Subgroups — C₇×#19767 #13196983

Order 1792.

658 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3392	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3386	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3298	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3284	C₂	1	1	yes	maximal, characteristic
5	C₇×#1714	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1686	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1673	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1665	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1654	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1653	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1642	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1627	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1582	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×#135	—	2	1	no	—
17	C₇×#557	C₂²	1	1	yes	characteristic
18	C₇×#550	C₂²	1	1	yes	characteristic
19	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
20	C₇×#547	C₂²	1	1	yes	characteristic
21	C₇×#501	—	2	1	no	—
22	C₇×#500	C₂²	1	1	yes	characteristic
23	C₇×#498	C₂²	1	1	yes	characteristic
24	C₇×#492	—	2	1	no	—
25	C₇×#492	C₂²	1	1	yes	characteristic
26	C₇×#492	C₂²	1	1	yes	characteristic
27	C₇×#487	C₂²	1	1	yes	characteristic
28	C₇×#486	C₂²	1	1	yes	characteristic
29	C₇×#483	—	2	1	no	—
30	C₇×#478	—	2	1	no	—
31	C₇×#477	—	2	1	no	—
32	C₇×#475	C₂²	1	1	yes	characteristic
33	C₇×#475	C₂²	1	1	yes	characteristic
34	C₇×#474	C₂²	1	1	yes	characteristic
35	C₇×#465	—	2	1	no	—
36	C₇×#465	—	2	1	no	—
37	C₇×#462	C₂²	1	1	yes	characteristic
38	C₇×#461	—	2	1	no	—
39	C₇×#461	C₂²	1	1	yes	characteristic
40	C₇×#461	C₂²	1	1	yes	characteristic
41	C₇×#460	—	2	1	no	—
42	C₇×#460	C₂²	1	1	yes	characteristic
43	C₇×#459	C₂²	1	1	yes	characteristic
44	C₇×#452	—	2	1	no	—
45	C₇×#449	—	2	1	no	—
46	C₇×#448	—	2	1	no	—
47	C₇×#446	C₂²	1	1	yes	characteristic
48	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
49	C₇×#440	—	2	1	no	—
50	C₄×C₇×SD₁₆	—	2	1	no	—
51	C₄×C₇×SD₁₆	C₂²	1	1	yes	characteristic
52	C₄×C₇×SD₁₆	C₂²	1	1	yes	characteristic
53	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
54	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
55	C₂×C₇×#103	—	2	1	no	—
56	C₇×#335	C₂²	1	1	yes	characteristic
57	C₇×#334	C₂²	1	1	yes	characteristic
58	C₇×#332	C₂²	1	1	yes	characteristic
59	C₇×#332	C₂²	1	1	yes	characteristic
60	C₇×#332	C₂²	1	1	yes	characteristic
61	C₇×#331	C₂²	1	1	yes	characteristic
62	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₇×#322	C₂²	1	1	yes	characteristic
67	#19767	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₇×#41	—	2	1	no	—
69	C₂×C₇×#41	—	2	1	no	—
70	C₂²×C₇×Q₈	—	2	1	no	—
