Subgroups — C₇×#60704 #13237920

Order 1792.

1974 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#540	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3179	C₂	1	8	yes	maximal, complemented
4	C₇×#3176	C₂	1	4	yes	maximal, complemented
5	C₇×#3174	C₂	1	4	yes	maximal, complemented
6	C₇×#3166	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3142	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3055	C₂	1	4	yes	maximal, complemented
11	C₄×C₇×AGL(1,ℤ/8)	C₂	1	4	yes	maximal, complemented
12	C₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	4	yes	—
14	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
15	C₇×#542	C₂²	1	4	yes	complemented
16	C₇×#541	C₂²	1	2	yes	complemented
17	C₇×#540	—	2	4	no	—
18	C₇×#540	C₂²	1	4	yes	—
19	C₇×#540	C₂²	1	4	yes	—
20	C₇×#540	C₂²	1	2	yes	complemented
21	C₇×#539	—	2	2	no	—
22	C₇×#538	—	2	2	no	—
23	C₇×#533	C₂²	1	1	yes	characteristic
24	C₇×#532	C₂²	1	1	yes	characteristic
25	C₂×C₇×#127	C₂²	1	1	yes	characteristic
26	C₇×#529	—	2	4	no	—
27	C₂×C₇×#125	C₂²	1	1	yes	characteristic
28	C₇×#525	C₂²	1	2	yes	—
29	C₂×C₇×#123	—	2	2	no	—
30	C₂×C₇×#121	—	2	2	no	—
31	C₂×C₇×#121	C₂²	1	2	yes	—
32	C₄×C₇×#41	C₂²	1	2	yes	—
33	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
34	C₂×C₇×#117	C₂²	1	1	yes	characteristic
35	C₇×#501	C₂²	1	4	yes	complemented
36	C₇×#499	C₂²	1	4	yes	complemented
37	C₇×#496	C₂²	1	4	yes	complemented
38	C₇×#493	C₂²	1	2	yes	complemented
39	C₇×#492	C₂²	1	2	yes	complemented
40	C₇×#491	C₂²	1	2	yes	complemented
41	C₇×#490	C₂²	1	4	yes	complemented
42	C₇×#490	C₂²	1	2	yes	complemented
43	C₇×#488	C₂²	1	4	yes	complemented
44	C₇×#486	C₂²	1	4	yes	complemented
45	C₇×#469	C₂²	1	8	yes	complemented
46	C₇×#468	C₂²	1	8	yes	complemented
47	C₇×#466	C₂²	1	8	yes	complemented
48	C₇×#465	C₂²	1	8	yes	complemented
49	C₇×#442	C₂²	1	8	yes	complemented
50	C₇×#440	C₂²	1	8	yes	complemented
51	C₄×C₇×SD₁₆	C₂²	1	8	yes	complemented
52	C₄×C₇×D₈	C₂²	1	8	yes	complemented
53	C₇×#428	C₂²	1	4	yes	—
54	C₇×#418	C₂²	1	4	yes	—
55	C₇×#417	C₂²	1	4	yes	—
56	C₇×#405	C₂²	1	4	yes	complemented
57	C₇×#405	C₂²	1	4	yes	complemented
58	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
59	C₄×C₇×M₁₆	C₂²	1	2	yes	—
60	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
61	#60704	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₇×#41	C₂³	1	2	yes	—
63	C₂²×C₇×D₄	—	2	2	no	—
64	C₂²×C₇×D₄	—	2	2	no	—
65	C₂²×C₇×D₄	C₂³	1	2	yes	—
66	C₂³×C₄×C₇	—	2	2	no	—
67	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
68	C₇×AGL(1,ℤ/8)	—	2	16	no	—
69	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
70	C₂×C₇×D₈	C₂³	1	8	yes	—
71	C₂×C₇×M₁₆	—	2	8	no	—
72	C₂×C₇×M₁₆	C₂³	1	4	yes	—
