Subgroups — C₇×#19666 #13196882

Order 1792.

712 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3228	C₂	1	2	yes	maximal, complemented
2	C₇×#3210	C₂	1	2	yes	maximal, complemented
3	C₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3169	C₂	1	1	yes	maximal, characteristic
5	C₇×#1720	C₂	1	2	yes	maximal, complemented
6	C₇×#1719	C₂	1	2	yes	maximal, complemented
7	C₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1709	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1450	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	2	yes	—
16	C₇×#500	C₂²	1	1	yes	characteristic
17	C₇×#498	C₂²	1	2	yes	—
18	C₇×#494	C₂²	1	1	yes	characteristic
19	C₇×#494	C₂²	1	2	yes	—
20	C₇×#493	C₂²	1	1	yes	characteristic
21	C₇×#492	C₂²	1	2	yes	—
22	C₇×#487	C₂²	1	2	yes	—
23	C₇×#487	C₂²	1	2	yes	—
24	C₇×#486	C₂²	1	2	yes	—
25	C₇×#486	C₂²	1	2	yes	—
26	C₇×#486	C₂²	1	2	yes	—
27	C₇×#486	C₂²	1	2	yes	—
28	C₇×#482	—	2	4	no	—
29	C₇×#478	—	2	4	no	—
30	C₇×#467	—	2	2	no	—
31	C₇×#466	—	2	2	no	—
32	C₇×#464	—	2	4	no	—
33	C₇×#452	—	2	4	no	—
34	C₄×C₇×Q₁₆	—	2	2	no	—
35	C₄×C₇×D₈	—	2	2	no	—
36	C₇×#431	C₂²	1	1	yes	characteristic
37	C₇×#431	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	#19666	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×#41	—	2	2	no	—
45	C₂×C₇×Q₁₆	—	2	2	no	—
46	C₂×C₇×Q₁₆	—	2	4	no	—
47	C₂×C₇×SD₁₆	—	2	4	no	—
48	C₂×C₇×D₈	—	2	2	no	—
49	C₂²×C₇×C₈	D₄	1	2	yes	—
50	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₇×#127	C₂³	1	2	yes	—
54	C₇×#125	—	2	2	no	—
55	C₇×#124	—	2	2	no	—
56	C₇×C₂²⋊Q₈	—	2	2	no	—
57	C₇×C₂²⋊Q₈	—	2	2	no	—
58	C₇×#121	—	2	2	no	—
59	C₇×#121	—	2	2	no	—
60	C₄×C₇×Q₈	—	2	2	no	—
61	C₄×C₇×Q₈	—	2	2	no	—
62	C₄×C₇×D₄	—	2	2	no	—
63	C₄×C₇×D₄	—	2	2	no	—
64	C₇×#117	—	2	2	no	—
65	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
66	C₇×#107	—	2	2	no	—
67	C₇×#107	—	2	2	no	—
68	C₇×#107	—	2	4	no	—
69	C₇×#106	—	2	4	no	—
70	C₇×C₄⋊C₈	D₄	1	2	yes	—
71	C₇×C₄⋊C₈	D₄	1	2	yes	—
72	C₇×#103	—	2	4	no	—
73	C₇×#103	—	2	4	no	—
74	C₇×#102	—	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×C₂²⋊C₈	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	#3228	C₂×C₇	1	2	yes	complemented
88	#3210	C₂×C₇	1	2	yes	complemented
89	#3170	C₂×C₇	1	1	yes	characteristic, complemented
90	#3169	C₂×C₇	1	1	yes	characteristic
91	#1720	C₂×C₇	1	2	yes	complemented
92	#1719	C₂×C₇	1	2	yes	complemented
93	#1713	C₂×C₇	1	1	yes	characteristic, complemented
94	#1709	C₂×C₇	1	1	yes	characteristic, complemented
95	#1707	C₂×C₇	1	1	yes	characteristic, complemented
96	#1706	C₂×C₇	1	1	yes	characteristic, complemented
97	#1450	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	—	4	4	no	—
111	C₇×Q₁₆	—	8	4	no	—
112	C₇×SD₁₆	—	8	4	no	—
113	C₇×D₈	—	4	4	no	—
114	C₂×C₇×C₈	—	2	4	no	—
115	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
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	C₇×C₄⋊C₄	—	2	2	no	—
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	—	4	2	no	—
128	C₇×C₄⋊C₄	—	4	2	no	—
