Subgroups — C₇×#19796 #13197012

Order 1792.

598 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3369	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3302	C₂	1	1	yes	maximal, characteristic
4	C₇×#3298	C₂	1	1	yes	maximal, characteristic
5	C₇×#1718	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1700	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1680	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1671	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1650	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1649	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1645	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1635	C₂	1	1	yes	maximal, characteristic
14	C₇×#1582	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×Q₈²	C₂²	1	1	yes	characteristic
17	C₇×#556	C₂²	1	1	yes	characteristic
18	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
19	C₇×#547	C₂²	1	1	yes	characteristic
20	C₇×#500	C₂²	1	1	yes	characteristic
21	C₇×#498	—	2	1	no	—
22	C₇×#498	C₂²	1	1	yes	characteristic
23	C₇×#489	—	2	1	no	—
24	C₇×#487	C₂²	1	1	yes	characteristic
25	C₇×#487	C₂²	1	1	yes	characteristic
26	C₇×#487	C₂²	1	1	yes	characteristic
27	C₇×#487	C₂²	1	1	yes	characteristic
28	C₇×#485	—	2	1	no	—
29	C₇×#483	—	2	1	no	—
30	C₇×#476	—	2	1	no	—
31	C₇×#476	C₂²	1	1	yes	characteristic
32	C₇×#476	C₂²	1	1	yes	characteristic
33	C₇×#475	—	2	1	no	—
34	C₇×#475	—	2	1	no	—
35	C₇×#475	C₂²	1	1	yes	characteristic
36	C₇×#475	C₂²	1	1	yes	characteristic
37	C₇×#474	C₂²	1	1	yes	characteristic
38	C₇×#467	—	2	1	no	—
39	C₇×#465	—	2	1	no	—
40	C₇×#464	—	2	1	no	—
41	C₇×#462	C₂²	1	1	yes	characteristic
42	C₇×#462	C₂²	1	1	yes	characteristic
43	C₇×#461	C₂²	1	1	yes	characteristic
44	C₇×#452	—	2	1	no	—
45	C₇×#448	—	2	1	no	—
46	C₇×#446	C₂²	1	1	yes	characteristic
47	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
48	C₇×#441	—	2	1	no	—
49	C₄×C₇×SD₁₆	—	2	1	no	—
50	C₄×C₇×SD₁₆	C₂²	1	1	yes	characteristic
51	C₄×C₇×SD₁₆	C₂²	1	1	yes	characteristic
52	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
53	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
54	C₇×#416	—	2	1	no	—
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	#19796	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₇×#41	—	2	1	no	—
69	C₂²×C₇×Q₈	—	2	1	no	—
70	C₂×C₇×Q₁₆	—	2	1	no	—
71	C₂×C₇×Q₁₆	—	2	1	no	—
72	C₂×C₇×SD₁₆	—	2	1	no	—
73	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
74	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
75	C₂²×C₇×C₈	—	2	1	no	—
76	C₂²×C₇×C₈	—	2	1	no	—
77	C₇×C₄⋊Q₈	—	2	1	no	—
78	C₇×C₄⋊Q₈	—	2	1	no	—
79	C₇×C₄⋊Q₈	—	2	1	no	—
80	C₇×C₄⋊Q₈	—	4	1	no	—
81	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₇×#126	—	4	1	no	—
86	C₇×#124	—	2	1	no	—
87	C₇×#123	—	4	1	no	—
88	C₇×C₂²⋊Q₈	—	2	1	no	—
89	C₇×C₂²⋊Q₈	—	2	1	no	—
90	C₇×C₂²⋊Q₈	—	2	1	no	—
