Subgroups — C₇×#19195 #13196411

Order 1792.

638 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3246	C₂	1	2	yes	maximal
2	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₇×#1691	C₂	1	4	yes	maximal, complemented
5	C₇×#1675	C₂	1	2	yes	maximal, complemented
6	C₇×#1673	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1550	C₂	1	2	yes	maximal, complemented
9	C₂×C₇×#131	—	2	1	no	—
10	C₇×#533	C₂²	1	2	yes	—
11	C₇×#532	C₂²	1	1	yes	characteristic
12	C₇×#500	C₂²	1	1	yes	characteristic
13	C₇×#500	C₂²	1	1	yes	characteristic
14	C₇×#499	—	2	1	no	—
15	C₇×#499	—	2	1	no	—
16	C₇×#498	C₂²	1	2	yes	—
17	C₇×#476	C₂²	1	4	yes	—
18	C₇×#475	—	2	2	no	—
19	C₇×#465	—	2	4	no	—
20	C₇×#461	C₂²	1	4	yes	—
21	C₇×#460	—	2	2	no	—
22	C₇×#436	—	2	1	no	—
23	C₇×#436	C₂²	1	2	yes	—
24	C₇×C₈×D₄	—	2	1	no	—
25	C₇×C₈×D₄	C₂²	1	2	yes	—
26	C₇×#427	—	2	1	no	—
27	C₇×#427	—	2	1	no	—
28	C₇×#423	C₂²	1	1	yes	characteristic
29	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
30	C₇×#408	—	2	1	no	—
31	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	C₇×#335	C₂²	1	2	yes	—
33	C₇×#335	C₂²	1	2	yes	—
34	C₇×#333	C₂²	1	4	yes	—
35	C₇×#332	C₂²	1	4	yes	—
36	#19195	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₇×#41	—	2	1	no	—
38	C₂×C₇×SD₁₆	—	2	4	no	—
39	C₇×#131	—	4	4	no	—
40	C₂×C₇×M₁₆	—	2	1	no	—
41	C₂×C₇×M₁₆	—	2	2	no	—
42	C₂²×C₇×C₈	—	2	2	no	—
43	C₂²×C₇×C₈	—	2	2	no	—
44	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
46	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₇×#127	D₄	1	1	yes	characteristic
50	C₇×#125	—	2	2	no	—
51	C₇×#124	—	2	1	no	—
52	C₇×C₂²⋊Q₈	—	2	2	no	—
53	C₇×C₂²⋊Q₈	—	2	2	no	—
54	C₇×#121	—	2	1	no	—
55	C₇×#121	D₄	1	2	yes	—
56	C₄×C₇×Q₈	—	2	2	no	—
57	C₄×C₇×Q₈	—	2	2	no	—
58	C₄×C₇×D₄	—	2	1	no	—
59	C₄×C₇×D₄	C₂³	1	2	yes	—
60	C₇×#117	—	2	2	no	—
61	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
62	C₇×#107	—	2	1	no	—
63	C₇×#107	—	2	1	no	—
64	C₇×#107	—	2	1	no	—
65	C₇×#107	—	2	1	no	—
66	C₇×#106	—	2	2	no	—
67	C₇×#106	—	2	2	no	—
68	C₇×#106	—	2	4	no	—
69	C₇×C₄⋊C₈	—	2	2	no	—
70	C₇×C₄⋊C₈	—	2	1	no	—
71	C₇×C₄⋊C₈	C₂³	1	4	yes	—
72	C₇×C₄⋊C₈	C₂³	1	2	yes	—
73	C₇×#103	—	2	4	no	—
74	C₇×#102	—	2	4	no	—
75	C₇×C₂²⋊C₈	—	2	2	no	—
76	C₇×C₂²⋊C₈	—	2	2	no	—
77	C₄×C₇×C₈	D₄	1	1	yes	characteristic
78	C₄×C₇×C₈	D₄	1	1	yes	characteristic
79	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
81	#3246	C₂×C₇	1	2	yes	—
82	#3160	C₂×C₇	1	1	yes	characteristic, complemented
83	#2931	C₂×C₇	1	1	yes	characteristic
84	#1691	C₂×C₇	1	4	yes	complemented
85	#1675	C₂×C₇	1	2	yes	complemented
86	#1673	C₂×C₇	1	2	yes	complemented
87	C₂×#335	C₂×C₇	1	1	yes	characteristic, complemented
88	#1550	C₂×C₇	1	2	yes	complemented
89	C₇×#41	—	4	2	no	—
90	C₇×#41	—	4	2	no	—
91	C₂×C₇×Q₈	—	2	1	no	—
92	C₂×C₇×Q₈	—	2	2	no	—
93	C₂×C₇×Q₈	—	2	2	no	—
94	C₂×C₇×D₄	—	2	2	no	—
95	C₂×C₇×D₄	—	2	1	no	—
96	C₂×C₇×D₄	—	2	1	no	—
97	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₇	—	2	2	no	—
99	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₇	#41	1	1	yes	characteristic
102	C₇×SD₁₆	—	8	4	no	—
103	C₇×M₁₆	—	4	2	no	—
104	C₇×M₁₆	—	4	4	no	—
105	C₂×C₇×C₈	—	2	2	no	—
106	C₂×C₇×C₈	—	2	2	no	—
107	C₂×C₇×C₈	—	2	4	no	—
108	C₂×C₇×C₈	—	4	4	no	—
109	C₂×C₇×C₈	—	8	2	no	—
110	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
111	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
