Subgroups — C₇×#22466 #13199682

Order 1792.

1384 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#576	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#443	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2951	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2897	C₂	1	2	yes	maximal, complemented
5	C₇×#1978	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1970	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1907	C₂	1	2	yes	maximal, complemented
8	C₇×#1883	C₂	1	2	yes	maximal, complemented
9	C₇×#1816	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1792	C₂	1	2	yes	maximal, complemented
11	C₇×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×H(5,𝔽₂)	C₂²	1	2	yes	—
13	C₇×#576	—	2	4	no	—
14	C₇×#576	C₄	1	8	yes	complemented
15	C₂×C₇×AGL(1,ℤ/8)	—	2	2	no	—
16	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	—
17	C₂×C₇×#135	—	2	1	no	—
18	C₂×C₇×#135	C₂²	1	1	yes	characteristic
19	C₂²×C₇×D₈	—	2	1	no	—
20	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
21	C₂×C₇×#131	C₂²	1	1	yes	characteristic
22	C₂×C₇×#131	C₂²	1	2	yes	complemented
23	C₇×#520	C₂²	1	2	yes	complemented
24	C₇×#443	—	2	4	no	—
25	C₇×#440	—	2	2	no	—
26	C₄×C₇×Q₁₆	—	2	1	no	—
27	C₄×C₇×D₈	—	2	1	no	—
28	C₂×C₇×#108	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₇×#107	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₇×#104	C₂²	1	1	yes	characteristic
31	C₂×C₇×#104	C₂²	1	2	yes	complemented
32	C₇×#418	—	2	2	no	—
33	C₇×#418	C₂²	1	2	yes	complemented
34	C₇×#417	—	2	2	no	—
35	C₇×#417	C₂²	1	2	yes	—
36	C₇×#417	C₂²	1	2	yes	complemented
37	C₇×#416	—	2	2	no	—
38	C₇×#416	C₂²	1	1	yes	characteristic
39	C₂×C₇×#102	—	2	2	no	—
40	C₂×C₇×#102	C₂²	1	2	yes	complemented
41	C₂×C₇×#100	C₂²	1	2	yes	complemented
42	C₇×#405	C₂²	1	1	yes	characteristic, complemented
43	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
44	C₇×#390	C₂²	1	2	yes	complemented
45	C₇×#343	C₂²	1	2	yes	complemented
46	C₇×#341	C₂²	1	1	yes	characteristic, complemented
47	C₇×#339	C₂²	1	1	yes	characteristic, complemented
48	#22466	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₇×H(5,𝔽₂)	—	2	4	no	—
50	C₇×H(5,𝔽₂)	—	2	4	no	—
51	C₇×H(5,𝔽₂)	—	4	2	no	—
52	C₇×H(5,𝔽₂)	C₂×C₄	1	8	yes	complemented
53	C₂×C₇×#41	—	2	2	no	—
54	C₂×C₇×#41	—	2	2	no	—
55	C₂×C₇×#41	C₂³	1	2	yes	—
56	C₂×C₇×#41	C₂³	1	1	yes	characteristic
57	C₂²×C₇×D₄	—	2	2	no	—
58	C₂²×C₇×D₄	—	2	2	no	—
59	C₂²×C₇×D₄	—	4	2	no	—
60	C₂²×C₇×D₄	C₂³	1	2	yes	—
61	C₇×AGL(1,ℤ/8)	—	2	4	no	—