91	C₇×C₂²⋊Q₈	—	4	1	no	—
92	C₇×#121	—	2	1	no	—
93	C₄×C₇×Q₈	—	2	1	no	—
94	C₄×C₇×Q₈	—	2	1	no	—
95	C₄×C₇×Q₈	—	2	1	no	—
96	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
97	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
98	C₄×C₇×D₄	—	2	1	no	—
99	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
100	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
101	C₇×#117	—	2	1	no	—
102	C₂×C₇×C₄⋊C₄	—	2	1	no	—
103	C₇×#107	—	2	1	no	—
104	C₇×#107	—	4	1	no	—
105	C₇×#106	—	2	1	no	—
106	C₇×#106	—	2	1	no	—
107	C₇×#106	—	2	1	no	—
108	C₇×#106	—	2	1	no	—
109	C₇×#106	—	2	1	no	—
110	C₇×#106	—	2	1	no	—
111	C₇×#106	D₄	1	1	yes	characteristic
112	C₇×#106	D₄	1	1	yes	characteristic
113	C₇×C₄⋊C₈	—	2	1	no	—
114	C₇×C₄⋊C₈	—	2	1	no	—
115	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₇×#103	—	2	1	no	—
120	C₇×#103	—	2	1	no	—
121	C₇×#103	—	2	1	no	—
122	C₇×#103	—	2	1	no	—
123	C₇×#103	—	2	1	no	—
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	D₄	1	1	yes	characteristic
137	C₇×#102	D₄	1	1	yes	characteristic
138	C₇×C₂²⋊C₈	—	2	1	no	—
139	C₇×C₂²⋊C₈	—	2	1	no	—
140	C₇×#97	—	2	1	no	—
141	C₄×C₇×C₈	—	2	1	no	—
142	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
143	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
145	Q₈×SD₁₆	C₂×C₇	1	1	yes	characteristic, complemented
146	#3369	C₂×C₇	1	1	yes	characteristic, complemented
147	#3302	C₂×C₇	1	1	yes	characteristic
148	#3298	C₂×C₇	1	1	yes	characteristic
149	#1718	C₂×C₇	1	1	yes	characteristic, complemented
150	#1700	C₂×C₇	1	1	yes	characteristic, complemented
151	#1683	C₂×C₇	1	1	yes	characteristic, complemented
152	#1680	C₂×C₇	1	1	yes	characteristic, complemented
153	#1671	C₂×C₇	1	1	yes	characteristic, complemented
154	#1650	C₂×C₇	1	1	yes	characteristic, complemented
155	#1649	C₂×C₇	1	1	yes	characteristic, complemented
156	#1645	C₂×C₇	1	1	yes	characteristic, complemented
157	#1635	C₂×C₇	1	1	yes	characteristic
158	#1582	C₂×C₇	1	1	yes	characteristic, complemented
159	SD₁₆⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
160	C₇×#41	—	4	2	no	—
161	C₇×#41	—	8	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	—	2	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	—	8	1	no	—
170	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
171	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
172	C₂×C₇×D₄	—	4	1	no	—
173	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
174	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₇	—	2	1	no	—
176	C₂²×C₄×C₇	—	2	1	no	—
177	C₂²×C₄×C₇	—	4	1	no	—
178	C₂²×C₄×C₇	—	4	1	no	—
179	C₇×Q₁₆	—	4	2	no	—
180	C₇×Q₁₆	—	8	1	no	—
181	C₇×SD₁₆	—	2	2	no	—
182	C₇×SD₁₆	—	2	2	no	—
183	C₇×SD₁₆	—	8	1	no	—
184	C₂×C₇×C₈	—	2	1	no	—
185	C₂×C₇×C₈	—	2	1	no	—
186	C₂×C₇×C₈	—	2	1	no	—
187	C₂×C₇×C₈	—	2	1	no	—
188	C₂×C₇×C₈	—	2	1	no	—
189	C₂×C₇×C₈	—	8	1	no	—
190	C₂×C₇×C₈	—	8	1	no	—
