Subgroups — C₇×#19646 #13196862

Order 1792.

732 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3210	C₂	1	2	yes	maximal, complemented
2	C₇×#3182	C₂	1	2	yes	maximal, complemented
3	C₇×#3169	C₂	1	1	yes	maximal, characteristic
4	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1720	C₂	1	2	yes	maximal, complemented
6	C₇×#1714	C₂	1	2	yes	maximal, complemented
7	C₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#533	C₂²	1	2	yes	—
13	C₇×#532	C₂²	1	2	yes	—
14	C₇×#500	C₂²	1	1	yes	characteristic
15	C₇×#500	C₂²	1	1	yes	characteristic
16	C₇×#500	C₂²	1	1	yes	characteristic
17	C₇×#500	C₂²	1	2	yes	—
18	C₇×#499	—	2	2	no	—
19	C₇×#498	C₂²	1	2	yes	—
20	C₇×#494	C₂²	1	2	yes	—
21	C₇×#493	C₂²	1	1	yes	characteristic
22	C₇×#492	C₂²	1	2	yes	—
23	C₇×#492	C₂²	1	2	yes	—
24	C₇×#487	C₂²	1	2	yes	—
25	C₇×#487	C₂²	1	2	yes	—
26	C₇×#486	C₂²	1	2	yes	—
27	C₇×#486	C₂²	1	2	yes	—
28	C₇×#486	C₂²	1	2	yes	—
29	C₇×#468	—	2	4	no	—
30	C₇×#466	—	2	2	no	—
31	C₇×#464	—	2	4	no	—
32	C₇×#452	—	2	4	no	—
33	C₇×#440	—	2	4	no	—
34	C₄×C₇×D₈	—	2	2	no	—
35	C₇×#431	C₂²	1	1	yes	characteristic
36	C₇×#427	—	2	2	no	—
37	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
39	C₇×#322	C₂²	1	1	yes	characteristic
40	C₇×#322	C₂²	1	1	yes	characteristic
41	C₇×C₈²	C₂²	1	1	yes	characteristic
42	C₇×C₈²	C₂²	1	1	yes	characteristic
43	#19646	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×#41	—	2	2	no	—
45	C₂×C₇×Q₁₆	—	2	4	no	—
46	C₂×C₇×SD₁₆	—	2	4	no	—
47	C₂×C₇×SD₁₆	—	2	4	no	—
48	C₂×C₇×D₈	—	2	2	no	—
49	C₂×C₇×M₁₆	D₄	1	2	yes	—
50	C₂²×C₇×C₈	D₄	1	2	yes	—
51	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₇×#127	C₂³	1	2	yes	—
55	C₇×#125	—	2	2	no	—
56	C₇×#124	—	2	2	no	—
57	C₇×C₂²⋊Q₈	—	2	2	no	—
58	C₇×C₂²⋊Q₈	—	2	2	no	—
59	C₇×#121	—	2	2	no	—
60	C₇×#121	—	2	2	no	—
61	C₄×C₇×Q₈	—	2	2	no	—
62	C₄×C₇×Q₈	—	2	2	no	—
63	C₄×C₇×D₄	—	2	2	no	—
64	C₄×C₇×D₄	—	2	2	no	—
65	C₇×#117	—	2	2	no	—
66	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
67	C₇×#107	—	2	2	no	—
68	C₇×#107	—	2	4	no	—
69	C₇×#107	—	2	2	no	—
70	C₇×#107	—	2	2	no	—
71	C₇×#106	—	2	4	no	—
72	C₇×C₄⋊C₈	D₄	1	2	yes	—
73	C₇×#103	—	2	4	no	—
74	C₇×#103	—	2	4	no	—
75	C₇×#102	—	2	4	no	—
76	C₇×#102	—	2	4	no	—
77	C₇×#102	—	2	4	no	—
78	C₇×C₂²⋊C₈	D₄	1	2	yes	—
79	C₇×#97	D₄	1	2	yes	—
80	C₄×C₇×C₈	D₄	1	2	yes	—
81	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
83	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
84	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
87	#3210	C₂×C₇	1	2	yes	complemented
88	#3182	C₂×C₇	1	2	yes	complemented
89	#3169	C₂×C₇	1	1	yes	characteristic
90	#3160	C₂×C₇	1	1	yes	characteristic, complemented
91	#1720	C₂×C₇	1	2	yes	complemented
92	#1714	C₂×C₇	1	2	yes	complemented
93	#1713	C₂×C₇	1	1	yes	characteristic, complemented
94	#1710	C₂×C₇	1	1	yes	characteristic, complemented
95	#1706	C₂×C₇	1	1	yes	characteristic, complemented
96	#1706	C₂×C₇	1	1	yes	characteristic, complemented
97	C₈×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
98	C₇×#41	—	8	4	no	—
99	C₂×C₇×Q₈	—	2	2	no	—
100	C₂×C₇×Q₈	—	2	2	no	—
101	C₂×C₇×Q₈	—	2	2	no	—
102	C₂×C₇×D₄	—	2	2	no	—
103	C₂×C₇×D₄	—	2	2	no	—
104	C₂×C₇×D₄	—	2	2	no	—
105	C₂×C₇×D₄	—	4	2	no	—
106	C₂²×C₄×C₇	—	4	2	no	—
107	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
110	C₇×Q₁₆	—	8	4	no	—
