristic, complemented
289	#424	C₂²×C₇	1	1	yes	characteristic, complemented
290	C₂×#104	C₂²×C₇	1	1	yes	characteristic
291	C₂×#104	C₂²×C₇	1	1	yes	characteristic
292	#418	—	2	1	no	—
293	#418	—	2	1	no	—
294	#418	C₂²×C₇	1	1	yes	characteristic
295	#418	C₂²×C₇	1	1	yes	characteristic
296	#417	—	2	1	no	—
297	#417	—	2	1	no	—
298	#417	C₂²×C₇	1	1	yes	characteristic
299	#417	C₂²×C₇	1	1	yes	characteristic
300	#417	C₂²×C₇	1	1	yes	characteristic
301	#417	C₂²×C₇	1	1	yes	characteristic
302	#416	—	2	1	no	—
303	#416	—	2	1	no	—
304	#416	C₂²×C₇	1	1	yes	characteristic
305	#416	C₂²×C₇	1	1	yes	characteristic
306	C₂×#102	—	2	1	no	—
307	C₂×#102	—	2	1	no	—
308	C₂×#102	C₂²×C₇	1	1	yes	characteristic
309	C₂×#102	C₂²×C₇	1	1	yes	characteristic
310	C₂×#100	C₂²×C₇	1	1	yes	characteristic
311	#390	C₂²×C₇	1	1	yes	characteristic
312	#382	C₂²×C₇	1	1	yes	characteristic, complemented
313	#343	C₂²×C₇	1	1	yes	characteristic, complemented
314	#343	C₂²×C₇	1	1	yes	characteristic, complemented
315	#343	C₂²×C₇	1	1	yes	characteristic, complemented
316	#339	C₂²×C₇	1	1	yes	characteristic, complemented
317	#339	C₂²×C₇	1	1	yes	characteristic, complemented
318	#336	C₂²×C₇	1	1	yes	characteristic, complemented
319	C₂³×C₇	—	2	1	no	—
320	C₂³×C₇	—	2	1	no	—
321	C₂³×C₇	—	2	2	no	—
322	C₂³×C₇	—	4	4	no	—
323	C₂³×C₇	—	4	1	no	—
324	C₂³×C₇	—	4	1	no	—
325	C₂³×C₇	—	8	1	no	—
326	C₂³×C₇	—	8	1	no	—
327	C₂³×C₇	—	16	1	no	—
328	C₂³×C₇	—	16	1	no	—
329	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
330	C₇×Q₈	—	2	1	no	—
331	C₇×Q₈	—	2	1	no	—
332	C₇×Q₈	—	16	1	no	—
333	C₇×Q₈	C₂²⋊Q₈	1	2	yes	—
334	C₇×Q₈	C₂²⋊Q₈	1	2	yes	—
335	C₇×D₄	—	2	2	no	—
336	C₇×D₄	—	2	2	no	—
337	C₇×D₄	—	2	2	no	—
338	C₇×D₄	—	2	2	no	—
339	C₇×D₄	—	2	2	no	—
340	C₇×D₄	—	2	2	no	—
341	C₇×D₄	—	2	1	no	—
342	C₇×D₄	—	2	1	no	—
343	C₇×D₄	—	4	4	no	—
344	C₇×D₄	—	4	4	no	—
345	C₇×D₄	—	4	1	no	—
346	C₇×D₄	—	4	1	no	—
347	C₇×D₄	C₂²⋊Q₈	1	2	yes	—
348	C₇×D₄	C₂²⋊Q₈	1	2	yes	—
349	C₂×C₄×C₇	—	2	1	no	—
350	C₂×C₄×C₇	—	2	1	no	—
351	C₂×C₄×C₇	—	2	2	no	—
352	C₂×C₄×C₇	—	2	2	no	—
353	C₂×C₄×C₇	—	2	2	no	—
354	C₂×C₄×C₇	—	2	2	no	—
355	C₂×C₄×C₇	—	2	1	no	—
356	C₂×C₄×C₇	—	2	1	no	—
357	C₂×C₄×C₇	—	2	1	no	—
358	C₂×C₄×C₇	—	2	1	no	—
359	C₂×C₄×C₇	—	4	4	no	—
360	C₂×C₄×C₇	—	4	1	no	—
361	C₂×C₄×C₇	—	4	1	no	—
362	C₂×C₄×C₇	—	4	1	no	—
363	C₂×C₄×C₇	—	4	1	no	—
364	C₂×C₄×C₇	—	4	1	no	—
365	C₂×C₄×C₇	—	4	1	no	—
366	C₂×C₄×C₇	—	8	1	no	—
367	C₂×C₄×C₇	—	8	1	no	—
368	C₂×C₄×C₇	—	8	1	no	—
369	C₂×C₄×C₇	—	16	1	no	—
370	C₂×C₄×C₇	—	16	1	no	—
371	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
372	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
373	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