71	C₇×#135	—	4	4	no	—
72	C₂×C₇×Q₁₆	—	2	1	no	—
73	C₂×C₇×SD₁₆	—	2	1	no	—
74	C₂×C₇×SD₁₆	—	2	1	no	—
75	C₂×C₇×SD₁₆	—	2	1	no	—
76	C₂×C₇×SD₁₆	—	2	1	no	—
77	C₂×C₇×SD₁₆	—	2	1	no	—
78	C₂×C₇×SD₁₆	—	2	1	no	—
79	C₂×C₇×SD₁₆	—	2	1	no	—
80	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
81	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
82	C₂×C₇×D₈	—	2	1	no	—
83	C₂²×C₇×C₈	—	2	1	no	—
84	C₂²×C₇×C₈	—	2	1	no	—
85	C₇×C₄⋊Q₈	—	2	1	no	—
86	C₇×C₄⋊Q₈	—	2	1	no	—
87	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₇×#127	—	2	1	no	—
91	C₇×#127	C₂³	1	1	yes	characteristic
92	C₇×#125	—	2	1	no	—
93	C₇×#125	—	4	1	no	—
94	C₇×#123	—	4	1	no	—
95	C₇×C₂²⋊Q₈	—	2	1	no	—
96	C₇×C₂²⋊Q₈	—	2	1	no	—
97	C₇×C₂²⋊Q₈	—	4	1	no	—
98	C₇×#121	—	2	1	no	—
99	C₇×#121	—	2	1	no	—
100	C₇×#121	—	4	1	no	—
101	C₄×C₇×Q₈	—	2	1	no	—
102	C₄×C₇×Q₈	—	2	1	no	—
103	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
104	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
105	C₄×C₇×D₄	—	2	1	no	—
106	C₄×C₇×D₄	—	2	1	no	—
107	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
108	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
109	C₂×C₇×C₄⋊C₄	—	2	1	no	—
110	C₇×#107	—	2	1	no	—
111	C₇×#107	—	2	1	no	—
112	C₇×#107	—	2	1	no	—
113	C₇×#106	—	2	1	no	—
114	C₇×#106	—	2	1	no	—
115	C₇×#106	—	4	1	no	—
116	C₇×#106	D₄	1	1	yes	characteristic
117	C₇×#106	D₄	1	1	yes	characteristic
118	C₇×C₄⋊C₈	—	2	1	no	—
119	C₇×C₄⋊C₈	—	2	1	no	—
120	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
124	C₇×#103	—	2	1	no	—
125	C₇×#103	—	2	1	no	—
126	C₇×#103	—	2	1	no	—
127	C₇×#103	—	2	1	no	—
128	C₇×#103	—	2	1	no	—
129	C₇×#103	—	2	1	no	—
130	C₇×#103	—	2	1	no	—
131	C₇×#103	D₄	1	1	yes	characteristic
132	C₇×#103	D₄	1	1	yes	characteristic
133	C₇×#102	—	2	1	no	—
134	C₇×#102	—	2	1	no	—
135	C₇×#102	—	2	1	no	—
136	C₇×#102	—	4	1	no	—
137	C₇×#102	D₄	1	1	yes	characteristic
138	C₇×#102	D₄	1	1	yes	characteristic
139	C₇×C₂²⋊C₈	—	2	1	no	—
140	C₇×C₂²⋊C₈	—	2	1	no	—
141	C₇×#97	—	2	1	no	—
142	C₄×C₇×C₈	—	2	1	no	—
143	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
145	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
146	#3392	C₂×C₇	1	1	yes	characteristic, complemented
147	#3386	C₂×C₇	1	1	yes	characteristic, complemented
148	#3298	C₂×C₇	1	1	yes	characteristic, complemented
149	#3284	C₂×C₇	1	1	yes	characteristic
150	#1714	C₂×C₇	1	1	yes	characteristic, complemented
151	#1686	C₂×C₇	1	1	yes	characteristic, complemented
152	#1680	C₂×C₇	1	1	yes	characteristic, complemented
153	#1673	C₂×C₇	1	1	yes	characteristic, complemented
154	#1665	C₂×C₇	1	1	yes	characteristic, complemented
155	#1654	C₂×C₇	1	1	yes	characteristic, complemented
156	#1653	C₂×C₇	1	1	yes	characteristic, complemented