62	C₇×AGL(1,ℤ/8)	—	2	16	no	—
63	C₇×AGL(1,ℤ/8)	C₂×C₄	1	8	yes	complemented
64	C₇×#135	—	2	8	no	—
65	C₇×#135	C₂×C₄	1	4	yes	—
66	C₇×#135	D₄	1	4	yes	—
67	C₂×C₇×Q₁₆	D₄	1	1	yes	characteristic
68	C₂×C₇×SD₁₆	—	2	2	no	—
69	C₂×C₇×SD₁₆	D₄	1	2	yes	—
70	C₂×C₇×D₈	—	2	2	no	—
71	C₂×C₇×D₈	—	2	8	no	—
72	C₂×C₇×D₈	—	2	1	no	—
73	C₂×C₇×D₈	—	2	1	no	—
74	C₂×C₇×D₈	—	2	2	no	—
75	C₂×C₇×D₈	C₂×C₄	1	4	yes	complemented
76	C₂×C₇×D₈	D₄	1	1	yes	characteristic
77	C₇×#131	—	2	2	no	—
78	C₇×#131	—	2	4	no	—
79	C₇×#131	—	4	2	no	—
80	C₇×#131	C₂×C₄	1	4	yes	complemented
81	C₂×C₇×M₁₆	—	2	1	no	—
82	C₂×C₇×M₁₆	—	2	2	no	—
83	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
84	C₂×C₇×M₁₆	C₂³	1	2	yes	—
85	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
86	C₂²×C₇×C₈	—	2	1	no	—
87	C₂²×C₇×C₈	—	2	2	no	—
88	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
89	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
90	C₄×C₇×Q₈	—	2	2	no	—
91	C₄×C₇×D₄	—	2	2	no	—
92	C₄×C₇×D₄	—	4	2	no	—
93	C₇×#117	—	2	2	no	—
94	C₇×#117	C₂³	1	1	yes	characteristic
95	C₂×C₇×C₄⋊C₄	—	2	2	no	—
96	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
97	C₂×C₇×C₂²⋊C₄	—	4	2	no	—
98	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
99	C₇×#108	—	2	2	no	—
100	C₇×#107	—	2	1	no	—
101	C₇×#107	—	2	1	no	—
102	C₇×#104	—	2	4	no	—
103	C₇×#104	—	2	2	no	—
104	C₇×#103	—	2	2	no	—
105	C₇×#103	—	2	2	no	—
106	C₇×#103	D₄	1	1	yes	characteristic
107	C₇×#103	D₄	1	1	yes	characteristic
108	C₇×#102	—	2	2	no	—
109	C₇×#102	—	2	2	no	—
110	C₇×#102	—	2	2	no	—
111	C₇×#102	—	2	2	no	—
112	C₇×#102	—	2	2	no	—
113	C₇×#102	—	2	2	no	—
114	C₇×#102	D₄	1	1	yes	characteristic
115	C₇×#102	D₄	1	1	yes	characteristic
116	C₇×#100	—	4	2	no	—
117	C₇×#97	—	2	1	no	—
118	C₄×C₇×C₈	—	2	2	no	—
119	C₄×C₇×C₈	—	2	1	no	—
120	C₂×#576	C₂×C₇	1	1	yes	characteristic
121	C₂×#443	C₂×C₇	1	1	yes	characteristic, complemented
122	#2951	C₂×C₇	1	1	yes	characteristic, complemented
123	#2897	C₂×C₇	1	2	yes	complemented
124	#1978	C₂×C₇	1	1	yes	characteristic, complemented
125	#1970	C₂×C₇	1	1	yes	characteristic, complemented
126	#1907	C₂×C₇	1	2	yes	complemented
127	#1883	C₂×C₇	1	2	yes	complemented
128	#1816	C₂×C₇	1	1	yes	characteristic, complemented
129	#1792	C₂×C₇	1	2	yes	complemented
130	#1764	C₂×C₇	1	1	yes	characteristic, complemented
131	C₂⁴×C₇	—	2	2	no	—
132	C₂⁴×C₇	—	4	2	no	—
133	C₇×#41	—	2	1	no	—
134	C₇×#41	—	2	2	no	—
135	C₇×#41	—	2	2	no	—
136	C₇×#41	—	2	1	no	—
137	C₇×#41	—	2	8	no	—
138	C₇×#41	—	2	8	no	—