73	C₂×C₇×M₁₆	C₂³	1	2	yes	—
74	C₂×C₇×M₁₆	C₂³	1	2	yes	—
75	C₂²×C₇×C₈	C₂³	1	4	yes	—
76	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
77	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
78	C₇×#127	—	2	2	no	—
79	C₇×#127	C₂³	1	2	yes	—
80	C₇×#127	C₂³	1	2	yes	—
81	C₇×#127	C₂³	1	1	yes	characteristic
82	C₇×#126	—	2	4	no	—
83	C₇×#126	—	2	4	no	—
84	C₇×#125	—	2	4	no	—
85	C₇×#125	—	2	2	no	—
86	C₇×#125	C₂³	1	2	yes	—
87	C₇×#125	C₂³	1	2	yes	—
88	C₇×#125	C₂³	1	2	yes	—
89	C₇×#124	—	2	4	no	—
90	C₇×#124	C₂³	1	2	yes	—
91	C₇×#123	—	2	2	no	—
92	C₇×#123	—	2	8	no	—
93	C₇×#123	—	2	8	no	—
94	C₇×#123	—	2	4	no	—
95	C₇×#123	—	2	4	no	—
96	C₇×C₂²⋊Q₈	—	2	2	no	—
97	C₇×C₂²⋊Q₈	—	2	4	no	—
98	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
99	C₇×#121	—	2	4	no	—
100	C₇×#121	—	2	2	no	—
101	C₇×#121	—	2	8	no	—
102	C₇×#121	—	2	8	no	—
103	C₇×#121	—	2	4	no	—
104	C₇×#121	C₂³	1	4	yes	—
105	C₇×#121	C₂³	1	8	yes	—
106	C₇×#120	—	2	2	no	—
107	C₄×C₇×Q₈	—	2	2	no	—
108	C₄×C₇×Q₈	C₂³	1	4	yes	—
109	C₄×C₇×D₄	—	2	4	no	—
110	C₄×C₇×D₄	—	2	4	no	—
111	C₄×C₇×D₄	—	2	4	no	—
112	C₄×C₇×D₄	—	2	2	no	—
113	C₄×C₇×D₄	C₂³	1	8	yes	—
114	C₄×C₇×D₄	C₂³	1	4	yes	—
115	C₇×#117	—	2	2	no	—
116	C₇×#117	D₄	1	4	yes	—
117	C₇×#117	C₂³	1	2	yes	—
118	C₇×#117	C₂³	1	2	yes	—
119	C₇×#117	C₂³	1	2	yes	—
120	C₂×C₇×C₄⋊C₄	—	2	2	no	—
121	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
122	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
123	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
124	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
125	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
126	C₂×C₄²×C₇	—	2	2	no	—
127	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
129	C₇×#107	C₂³	1	8	yes	—
130	C₇×#106	C₂³	1	8	yes	—
131	C₇×#103	C₂³	1	8	yes	—
132	C₇×#102	C₂³	1	8	yes	—
133	C₇×#102	C₂³	1	8	yes	—
134	C₇×#102	C₂³	1	8	yes	—
135	C₇×#97	C₂³	1	2	yes	—
136	C₇×#97	C₂³	1	2	yes	—
137	C₇×#97	C₂³	1	4	yes	—
138	C₄×C₇×C₈	C₂³	1	2	yes	—
139	C₄×C₇×C₈	C₂³	1	2	yes	—
140	C₄×C₇×C₈	C₂³	1	4	yes	—
141	#3455	C₂×C₇	1	1	yes	characteristic, complemented
142	C₂×#540	C₂×C₇	1	1	yes	characteristic, complemented
143	#3179	C₂×C₇	1	8	yes	complemented
144	#3176	C₂×C₇	1	4	yes	complemented
145	#3174	C₂×C₇	1	4	yes	complemented
146	#3166	C₂×C₇	1	1	yes	characteristic, complemented
147	#3151	C₂×C₇	1	1	yes	characteristic, complemented
148	#3142	C₂×C₇	1	1	yes	characteristic, complemented
149	#3138	C₂×C₇	1	1	yes	characteristic, complemented
150	#3055	C₂×C₇	1	4	yes	complemented
151	C₄×AGL(1,ℤ/8)	C₂×C₇	1	4	yes	complemented
152	#2873	C₂×C₇	1	1	yes	characteristic, complemented
153	C₂⁴×C₇	—	2	2	no	—