157	#1642	C₂×C₇	1	1	yes	characteristic, complemented
158	#1627	C₂×C₇	1	1	yes	characteristic, complemented
159	#1582	C₂×C₇	1	1	yes	characteristic, complemented
160	SD₁₆⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
161	C₇×#41	—	4	2	no	—
162	C₇×#41	—	4	2	no	—
163	C₇×#41	—	8	1	no	—
164	C₇×#41	—	8	1	no	—
165	C₂×C₇×Q₈	—	2	1	no	—
166	C₂×C₇×Q₈	—	2	1	no	—
167	C₂×C₇×Q₈	—	2	1	no	—
168	C₂×C₇×Q₈	—	4	1	no	—
169	C₂×C₇×Q₈	—	4	2	no	—
170	C₂×C₇×Q₈	—	8	1	no	—
171	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
172	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
173	C₂×C₇×D₄	—	2	1	no	—
174	C₂×C₇×D₄	—	2	1	no	—
175	C₂×C₇×D₄	—	2	1	no	—
176	C₂×C₇×D₄	—	4	1	no	—
177	C₂×C₇×D₄	—	4	1	no	—
178	C₂×C₇×D₄	—	4	1	no	—
179	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
180	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₇	—	2	1	no	—
182	C₂²×C₄×C₇	—	2	1	no	—
183	C₂²×C₄×C₇	—	4	1	no	—
184	C₂²×C₄×C₇	—	4	1	no	—
185	C₂²×C₄×C₇	—	4	1	no	—
186	C₇×Q₁₆	—	4	2	no	—
187	C₇×SD₁₆	—	4	1	no	—
188	C₇×SD₁₆	—	4	2	no	—
189	C₇×SD₁₆	—	4	2	no	—
190	C₇×SD₁₆	—	8	1	no	—
191	C₇×SD₁₆	—	8	1	no	—
192	C₇×SD₁₆	—	8	1	no	—
193	C₇×SD₁₆	—	8	1	no	—
194	C₇×SD₁₆	—	8	1	no	—
195	C₇×SD₁₆	SD₁₆	1	4	yes	complemented
196	C₇×D₈	—	4	2	no	—
197	C₂×C₇×C₈	—	2	1	no	—
198	C₂×C₇×C₈	—	2	1	no	—
199	C₂×C₇×C₈	—	2	1	no	—
200	C₂×C₇×C₈	—	2	1	no	—
201	C₂×C₇×C₈	—	2	1	no	—
202	C₂×C₇×C₈	—	4	2	no	—
203	C₂×C₇×C₈	—	8	1	no	—
204	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
205	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
206	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
207	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
208	C₇×C₄⋊C₄	—	2	1	no	—
209	C₇×C₄⋊C₄	—	2	1	no	—
210	C₇×C₄⋊C₄	—	2	1	no	—
211	C₇×C₄⋊C₄	—	2	1	no	—
212	C₇×C₄⋊C₄	—	2	1	no	—
213	C₇×C₄⋊C₄	—	2	1	no	—
214	C₇×C₄⋊C₄	—	2	1	no	—
215	C₇×C₄⋊C₄	—	2	1	no	—
216	C₇×C₄⋊C₄	—	4	1	no	—
217	C₇×C₄⋊C₄	—	4	1	no	—
218	C₇×C₄⋊C₄	—	4	1	no	—
219	C₇×C₄⋊C₄	—	4	1	no	—
220	C₇×C₄⋊C₄	—	4	1	no	—
221	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
222	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
223	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
224	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
225	C₇×C₂²⋊C₄	—	2	1	no	—
226	C₇×C₂²⋊C₄	—	2	1	no	—
227	C₇×C₂²⋊C₄	—	4	1	no	—
228	C₇×C₂²⋊C₄	—	4	1	no	—
229	C₇×C₂²⋊C₄	—	4	1	no	—
230	C₄²×C₇	—	2	1	no	—
231	C₄²×C₇	—	2	1	no	—
232	C₄²×C₇	—	4	1	no	—
233	C₄²×C₇	C₂⁴	1	1	yes	characteristic
234	C₂×#135	—	2	1	no	—
235	#557	C₂²×C₇	1	1	yes	characteristic
236	#550	C₂²×C₇	1	1	yes	characteristic
237	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
238	#547	C₂²×C₇	1	1	yes	characteristic
239	#501	—	2	1	no	—
240	#500	C₂²×C₇	1	1	yes	characteristic