191	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
192	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
193	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
194	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
195	C₇×C₄⋊C₄	—	2	1	no	—
196	C₇×C₄⋊C₄	—	2	1	no	—
197	C₇×C₄⋊C₄	—	2	1	no	—
198	C₇×C₄⋊C₄	—	2	1	no	—
199	C₇×C₄⋊C₄	—	2	1	no	—
200	C₇×C₄⋊C₄	—	2	1	no	—
201	C₇×C₄⋊C₄	—	2	1	no	—
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	—	4	1	no	—
206	C₇×C₄⋊C₄	—	4	1	no	—
207	C₇×C₄⋊C₄	—	4	1	no	—
208	C₇×C₄⋊C₄	—	4	1	no	—
209	C₇×C₄⋊C₄	—	4	1	no	—
210	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
211	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
212	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
213	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
214	C₇×C₂²⋊C₄	—	2	1	no	—
215	C₇×C₂²⋊C₄	—	2	1	no	—
216	C₇×C₂²⋊C₄	—	4	1	no	—
217	C₇×C₂²⋊C₄	—	4	1	no	—
218	C₇×C₂²⋊C₄	—	4	1	no	—
219	C₇×C₂²⋊C₄	—	4	1	no	—
220	C₄²×C₇	—	2	1	no	—
221	C₄²×C₇	—	2	1	no	—
222	C₄²×C₇	—	4	1	no	—
223	C₄²×C₇	—	4	1	no	—
224	C₄²×C₇	C₂⁴	1	1	yes	characteristic
225	Q₈²	C₂²×C₇	1	1	yes	characteristic
226	#556	C₂²×C₇	1	1	yes	characteristic
227	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
228	#547	C₂²×C₇	1	1	yes	characteristic
229	#500	C₂²×C₇	1	1	yes	characteristic
230	#498	—	2	1	no	—
231	#498	C₂²×C₇	1	1	yes	characteristic
232	#489	—	2	1	no	—
233	#487	C₂²×C₇	1	1	yes	characteristic
234	#487	C₂²×C₇	1	1	yes	characteristic
235	#487	C₂²×C₇	1	1	yes	characteristic
236	#487	C₂²×C₇	1	1	yes	characteristic
237	#485	—	2	1	no	—
238	#483	—	2	1	no	—
239	#476	—	2	1	no	—
240	#476	C₂²×C₇	1	1	yes	characteristic
241	#476	C₂²×C₇	1	1	yes	characteristic
242	#475	—	2	1	no	—
243	#475	—	2	1	no	—
244	#475	C₂²×C₇	1	1	yes	characteristic
245	#475	C₂²×C₇	1	1	yes	characteristic
246	#474	C₂²×C₇	1	1	yes	characteristic
247	#467	—	2	1	no	—
248	#465	—	2	1	no	—
249	#464	—	2	1	no	—
250	#462	C₂²×C₇	1	1	yes	characteristic
251	#462	C₂²×C₇	1	1	yes	characteristic
252	#461	C₂²×C₇	1	1	yes	characteristic
253	#452	—	2	1	no	—
254	#448	—	2	1	no	—
255	#446	C₂²×C₇	1	1	yes	characteristic
256	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
257	#441	—	2	1	no	—
258	C₄×SD₁₆	—	2	1	no	—
259	C₄×SD₁₆	C₂²×C₇	1	1	yes	characteristic
260	C₄×SD₁₆	C₂²×C₇	1	1	yes	characteristic
261	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
262	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
263	#416	—	2	1	no	—
264	C₂×#103	—	2	1	no	—
265	#335	C₂²×C₇	1	1	yes	characteristic
266	#334	C₂²×C₇	1	1	yes	characteristic
267	#333	C₂²×C₇	1	1	yes	characteristic
268	#332	C₂²×C₇	1	1	yes	characteristic
269	#332	C₂²×C₇	1	1	yes	characteristic
270	#332	C₂²×C₇	1	1	yes	characteristic
271	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
272	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