115	C₇×C₄⋊C₄	—	2	1	no	—
116	C₇×C₄⋊C₄	—	2	2	no	—
117	C₇×C₄⋊C₄	—	2	2	no	—
118	C₇×C₄⋊C₄	—	2	2	no	—
119	C₇×C₄⋊C₄	—	2	2	no	—
120	C₇×C₄⋊C₄	—	4	2	no	—
121	C₇×C₄⋊C₄	—	4	2	no	—
122	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
123	C₇×C₂²⋊C₄	—	2	2	no	—
124	C₇×C₂²⋊C₄	—	2	2	no	—
125	C₇×C₂²⋊C₄	—	4	2	no	—
126	C₄²×C₇	—	4	2	no	—
127	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₇	#41	1	1	yes	characteristic
130	C₄²×C₇	C₂⁴	1	1	yes	characteristic
131	C₂×#131	—	2	1	no	—
132	#533	C₂²×C₇	1	2	yes	—
133	#532	C₂²×C₇	1	1	yes	characteristic
134	#500	C₂²×C₇	1	1	yes	characteristic
135	#500	C₂²×C₇	1	1	yes	characteristic
136	#499	—	2	1	no	—
137	#499	—	2	1	no	—
138	#498	C₂²×C₇	1	2	yes	—
139	#476	C₂²×C₇	1	4	yes	—
140	#475	—	2	2	no	—
141	#465	—	2	4	no	—
142	#461	C₂²×C₇	1	4	yes	—
143	#460	—	2	2	no	—
144	#436	—	2	1	no	—
145	#436	C₂²×C₇	1	2	yes	—
146	C₈×D₄	—	2	1	no	—
147	C₈×D₄	C₂²×C₇	1	2	yes	—
148	#427	—	2	1	no	—
149	#427	—	2	1	no	—
150	#423	C₂²×C₇	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
152	#408	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
154	#335	C₂²×C₇	1	2	yes	—
155	#335	C₂²×C₇	1	2	yes	—
156	#333	C₂²×C₇	1	4	yes	—
157	#332	C₂²×C₇	1	4	yes	—
158	C₂³×C₇	—	2	2	no	—
159	C₂³×C₇	#121	1	1	yes	characteristic
160	C₇×Q₈	—	4	2	no	—
161	C₇×Q₈	—	4	2	no	—
162	C₇×Q₈	—	4	2	no	—
163	C₇×Q₈	—	8	2	no	—
164	C₇×D₄	—	2	2	no	—
165	C₇×D₄	—	2	2	no	—
166	C₇×D₄	—	4	2	no	—
167	C₇×D₄	—	4	4	no	—
168	C₇×D₄	—	4	2	no	—
169	C₂×C₄×C₇	—	2	1	no	—
170	C₂×C₄×C₇	—	2	1	no	—
171	C₂×C₄×C₇	—	2	2	no	—
172	C₂×C₄×C₇	—	4	1	no	—
173	C₂×C₄×C₇	—	4	2	no	—
174	C₂×C₄×C₇	—	4	2	no	—
175	C₂×C₄×C₇	—	4	2	no	—
176	C₂×C₄×C₇	—	4	2	no	—
177	C₂×C₄×C₇	#121	1	1	yes	characteristic
178	C₂×C₄×C₇	#121	1	1	yes	characteristic
179	C₂×C₄×C₇	#121	1	1	yes	characteristic
180	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
181	C₂×C₄×C₇	#137	1	2	yes	—
182	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
185	C₇×C₈	—	2	1	no	—
186	C₇×C₈	—	2	1	no	—
187	C₇×C₈	—	4	2	no	—
188	C₇×C₈	—	4	4	no	—
189	C₇×C₈	C₂×SD₁₆	1	2	yes	—
190	C₇×C₈	C₂×SD₁₆	1	2	yes	—
191	C₂×#41	—	2	1	no	—
192	C₂×SD₁₆	—	2	4	no	—
193	#131	—	4	4	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
199	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
200	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
201	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
202	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
203	#127	C₇×D₄	1	1	yes	characteristic
204	#125	—	2	2	no	—
205	#124	—	2	1	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	#121	—	2	1	no	—
209	#121	C₇×D₄	1	2	yes	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	C₂³×C₇	1	2	yes	—
214	#117	—	2	2	no	—
215	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
216	#107	—	2	1	no	—
217	#107	—	2	1	no	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#106	—	2	2	no	—
221	#106	—	2	2	no	—
222	#106	—	2	4	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	C₂³×C₇	1	2	yes	—
226	C₄⋊C₈	C₂³×C₇	1	4	yes	—
227	#103	—	2	4	no	—
228	#102	—	2	4	no	—
229	C₂²⋊C₈	—	2	2	no	—
230	C₂²⋊C₈	—	2	2	no	—
231	C₄×C₈	C₇×D₄	1	1	yes	characteristic
232	C₄×C₈	C₇×D₄	1	1	yes	characteristic
233	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
235	C₂²×C₇	—	2	1	no	—
236	C₂²×C₇	—	2	1	no	—
237	C₂²×C₇	—	4	2	no	—
238	C₂²×C₇	—	4	2	no	—
239	C₂²×C₇	—	4	2	no	—