374	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
375	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
376	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
377	C₇×C₈	—	2	1	no	—
378	C₇×C₈	—	2	1	no	—
379	C₇×C₈	—	2	1	no	—
380	C₇×C₈	—	2	1	no	—
381	C₇×C₈	—	4	1	no	—
382	C₇×C₈	—	4	1	no	—
383	C₇×C₈	—	4	1	no	—
384	C₇×C₈	—	4	1	no	—
385	H(5,𝔽₂)	—	2	2	no	—
386	H(5,𝔽₂)	—	2	2	no	—
387	H(5,𝔽₂)	—	4	1	no	—
388	H(5,𝔽₂)	C₇×Q₈	1	4	yes	complemented
389	C₂×#41	—	2	1	no	—
390	C₂×#41	—	2	1	no	—
391	C₂×#41	C₂³×C₇	1	1	yes	characteristic
392	C₂×#41	C₂³×C₇	1	1	yes	characteristic
393	C₂²×D₄	—	2	1	no	—
394	C₂²×D₄	—	2	1	no	—
395	C₂²×D₄	—	4	1	no	—
396	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
397	#131	—	2	2	no	—
398	#131	—	2	2	no	—
399	#131	—	4	1	no	—
400	#131	—	4	1	no	—
401	C₂×M₁₆	—	2	1	no	—
402	C₂×M₁₆	—	2	1	no	—
403	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
404	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
405	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
406	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
407	C₂²×C₈	—	2	1	no	—
408	C₂²×C₈	—	2	1	no	—
409	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
410	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
411	C₄⋊Q₈	—	2	1	no	—
412	C₄⋊Q₈	—	2	1	no	—
413	#125	—	2	1	no	—
414	#125	—	2	1	no	—
415	C₂²⋊Q₈	—	4	1	no	—
416	C₄×Q₈	—	2	1	no	—
417	C₄×Q₈	—	2	1	no	—
418	C₄×D₄	—	2	1	no	—
419	C₄×D₄	—	2	1	no	—
420	C₄×D₄	—	4	1	no	—
421	C₄×D₄	—	4	1	no	—
422	#117	—	2	1	no	—
423	#117	—	2	1	no	—
424	#117	C₂³×C₇	1	1	yes	characteristic
425	#117	C₂³×C₇	1	1	yes	characteristic
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
431	C₂×C₂²⋊C₄	—	4	1	no	—
432	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
433	#108	—	4	1	no	—
434	#108	—	4	1	no	—
435	#107	—	2	1	no	—
436	#107	—	2	1	no	—
437	#107	—	2	1	no	—
438	#107	—	2	1	no	—
439	#106	—	2	1	no	—
440	#106	—	2	1	no	—
441	#106	—	2	1	no	—
442	#106	—	2	1	no	—
443	C₄⋊C₈	—	2	1	no	—
444	C₄⋊C₈	—	2	1	no	—
445	C₄⋊C₈	—	2	1	no	—
446	C₄⋊C₈	—	2	1	no	—
447	#104	—	2	2	no	—
448	#104	—	2	2	no	—
449	#103	—	2	1	no	—
450	#103	—	2	1	no	—
451	#103	—	2	1	no	—
452	#103	—	2	1	no	—
453	#103	C₇×D₄	1	1	yes	characteristic
454	#103	C₇×D₄	1	1	yes	characteristic
455	#103	C₇×D₄	1	1	yes	characteristic
456	#103	C₇×D₄	1	1	yes	characteristic
457	#102	—	2	1	no	—
458	#102	—	2	1	no	—
459	#102	—	2	1	no	—
460	#102	—	2	1	no	—
461	#102	—	2	1	no	—
462	#102	—	2	1	no	—
463	#102	—	2	1	no	—
464	#102	—	2	1	no	—
465	#102	—	2	1	no	—
466	#102	—	2	1	no	—
467	#102	—	2	1	no	—
468	#102	—	2	1	no	—
469	#102	C₇×D₄	1	1	yes	characteristic
470	#102	C₇×D₄	1	1	yes	characteristic
471	#102	C₇×D₄	1	1	yes	characteristic
472	#102	C₇×D₄	1	1	yes	characteristic