111	C₇×SD₁₆	—	4	8	no	—
112	C₇×SD₁₆	—	8	4	no	—
113	C₇×D₈	—	4	4	no	—
114	C₇×M₁₆	—	2	4	no	—
115	C₂×C₇×C₈	—	2	4	no	—
116	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
117	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
118	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
119	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
120	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
121	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
122	C₇×C₄⋊C₄	—	2	2	no	—
123	C₇×C₄⋊C₄	—	2	2	no	—
124	C₇×C₄⋊C₄	—	2	2	no	—
125	C₇×C₄⋊C₄	—	2	2	no	—
126	C₇×C₄⋊C₄	—	2	2	no	—
127	C₇×C₄⋊C₄	—	2	2	no	—
128	C₇×C₄⋊C₄	—	4	2	no	—
129	C₇×C₄⋊C₄	—	4	2	no	—
130	C₇×C₂²⋊C₄	—	4	2	no	—
131	C₇×C₂²⋊C₄	—	4	2	no	—
132	C₇×C₂²⋊C₄	—	4	2	no	—
133	C₄²×C₇	—	4	2	no	—
134	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₇	C₂⁴	1	1	yes	characteristic
138	#533	C₂²×C₇	1	2	yes	—
139	#532	C₂²×C₇	1	2	yes	—
140	#500	C₂²×C₇	1	1	yes	characteristic
141	#500	C₂²×C₇	1	1	yes	characteristic
142	#500	C₂²×C₇	1	1	yes	characteristic
143	#500	C₂²×C₇	1	2	yes	—
144	#499	—	2	2	no	—
145	#498	C₂²×C₇	1	2	yes	—
146	#494	C₂²×C₇	1	2	yes	—
147	#493	C₂²×C₇	1	1	yes	characteristic
148	#492	C₂²×C₇	1	2	yes	—
149	#492	C₂²×C₇	1	2	yes	—
150	#487	C₂²×C₇	1	2	yes	—
151	#487	C₂²×C₇	1	2	yes	—
152	#486	C₂²×C₇	1	2	yes	—
153	#486	C₂²×C₇	1	2	yes	—
154	#486	C₂²×C₇	1	2	yes	—
155	#468	—	2	4	no	—
156	#466	—	2	2	no	—
157	#464	—	2	4	no	—
158	#452	—	2	4	no	—
159	#440	—	2	4	no	—
160	C₄×D₈	—	2	2	no	—
161	#431	C₂²×C₇	1	1	yes	characteristic
162	#427	—	2	2	no	—
163	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
165	#322	C₂²×C₇	1	1	yes	characteristic
166	#322	C₂²×C₇	1	1	yes	characteristic
167	C₈²	C₂²×C₇	1	1	yes	characteristic
168	C₈²	C₂²×C₇	1	1	yes	characteristic
169	C₂³×C₇	—	4	2	no	—
170	C₂³×C₇	#127	1	1	yes	characteristic
171	C₇×Q₈	—	4	4	no	—
172	C₇×Q₈	—	4	4	no	—
173	C₇×Q₈	—	4	4	no	—
174	C₇×D₄	—	4	4	no	—
175	C₇×D₄	—	4	4	no	—
176	C₇×D₄	—	4	4	no	—
177	C₇×D₄	—	8	4	no	—
178	C₂×C₄×C₇	—	2	2	no	—
179	C₂×C₄×C₇	—	2	2	no	—
180	C₂×C₄×C₇	—	2	2	no	—
181	C₂×C₄×C₇	—	4	2	no	—
182	C₂×C₄×C₇	—	4	2	no	—
183	C₂×C₄×C₇	—	4	2	no	—
184	C₂×C₄×C₇	—	8	4	no	—
185	C₂×C₄×C₇	#127	1	1	yes	characteristic
186	C₂×C₄×C₇	#127	1	1	yes	characteristic
187	C₂×C₄×C₇	#127	1	1	yes	characteristic
188	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
191	C₇×C₈	—	2	2	no	—
192	C₇×C₈	—	2	2	no	—
193	C₇×C₈	—	2	2	no	—
194	C₇×C₈	—	2	2	no	—
195	C₇×C₈	#135	1	4	yes	—
196	C₇×C₈	#137	1	4	yes	—
197	C₂×#41	—	2	2	no	—
198	C₂×Q₁₆	—	2	4	no	—
199	C₂×SD₁₆	—	2	4	no	—
200	C₂×SD₁₆	—	2	4	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂×M₁₆	C₇×D₄	1	2	yes	—
203	C₂²×C₈	C₇×D₄	1	2	yes	—
204	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
205	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
206	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
207	#127	C₂³×C₇	1	2	yes	—
208	#125	—	2	2	no	—
209	#124	—	2	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	#121	—	2	2	no	—
213	#121	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	#117	—	2	2	no	—
219	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
220	#107	—	2	2	no	—
221	#107	—	2	2	no	—
222	#107	—	2	2	no	—
223	#107	—	2	4	no	—
224	#106	—	2	4	no	—
225	C₄⋊C₈	C₇×D₄	1	2	yes	—
226	#103	—	2	4	no	—
227	#103	—	2	4	no	—