273	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
274	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
275	#322	C₂²×C₇	1	1	yes	characteristic
276	C₂³×C₇	—	2	1	no	—
277	C₂³×C₇	—	2	1	no	—
278	C₇×Q₈	—	2	1	no	—
279	C₇×Q₈	—	2	1	no	—
280	C₇×Q₈	—	2	1	no	—
281	C₇×Q₈	—	4	1	no	—
282	C₇×Q₈	—	4	1	no	—
283	C₇×Q₈	—	4	1	no	—
284	C₇×Q₈	—	4	1	no	—
285	C₇×Q₈	—	4	1	no	—
286	C₇×Q₈	—	4	1	no	—
287	C₇×Q₈	—	8	2	no	—
288	C₇×Q₈	—	8	1	no	—
289	C₇×Q₈	—	16	1	no	—
290	C₇×Q₈	#135	1	2	yes	—
291	C₇×D₄	—	2	1	no	—
292	C₇×D₄	—	2	1	no	—
293	C₇×D₄	—	2	1	no	—
294	C₇×D₄	—	8	2	no	—
295	C₇×D₄	#137	1	2	yes	—
296	C₂×C₄×C₇	—	2	1	no	—
297	C₂×C₄×C₇	—	2	1	no	—
298	C₂×C₄×C₇	—	2	1	no	—
299	C₂×C₄×C₇	—	2	1	no	—
300	C₂×C₄×C₇	—	2	1	no	—
301	C₂×C₄×C₇	—	2	1	no	—
302	C₂×C₄×C₇	—	4	1	no	—
303	C₂×C₄×C₇	—	4	1	no	—
304	C₂×C₄×C₇	—	4	1	no	—
305	C₂×C₄×C₇	—	4	1	no	—
306	C₂×C₄×C₇	—	4	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	—	8	2	no	—
311	C₂×C₄×C₇	—	8	2	no	—
312	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
313	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
315	C₇×C₈	—	2	1	no	—
316	C₇×C₈	—	2	1	no	—
317	C₇×C₈	—	2	1	no	—
318	C₇×C₈	—	2	1	no	—
319	C₇×C₈	—	4	1	no	—
320	C₇×C₈	—	4	1	no	—
321	C₇×C₈	—	4	1	no	—
322	C₇×C₈	—	4	1	no	—
323	C₇×C₈	—	4	1	no	—
324	C₂×#41	—	2	1	no	—
325	C₂²×Q₈	—	2	1	no	—
326	C₂×Q₁₆	—	2	1	no	—
327	C₂×Q₁₆	—	2	1	no	—
328	C₂×SD₁₆	—	2	1	no	—
329	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
330	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
331	C₂²×C₈	—	2	1	no	—
332	C₂²×C₈	—	2	1	no	—
333	C₄⋊Q₈	—	2	1	no	—
334	C₄⋊Q₈	—	2	1	no	—
335	C₄⋊Q₈	—	2	1	no	—
336	C₄⋊Q₈	—	4	1	no	—
337	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
338	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
339	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
340	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
341	#126	—	4	1	no	—
342	#124	—	2	1	no	—
343	#123	—	4	1	no	—
344	C₂²⋊Q₈	—	2	1	no	—
345	C₂²⋊Q₈	—	2	1	no	—
346	C₂²⋊Q₈	—	2	1	no	—
347	C₂²⋊Q₈	—	4	1	no	—
348	#121	—	2	1	no	—
349	C₄×Q₈	—	2	1	no	—
350	C₄×Q₈	—	2	1	no	—
351	C₄×Q₈	—	2	1	no	—
352	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
353	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
354	C₄×D₄	—	2	1	no	—
355	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
356	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
357	#117	—	2	1	no	—
358	C₂×C₄⋊C₄	—	2	1	no	—
359	#107	—	2	1	no	—
360	#107	—	4	1	no	—
361	#106	—	2	1	no	—
362	#106	—	2	1	no	—
363	#106	—	2	1	no	—
364	#106	—	2	1	no	—
365	#106	—	2	1	no	—
366	#106	—	2	1	no	—
367	#106	C₇×D₄	1	1	yes	characteristic
368	#106	C₇×D₄	1	1	yes	characteristic
369	C₄⋊C₈	—	2	1	no	—
370	C₄⋊C₈	—	2	1	no	—
371	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
372	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