473	#100	—	4	1	no	—
474	#95	—	4	1	no	—
475	C₂²×C₇	—	2	1	no	—
476	C₂²×C₇	—	2	1	no	—
477	C₂²×C₇	—	2	2	no	—
478	C₂²×C₇	—	2	2	no	—
479	C₂²×C₇	—	4	1	no	—
480	C₂²×C₇	—	4	1	no	—
481	C₂²×C₇	—	4	1	no	—
482	C₂²×C₇	—	4	1	no	—
483	C₂²×C₇	—	4	2	no	—
484	C₂²×C₇	—	8	2	no	—
485	C₂²×C₇	—	8	1	no	—
486	C₂²×C₇	—	8	1	no	—
487	C₂²×C₇	—	16	1	no	—
488	C₂²×C₇	—	16	4	no	—
489	C₂²×C₇	#543	1	1	yes	characteristic
490	C₂²×C₇	D₄²	1	1	yes	characteristic
491	C₂²×C₇	#551	1	1	yes	characteristic
492	C₄×C₇	—	2	2	no	—
493	C₄×C₇	—	2	2	no	—
494	C₄×C₇	—	4	1	no	—
495	C₄×C₇	—	4	1	no	—
496	C₄×C₇	—	4	1	no	—
497	C₄×C₇	—	4	1	no	—
498	C₄×C₇	—	8	1	no	—
499	C₄×C₇	—	8	1	no	—
500	C₄×C₇	—	8	1	no	—
501	C₄×C₇	—	8	1	no	—
502	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
503	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
504	C₄×C₇	#546	1	1	yes	characteristic
505	C₄×C₇	#546	1	1	yes	characteristic
506	C₂⁴	—	2	1	no	—
507	C₂⁴	—	4	1	no	—
508	#41	—	2	1	no	—
509	#41	—	2	1	no	—
510	#41	—	2	1	no	—
511	#41	—	2	1	no	—
512	#41	—	2	4	no	—
513	#41	—	2	4	no	—
514	#41	—	4	2	no	—
515	#41	—	4	2	no	—
516	#41	C₂×C₇×Q₈	1	2	yes	—
517	#41	C₂×C₇×Q₈	1	2	yes	—
518	#41	C₇×#41	1	2	yes	—
519	#41	C₇×#41	1	2	yes	—
520	C₂×Q₈	—	4	1	no	—
521	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
522	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
523	C₂×D₄	—	2	1	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	4	no	—
526	C₂×D₄	—	2	4	no	—
527	C₂×D₄	—	2	1	no	—
528	C₂×D₄	—	2	1	no	—
529	C₂×D₄	—	2	1	no	—
530	C₂×D₄	—	2	1	no	—
531	C₂×D₄	—	2	1	no	—
532	C₂×D₄	—	4	2	no	—
533	C₂×D₄	—	4	2	no	—
534	C₂×D₄	—	4	1	no	—
535	C₂×D₄	—	4	1	no	—
536	C₂×D₄	—	4	4	no	—
537	C₂×D₄	—	4	4	no	—
538	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
539	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
540	C₂×D₄	C₂×C₇×Q₈	1	2	yes	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	1	no	—
544	C₂²×C₄	—	2	1	no	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	1	no	—
547	C₂²×C₄	—	4	1	no	—
548	C₂²×C₄	—	4	1	no	—
549	C₂²×C₄	—	4	1	no	—
550	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
551	M₁₆	—	2	1	no	—
552	M₁₆	—	2	1	no	—
553	M₁₆	—	2	1	no	—
554	M₁₆	—	2	1	no	—
555	M₁₆	—	2	1	no	—
556	M₁₆	—	2	1	no	—
557	M₁₆	—	2	1	no	—
558	M₁₆	—	2	1	no	—
559	M₁₆	—	4	2	no	—
560	M₁₆	—	4	2	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	—	2	1	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	2	1	no	—
568	C₂×C₈	—	2	1	no	—
569	C₂×C₈	—	4	2	no	—
570	C₂×C₈	—	4	2	no	—
571	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
572	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
573	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