241	#498	C₂²×C₇	1	1	yes	characteristic
242	#492	—	2	1	no	—
243	#492	C₂²×C₇	1	1	yes	characteristic
244	#492	C₂²×C₇	1	1	yes	characteristic
245	#487	C₂²×C₇	1	1	yes	characteristic
246	#486	C₂²×C₇	1	1	yes	characteristic
247	#483	—	2	1	no	—
248	#478	—	2	1	no	—
249	#477	—	2	1	no	—
250	#475	C₂²×C₇	1	1	yes	characteristic
251	#475	C₂²×C₇	1	1	yes	characteristic
252	#474	C₂²×C₇	1	1	yes	characteristic
253	#465	—	2	1	no	—
254	#465	—	2	1	no	—
255	#462	C₂²×C₇	1	1	yes	characteristic
256	#461	—	2	1	no	—
257	#461	C₂²×C₇	1	1	yes	characteristic
258	#461	C₂²×C₇	1	1	yes	characteristic
259	#460	—	2	1	no	—
260	#460	C₂²×C₇	1	1	yes	characteristic
261	#459	C₂²×C₇	1	1	yes	characteristic
262	#452	—	2	1	no	—
263	#449	—	2	1	no	—
264	#448	—	2	1	no	—
265	#446	C₂²×C₇	1	1	yes	characteristic
266	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
267	#440	—	2	1	no	—
268	C₄×SD₁₆	—	2	1	no	—
269	C₄×SD₁₆	C₂²×C₇	1	1	yes	characteristic
270	C₄×SD₁₆	C₂²×C₇	1	1	yes	characteristic
271	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
272	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
273	C₂×#103	—	2	1	no	—
274	#335	C₂²×C₇	1	1	yes	characteristic
275	#334	C₂²×C₇	1	1	yes	characteristic
276	#332	C₂²×C₇	1	1	yes	characteristic
277	#332	C₂²×C₇	1	1	yes	characteristic
278	#332	C₂²×C₇	1	1	yes	characteristic
279	#331	C₂²×C₇	1	1	yes	characteristic
280	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
281	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
282	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
283	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
284	#322	C₂²×C₇	1	1	yes	characteristic
285	C₂³×C₇	—	2	1	no	—
286	C₂³×C₇	—	2	1	no	—
287	C₂³×C₇	—	4	1	no	—
288	C₇×Q₈	—	2	1	no	—
289	C₇×Q₈	—	2	1	no	—
290	C₇×Q₈	—	2	1	no	—
291	C₇×Q₈	—	4	1	no	—
292	C₇×Q₈	—	4	1	no	—
293	C₇×Q₈	—	4	1	no	—
294	C₇×Q₈	—	4	1	no	—
295	C₇×Q₈	—	4	1	no	—
296	C₇×Q₈	—	4	1	no	—
297	C₇×Q₈	—	8	2	no	—
298	C₇×Q₈	C₂×SD₁₆	1	2	yes	—
299	C₇×D₄	—	2	1	no	—
300	C₇×D₄	—	2	1	no	—
301	C₇×D₄	—	2	1	no	—
302	C₇×D₄	—	4	1	no	—
303	C₇×D₄	—	4	1	no	—
304	C₇×D₄	—	4	1	no	—
305	C₇×D₄	—	4	1	no	—
306	C₇×D₄	—	8	2	no	—
307	C₇×D₄	—	8	1	no	—
308	C₇×D₄	—	8	2	no	—
309	C₇×D₄	—	16	1	no	—
310	C₇×D₄	C₂×SD₁₆	1	2	yes	—
311	C₂×C₄×C₇	—	2	1	no	—
312	C₂×C₄×C₇	—	2	1	no	—
313	C₂×C₄×C₇	—	2	1	no	—
314	C₂×C₄×C₇	—	2	1	no	—
315	C₂×C₄×C₇	—	2	1	no	—
316	C₂×C₄×C₇	—	2	1	no	—
317	C₂×C₄×C₇	—	4	1	no	—
318	C₂×C₄×C₇	—	4	1	no	—
319	C₂×C₄×C₇	—	4	1	no	—
320	C₂×C₄×C₇	—	4	1	no	—
321	C₂×C₄×C₇	—	4	1	no	—
322	C₂×C₄×C₇	—	4	1	no	—
323	C₂×C₄×C₇	—	4	1	no	—
324	C₂×C₄×C₇	—	8	2	no	—
325	C₂×C₄×C₇	—	8	2	no	—
326	C₂×C₄×C₇	—	8	2	no	—
327	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
328	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