154	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
155	C₇×#41	—	2	8	no	—
156	C₂×C₇×Q₈	—	2	2	no	—
157	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
158	C₂×C₇×D₄	—	2	2	no	—
159	C₂×C₇×D₄	—	2	4	no	—
160	C₂×C₇×D₄	—	2	8	no	—
161	C₂×C₇×D₄	—	2	2	no	—
162	C₂×C₇×D₄	—	2	2	no	—
163	C₂×C₇×D₄	—	2	4	no	—
164	C₂×C₇×D₄	—	2	4	no	—
165	C₂×C₇×D₄	—	2	8	no	—
166	C₂×C₇×D₄	—	8	4	no	—
167	C₂×C₇×D₄	—	8	4	no	—
168	C₂×C₇×D₄	C₂⁴	1	2	yes	—
169	C₂×C₇×D₄	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₇	—	2	4	no	—
171	C₂²×C₄×C₇	—	2	2	no	—
172	C₂²×C₄×C₇	—	2	2	no	—
173	C₂²×C₄×C₇	—	2	4	no	—
174	C₂²×C₄×C₇	—	2	8	no	—
175	C₂²×C₄×C₇	—	8	4	no	—
176	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
182	C₇×SD₁₆	—	2	16	no	—
183	C₇×D₈	—	2	16	no	—
184	C₇×M₁₆	—	2	4	no	—
185	C₇×M₁₆	—	2	8	no	—
186	C₇×M₁₆	#41	1	8	yes	—
187	C₂×C₇×C₈	—	2	8	no	—
188	C₂×C₇×C₈	C₂⁴	1	4	yes	—
189	C₂×C₇×C₈	C₂⁴	1	4	yes	—
190	C₇×C₄⋊C₄	—	2	4	no	—
191	C₇×C₄⋊C₄	—	2	4	no	—
192	C₇×C₄⋊C₄	—	2	2	no	—
193	C₇×C₄⋊C₄	—	2	4	no	—
194	C₇×C₄⋊C₄	—	2	2	no	—
195	C₇×C₄⋊C₄	—	2	4	no	—
196	C₇×C₄⋊C₄	—	2	4	no	—
197	C₇×C₄⋊C₄	—	2	2	no	—
198	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
199	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
200	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
201	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
202	C₇×C₂²⋊C₄	—	2	2	no	—
203	C₇×C₂²⋊C₄	—	2	4	no	—
204	C₇×C₂²⋊C₄	—	2	8	no	—
205	C₇×C₂²⋊C₄	—	2	8	no	—
206	C₇×C₂²⋊C₄	—	2	4	no	—
207	C₇×C₂²⋊C₄	—	2	4	no	—
208	C₇×C₂²⋊C₄	—	2	4	no	—
209	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
210	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
211	C₄²×C₇	—	2	2	no	—
212	C₄²×C₇	—	2	4	no	—
213	C₄²×C₇	C₂×D₄	1	2	yes	—
214	C₄²×C₇	C₂×D₄	1	2	yes	—
215	C₄²×C₇	C₂⁴	1	2	yes	—
216	C₄²×C₇	C₂⁴	1	1	yes	characteristic
217	C₄²×C₇	C₂⁴	1	1	yes	characteristic
218	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	4	yes	—
219	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
220	#542	C₂²×C₇	1	4	yes	complemented
221	#541	C₂²×C₇	1	2	yes	complemented
222	#540	—	2	4	no	—
223	#540	C₂²×C₇	1	4	yes	—
224	#540	C₂²×C₇	1	4	yes	—
225	#540	C₂²×C₇	1	2	yes	complemented
226	#539	—	2	2	no	—
227	#538	—	2	2	no	—
228	#533	C₂²×C₇	1	1	yes	characteristic
229	#532	C₂²×C₇	1	1	yes	characteristic
230	C₂×#127	C₂²×C₇	1	1	yes	characteristic
231	#529	—	2	4	no	—
232	C₂×#125	C₂²×C₇	1	1	yes	characteristic
233	#525	C₂²×C₇	1	2	yes	—
234	C₂×#123	—	2	2	no	—
235	C₂×#121	—	2	2	no	—
236	C₂×#121	C₂²×C₇	1	2	yes	—
237	C₄×#41	C₂²×C₇	1	2	yes	—
238	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
239	C₂×#117	C₂²×C₇	1	1	yes	characteristic
240	#501	C₂²×C₇	1	4	yes	complemented
241	#499	C₂²×C₇	1	4	yes	complemented