373	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
374	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
375	#103	—	2	1	no	—
376	#103	—	2	1	no	—
377	#103	—	2	1	no	—
378	#103	—	2	1	no	—
379	#103	—	2	1	no	—
380	#103	—	2	1	no	—
381	#103	—	2	1	no	—
382	#103	—	2	1	no	—
383	#103	—	2	1	no	—
384	#103	—	2	1	no	—
385	#103	—	2	1	no	—
386	#103	—	4	1	no	—
387	#103	C₇×D₄	1	1	yes	characteristic
388	#103	C₇×D₄	1	1	yes	characteristic
389	#102	—	2	1	no	—
390	#102	—	2	1	no	—
391	#102	—	2	1	no	—
392	#102	C₇×D₄	1	1	yes	characteristic
393	#102	C₇×D₄	1	1	yes	characteristic
394	C₂²⋊C₈	—	2	1	no	—
395	C₂²⋊C₈	—	2	1	no	—
396	#97	—	2	1	no	—
397	C₄×C₈	—	2	1	no	—
398	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
399	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
400	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
401	C₂²×C₇	—	2	2	no	—
402	C₂²×C₇	—	4	1	no	—
403	C₂²×C₇	—	4	1	no	—
404	C₂²×C₇	—	4	1	no	—
405	C₂²×C₇	—	4	1	no	—
406	C₂²×C₇	—	4	1	no	—
407	C₂²×C₇	D₄²	1	1	yes	characteristic
408	C₄×C₇	—	2	2	no	—
409	C₄×C₇	—	2	1	no	—
410	C₄×C₇	—	4	1	no	—
411	C₄×C₇	—	4	1	no	—
412	C₄×C₇	—	4	1	no	—
413	C₄×C₇	—	4	1	no	—
414	C₄×C₇	—	4	1	no	—
415	C₄×C₇	—	8	1	no	—
416	C₄×C₇	—	8	1	no	—
417	C₄×C₇	—	8	1	no	—
418	C₄×C₇	—	8	1	no	—
419	C₄×C₇	C₂×#135	1	1	yes	characteristic
420	C₄×C₇	C₂×#137	1	1	yes	characteristic
421	C₄×C₇	#577	1	1	yes	characteristic
422	C₄×C₇	#577	1	1	yes	characteristic
423	#41	—	4	2	no	—
424	#41	—	8	1	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×Q₈	—	2	1	no	—
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	4	1	no	—
430	C₂×Q₈	—	4	1	no	—
431	C₂×Q₈	—	4	2	no	—
432	C₂×Q₈	—	8	1	no	—
433	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
434	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
435	C₂×D₄	—	4	1	no	—
436	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
437	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	4	1	no	—
441	C₂²×C₄	—	4	1	no	—
442	Q₁₆	—	4	2	no	—
443	Q₁₆	—	8	1	no	—
444	SD₁₆	—	2	2	no	—
445	SD₁₆	—	2	2	no	—
446	SD₁₆	—	8	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	8	1	no	—
453	C₂×C₈	—	8	1	no	—
454	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
455	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
456	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
457	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	4	1	no	—
469	C₄⋊C₄	—	4	1	no	—
470	C₄⋊C₄	—	4	1	no	—
471	C₄⋊C₄	—	4	1	no	—
472	C₄⋊C₄	—	4	1	no	—
473	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
474	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
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	—	2	1	no	—
479	C₂²⋊C₄	—	4	1	no	—
480	C₂²⋊C₄	—	4	1	no	—
481	C₂²⋊C₄	—	4	1	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	4	1	no	—
486	C₄²	—	4	1	no	—
487	C₄²	C₂⁴×C₇	1	1	yes	characteristic