329	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
330	C₇×C₈	—	2	1	no	—
331	C₇×C₈	—	2	1	no	—
332	C₇×C₈	—	2	1	no	—
333	C₇×C₈	—	4	1	no	—
334	C₇×C₈	—	4	1	no	—
335	C₇×C₈	—	4	1	no	—
336	C₇×C₈	—	4	1	no	—
337	C₇×C₈	—	4	1	no	—
338	C₇×C₈	C₂×SD₁₆	1	2	yes	—
339	C₂×#41	—	2	1	no	—
340	C₂×#41	—	2	1	no	—
341	C₂²×Q₈	—	2	1	no	—
342	#135	—	4	4	no	—
343	C₂×Q₁₆	—	2	1	no	—
344	C₂×SD₁₆	—	2	1	no	—
345	C₂×SD₁₆	—	2	1	no	—
346	C₂×SD₁₆	—	2	1	no	—
347	C₂×SD₁₆	—	2	1	no	—
348	C₂×SD₁₆	—	2	1	no	—
349	C₂×SD₁₆	—	2	1	no	—
350	C₂×SD₁₆	—	2	1	no	—
351	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
352	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
353	C₂×D₈	—	2	1	no	—
354	C₂²×C₈	—	2	1	no	—
355	C₂²×C₈	—	2	1	no	—
356	C₄⋊Q₈	—	2	1	no	—
357	C₄⋊Q₈	—	2	1	no	—
358	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
359	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
360	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
361	#127	—	2	1	no	—
362	#127	C₂³×C₇	1	1	yes	characteristic
363	#125	—	2	1	no	—
364	#125	—	4	1	no	—
365	#123	—	4	1	no	—
366	C₂²⋊Q₈	—	2	1	no	—
367	C₂²⋊Q₈	—	2	1	no	—
368	C₂²⋊Q₈	—	4	1	no	—
369	#121	—	2	1	no	—
370	#121	—	2	1	no	—
371	#121	—	4	1	no	—
372	C₄×Q₈	—	2	1	no	—
373	C₄×Q₈	—	2	1	no	—
374	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
375	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
379	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
380	C₂×C₄⋊C₄	—	2	1	no	—
381	#107	—	2	1	no	—
382	#107	—	2	1	no	—
383	#107	—	2	1	no	—
384	#106	—	2	1	no	—
385	#106	—	2	1	no	—
386	#106	—	4	1	no	—
387	#106	C₇×D₄	1	1	yes	characteristic
388	#106	C₇×D₄	1	1	yes	characteristic
389	C₄⋊C₈	—	2	1	no	—
390	C₄⋊C₈	—	2	1	no	—
391	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
392	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
393	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
394	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
395	#103	—	2	1	no	—
396	#103	—	2	1	no	—
397	#103	—	2	1	no	—
398	#103	—	2	1	no	—
399	#103	—	2	1	no	—
400	#103	—	2	1	no	—
401	#103	—	2	1	no	—
402	#103	C₇×D₄	1	1	yes	characteristic
403	#103	C₇×D₄	1	1	yes	characteristic
404	#102	—	2	1	no	—
405	#102	—	2	1	no	—
406	#102	—	2	1	no	—
407	#102	—	4	1	no	—
408	#102	C₇×D₄	1	1	yes	characteristic
409	#102	C₇×D₄	1	1	yes	characteristic
410	C₂²⋊C₈	—	2	1	no	—
411	C₂²⋊C₈	—	2	1	no	—
412	#97	—	2	1	no	—
413	C₄×C₈	—	2	1	no	—
414	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
415	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
416	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
417	C₂²×C₇	—	2	2	no	—
418	C₂²×C₇	—	4	1	no	—
419	C₂²×C₇	—	4	1	no	—
420	C₂²×C₇	—	4	1	no	—
421	C₂²×C₇	—	4	1	no	—
422	C₂²×C₇	—	4	1	no	—
