Subgroups — C₇×#19811 #13197027

Order 1792.

642 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3384	C₂	1	1	yes	maximal, characteristic
2	C₇×Q₈×D₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#3299	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1723	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1694	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1672	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1669	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1649	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1649	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1644	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1637	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1584	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1580	C₂	1	1	yes	maximal, characteristic
16	C₂²×C₇×Q₁₆	—	2	1	no	—
17	C₇×#554	C₂²	1	1	yes	characteristic
18	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
19	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
20	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
21	C₇×#501	—	2	1	no	—
22	C₇×#500	—	2	1	no	—
23	C₇×#500	C₂²	1	1	yes	characteristic
24	C₇×#500	C₂²	1	1	yes	characteristic
25	C₇×#498	C₂²	1	1	yes	characteristic
26	C₇×#494	C₂²	1	1	yes	characteristic
27	C₇×#487	C₂²	1	1	yes	characteristic
28	C₇×#487	C₂²	1	1	yes	characteristic
29	C₇×#484	—	2	1	no	—
30	C₇×#478	—	2	1	no	—
31	C₇×#477	C₂²	1	1	yes	characteristic
32	C₇×#476	—	2	1	no	—
33	C₇×#476	C₂²	1	1	yes	characteristic
34	C₇×#474	—	2	1	no	—
35	C₇×#474	—	2	1	no	—
36	C₇×#474	C₂²	1	1	yes	characteristic
37	C₇×#474	C₂²	1	1	yes	characteristic
38	C₇×#474	C₂²	1	1	yes	characteristic
39	C₇×#467	—	2	1	no	—
40	C₇×#467	—	2	1	no	—
41	C₇×#462	C₂²	1	1	yes	characteristic
42	C₇×#462	C₂²	1	1	yes	characteristic
43	C₇×#462	C₂²	1	1	yes	characteristic
44	C₇×#451	—	2	1	no	—
45	C₇×#451	—	2	1	no	—
46	C₇×#451	—	2	1	no	—
47	C₇×#446	C₂²	1	1	yes	characteristic
48	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
49	C₇×#442	—	2	1	no	—
50	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
51	C₄×C₇×D₈	—	2	1	no	—
52	C₄×C₇×D₈	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×#333	C₂²	1	1	yes	characteristic
59	C₇×#332	C₂²	1	1	yes	characteristic
60	C₇×#332	C₂²	1	1	yes	characteristic
61	C₇×#332	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	#19811	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₇×Q₈	—	2	1	no	—
69	C₂²×C₇×Q₈	—	2	1	no	—
70	C₂²×C₇×Q₈	—	2	1	no	—
71	C₂×C₇×Q₁₆	—	2	1	no	—
72	C₂×C₇×Q₁₆	—	2	1	no	—
73	C₂×C₇×Q₁₆	—	2	1	no	—
74	C₂×C₇×Q₁₆	—	2	1	no	—
75	C₂×C₇×Q₁₆	—	2	1	no	—
76	C₂×C₇×Q₁₆	—	4	4	no	—
77	C₂×C₇×Q₁₆	D₄	1	1	yes	characteristic
78	C₂×C₇×D₈	D₄	1	1	yes	characteristic
79	C₂²×C₇×C₈	—	2	1	no	—
80	C₂²×C₇×C₈	—	2	1	no	—
81	C₇×C₄⋊Q₈	—	2	1	no	—
82	C₇×C₄⋊Q₈	—	2	1	no	—
83	C₇×C₄⋊Q₈	—	2	1	no	—
84	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₇×#125	—	2	1	no	—
89	C₇×C₂²⋊Q₈	—	2	1	no	—