139	C₇×#41	—	4	4	no	—
140	C₇×#41	—	4	4	no	—
141	C₇×#41	C₂²×C₄	1	4	yes	—
142	C₇×#41	C₂²×C₄	1	2	yes	—
143	C₇×#41	C₂×D₄	1	4	yes	—
144	C₇×#41	#41	1	2	yes	—
145	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
146	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
147	C₂×C₇×D₄	—	2	2	no	—
148	C₂×C₇×D₄	—	2	1	no	—
149	C₂×C₇×D₄	—	2	8	no	—
150	C₂×C₇×D₄	—	2	8	no	—
151	C₂×C₇×D₄	—	2	2	no	—
152	C₂×C₇×D₄	—	2	2	no	—
153	C₂×C₇×D₄	—	2	2	no	—
154	C₂×C₇×D₄	—	2	2	no	—
155	C₂×C₇×D₄	—	2	2	no	—
156	C₂×C₇×D₄	—	4	4	no	—
157	C₂×C₇×D₄	—	4	4	no	—
158	C₂×C₇×D₄	—	4	2	no	—
159	C₂×C₇×D₄	—	4	2	no	—
160	C₂×C₇×D₄	—	4	8	no	—
161	C₂×C₇×D₄	—	4	8	no	—
162	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
163	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
164	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₇	—	2	2	no	—
166	C₂²×C₄×C₇	—	2	1	no	—
167	C₂²×C₄×C₇	—	2	1	no	—
168	C₂²×C₄×C₇	—	2	2	no	—
169	C₂²×C₄×C₇	—	4	2	no	—
170	C₂²×C₄×C₇	—	4	2	no	—
171	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
172	C₇×Q₁₆	C₂²⋊C₄	1	4	yes	complemented
173	C₇×SD₁₆	—	2	8	no	—
174	C₇×SD₁₆	C₂²⋊C₄	1	8	yes	complemented
175	C₇×D₈	—	2	8	no	—
176	C₇×D₈	—	2	4	no	—
177	C₇×D₈	—	2	4	no	—
178	C₇×D₈	C₂²⋊C₄	1	4	yes	complemented
179	C₇×M₁₆	—	2	1	no	—
180	C₇×M₁₆	—	2	4	no	—
181	C₇×M₁₆	—	2	2	no	—
182	C₇×M₁₆	—	2	1	no	—
183	C₇×M₁₆	—	2	1	no	—
184	C₇×M₁₆	—	2	2	no	—
185	C₇×M₁₆	—	4	4	no	—
186	C₇×M₁₆	C₂²×C₄	1	2	yes	—
187	C₂×C₇×C₈	—	2	1	no	—
188	C₂×C₇×C₈	—	2	1	no	—
189	C₂×C₇×C₈	—	2	1	no	—
190	C₂×C₇×C₈	—	2	4	no	—
191	C₂×C₇×C₈	—	2	1	no	—
192	C₂×C₇×C₈	—	2	2	no	—
193	C₂×C₇×C₈	—	4	4	no	—
194	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
195	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
196	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
197	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
198	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
199	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
200	C₇×C₄⋊C₄	—	2	2	no	—
201	C₇×C₄⋊C₄	—	2	2	no	—
202	C₇×C₄⋊C₄	—	4	2	no	—
203	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
204	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
205	C₇×C₂²⋊C₄	—	2	1	no	—
206	C₇×C₂²⋊C₄	—	4	2	no	—
207	C₇×C₂²⋊C₄	—	8	4	no	—
208	C₄²×C₇	—	2	1	no	—
209	C₄²×C₇	—	2	1	no	—
210	C₄²×C₇	—	2	1	no	—
211	C₄²×C₇	—	4	2	no	—
212	C₂×H(5,𝔽₂)	C₂²×C₇	1	2	yes	—
213	#576	—	2	4	no	—
214	#576	C₄×C₇	1	8	yes	complemented
215	C₂×AGL(1,ℤ/8)	—	2	2	no	—