129	C₇×C₂²⋊C₄	—	4	2	no	—
130	C₇×C₂²⋊C₄	—	4	2	no	—
131	C₇×C₂²⋊C₄	—	4	2	no	—
132	C₄²×C₇	—	4	2	no	—
133	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₇	C₂⁴	1	1	yes	characteristic
137	#533	C₂²×C₇	1	2	yes	—
138	#532	C₂²×C₇	1	2	yes	—
139	#500	C₂²×C₇	1	1	yes	characteristic
140	#500	C₂²×C₇	1	1	yes	characteristic
141	#500	C₂²×C₇	1	2	yes	—
142	#498	C₂²×C₇	1	2	yes	—
143	#494	C₂²×C₇	1	1	yes	characteristic
144	#494	C₂²×C₇	1	2	yes	—
145	#493	C₂²×C₇	1	1	yes	characteristic
146	#492	C₂²×C₇	1	2	yes	—
147	#487	C₂²×C₇	1	2	yes	—
148	#487	C₂²×C₇	1	2	yes	—
149	#486	C₂²×C₇	1	2	yes	—
150	#486	C₂²×C₇	1	2	yes	—
151	#486	C₂²×C₇	1	2	yes	—
152	#486	C₂²×C₇	1	2	yes	—
153	#482	—	2	4	no	—
154	#478	—	2	4	no	—
155	#467	—	2	2	no	—
156	#466	—	2	2	no	—
157	#464	—	2	4	no	—
158	#452	—	2	4	no	—
159	C₄×Q₁₆	—	2	2	no	—
160	C₄×D₈	—	2	2	no	—
161	#431	C₂²×C₇	1	1	yes	characteristic
162	#431	C₂²×C₇	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
164	#322	C₂²×C₇	1	1	yes	characteristic
165	#322	C₂²×C₇	1	1	yes	characteristic
166	C₈²	C₂²×C₇	1	1	yes	characteristic
167	C₈²	C₂²×C₇	1	1	yes	characteristic
168	C₂³×C₇	—	4	2	no	—
169	C₂³×C₇	#127	1	1	yes	characteristic
170	C₇×Q₈	—	4	4	no	—
171	C₇×Q₈	—	4	4	no	—
172	C₇×Q₈	—	4	4	no	—
173	C₇×D₄	—	4	4	no	—
174	C₇×D₄	—	4	4	no	—
175	C₇×D₄	—	4	4	no	—
176	C₇×D₄	—	8	4	no	—
177	C₂×C₄×C₇	—	2	2	no	—
178	C₂×C₄×C₇	—	2	2	no	—
179	C₂×C₄×C₇	—	2	2	no	—
180	C₂×C₄×C₇	—	4	2	no	—
181	C₂×C₄×C₇	—	4	2	no	—
182	C₂×C₄×C₇	—	4	2	no	—
183	C₂×C₄×C₇	—	8	4	no	—
184	C₂×C₄×C₇	#127	1	1	yes	characteristic
185	C₂×C₄×C₇	#127	1	1	yes	characteristic
186	C₂×C₄×C₇	#127	1	1	yes	characteristic
187	C₂×C₄×C₇	C₂²×D₄	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	C₇×C₈	—	2	2	no	—
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	C₂×#41	—	2	2	no	—
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×Q₁₆	—	2	4	no	—
199	C₂×SD₁₆	—	2	4	no	—
200	C₂×D₈	—	2	2	no	—
201	C₂²×C₈	C₇×D₄	1	2	yes	—
202	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
203	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
204	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
205	#127	C₂³×C₇	1	2	yes	—
206	#125	—	2	2	no	—
207	#124	—	2	2	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	#121	—	2	2	no	—
211	#121	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	—	2	2	no	—
216	#117	—	2	2	no	—
217	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
218	#107	—	2	2	no	—
219	#107	—	2	2	no	—
220	#107	—	2	4	no	—
221	#106	—	2	4	no	—
222	C₄⋊C₈	C₇×D₄	1	2	yes	—
223	C₄⋊C₈	C₇×D₄	1	2	yes	—
224	#103	—	2	4	no	—
225	#103	—	2	4	no	—
226	#102	—	2	4	no	—
227	#102	—	2	4	no	—
228	#102	—	2	4	no	—
229	#102	—	2	4	no	—
230	C₂²⋊C₈	C₇×D₄	1	2	yes	—
231	C₂²⋊C₈	C₇×D₄	1	2	yes	—
232	C₄×C₈	C₇×D₄	1	2	yes	—
233	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
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	C₂²×C₇	—	2	1	no	—
240	C₂²×C₇	—	2	1	no	—
241	C₂²×C₇	—	2	1	no	—
242	C₂²×C₇	—	8	2	no	—