90	C₇×C₂²⋊Q₈	—	2	1	no	—
91	C₇×C₂²⋊Q₈	—	2	1	no	—
92	C₇×C₂²⋊Q₈	—	2	1	no	—
93	C₇×C₂²⋊Q₈	—	4	1	no	—
94	C₇×C₂²⋊Q₈	—	4	1	no	—
95	C₇×C₂²⋊Q₈	—	4	1	no	—
96	C₇×C₂²⋊Q₈	—	4	1	no	—
97	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
98	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
99	C₄×C₇×D₄	—	2	1	no	—
100	C₄×C₇×D₄	—	2	1	no	—
101	C₄×C₇×D₄	—	2	1	no	—
102	C₄×C₇×D₄	—	2	1	no	—
103	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
104	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
105	C₂×C₇×C₄⋊C₄	—	2	1	no	—
106	C₇×#107	—	2	1	no	—
107	C₇×#107	—	2	1	no	—
108	C₇×#107	—	2	1	no	—
109	C₇×#107	—	2	1	no	—
110	C₇×#107	—	2	1	no	—
111	C₇×#107	—	2	1	no	—
112	C₇×#107	—	4	1	no	—
113	C₇×#107	D₄	1	1	yes	characteristic
114	C₇×#107	D₄	1	1	yes	characteristic
115	C₇×#106	—	2	1	no	—
116	C₇×#106	—	2	1	no	—
117	C₇×#106	—	2	1	no	—
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	—	4	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	—	2	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	#3384	C₂×C₇	1	1	yes	characteristic
147	Q₈×D₈	C₂×C₇	1	1	yes	characteristic
148	#3299	C₂×C₇	1	1	yes	characteristic, complemented
149	D₄×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
150	#1723	C₂×C₇	1	1	yes	characteristic, complemented
151	#1694	C₂×C₇	1	1	yes	characteristic, complemented
152	#1683	C₂×C₇	1	1	yes	characteristic, complemented
153	#1672	C₂×C₇	1	1	yes	characteristic, complemented
154	#1669	C₂×C₇	1	1	yes	characteristic, complemented
155	#1649	C₂×C₇	1	1	yes	characteristic, complemented
156	#1649	C₂×C₇	1	1	yes	characteristic, complemented
157	#1644	C₂×C₇	1	1	yes	characteristic, complemented
158	#1637	C₂×C₇	1	1	yes	characteristic, complemented
159	#1584	C₂×C₇	1	1	yes	characteristic, complemented
160	#1580	C₂×C₇	1	1	yes	characteristic
161	C₂×C₇×Q₈	—	2	1	no	—
162	C₂×C₇×Q₈	—	2	1	no	—
163	C₂×C₇×Q₈	—	2	1	no	—
164	C₂×C₇×Q₈	—	2	1	no	—
165	C₂×C₇×Q₈	—	4	1	no	—
166	C₂×C₇×Q₈	—	4	1	no	—
167	C₂×C₇×Q₈	—	4	1	no	—
168	C₂×C₇×Q₈	—	4	2	no	—
169	C₂×C₇×Q₈	—	4	2	no	—
170	C₂×C₇×Q₈	—	4	2	no	—
171	C₂×C₇×Q₈	—	8	1	no	—
172	C₂×C₇×Q₈	—	8	1	no	—
173	C₂×C₇×Q₈	—	8	1	no	—
174	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
175	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
176	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
177	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₇	—	2	1	no	—
179	C₂²×C₄×C₇	—	2	1	no	—
180	C₂²×C₄×C₇	—	4	1	no	—
181	C₂²×C₄×C₇	—	4	1	no	—
182	C₂²×C₄×C₇	—	4	1	no	—
183	C₂²×C₄×C₇	—	4	1	no	—
184	C₇×Q₁₆	—	2	2	no	—
185	C₇×Q₁₆	—	4	2	no	—
186	C₇×Q₁₆	—	4	2	no	—
187	C₇×Q₁₆	—	4	2	no	—
188	C₇×Q₁₆	—	8	1	no	—
189	C₇×Q₁₆	—	8	1	no	—
190	C₇×D₈	Q₁₆	1	4	yes	complemented
191	C₂×C₇×C₈	—	2	1	no	—
192	C₂×C₇×C₈	—	2	1	no	—
193	C₂×C₇×C₈	—	2	1	no	—
194	C₂×C₇×C₈	—	2	1	no	—
195	C₂×C₇×C₈	—	2	1	no	—
196	C₂×C₇×C₈	—	4	2	no	—
197	C₂×C₇×C₈	—	8	1	no	—
198	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