216	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	—
217	C₂×#135	—	2	1	no	—
218	C₂×#135	C₂²×C₇	1	1	yes	characteristic
219	C₂²×D₈	—	2	1	no	—
220	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
221	C₂×#131	C₂²×C₇	1	1	yes	characteristic
222	C₂×#131	C₂²×C₇	1	2	yes	complemented
223	#520	C₂²×C₇	1	2	yes	complemented
224	#443	—	2	4	no	—
225	#440	—	2	2	no	—
226	C₄×Q₁₆	—	2	1	no	—
227	C₄×D₈	—	2	1	no	—
228	C₂×#108	C₂²×C₇	1	1	yes	characteristic, complemented
229	C₂×#107	C₂²×C₇	1	1	yes	characteristic, complemented
230	C₂×#104	C₂²×C₇	1	1	yes	characteristic
231	C₂×#104	C₂²×C₇	1	2	yes	complemented
232	#418	—	2	2	no	—
233	#418	C₂²×C₇	1	2	yes	complemented
234	#417	—	2	2	no	—
235	#417	C₂²×C₇	1	2	yes	—
236	#417	C₂²×C₇	1	2	yes	complemented
237	#416	—	2	2	no	—
238	#416	C₂²×C₇	1	1	yes	characteristic
239	C₂×#102	—	2	2	no	—
240	C₂×#102	C₂²×C₇	1	2	yes	complemented
241	C₂×#100	C₂²×C₇	1	2	yes	complemented
242	#405	C₂²×C₇	1	1	yes	characteristic, complemented
243	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
244	#390	C₂²×C₇	1	2	yes	complemented
245	#343	C₂²×C₇	1	2	yes	complemented
246	#341	C₂²×C₇	1	1	yes	characteristic, complemented
247	#339	C₂²×C₇	1	1	yes	characteristic, complemented
248	C₂³×C₇	—	2	2	no	—
249	C₂³×C₇	—	2	1	no	—
250	C₂³×C₇	—	2	4	no	—
251	C₂³×C₇	—	4	8	no	—
252	C₂³×C₇	—	4	2	no	—
253	C₂³×C₇	—	4	2	no	—
254	C₂³×C₇	—	8	2	no	—
255	C₂³×C₇	—	8	2	no	—
256	C₂³×C₇	—	16	2	no	—
257	C₂³×C₇	—	16	2	no	—
258	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
259	C₇×Q₈	—	2	1	no	—
260	C₇×Q₈	—	2	2	no	—
261	C₇×Q₈	C₂×C₂²⋊C₄	1	4	yes	—
262	C₇×Q₈	C₄×D₄	1	2	yes	—
263	C₇×D₄	—	2	2	no	—
264	C₇×D₄	—	2	4	no	—
265	C₇×D₄	—	2	2	no	—
266	C₇×D₄	—	2	4	no	—
267	C₇×D₄	—	2	4	no	—
268	C₇×D₄	—	2	4	no	—
269	C₇×D₄	—	2	1	no	—
270	C₇×D₄	—	2	2	no	—
271	C₇×D₄	—	4	8	no	—
272	C₇×D₄	—	4	8	no	—
273	C₇×D₄	—	4	2	no	—
274	C₇×D₄	—	4	2	no	—
275	C₇×D₄	C₂×C₂²⋊C₄	1	4	yes	—
276	C₇×D₄	C₄×D₄	1	2	yes	—
277	C₂×C₄×C₇	—	2	2	no	—
278	C₂×C₄×C₇	—	2	1	no	—
279	C₂×C₄×C₇	—	2	2	no	—
280	C₂×C₄×C₇	—	2	4	no	—
281	C₂×C₄×C₇	—	2	4	no	—
282	C₂×C₄×C₇	—	2	2	no	—
283	C₂×C₄×C₇	—	2	2	no	—
284	C₂×C₄×C₇	—	2	1	no	—
285	C₂×C₄×C₇	—	2	1	no	—
286	C₂×C₄×C₇	—	4	8	no	—
287	C₂×C₄×C₇	—	4	1	no	—
288	C₂×C₄×C₇	—	4	1	no	—
289	C₂×C₄×C₇	—	4	2	no	—
290	C₂×C₄×C₇	—	8	2	no	—
291	C₂×C₄×C₇	—	16	2	no	—
292	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
293	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
295	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