240	C₂²×C₇	#461	1	2	yes	—
241	C₂²×C₇	C₂×#121	1	1	yes	characteristic
242	C₄×C₇	—	2	1	no	—
243	C₄×C₇	—	2	1	no	—
244	C₄×C₇	—	2	1	no	—
245	C₄×C₇	—	4	2	no	—
246	C₄×C₇	—	8	2	no	—
247	C₄×C₇	—	8	2	no	—
248	C₄×C₇	#461	1	2	yes	—
249	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
250	C₄×C₇	C₂×#137	1	1	yes	characteristic
251	C₄×C₇	#576	1	1	yes	characteristic
252	C₄×C₇	#578	1	1	yes	characteristic
253	#41	—	4	2	no	—
254	#41	—	4	2	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
265	C₂²×C₄	C₇×#41	1	1	yes	characteristic
266	SD₁₆	—	8	4	no	—
267	M₁₆	—	4	2	no	—
268	M₁₆	—	4	4	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	4	4	no	—
273	C₂×C₈	—	8	2	no	—
274	C₂×C₈	C₇×SD₁₆	1	4	yes	complemented
275	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
276	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
277	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
278	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	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	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
292	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
293	C₄²	C₇×#41	1	1	yes	characteristic
294	C₄²	C₂⁴×C₇	1	1	yes	characteristic
295	C₂×C₇	—	2	2	no	—
296	C₂×C₇	—	8	2	no	—
297	C₂×C₇	C₂×#461	1	1	yes	characteristic
298	C₂×C₇	#3061	1	1	yes	characteristic
299	C₂×C₇	#3067	1	1	yes	characteristic
300	C₂³	—	2	2	no	—
301	C₂³	C₇×#121	1	1	yes	characteristic
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	8	2	no	—
306	D₄	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	C₇×#121	1	1	yes	characteristic
320	C₂×C₄	C₇×#121	1	1	yes	characteristic
321	C₂×C₄	C₇×#121	1	1	yes	characteristic
322	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
323	C₂×C₄	C₇×#137	1	2	yes	—
324	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
326	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	4	2	no	—
330	C₈	—	4	4	no	—
331	C₈	C₂×C₇×SD₁₆	1	2	yes	—
332	C₈	C₂×C₇×SD₁₆	1	2	yes	—
333	C₇	#19195	1	1	yes	characteristic, direct product factor, Sylow
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	C₇×#461	1	2	yes	—
340	C₂²	C₂×C₇×#121	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	C₇×#461	1	2	yes	—
348	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
349	C₄	C₂×C₇×#137	1	1	yes	characteristic
350	C₄	C₇×#576	1	1	yes	characteristic
351	C₄	C₇×#578	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	8	2	no	—
354	C₂	C₂×C₇×#461	1	1	yes	characteristic
355	C₂	C₇×#3061	1	1	yes	characteristic
356	C₂	C₇×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#532	C₂²
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
2. C₇×#3160	C₂	semidirect
4. C₇×#1691	C₂	semidirect
5. C₇×#1675	C₂	semidirect
6. C₇×#1673	C₂	semidirect
7. C₂×C₇×#335	C₂	semidirect
8. C₇×#1550	C₂	semidirect
36. #19195	C₇	direct (N × H)
82. #3160	C₂×C₇	semidirect
84. #1691	C₂×C₇	semidirect
85. #1675	C₂×C₇	semidirect
86. #1673	C₂×C₇	semidirect
87. C₂×#335	C₂×C₇	semidirect
88. #1550	C₂×C₇	semidirect
110. C₂×C₇×C₈	SD₁₆	semidirect
274. C₂×C₈	C₇×SD₁₆	semidirect
333. C₇	#19195	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×#3246	2
C₇×#3160	2
C₇×#2931	2
C₇×#1691	2
C₇×#1675	2
C₇×#1673	2
C₂×C₇×#335	2
C₇×#1550	2
#19195	7

Subgroups — C7×#19195 #13196411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19195 #13196411