199	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
200	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
201	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
202	C₇×C₄⋊C₄	—	2	1	no	—
203	C₇×C₄⋊C₄	—	2	1	no	—
204	C₇×C₄⋊C₄	—	2	1	no	—
205	C₇×C₄⋊C₄	—	2	1	no	—
206	C₇×C₄⋊C₄	—	2	1	no	—
207	C₇×C₄⋊C₄	—	2	1	no	—
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	—	4	1	no	—
213	C₇×C₄⋊C₄	—	4	1	no	—
214	C₇×C₄⋊C₄	—	4	1	no	—
215	C₇×C₄⋊C₄	—	4	1	no	—
216	C₇×C₄⋊C₄	—	4	1	no	—
217	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
218	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
219	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
220	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
221	C₇×C₂²⋊C₄	—	2	1	no	—
222	C₇×C₂²⋊C₄	—	2	1	no	—
223	C₇×C₂²⋊C₄	—	4	1	no	—
224	C₇×C₂²⋊C₄	—	4	1	no	—
225	C₇×C₂²⋊C₄	—	4	1	no	—
226	C₇×C₂²⋊C₄	—	4	1	no	—
227	C₄²×C₇	—	2	1	no	—
228	C₄²×C₇	—	2	1	no	—
229	C₄²×C₇	C₂⁴	1	1	yes	characteristic
230	C₂²×Q₁₆	—	2	1	no	—
231	#554	C₂²×C₇	1	1	yes	characteristic
232	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
233	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
234	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
235	#501	—	2	1	no	—
236	#500	—	2	1	no	—
237	#500	C₂²×C₇	1	1	yes	characteristic
238	#500	C₂²×C₇	1	1	yes	characteristic
239	#498	C₂²×C₇	1	1	yes	characteristic
240	#494	C₂²×C₇	1	1	yes	characteristic
241	#487	C₂²×C₇	1	1	yes	characteristic
242	#487	C₂²×C₇	1	1	yes	characteristic
243	#484	—	2	1	no	—
244	#478	—	2	1	no	—
245	#477	C₂²×C₇	1	1	yes	characteristic
246	#476	—	2	1	no	—
247	#476	C₂²×C₇	1	1	yes	characteristic
248	#474	—	2	1	no	—
249	#474	—	2	1	no	—
250	#474	C₂²×C₇	1	1	yes	characteristic
251	#474	C₂²×C₇	1	1	yes	characteristic
252	#474	C₂²×C₇	1	1	yes	characteristic
253	#467	—	2	1	no	—
254	#467	—	2	1	no	—
255	#462	C₂²×C₇	1	1	yes	characteristic
256	#462	C₂²×C₇	1	1	yes	characteristic
257	#462	C₂²×C₇	1	1	yes	characteristic
258	#451	—	2	1	no	—
259	#451	—	2	1	no	—
260	#451	—	2	1	no	—
261	#446	C₂²×C₇	1	1	yes	characteristic
262	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
263	#442	—	2	1	no	—
264	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
265	C₄×D₈	—	2	1	no	—
266	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
267	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
268	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
269	C₂×#103	—	2	1	no	—
270	#335	C₂²×C₇	1	1	yes	characteristic
271	#334	C₂²×C₇	1	1	yes	characteristic
272	#333	C₂²×C₇	1	1	yes	characteristic
273	#332	C₂²×C₇	1	1	yes	characteristic
274	#332	C₂²×C₇	1	1	yes	characteristic
275	#332	C₂²×C₇	1	1	yes	characteristic
276	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
277	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
278	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
279	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
280	#322	C₂²×C₇	1	1	yes	characteristic