574	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	4	1	no	—
582	C₄⋊C₄	—	4	1	no	—
583	C₄⋊C₄	—	4	1	no	—
584	C₄⋊C₄	—	4	1	no	—
585	C₄⋊C₄	—	4	1	no	—
586	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
587	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
588	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
589	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
590	C₂²⋊C₄	—	2	1	no	—
591	C₂²⋊C₄	—	2	1	no	—
592	C₂²⋊C₄	—	4	1	no	—
593	C₂²⋊C₄	—	4	1	no	—
594	C₂²⋊C₄	—	8	2	no	—
595	C₄²	—	2	1	no	—
596	C₄²	—	2	1	no	—
597	C₄²	—	2	1	no	—
598	C₄²	—	2	1	no	—
599	C₄²	—	4	1	no	—
600	C₄²	—	4	1	no	—
601	C₂×C₇	—	2	1	no	—
602	C₂×C₇	—	2	1	no	—
603	C₂×C₇	—	4	2	no	—
604	C₂×C₇	—	4	2	no	—
605	C₂×C₇	—	8	2	no	—
606	C₂×C₇	#2437	1	1	yes	characteristic
607	C₂×C₇	#3288	1	1	yes	characteristic
608	C₂×C₇	#3290	1	1	yes	characteristic
609	C₂³	—	2	1	no	—
610	C₂³	—	2	1	no	—
611	C₂³	—	2	2	no	—
612	C₂³	—	4	4	no	—
613	C₂³	—	4	1	no	—
614	C₂³	—	4	1	no	—
615	C₂³	—	8	1	no	—
616	C₂³	—	8	1	no	—
617	C₂³	—	16	1	no	—
618	C₂³	—	16	1	no	—
619	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
620	Q₈	—	2	1	no	—
621	Q₈	—	2	1	no	—
622	Q₈	—	16	1	no	—
623	Q₈	C₇×C₂²⋊Q₈	1	2	yes	—
624	Q₈	C₇×C₂²⋊Q₈	1	2	yes	—
625	D₄	—	2	2	no	—
626	D₄	—	2	2	no	—
627	D₄	—	2	2	no	—
628	D₄	—	2	2	no	—
629	D₄	—	2	2	no	—
630	D₄	—	2	2	no	—
631	D₄	—	2	1	no	—
632	D₄	—	2	1	no	—
633	D₄	—	4	4	no	—
634	D₄	—	4	4	no	—
635	D₄	—	4	1	no	—
636	D₄	—	4	1	no	—
637	D₄	C₇×C₂²⋊Q₈	1	2	yes	—
638	D₄	C₇×C₂²⋊Q₈	1	2	yes	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	4	4	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	1	no	—
653	C₂×C₄	—	4	1	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	8	1	no	—
657	C₂×C₄	—	8	1	no	—
658	C₂×C₄	—	8	1	no	—
659	C₂×C₄	—	16	1	no	—
660	C₂×C₄	—	16	1	no	—
661	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
662	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
663	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
664	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
665	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
666	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
667	C₈	—	2	1	no	—
668	C₈	—	2	1	no	—
669	C₈	—	2	1	no	—
670	C₈	—	2	1	no	—
671	C₈	—	4	1	no	—
672	C₈	—	4	1	no	—
673	C₈	—	4	1	no	—
674	C₈	—	4	1	no	—
675	C₇	#25984	1	1	yes	characteristic, direct product factor, Sylow
676	C₂²	—	2	1	no	—
677	C₂²	—	2	1	no	—
678	C₂²	—	2	2	no	—
679	C₂²	—	2	2	no	—
680	C₂²	—	4	1	no	—
681	C₂²	—	4	1	no	—
682	C₂²	—	4	1	no	—
683	C₂²	—	4	1	no	—
684	C₂²	—	4	2	no	—
685	C₂²	—	8	2	no	—
686	C₂²	—	8	1	no	—
687	C₂²	—	8	1	no	—
688	C₂²	—	16	1	no	—
689	C₂²	—	16	4	no	—
690	C₂²	C₇×#543	1	1	yes	characteristic