242	#496	C₂²×C₇	1	4	yes	complemented
243	#493	C₂²×C₇	1	2	yes	complemented
244	#492	C₂²×C₇	1	2	yes	complemented
245	#491	C₂²×C₇	1	2	yes	complemented
246	#490	C₂²×C₇	1	4	yes	complemented
247	#490	C₂²×C₇	1	2	yes	complemented
248	#488	C₂²×C₇	1	4	yes	complemented
249	#486	C₂²×C₇	1	4	yes	complemented
250	#469	C₂²×C₇	1	8	yes	complemented
251	#468	C₂²×C₇	1	8	yes	complemented
252	#466	C₂²×C₇	1	8	yes	complemented
253	#465	C₂²×C₇	1	8	yes	complemented
254	#442	C₂²×C₇	1	8	yes	complemented
255	#440	C₂²×C₇	1	8	yes	complemented
256	C₄×SD₁₆	C₂²×C₇	1	8	yes	complemented
257	C₄×D₈	C₂²×C₇	1	8	yes	complemented
258	#428	C₂²×C₇	1	4	yes	—
259	#418	C₂²×C₇	1	4	yes	—
260	#417	C₂²×C₇	1	4	yes	—
261	#405	C₂²×C₇	1	4	yes	complemented
262	#405	C₂²×C₇	1	4	yes	complemented
263	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
264	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
265	C₄×M₁₆	C₂²×C₇	1	2	yes	—
266	C₂³×C₇	—	2	2	no	—
267	C₂³×C₇	—	2	2	no	—
268	C₂³×C₇	—	2	4	no	—
269	C₂³×C₇	—	4	2	no	—
270	C₂³×C₇	—	4	4	no	—
271	C₂³×C₇	—	4	4	no	—
272	C₂³×C₇	—	4	4	no	—
273	C₂³×C₇	C₂²×D₄	1	2	yes	—
274	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
275	C₇×Q₈	—	2	4	no	—
276	C₇×Q₈	—	8	2	no	—
277	C₇×D₄	—	2	4	no	—
278	C₇×D₄	—	2	8	no	—
279	C₇×D₄	—	4	4	no	—
280	C₇×D₄	—	4	4	no	—
281	C₇×D₄	—	8	2	no	—
282	C₇×D₄	—	8	4	no	—
283	C₇×D₄	—	8	4	no	—
284	C₇×D₄	—	8	4	no	—
285	C₇×D₄	—	8	8	no	—
286	C₂×C₄×C₇	—	2	2	no	—
287	C₂×C₄×C₇	—	2	4	no	—
288	C₂×C₄×C₇	—	2	2	no	—
289	C₂×C₄×C₇	—	2	2	no	—
290	C₂×C₄×C₇	—	2	2	no	—
291	C₂×C₄×C₇	—	2	4	no	—
292	C₂×C₄×C₇	—	2	4	no	—
293	C₂×C₄×C₇	—	4	2	no	—
294	C₂×C₄×C₇	—	4	2	no	—
295	C₂×C₄×C₇	—	4	4	no	—
296	C₂×C₄×C₇	—	4	4	no	—
297	C₂×C₄×C₇	—	8	4	no	—
298	C₂×C₄×C₇	—	8	8	no	—
299	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
300	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
301	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
302	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
303	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₇	C₂×#41	1	4	yes	—
305	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
306	C₇×C₈	—	2	4	no	—
307	C₇×C₈	C₂×#41	1	8	yes	—
308	C₂×#41	C₂³×C₇	1	2	yes	—
309	C₂²×D₄	—	2	2	no	—
310	C₂²×D₄	—	2	2	no	—
311	C₂²×D₄	C₂³×C₇	1	2	yes	—
312	C₂³×C₄	—	2	2	no	—
313	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
314	AGL(1,ℤ/8)	—	2	16	no	—
315	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
316	C₂×D₈	C₂³×C₇	1	8	yes	—
317	C₂×M₁₆	—	2	8	no	—
318	C₂×M₁₆	C₂³×C₇	1	2	yes	—
319	C₂×M₁₆	C₂³×C₇	1	4	yes	—
320	C₂×M₁₆	C₂³×C₇	1	2	yes	—
321	C₂²×C₈	C₂³×C₇	1	4	yes	—
322	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