281	C₂³×C₇	—	2	1	no	—
282	C₂³×C₇	—	2	1	no	—
283	C₇×Q₈	—	2	1	no	—
284	C₇×Q₈	—	2	1	no	—
285	C₇×Q₈	—	2	1	no	—
286	C₇×Q₈	—	4	1	no	—
287	C₇×Q₈	—	4	1	no	—
288	C₇×Q₈	—	4	1	no	—
289	C₇×Q₈	—	4	1	no	—
290	C₇×Q₈	—	4	1	no	—
291	C₇×Q₈	—	4	1	no	—
292	C₇×Q₈	—	8	2	no	—
293	C₇×Q₈	—	8	2	no	—
294	C₇×Q₈	—	8	1	no	—
295	C₇×Q₈	—	8	2	no	—
296	C₇×Q₈	AGL(1,ℤ/8)	1	2	yes	—
297	C₇×D₄	—	2	1	no	—
298	C₇×D₄	—	2	1	no	—
299	C₇×D₄	C₂×Q₁₆	1	2	yes	—
300	C₇×D₄	C₂×Q₁₆	1	2	yes	complemented
301	C₂×C₄×C₇	—	2	1	no	—
302	C₂×C₄×C₇	—	2	1	no	—
303	C₂×C₄×C₇	—	2	1	no	—
304	C₂×C₄×C₇	—	2	1	no	—
305	C₂×C₄×C₇	—	2	1	no	—
306	C₂×C₄×C₇	—	2	1	no	—
307	C₂×C₄×C₇	—	4	1	no	—
308	C₂×C₄×C₇	—	4	1	no	—
309	C₂×C₄×C₇	—	4	1	no	—
310	C₂×C₄×C₇	—	4	1	no	—
311	C₂×C₄×C₇	—	4	1	no	—
312	C₂×C₄×C₇	—	4	1	no	—
313	C₂×C₄×C₇	—	4	1	no	—
314	C₂×C₄×C₇	—	4	1	no	—
315	C₂×C₄×C₇	—	8	2	no	—
316	C₂×C₄×C₇	—	8	2	no	—
317	C₂×C₄×C₇	—	8	2	no	—
318	C₂×C₄×C₇	—	16	1	no	—
319	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
322	C₇×C₈	—	2	1	no	—
323	C₇×C₈	—	2	1	no	—
324	C₇×C₈	—	2	1	no	—
325	C₇×C₈	—	4	1	no	—
326	C₇×C₈	—	4	1	no	—
327	C₇×C₈	—	4	1	no	—
328	C₇×C₈	—	4	1	no	—
329	C₇×C₈	—	4	1	no	—
330	C₇×C₈	C₂×Q₁₆	1	2	yes	complemented
331	C₂²×Q₈	—	2	1	no	—
332	C₂²×Q₈	—	2	1	no	—
333	C₂²×Q₈	—	2	1	no	—
334	C₂×Q₁₆	—	2	1	no	—
335	C₂×Q₁₆	—	2	1	no	—
336	C₂×Q₁₆	—	2	1	no	—
337	C₂×Q₁₆	—	2	1	no	—
338	C₂×Q₁₆	—	2	1	no	—
339	C₂×Q₁₆	—	4	4	no	—
340	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
341	C₂×D₈	C₇×D₄	1	1	yes	characteristic
342	C₂²×C₈	—	2	1	no	—
343	C₂²×C₈	—	2	1	no	—
344	C₄⋊Q₈	—	2	1	no	—
345	C₄⋊Q₈	—	2	1	no	—
346	C₄⋊Q₈	—	2	1	no	—
347	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
348	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
349	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
350	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
351	#125	—	2	1	no	—
352	C₂²⋊Q₈	—	2	1	no	—
353	C₂²⋊Q₈	—	2	1	no	—
354	C₂²⋊Q₈	—	2	1	no	—
355	C₂²⋊Q₈	—	2	1	no	—
356	C₂²⋊Q₈	—	4	1	no	—
357	C₂²⋊Q₈	—	4	1	no	—
358	C₂²⋊Q₈	—	4	1	no	—
359	C₂²⋊Q₈	—	4	1	no	—
360	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
361	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
362	C₄×D₄	—	2	1	no	—
363	C₄×D₄	—	2	1	no	—
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
367	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
368	C₂×C₄⋊C₄	—	2	1	no	—
369	#107	—	2	1	no	—
370	#107	—	2	1	no	—
371	#107	—	2	1	no	—
372	#107	—	2	1	no	—
373	#107	—	2	1	no	—
374	#107	—	2	1	no	—
375	#107	—	4	1	no	—
376	#107	C₇×D₄	1	1	yes	characteristic
377	#107	C₇×D₄	1	1	yes	characteristic
378	#106	—	2	1	no	—
379	#106	—	2	1	no	—
380	#106	—	2	1	no	—
381	C₄⋊C₈	—	2	1	no	—
382	C₄⋊C₈	—	2	1	no	—
383	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
384	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
385	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
386	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
387	#103	—	2	1	no	—
388	#103	—	2	1	no	—