423	C₂²×C₇	—	8	1	no	—
424	C₂²×C₇	—	8	1	no	—
425	C₂²×C₇	—	8	1	no	—
426	C₂²×C₇	D₄²	1	1	yes	characteristic
427	C₄×C₇	—	2	2	no	—
428	C₄×C₇	—	2	1	no	—
429	C₄×C₇	—	4	1	no	—
430	C₄×C₇	—	4	1	no	—
431	C₄×C₇	—	4	1	no	—
432	C₄×C₇	—	4	1	no	—
433	C₄×C₇	—	4	1	no	—
434	C₄×C₇	—	8	1	no	—
435	C₄×C₇	—	8	1	no	—
436	C₄×C₇	—	8	1	no	—
437	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
438	C₄×C₇	#575	1	1	yes	characteristic
439	C₄×C₇	#576	1	1	yes	characteristic
440	C₄×C₇	#577	1	1	yes	characteristic
441	#41	—	4	2	no	—
442	#41	—	4	2	no	—
443	#41	—	8	1	no	—
444	#41	—	8	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	—	4	1	no	—
449	C₂×Q₈	—	4	2	no	—
450	C₂×Q₈	—	8	1	no	—
451	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
452	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	—	4	1	no	—
457	C₂×D₄	—	4	1	no	—
458	C₂×D₄	—	4	1	no	—
459	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
460	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	4	1	no	—
464	C₂²×C₄	—	4	1	no	—
465	C₂²×C₄	—	4	1	no	—
466	Q₁₆	—	4	2	no	—
467	SD₁₆	—	4	1	no	—
468	SD₁₆	—	4	2	no	—
469	SD₁₆	—	4	2	no	—
470	SD₁₆	—	8	1	no	—
471	SD₁₆	—	8	1	no	—
472	SD₁₆	—	8	1	no	—
473	SD₁₆	—	8	1	no	—
474	SD₁₆	—	8	1	no	—
475	SD₁₆	C₇×SD₁₆	1	4	yes	complemented
476	D₈	—	4	2	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	4	2	no	—
483	C₂×C₈	—	8	1	no	—
484	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
485	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
486	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
487	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	4	1	no	—
497	C₄⋊C₄	—	4	1	no	—
498	C₄⋊C₄	—	4	1	no	—
499	C₄⋊C₄	—	4	1	no	—
500	C₄⋊C₄	—	4	1	no	—
501	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
502	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
503	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
504	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
505	C₂²⋊C₄	—	2	1	no	—
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	4	1	no	—
508	C₂²⋊C₄	—	4	1	no	—
509	C₂²⋊C₄	—	4	1	no	—
510	C₄²	—	2	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	4	1	no	—
513	C₄²	C₂⁴×C₇	1	1	yes	characteristic
514	C₂×C₇	—	4	2	no	—
515	C₂×C₇	—	8	1	no	—
516	C₂×C₇	—	16	1	no	—
517	C₂×C₇	#3268	1	1	yes	characteristic
518	C₂×C₇	D₄×SD₁₆	1	1	yes	characteristic
519	C₂×C₇	#3289	1	1	yes	characteristic
520	C₂³	—	2	1	no	—
521	C₂³	—	2	1	no	—
522	C₂³	—	4	1	no	—
523	Q₈	—	2	1	no	—
524	Q₈	—	2	1	no	—
525	Q₈	—	2	1	no	—
526	Q₈	—	4	1	no	—
527	Q₈	—	4	1	no	—
528	Q₈	—	4	1	no	—
529	Q₈	—	4	1	no	—
530	Q₈	—	4	1	no	—
531	Q₈	—	4	1	no	—
532	Q₈	—	8	2	no	—
533	Q₈	C₂×C₇×SD₁₆	1	2	yes	—
534	D₄	—	2	1	no	—
535	D₄	—	2	1	no	—
536	D₄	—	2	1	no	—
537	D₄	—	4	1	no	—