297	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
298	C₇×C₈	—	2	2	no	—
299	C₇×C₈	—	2	1	no	—
300	C₇×C₈	—	2	1	no	—
301	C₇×C₈	—	4	2	no	—
302	C₇×C₈	—	4	1	no	—
303	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
304	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
305	H(5,𝔽₂)	—	2	4	no	—
306	H(5,𝔽₂)	—	2	4	no	—
307	H(5,𝔽₂)	—	4	2	no	—
308	H(5,𝔽₂)	C₂×C₄×C₇	1	8	yes	complemented
309	C₂×#41	—	2	2	no	—
310	C₂×#41	—	2	2	no	—
311	C₂×#41	C₂³×C₇	1	2	yes	—
312	C₂×#41	C₂³×C₇	1	1	yes	characteristic
313	C₂²×D₄	—	2	2	no	—
314	C₂²×D₄	—	2	2	no	—
315	C₂²×D₄	—	4	2	no	—
316	C₂²×D₄	C₂³×C₇	1	2	yes	—
317	AGL(1,ℤ/8)	—	2	4	no	—
318	AGL(1,ℤ/8)	—	2	16	no	—
319	AGL(1,ℤ/8)	C₂×C₄×C₇	1	8	yes	complemented
320	#135	—	2	8	no	—
321	#135	C₂×C₄×C₇	1	4	yes	—
322	#135	C₇×D₄	1	4	yes	—
323	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
324	C₂×SD₁₆	—	2	2	no	—
325	C₂×SD₁₆	C₇×D₄	1	2	yes	—
326	C₂×D₈	—	2	2	no	—
327	C₂×D₈	—	2	8	no	—
328	C₂×D₈	—	2	1	no	—
329	C₂×D₈	—	2	1	no	—
330	C₂×D₈	—	2	2	no	—
331	C₂×D₈	C₂×C₄×C₇	1	4	yes	complemented
332	C₂×D₈	C₇×D₄	1	1	yes	characteristic
333	#131	—	2	2	no	—
334	#131	—	2	4	no	—
335	#131	—	4	2	no	—
336	#131	C₂×C₄×C₇	1	4	yes	complemented
337	C₂×M₁₆	—	2	1	no	—
338	C₂×M₁₆	—	2	2	no	—
339	C₂×M₁₆	C₂³×C₇	1	2	yes	—
340	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
341	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
342	C₂²×C₈	—	2	1	no	—
343	C₂²×C₈	—	2	2	no	—
344	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
346	C₄×Q₈	—	2	2	no	—
347	C₄×D₄	—	2	2	no	—
348	C₄×D₄	—	4	2	no	—
349	#117	—	2	2	no	—
350	#117	C₂³×C₇	1	1	yes	characteristic
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
353	C₂×C₂²⋊C₄	—	4	2	no	—
354	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
355	#108	—	2	2	no	—
356	#107	—	2	1	no	—
357	#107	—	2	1	no	—
358	#104	—	2	4	no	—
359	#104	—	2	2	no	—
360	#103	—	2	2	no	—
361	#103	—	2	2	no	—
362	#103	C₇×D₄	1	1	yes	characteristic
363	#103	C₇×D₄	1	1	yes	characteristic
364	#102	—	2	2	no	—
365	#102	—	2	2	no	—
366	#102	—	2	2	no	—
367	#102	—	2	2	no	—
368	#102	—	2	2	no	—
369	#102	—	2	2	no	—
370	#102	C₇×D₄	1	1	yes	characteristic
371	#102	C₇×D₄	1	1	yes	characteristic
372	#100	—	4	2	no	—
373	#97	—	2	1	no	—
374	C₄×C₈	—	2	2	no	—
375	C₄×C₈	—	2	1	no	—
376	C₂²×C₇	—	2	1	no	—
377	C₂²×C₇	—	2	1	no	—
378	C₂²×C₇	—	2	2	no	—
379	C₂²×C₇	—	2	4	no	—
380	C₂²×C₇	—	4	2	no	—
381	C₂²×C₇	—	4	1	no	—
382	C₂²×C₇	—	4	2	no	—
383	C₂²×C₇	—	4	1	no	—
384	C₂²×C₇	—	4	4	no	—
385	C₂²×C₇	—	8	4	no	—