389	#103	—	2	1	no	—
390	#103	—	2	1	no	—
391	#103	—	2	1	no	—
392	#103	—	2	1	no	—
393	#103	—	4	1	no	—
394	#103	C₇×D₄	1	1	yes	characteristic
395	#103	C₇×D₄	1	1	yes	characteristic
396	#102	—	2	1	no	—
397	#102	—	2	1	no	—
398	#102	—	2	1	no	—
399	#102	—	2	1	no	—
400	#102	C₇×D₄	1	1	yes	characteristic
401	#102	C₇×D₄	1	1	yes	characteristic
402	C₂²⋊C₈	—	2	1	no	—
403	C₂²⋊C₈	—	2	1	no	—
404	#97	—	2	1	no	—
405	C₄×C₈	—	2	1	no	—
406	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
407	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
408	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
409	C₂²×C₇	—	2	2	no	—
410	C₂²×C₇	—	2	2	no	—
411	C₂²×C₇	—	4	1	no	—
412	C₂²×C₇	—	4	1	no	—
413	C₂²×C₇	—	4	1	no	—
414	C₂²×C₇	—	4	1	no	—
415	C₂²×C₇	D₄²	1	1	yes	characteristic
416	C₄×C₇	—	2	2	no	—
417	C₄×C₇	—	2	1	no	—
418	C₄×C₇	—	4	1	no	—
419	C₄×C₇	—	4	1	no	—
420	C₄×C₇	—	4	1	no	—
421	C₄×C₇	—	4	1	no	—
422	C₄×C₇	—	4	1	no	—
423	C₄×C₇	—	8	1	no	—
424	C₄×C₇	—	8	1	no	—
425	C₄×C₇	—	8	1	no	—
426	C₄×C₇	—	8	1	no	—
427	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
428	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
429	C₄×C₇	#576	1	1	yes	characteristic
430	C₄×C₇	#578	1	1	yes	characteristic
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	4	1	no	—
436	C₂×Q₈	—	4	1	no	—
437	C₂×Q₈	—	4	1	no	—
438	C₂×Q₈	—	4	2	no	—
439	C₂×Q₈	—	4	2	no	—
440	C₂×Q₈	—	4	2	no	—
441	C₂×Q₈	—	8	1	no	—
442	C₂×Q₈	—	8	1	no	—
443	C₂×Q₈	—	8	1	no	—
444	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
445	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
446	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
447	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	4	1	no	—
451	C₂²×C₄	—	4	1	no	—
452	C₂²×C₄	—	4	1	no	—
453	C₂²×C₄	—	4	1	no	—
454	Q₁₆	—	2	2	no	—
455	Q₁₆	—	4	2	no	—
456	Q₁₆	—	4	2	no	—
457	Q₁₆	—	4	2	no	—
458	Q₁₆	—	8	1	no	—
459	Q₁₆	—	8	1	no	—
460	D₈	C₇×Q₁₆	1	4	yes	complemented
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	4	2	no	—
467	C₂×C₈	—	8	1	no	—
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	1	yes	characteristic
471	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	4	1	no	—
483	C₄⋊C₄	—	4	1	no	—
484	C₄⋊C₄	—	4	1	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₄⋊C₄	—	4	1	no	—
487	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
488	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
489	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
490	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
491	C₂²⋊C₄	—	2	1	no	—
492	C₂²⋊C₄	—	2	1	no	—
493	C₂²⋊C₄	—	4	1	no	—
494	C₂²⋊C₄	—	4	1	no	—
495	C₂²⋊C₄	—	4	1	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	C₂⁴×C₇	1	1	yes	characteristic
500	C₂×C₇	—	4	2	no	—
501	C₂×C₇	—	4	2	no	—
502	C₂×C₇	#3264	1	1	yes	characteristic
503	C₂×C₇	D₄×Q₁₆	1	1	yes	characteristic
504	C₂×C₇	#3289	1	1	yes	characteristic
505	C₂³	—	2	1	no	—
506	C₂³	—	2	1	no	—
507	Q₈	—	2	1	no	—