243	C₂²×C₇	—	8	2	no	—
244	C₂²×C₇	—	8	2	no	—
245	C₂²×C₇	C₂×#127	1	1	yes	characteristic
246	C₄×C₇	—	2	1	no	—
247	C₄×C₇	—	2	1	no	—
248	C₄×C₇	—	2	1	no	—
249	C₄×C₇	—	8	2	no	—
250	C₄×C₇	—	8	2	no	—
251	C₄×C₇	—	8	2	no	—
252	C₄×C₇	C₂×#135	1	2	yes	—
253	C₄×C₇	#576	1	2	yes	—
254	C₄×C₇	#578	1	2	yes	—
255	#41	—	8	4	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
267	Q₁₆	—	4	4	no	—
268	Q₁₆	—	8	4	no	—
269	SD₁₆	—	8	4	no	—
270	D₈	—	4	4	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
273	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
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	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
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	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₄²	—	4	2	no	—
290	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
291	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
292	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
293	C₄²	C₂⁴×C₇	1	1	yes	characteristic
294	C₂×C₇	—	4	1	no	—
295	C₂×C₇	—	16	2	no	—
296	C₂×C₇	#3147	1	1	yes	characteristic
297	C₂×C₇	#3155	1	1	yes	characteristic
298	C₂×C₇	#3156	1	1	yes	characteristic
299	C₂³	—	4	2	no	—
300	C₂³	C₇×#127	1	1	yes	characteristic
301	Q₈	—	4	4	no	—
302	Q₈	—	4	4	no	—
303	Q₈	—	4	4	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	4	no	—
307	D₄	—	8	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	C₇×#127	1	1	yes	characteristic
316	C₂×C₄	C₇×#127	1	1	yes	characteristic
317	C₂×C₄	C₇×#127	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	C₇×#135	1	4	yes	—
327	C₇	#19666	1	1	yes	characteristic, direct product factor, Sylow
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	C₂×C₇×#127	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	8	2	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	C₂×C₇×#135	1	2	yes	—
342	C₄	C₇×#576	1	2	yes	—
343	C₄	C₇×#578	1	2	yes	—
344	C₂	—	4	1	no	—
345	C₂	—	16	2	no	—
346	C₂	C₇×#3147	1	1	yes	characteristic
347	C₂	C₇×#3155	1	1	yes	characteristic
348	C₂	C₇×#3156	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×#3228	C₂	semidirect
2. C₇×#3210	C₂	semidirect
3. C₇×#3170	C₂	semidirect
5. C₇×#1720	C₂	semidirect
6. C₇×#1719	C₂	semidirect
7. C₇×#1713	C₂	semidirect
8. C₇×#1709	C₂	semidirect
9. C₇×#1707	C₂	semidirect
10. C₇×#1706	C₂	semidirect
11. C₇×#1450	C₂	semidirect
43. #19666	C₇	direct (N × H)
87. #3228	C₂×C₇	semidirect
88. #3210	C₂×C₇	semidirect
89. #3170	C₂×C₇	semidirect
91. #1720	C₂×C₇	semidirect
92. #1719	C₂×C₇	semidirect
93. #1713	C₂×C₇	semidirect
94. #1709	C₂×C₇	semidirect
95. #1707	C₂×C₇	semidirect
96. #1706	C₂×C₇	semidirect
97. #1450	C₂×C₇	semidirect
327. C₇	#19666	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×#3228	2
C₇×#3210	2
C₇×#3170	2
C₇×#3169	2
C₇×#1720	2
C₇×#1719	2
C₇×#1713	2
C₇×#1709	2
C₇×#1707	2
C₇×#1706	2
C₇×#1450	2
#19666	7

Subgroups — C7×#19666 #13196882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19666 #13196882