386	C₂²×C₇	—	8	2	no	—
387	C₂²×C₇	—	8	2	no	—
388	C₂²×C₇	—	16	2	no	—
389	C₂²×C₇	—	16	8	no	—
390	C₂²×C₇	#518	1	1	yes	characteristic
391	C₂²×C₇	D₄²	1	1	yes	characteristic
392	C₂²×C₇	#546	1	1	yes	characteristic
393	C₄×C₇	—	2	2	no	—
394	C₄×C₇	—	2	4	no	—
395	C₄×C₇	—	4	2	no	—
396	C₄×C₇	—	4	1	no	—
397	C₄×C₇	—	4	1	no	—
398	C₄×C₇	—	8	2	no	—
399	C₄×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
400	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
401	C₄×C₇	D₄²	1	1	yes	characteristic
402	C₄×C₇	#546	1	1	yes	characteristic
403	C₂⁴	—	2	2	no	—
404	C₂⁴	—	4	2	no	—
405	#41	—	2	1	no	—
406	#41	—	2	2	no	—
407	#41	—	2	2	no	—
408	#41	—	2	1	no	—
409	#41	—	2	8	no	—
410	#41	—	2	8	no	—
411	#41	—	4	4	no	—
412	#41	—	4	4	no	—
413	#41	C₂²×C₄×C₇	1	4	yes	—
414	#41	C₂²×C₄×C₇	1	2	yes	—
415	#41	C₂×C₇×D₄	1	4	yes	—
416	#41	C₇×#41	1	2	yes	—
417	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
418	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	4	4	no	—
429	C₂×D₄	—	4	4	no	—
430	C₂×D₄	—	4	2	no	—
431	C₂×D₄	—	4	2	no	—
432	C₂×D₄	—	4	8	no	—
433	C₂×D₄	—	4	8	no	—
434	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
435	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
436	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
444	Q₁₆	C₇×C₂²⋊C₄	1	4	yes	complemented
445	SD₁₆	—	2	8	no	—
446	SD₁₆	C₇×C₂²⋊C₄	1	8	yes	complemented
447	D₈	—	2	8	no	—
448	D₈	—	2	4	no	—
449	D₈	—	2	4	no	—
450	D₈	C₇×C₂²⋊C₄	1	4	yes	complemented
451	M₁₆	—	2	1	no	—
452	M₁₆	—	2	4	no	—
453	M₁₆	—	2	2	no	—
454	M₁₆	—	2	1	no	—
455	M₁₆	—	2	1	no	—
456	M₁₆	—	2	2	no	—
457	M₁₆	—	4	4	no	—
458	M₁₆	C₂²×C₄×C₇	1	2	yes	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	4	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	2	no	—
465	C₂×C₈	—	4	4	no	—
466	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
467	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
468	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
469	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
470	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
471	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	4	2	no	—
475	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
476	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
477	C₂²⋊C₄	—	2	1	no	—
478	C₂²⋊C₄	—	4	2	no	—
479	C₂²⋊C₄	—	8	4	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	4	2	no	—
484	C₂×C₇	—	2	1	no	—
485	C₂×C₇	—	2	1	no	—
486	C₂×C₇	—	4	4	no	—
487	C₂×C₇	—	4	2	no	—