691	C₂²	C₇×D₄²	1	1	yes	characteristic
692	C₂²	C₇×#551	1	1	yes	characteristic
693	C₄	—	2	2	no	—
694	C₄	—	2	2	no	—
695	C₄	—	4	1	no	—
696	C₄	—	4	1	no	—
697	C₄	—	4	1	no	—
698	C₄	—	4	1	no	—
699	C₄	—	8	1	no	—
700	C₄	—	8	1	no	—
701	C₄	—	8	1	no	—
702	C₄	—	8	1	no	—
703	C₄	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
704	C₄	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
705	C₄	C₇×#546	1	1	yes	characteristic
706	C₄	C₇×#546	1	1	yes	characteristic
707	C₂	—	2	1	no	—
708	C₂	—	2	1	no	—
709	C₂	—	4	2	no	—
710	C₂	—	4	2	no	—
711	C₂	—	8	2	no	—
712	C₂	C₇×#2437	1	1	yes	characteristic
713	C₂	C₇×#3288	1	1	yes	characteristic
714	C₂	C₇×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇×#3084	C₂	semidirect
2. C₇×#3083	C₂	semidirect
5. C₇×#2092	C₂	semidirect
6. C₇×#2063	C₂	semidirect
7. C₇×#2063	C₂	semidirect
8. C₇×#2027	C₂	semidirect
9. C₇×#1866	C₂	semidirect
10. C₇×#1866	C₂	semidirect
11. C₇×#1860	C₂	semidirect
12. C₇×#1860	C₂	semidirect
13. C₇×#1816	C₂	semidirect
14. C₇×#1815	C₂	semidirect
19. C₇×#544	C₂²	semidirect
30. C₂×C₇×#108	C₂²	semidirect
31. C₂×C₇×#108	C₂²	semidirect
32. C₇×#428	C₂²	semidirect
33. C₇×#428	C₂²	semidirect
34. C₂×C₇×#107	C₂²	semidirect
35. C₂×C₇×#106	C₂²	semidirect
36. C₇×#424	C₂²	semidirect
37. C₇×#424	C₂²	semidirect
60. C₇×#382	C₂²	semidirect
61. C₇×#343	C₂²	semidirect
62. C₇×#343	C₂²	semidirect
63. C₇×#343	C₂²	semidirect
64. C₇×#339	C₂²	semidirect
65. C₇×#339	C₂²	semidirect
66. C₇×#336	C₂²	semidirect
67. #25984	C₇	direct (N × H)
71. C₇×H(5,𝔽₂)	Q₈	semidirect
158. #3084	C₂×C₇	semidirect
159. #3083	C₂×C₇	semidirect
162. #2092	C₂×C₇	semidirect
163. #2063	C₂×C₇	semidirect
164. #2063	C₂×C₇	semidirect
165. #2027	C₂×C₇	semidirect
166. #1866	C₂×C₇	semidirect
167. #1866	C₂×C₇	semidirect
168. #1860	C₂×C₇	semidirect
169. #1860	C₂×C₇	semidirect
170. #1816	C₂×C₇	semidirect
171. #1815	C₂×C₇	semidirect
271. #544	C₂²×C₇	semidirect
282. C₂×#108	C₂²×C₇	semidirect
283. C₂×#108	C₂²×C₇	semidirect
284. #428	C₂²×C₇	semidirect
285. #428	C₂²×C₇	semidirect
286. C₂×#107	C₂²×C₇	semidirect
287. C₂×#106	C₂²×C₇	semidirect
288. #424	C₂²×C₇	semidirect
289. #424	C₂²×C₇	semidirect
312. #382	C₂²×C₇	semidirect
313. #343	C₂²×C₇	semidirect
314. #343	C₂²×C₇	semidirect
315. #343	C₂²×C₇	semidirect
316. #339	C₂²×C₇	semidirect
317. #339	C₂²×C₇	semidirect
318. #336	C₂²×C₇	semidirect
388. H(5,𝔽₂)	C₇×Q₈	semidirect
675. C₇	#25984	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#3084	2
C₇×#3083	2
C₇×#2897	2
C₇×#2092	2
C₇×#2063	2
C₇×#2027	2
C₇×#1866	2
C₇×#1860	2
C₇×#1816	2
C₇×#1815	2
C₇×#1792	2
#25984	7

Subgroups — C7×#25984 #13203200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#25984 #13203200