323	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
324	#127	—	2	2	no	—
325	#127	C₂³×C₇	1	2	yes	—
326	#127	C₂³×C₇	1	2	yes	—
327	#127	C₂³×C₇	1	1	yes	characteristic
328	#126	—	2	4	no	—
329	#126	—	2	4	no	—
330	#125	—	2	4	no	—
331	#125	—	2	2	no	—
332	#125	C₂³×C₇	1	2	yes	—
333	#125	C₂³×C₇	1	2	yes	—
334	#125	C₂³×C₇	1	2	yes	—
335	#124	—	2	4	no	—
336	#124	C₂³×C₇	1	2	yes	—
337	#123	—	2	2	no	—
338	#123	—	2	8	no	—
339	#123	—	2	8	no	—
340	#123	—	2	4	no	—
341	#123	—	2	4	no	—
342	C₂²⋊Q₈	—	2	2	no	—
343	C₂²⋊Q₈	—	2	4	no	—
344	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
345	#121	—	2	4	no	—
346	#121	—	2	2	no	—
347	#121	—	2	8	no	—
348	#121	—	2	8	no	—
349	#121	—	2	4	no	—
350	#121	C₂³×C₇	1	8	yes	—
351	#121	C₂³×C₇	1	4	yes	—
352	#120	—	2	2	no	—
353	C₄×Q₈	—	2	2	no	—
354	C₄×Q₈	C₂³×C₇	1	4	yes	—
355	C₄×D₄	—	2	4	no	—
356	C₄×D₄	—	2	4	no	—
357	C₄×D₄	—	2	4	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	C₂³×C₇	1	8	yes	—
360	C₄×D₄	C₂³×C₇	1	4	yes	—
361	#117	—	2	2	no	—
362	#117	C₇×D₄	1	4	yes	—
363	#117	C₂³×C₇	1	2	yes	—
364	#117	C₂³×C₇	1	2	yes	—
365	#117	C₂³×C₇	1	2	yes	—
366	C₂×C₄⋊C₄	—	2	2	no	—
367	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
369	C₂×C₂²⋊C₄	—	2	2	no	—
370	C₂×C₂²⋊C₄	—	2	2	no	—
371	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
372	C₂×C₄²	—	2	2	no	—
373	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
374	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
375	#107	C₂³×C₇	1	8	yes	—
376	#106	C₂³×C₇	1	8	yes	—
377	#103	C₂³×C₇	1	8	yes	—
378	#102	C₂³×C₇	1	8	yes	—
379	#102	C₂³×C₇	1	8	yes	—
380	#102	C₂³×C₇	1	8	yes	—
381	#97	C₂³×C₇	1	4	yes	—
382	#97	C₂³×C₇	1	2	yes	—
383	#97	C₂³×C₇	1	2	yes	—
384	C₄×C₈	C₂³×C₇	1	4	yes	—
385	C₄×C₈	C₂³×C₇	1	2	yes	—
386	C₄×C₈	C₂³×C₇	1	2	yes	—
387	C₂²×C₇	—	2	2	no	—
388	C₂²×C₇	—	2	4	no	—
389	C₂²×C₇	—	2	2	no	—
390	C₂²×C₇	—	4	4	no	—
391	C₂²×C₇	—	4	4	no	—
392	C₂²×C₇	—	4	4	no	—
393	C₂²×C₇	—	4	2	no	—
394	C₂²×C₇	—	4	4	no	—
395	C₂²×C₇	—	8	2	no	—
396	C₂²×C₇	—	8	8	no	—
397	C₂²×C₇	#532	1	2	yes	—
398	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
399	C₄×C₇	—	2	4	no	—
400	C₄×C₇	—	2	2	no	—
401	C₄×C₇	—	2	2	no	—
402	C₄×C₇	—	2	2	no	—
403	C₄×C₇	—	4	4	no	—
404	C₄×C₇	—	4	4	no	—
405	C₄×C₇	—	4	2	no	—
406	C₄×C₇	#532	1	2	yes	—
407	C₄×C₇	C₂²×#41	1	2	yes	—
408	C₂⁴	—	2	2	no	—
409	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
410	#41	—	2	8	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	8	no	—
421	C₂×D₄	—	8	4	no	—
422	C₂×D₄	—	8	4	no	—
423	C₂×D₄	C₂⁴×C₇	1	2	yes	—
424	C₂×D₄	C₂⁴×C₇	1	4	yes	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	8	no	—
430	C₂²×C₄	—	8	4	no	—
431	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