488	C₂×C₇	—	8	4	no	—
489	C₂×C₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
490	C₂×C₇	#3290	1	1	yes	characteristic
491	C₂×C₇	#3292	1	1	yes	characteristic
492	C₂³	—	2	2	no	—
493	C₂³	—	2	1	no	—
494	C₂³	—	2	4	no	—
495	C₂³	—	4	8	no	—
496	C₂³	—	4	2	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	8	2	no	—
499	C₂³	—	8	2	no	—
500	C₂³	—	16	2	no	—
501	C₂³	—	16	2	no	—
502	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
503	Q₈	—	2	1	no	—
504	Q₈	—	2	2	no	—
505	Q₈	C₂×C₇×C₂²⋊C₄	1	4	yes	—
506	Q₈	C₄×C₇×D₄	1	2	yes	—
507	D₄	—	2	2	no	—
508	D₄	—	2	4	no	—
509	D₄	—	2	2	no	—
510	D₄	—	2	4	no	—
511	D₄	—	2	4	no	—
512	D₄	—	2	4	no	—
513	D₄	—	2	1	no	—
514	D₄	—	2	2	no	—
515	D₄	—	4	8	no	—
516	D₄	—	4	8	no	—
517	D₄	—	4	2	no	—
518	D₄	—	4	2	no	—
519	D₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
520	D₄	C₄×C₇×D₄	1	2	yes	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	4	8	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	8	2	no	—
535	C₂×C₄	—	16	2	no	—
536	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
537	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
538	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
539	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
540	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
541	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
542	C₈	—	2	2	no	—
543	C₈	—	2	1	no	—
544	C₈	—	2	1	no	—
545	C₈	—	4	2	no	—
546	C₈	—	4	1	no	—
547	C₈	C₂×C₇×C₂²⋊C₄	1	2	yes	—
548	C₈	C₂×C₇×C₂²⋊C₄	1	2	yes	—
549	C₇	#22466	1	1	yes	characteristic, direct product factor, Sylow
550	C₂²	—	2	1	no	—
551	C₂²	—	2	1	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	2	4	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	4	2	no	—
556	C₂²	—	4	2	no	—
557	C₂²	—	4	1	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	8	4	no	—
560	C₂²	—	8	2	no	—
561	C₂²	—	8	2	no	—
562	C₂²	—	16	2	no	—
563	C₂²	—	16	8	no	—
564	C₂²	C₇×#518	1	1	yes	characteristic
565	C₂²	C₇×D₄²	1	1	yes	characteristic
566	C₂²	C₇×#546	1	1	yes	characteristic
567	C₄	—	2	2	no	—
568	C₄	—	2	4	no	—
569	C₄	—	4	2	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	8	2	no	—
573	C₄	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
574	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
575	C₄	C₇×D₄²	1	1	yes	characteristic
576	C₄	C₇×#546	1	1	yes	characteristic
577	C₂	—	2	1	no	—
578	C₂	—	2	1	no	—
579	C₂	—	4	2	no	—
580	C₂	—	4	4	no	—
581	C₂	—	8	4	no	—
582	C₂	C₇×D₄×C₂²⋊C₄	1	1	yes	characteristic