508	Q₈	—	2	1	no	—
509	Q₈	—	2	1	no	—
510	Q₈	—	4	1	no	—
511	Q₈	—	4	1	no	—
512	Q₈	—	4	1	no	—
513	Q₈	—	4	1	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	4	1	no	—
516	Q₈	—	8	2	no	—
517	Q₈	—	8	2	no	—
518	Q₈	—	8	1	no	—
519	Q₈	—	8	2	no	—
520	Q₈	C₇×AGL(1,ℤ/8)	1	2	yes	—
521	D₄	—	2	1	no	—
522	D₄	—	2	1	no	—
523	D₄	C₂×C₇×Q₁₆	1	2	yes	—
524	D₄	C₂×C₇×Q₁₆	1	2	yes	complemented
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	8	2	no	—
540	C₂×C₄	—	8	2	no	—
541	C₂×C₄	—	8	2	no	—
542	C₂×C₄	—	16	1	no	—
543	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
544	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
545	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
546	C₈	—	2	1	no	—
547	C₈	—	2	1	no	—
548	C₈	—	2	1	no	—
549	C₈	—	4	1	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	1	no	—
552	C₈	—	4	1	no	—
553	C₈	—	4	1	no	—
554	C₈	C₂×C₇×Q₁₆	1	2	yes	complemented
555	C₇	#19811	1	1	yes	characteristic, direct product factor, Sylow
556	C₂²	—	2	2	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	4	1	no	—
559	C₂²	—	4	1	no	—
560	C₂²	—	4	1	no	—
561	C₂²	—	4	1	no	—
562	C₂²	C₇×D₄²	1	1	yes	characteristic
563	C₄	—	2	2	no	—
564	C₄	—	2	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	8	1	no	—
571	C₄	—	8	1	no	—
572	C₄	—	8	1	no	—
573	C₄	—	8	1	no	—
574	C₄	C₂²×C₇×Q₁₆	1	1	yes	characteristic
575	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
576	C₄	C₇×#576	1	1	yes	characteristic
577	C₄	C₇×#578	1	1	yes	characteristic
578	C₂	—	4	2	no	—
579	C₂	—	4	2	no	—
580	C₂	C₇×#3264	1	1	yes	characteristic
581	C₂	C₇×D₄×Q₁₆	1	1	yes	characteristic
582	C₂	C₇×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×D₈	D₄
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
3. C₇×#3299	C₂	semidirect
4. C₇×D₄×Q₁₆	C₂	semidirect
5. C₇×#1723	C₂	semidirect
6. C₇×#1694	C₂	semidirect
7. C₇×#1683	C₂	semidirect
8. C₇×#1672	C₂	semidirect
9. C₇×#1669	C₂	semidirect
10. C₇×#1649	C₂	semidirect
11. C₇×#1649	C₂	semidirect
12. C₇×#1644	C₂	semidirect
13. C₇×#1637	C₂	semidirect
14. C₇×#1584	C₂	semidirect
67. #19811	C₇	direct (N × H)
148. #3299	C₂×C₇	semidirect
149. D₄×Q₁₆	C₂×C₇	semidirect
150. #1723	C₂×C₇	semidirect
151. #1694	C₂×C₇	semidirect
152. #1683	C₂×C₇	semidirect
153. #1672	C₂×C₇	semidirect
154. #1669	C₂×C₇	semidirect
155. #1649	C₂×C₇	semidirect
156. #1649	C₂×C₇	semidirect
157. #1644	C₂×C₇	semidirect
158. #1637	C₂×C₇	semidirect
159. #1584	C₂×C₇	semidirect
190. C₇×D₈	Q₁₆	semidirect
300. C₇×D₄	C₂×Q₁₆	semidirect
330. C₇×C₈	C₂×Q₁₆	semidirect
460. D₈	C₇×Q₁₆	semidirect
524. D₄	C₂×C₇×Q₁₆	semidirect
554. C₈	C₂×C₇×Q₁₆	semidirect
555. C₇	#19811	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×#3384	2
C₇×Q₈×D₈	2
C₇×#3299	2
C₇×D₄×Q₁₆	2
C₇×#1723	2
C₇×#1694	2
C₇×#1683	2
C₇×#1672	2
C₇×#1669	2
C₇×#1649	2
C₇×#1644	2
C₇×#1637	2
C₇×#1584	2
C₇×#1580	2
#19811	7

Subgroups — C7×#19811 #13197027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19811 #13197027