432	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
433	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
434	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
437	SD₁₆	—	2	16	no	—
438	D₈	—	2	16	no	—
439	M₁₆	—	2	4	no	—
440	M₁₆	—	2	8	no	—
441	M₁₆	C₇×#41	1	8	yes	—
442	C₂×C₈	—	2	8	no	—
443	C₂×C₈	C₂⁴×C₇	1	4	yes	—
444	C₂×C₈	C₂⁴×C₇	1	4	yes	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	4	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
454	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
455	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
456	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
457	C₂²⋊C₄	—	2	2	no	—
458	C₂²⋊C₄	—	2	4	no	—
459	C₂²⋊C₄	—	2	8	no	—
460	C₂²⋊C₄	—	2	8	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	—	2	4	no	—
464	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
465	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
466	C₄²	—	2	2	no	—
467	C₄²	—	2	4	no	—
468	C₄²	C₂×C₇×D₄	1	2	yes	—
469	C₄²	C₂×C₇×D₄	1	2	yes	—
470	C₄²	C₂⁴×C₇	1	2	yes	—
471	C₄²	C₂⁴×C₇	1	1	yes	characteristic
472	C₄²	C₂⁴×C₇	1	1	yes	characteristic
473	C₂×C₇	—	2	2	no	—
474	C₂×C₇	—	4	2	no	—
475	C₂×C₇	—	8	2	no	—
476	C₂×C₇	—	8	4	no	—
477	C₂×C₇	C₂×#532	1	1	yes	characteristic
478	C₂×C₇	#3585	1	2	yes	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	4	no	—
482	C₂³	—	4	2	no	—
483	C₂³	—	4	4	no	—
484	C₂³	—	4	4	no	—
485	C₂³	—	4	4	no	—
486	C₂³	C₂²×C₇×D₄	1	2	yes	—
487	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
488	Q₈	—	2	4	no	—
489	Q₈	—	8	2	no	—
490	D₄	—	2	4	no	—
491	D₄	—	2	8	no	—
492	D₄	—	4	4	no	—
493	D₄	—	4	4	no	—
494	D₄	—	8	2	no	—
495	D₄	—	8	4	no	—
496	D₄	—	8	4	no	—
497	D₄	—	8	4	no	—
498	D₄	—	8	8	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	4	no	—
509	C₂×C₄	—	4	4	no	—
510	C₂×C₄	—	8	4	no	—
511	C₂×C₄	—	8	8	no	—
512	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
513	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
514	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
515	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
516	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
517	C₂×C₄	C₂×C₇×#41	1	4	yes	—
518	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
519	C₈	—	2	4	no	—
520	C₈	C₂×C₇×#41	1	8	yes	—
521	C₇	#60704	1	1	yes	characteristic, direct product factor, Sylow
522	C₂²	—	2	2	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	4	4	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	8	2	no	—
531	C₂²	—	8	8	no	—
532	C₂²	C₇×#532	1	2	yes	—
533	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
534	C₄	—	2	4	no	—
535	C₄	—	2	2	no	—
536	C₄	—	2	2	no	—