538	D₄	—	4	1	no	—
539	D₄	—	4	1	no	—
540	D₄	—	4	1	no	—
541	D₄	—	8	2	no	—
542	D₄	—	8	1	no	—
543	D₄	—	8	2	no	—
544	D₄	—	16	1	no	—
545	D₄	C₂×C₇×SD₁₆	1	2	yes	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	8	2	no	—
560	C₂×C₄	—	8	2	no	—
561	C₂×C₄	—	8	2	no	—
562	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
563	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
564	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
565	C₈	—	2	1	no	—
566	C₈	—	2	1	no	—
567	C₈	—	2	1	no	—
568	C₈	—	4	1	no	—
569	C₈	—	4	1	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	1	no	—
572	C₈	—	4	1	no	—
573	C₈	C₂×C₇×SD₁₆	1	2	yes	—
574	C₇	#19767	1	1	yes	characteristic, direct product factor, Sylow
575	C₂²	—	2	2	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	8	1	no	—
582	C₂²	—	8	1	no	—
583	C₂²	—	8	1	no	—
584	C₂²	C₇×D₄²	1	1	yes	characteristic
585	C₄	—	2	2	no	—
586	C₄	—	2	1	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	1	no	—
590	C₄	—	4	1	no	—
591	C₄	—	4	1	no	—
592	C₄	—	8	1	no	—
593	C₄	—	8	1	no	—
594	C₄	—	8	1	no	—
595	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
596	C₄	C₇×#575	1	1	yes	characteristic
597	C₄	C₇×#576	1	1	yes	characteristic
598	C₄	C₇×#577	1	1	yes	characteristic
599	C₂	—	4	2	no	—
600	C₂	—	8	1	no	—
601	C₂	—	16	1	no	—
602	C₂	C₇×#3268	1	1	yes	characteristic
603	C₂	C₇×D₄×SD₁₆	1	1	yes	characteristic
604	C₂	C₇×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3284	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3392	C₂	semidirect
2. C₇×#3386	C₂	semidirect
3. C₇×#3298	C₂	semidirect
5. C₇×#1714	C₂	semidirect
6. C₇×#1686	C₂	semidirect
7. C₇×#1680	C₂	semidirect
8. C₇×#1673	C₂	semidirect
9. C₇×#1665	C₂	semidirect
10. C₇×#1654	C₂	semidirect
11. C₇×#1653	C₂	semidirect
12. C₇×#1642	C₂	semidirect
13. C₇×#1627	C₂	semidirect
14. C₇×#1582	C₂	semidirect
15. C₇×SD₁₆⋊C₈	C₂	semidirect
67. #19767	C₇	direct (N × H)
146. #3392	C₂×C₇	semidirect
147. #3386	C₂×C₇	semidirect
148. #3298	C₂×C₇	semidirect
150. #1714	C₂×C₇	semidirect
151. #1686	C₂×C₇	semidirect
152. #1680	C₂×C₇	semidirect
153. #1673	C₂×C₇	semidirect
154. #1665	C₂×C₇	semidirect
155. #1654	C₂×C₇	semidirect
156. #1653	C₂×C₇	semidirect
157. #1642	C₂×C₇	semidirect
158. #1627	C₂×C₇	semidirect
159. #1582	C₂×C₇	semidirect
160. SD₁₆⋊C₈	C₂×C₇	semidirect
195. C₇×SD₁₆	SD₁₆	semidirect
475. SD₁₆	C₇×SD₁₆	semidirect
574. C₇	#19767	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×#3392	2
C₇×#3386	2
C₇×#3298	2
C₇×#3284	2
C₇×#1714	2
C₇×#1686	2
C₇×#1680	2
C₇×#1673	2
C₇×#1665	2
C₇×#1654	2
C₇×#1653	2
C₇×#1642	2
C₇×#1627	2
C₇×#1582	2
C₇×SD₁₆⋊C₈	2
#19767	7

Subgroups — C7×#19767 #13196983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19767 #13196983