488	C₂×C₇	—	4	2	no	—
489	C₂×C₇	—	8	1	no	—
490	C₂×C₇	#3266	1	1	yes	characteristic
491	C₂×C₇	#3282	1	1	yes	characteristic
492	C₂×C₇	#3289	1	1	yes	characteristic
493	C₂³	—	2	1	no	—
494	C₂³	—	2	1	no	—
495	Q₈	—	2	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	2	1	no	—
498	Q₈	—	4	1	no	—
499	Q₈	—	4	1	no	—
500	Q₈	—	4	1	no	—
501	Q₈	—	4	1	no	—
502	Q₈	—	4	1	no	—
503	Q₈	—	4	1	no	—
504	Q₈	—	8	2	no	—
505	Q₈	—	8	1	no	—
506	Q₈	—	16	1	no	—
507	Q₈	C₇×#135	1	2	yes	—
508	D₄	—	2	1	no	—
509	D₄	—	2	1	no	—
510	D₄	—	2	1	no	—
511	D₄	—	8	2	no	—
512	D₄	C₇×#137	1	2	yes	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	8	2	no	—
528	C₂×C₄	—	8	2	no	—
529	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
530	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
531	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
532	C₈	—	2	1	no	—
533	C₈	—	2	1	no	—
534	C₈	—	2	1	no	—
535	C₈	—	2	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	1	no	—
538	C₈	—	4	1	no	—
539	C₈	—	4	1	no	—
540	C₈	—	4	1	no	—
541	C₇	#19796	1	1	yes	characteristic, direct product factor, Sylow
542	C₂²	—	2	2	no	—
543	C₂²	—	4	1	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	4	1	no	—
548	C₂²	C₇×D₄²	1	1	yes	characteristic
549	C₄	—	2	2	no	—
550	C₄	—	2	1	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	8	1	no	—
557	C₄	—	8	1	no	—
558	C₄	—	8	1	no	—
559	C₄	—	8	1	no	—
560	C₄	C₂×C₇×#135	1	1	yes	characteristic
561	C₄	C₂×C₇×#137	1	1	yes	characteristic
562	C₄	C₇×#577	1	1	yes	characteristic
563	C₄	C₇×#577	1	1	yes	characteristic
564	C₂	—	4	2	no	—
565	C₂	—	8	1	no	—
566	C₂	C₇×#3266	1	1	yes	characteristic
567	C₂	C₇×#3282	1	1	yes	characteristic
568	C₂	C₇×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#547	C₂²
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×Q₈×SD₁₆	C₂	semidirect
2. C₇×#3369	C₂	semidirect
5. C₇×#1718	C₂	semidirect
6. C₇×#1700	C₂	semidirect
7. C₇×#1683	C₂	semidirect
8. C₇×#1680	C₂	semidirect
9. C₇×#1671	C₂	semidirect
10. C₇×#1650	C₂	semidirect
11. C₇×#1649	C₂	semidirect
12. C₇×#1645	C₂	semidirect
14. C₇×#1582	C₂	semidirect
15. C₇×SD₁₆⋊C₈	C₂	semidirect
67. #19796	C₇	direct (N × H)
145. Q₈×SD₁₆	C₂×C₇	semidirect
146. #3369	C₂×C₇	semidirect
149. #1718	C₂×C₇	semidirect
150. #1700	C₂×C₇	semidirect
151. #1683	C₂×C₇	semidirect
152. #1680	C₂×C₇	semidirect
153. #1671	C₂×C₇	semidirect
154. #1650	C₂×C₇	semidirect
155. #1649	C₂×C₇	semidirect
156. #1645	C₂×C₇	semidirect
158. #1582	C₂×C₇	semidirect
159. SD₁₆⋊C₈	C₂×C₇	semidirect
541. C₇	#19796	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×Q₈×SD₁₆	2
C₇×#3369	2
C₇×#3302	2
C₇×#3298	2
C₇×#1718	2
C₇×#1700	2
C₇×#1683	2
C₇×#1680	2
C₇×#1671	2
C₇×#1650	2
C₇×#1649	2
C₇×#1645	2
C₇×#1635	2
C₇×#1582	2
C₇×SD₁₆⋊C₈	2
#19796	7

Subgroups — C7×#19796 #13197012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19796 #13197012