583	C₂	C₇×#3290	1	1	yes	characteristic
584	C₂	C₇×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#576	C₂
LCS	G ▷ C₂×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
2. C₂×C₇×#443	C₂	semidirect
3. C₇×#2951	C₂	semidirect
4. C₇×#2897	C₂	semidirect
5. C₇×#1978	C₂	semidirect
6. C₇×#1970	C₂	semidirect
7. C₇×#1907	C₂	semidirect
8. C₇×#1883	C₂	semidirect
9. C₇×#1816	C₂	semidirect
10. C₇×#1792	C₂	semidirect
11. C₇×#1764	C₂	semidirect
14. C₇×#576	C₄	semidirect
22. C₂×C₇×#131	C₂²	semidirect
23. C₇×#520	C₂²	semidirect
28. C₂×C₇×#108	C₂²	semidirect
29. C₂×C₇×#107	C₂²	semidirect
31. C₂×C₇×#104	C₂²	semidirect
33. C₇×#418	C₂²	semidirect
36. C₇×#417	C₂²	semidirect
40. C₂×C₇×#102	C₂²	semidirect
41. C₂×C₇×#100	C₂²	semidirect
42. C₇×#405	C₂²	semidirect
43. C₂×C₄×C₇×C₈	C₂²	semidirect
44. C₇×#390	C₂²	semidirect
45. C₇×#343	C₂²	semidirect
46. C₇×#341	C₂²	semidirect
47. C₇×#339	C₂²	semidirect
48. #22466	C₇	direct (N × H)
52. C₇×H(5,𝔽₂)	C₂×C₄	semidirect
63. C₇×AGL(1,ℤ/8)	C₂×C₄	semidirect
75. C₂×C₇×D₈	C₂×C₄	semidirect
80. C₇×#131	C₂×C₄	semidirect
121. C₂×#443	C₂×C₇	semidirect
122. #2951	C₂×C₇	semidirect
123. #2897	C₂×C₇	semidirect
124. #1978	C₂×C₇	semidirect
125. #1970	C₂×C₇	semidirect
126. #1907	C₂×C₇	semidirect
127. #1883	C₂×C₇	semidirect
128. #1816	C₂×C₇	semidirect
129. #1792	C₂×C₇	semidirect
130. #1764	C₂×C₇	semidirect
172. C₇×Q₁₆	C₂²⋊C₄	semidirect
174. C₇×SD₁₆	C₂²⋊C₄	semidirect
178. C₇×D₈	C₂²⋊C₄	semidirect
214. #576	C₄×C₇	semidirect
222. C₂×#131	C₂²×C₇	semidirect
223. #520	C₂²×C₇	semidirect
228. C₂×#108	C₂²×C₇	semidirect
229. C₂×#107	C₂²×C₇	semidirect
231. C₂×#104	C₂²×C₇	semidirect
233. #418	C₂²×C₇	semidirect
236. #417	C₂²×C₇	semidirect
240. C₂×#102	C₂²×C₇	semidirect
241. C₂×#100	C₂²×C₇	semidirect
242. #405	C₂²×C₇	semidirect
243. C₂×C₄×C₈	C₂²×C₇	semidirect
244. #390	C₂²×C₇	semidirect
245. #343	C₂²×C₇	semidirect
246. #341	C₂²×C₇	semidirect
247. #339	C₂²×C₇	semidirect
308. H(5,𝔽₂)	C₂×C₄×C₇	semidirect
319. AGL(1,ℤ/8)	C₂×C₄×C₇	semidirect
331. C₂×D₈	C₂×C₄×C₇	semidirect
336. #131	C₂×C₄×C₇	semidirect
444. Q₁₆	C₇×C₂²⋊C₄	semidirect
446. SD₁₆	C₇×C₂²⋊C₄	semidirect
450. D₈	C₇×C₂²⋊C₄	semidirect
549. C₇	#22466	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#576	2
C₂×C₇×#443	2
C₇×#2951	2
C₇×#2897	2
C₇×#1978	2
C₇×#1970	2
C₇×#1907	2
C₇×#1883	2
C₇×#1816	2
C₇×#1792	2
C₇×#1764	2
#22466	7

Subgroups — C7×#22466 #13199682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22466 #13199682