537	C₄	—	2	2	no	—
538	C₄	—	4	4	no	—
539	C₄	—	4	4	no	—
540	C₄	—	4	2	no	—
541	C₄	C₇×#532	1	2	yes	—
542	C₄	C₂²×C₇×#41	1	2	yes	—
543	C₂	—	2	2	no	—
544	C₂	—	4	2	no	—
545	C₂	—	8	4	no	—
546	C₂	—	8	2	no	—
547	C₂	C₂×C₇×#532	1	1	yes	characteristic
548	C₂	C₇×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3455	C₂	semidirect
2. C₂×C₇×#540	C₂	semidirect
3. C₇×#3179	C₂	semidirect
4. C₇×#3176	C₂	semidirect
5. C₇×#3174	C₂	semidirect
6. C₇×#3166	C₂	semidirect
7. C₇×#3151	C₂	semidirect
8. C₇×#3142	C₂	semidirect
9. C₇×#3138	C₂	semidirect
10. C₇×#3055	C₂	semidirect
11. C₄×C₇×AGL(1,ℤ/8)	C₂	semidirect
12. C₇×#2873	C₂	semidirect
15. C₇×#542	C₂²	semidirect
16. C₇×#541	C₂²	semidirect
20. C₇×#540	C₂²	semidirect
35. C₇×#501	C₂²	semidirect
36. C₇×#499	C₂²	semidirect
37. C₇×#496	C₂²	semidirect
38. C₇×#493	C₂²	semidirect
39. C₇×#492	C₂²	semidirect
40. C₇×#491	C₂²	semidirect
41. C₇×#490	C₂²	semidirect
42. C₇×#490	C₂²	semidirect
43. C₇×#488	C₂²	semidirect
44. C₇×#486	C₂²	semidirect
45. C₇×#469	C₂²	semidirect
46. C₇×#468	C₂²	semidirect
47. C₇×#466	C₂²	semidirect
48. C₇×#465	C₂²	semidirect
49. C₇×#442	C₂²	semidirect
50. C₇×#440	C₂²	semidirect
51. C₄×C₇×SD₁₆	C₂²	semidirect
52. C₄×C₇×D₈	C₂²	semidirect
56. C₇×#405	C₂²	semidirect
57. C₇×#405	C₂²	semidirect
61. #60704	C₇	direct (N × H)
141. #3455	C₂×C₇	semidirect
142. C₂×#540	C₂×C₇	semidirect
143. #3179	C₂×C₇	semidirect
144. #3176	C₂×C₇	semidirect
145. #3174	C₂×C₇	semidirect
146. #3166	C₂×C₇	semidirect
147. #3151	C₂×C₇	semidirect
148. #3142	C₂×C₇	semidirect
149. #3138	C₂×C₇	semidirect
150. #3055	C₂×C₇	semidirect
151. C₄×AGL(1,ℤ/8)	C₂×C₇	semidirect
152. #2873	C₂×C₇	semidirect
220. #542	C₂²×C₇	semidirect
221. #541	C₂²×C₇	semidirect
225. #540	C₂²×C₇	semidirect
240. #501	C₂²×C₇	semidirect
241. #499	C₂²×C₇	semidirect
242. #496	C₂²×C₇	semidirect
243. #493	C₂²×C₇	semidirect
244. #492	C₂²×C₇	semidirect
245. #491	C₂²×C₇	semidirect
246. #490	C₂²×C₇	semidirect
247. #490	C₂²×C₇	semidirect
248. #488	C₂²×C₇	semidirect
249. #486	C₂²×C₇	semidirect
250. #469	C₂²×C₇	semidirect
251. #468	C₂²×C₇	semidirect
252. #466	C₂²×C₇	semidirect
253. #465	C₂²×C₇	semidirect
254. #442	C₂²×C₇	semidirect
255. #440	C₂²×C₇	semidirect
256. C₄×SD₁₆	C₂²×C₇	semidirect
257. C₄×D₈	C₂²×C₇	semidirect
261. #405	C₂²×C₇	semidirect
262. #405	C₂²×C₇	semidirect
521. C₇	#60704	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×#3455	2
C₂×C₇×#540	2
C₇×#3179	2
C₇×#3176	2
C₇×#3174	2
C₇×#3166	2
C₇×#3151	2
C₇×#3142	2
C₇×#3138	2
C₇×#3055	2
C₄×C₇×AGL(1,ℤ/8)	2
C₇×#2873	2
#60704	7

Subgroups — C7×#60704 #13237920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60704 #13237920