228	#102	—	2	4	no	—
229	#102	—	2	4	no	—
230	#102	—	2	4	no	—
231	C₂²⋊C₈	C₇×D₄	1	2	yes	—
232	#97	C₇×D₄	1	2	yes	—
233	C₄×C₈	C₇×D₄	1	2	yes	—
234	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
240	C₂²×C₇	—	2	1	no	—
241	C₂²×C₇	—	2	1	no	—
242	C₂²×C₇	—	2	1	no	—
243	C₂²×C₇	—	8	2	no	—
244	C₂²×C₇	—	8	2	no	—
245	C₂²×C₇	—	8	2	no	—
246	C₂²×C₇	C₂×#127	1	1	yes	characteristic
247	C₄×C₇	—	2	1	no	—
248	C₄×C₇	—	2	1	no	—
249	C₄×C₇	—	2	1	no	—
250	C₄×C₇	—	8	2	no	—
251	C₄×C₇	—	8	2	no	—
252	C₄×C₇	—	8	2	no	—
253	C₄×C₇	C₂×#135	1	2	yes	—
254	C₄×C₇	C₂×#137	1	2	yes	—
255	C₄×C₇	#576	1	2	yes	—
256	#41	—	8	4	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
268	Q₁₆	—	8	4	no	—
269	SD₁₆	—	4	8	no	—
270	SD₁₆	—	8	4	no	—
271	D₈	—	4	4	no	—
272	M₁₆	—	2	4	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
275	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
276	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
277	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
278	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
279	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₄²	—	4	2	no	—
292	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
293	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
294	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
295	C₄²	C₂⁴×C₇	1	1	yes	characteristic
296	C₂×C₇	—	4	1	no	—
297	C₂×C₇	—	16	2	no	—
298	C₂×C₇	#3147	1	1	yes	characteristic
299	C₂×C₇	#3152	1	1	yes	characteristic
300	C₂×C₇	#3155	1	1	yes	characteristic
301	C₂³	—	4	2	no	—
302	C₂³	C₇×#127	1	1	yes	characteristic
303	Q₈	—	4	4	no	—
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	D₄	—	4	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	4	no	—
309	D₄	—	8	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	C₇×#127	1	1	yes	characteristic
318	C₂×C₄	C₇×#127	1	1	yes	characteristic
319	C₂×C₄	C₇×#127	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
321	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	C₇×#135	1	4	yes	—
328	C₈	C₇×#137	1	4	yes	—
329	C₇	#19646	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	C₂×C₇×#127	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₂×C₇×#135	1	2	yes	—
344	C₄	C₂×C₇×#137	1	2	yes	—
345	C₄	C₇×#576	1	2	yes	—
346	C₂	—	4	1	no	—
347	C₂	—	16	2	no	—
348	C₂	C₇×#3147	1	1	yes	characteristic
349	C₂	C₇×#3152	1	1	yes	characteristic
350	C₂	C₇×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3210	C₂	semidirect
2. C₇×#3182	C₂	semidirect
4. C₇×#3160	C₂	semidirect
5. C₇×#1720	C₂	semidirect
6. C₇×#1714	C₂	semidirect
7. C₇×#1713	C₂	semidirect
8. C₇×#1710	C₂	semidirect
9. C₇×#1706	C₂	semidirect
10. C₇×#1706	C₂	semidirect
11. C₇×C₈×M₁₆	C₂	semidirect
43. #19646	C₇	direct (N × H)
87. #3210	C₂×C₇	semidirect
88. #3182	C₂×C₇	semidirect
90. #3160	C₂×C₇	semidirect
91. #1720	C₂×C₇	semidirect
92. #1714	C₂×C₇	semidirect
93. #1713	C₂×C₇	semidirect
94. #1710	C₂×C₇	semidirect
95. #1706	C₂×C₇	semidirect
96. #1706	C₂×C₇	semidirect
97. C₈×M₁₆	C₂×C₇	semidirect
329. C₇	#19646	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×#3210	2
C₇×#3182	2
C₇×#3169	2
C₇×#3160	2
C₇×#1720	2
C₇×#1714	2
C₇×#1713	2
C₇×#1710	2
C₇×#1706	2
C₇×C₈×M₁₆	2
#19646	7

Subgroups — C7×#19646 #13196862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19646 #13196862