Subgroups — C₇×#19146 #13196362

Order 1792.

694 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3147	C₂	1	1	yes	maximal, characteristic
2	C₇×#3103	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1690	C₂	1	2	yes	maximal, complemented
5	C₇×#1688	C₂	1	2	yes	maximal, complemented
6	C₇×#1686	C₂	1	4	yes	maximal, complemented
7	C₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1540	C₂	1	2	yes	maximal, complemented
9	C₂×C₇×#135	—	2	1	no	—
10	C₂×C₇×#135	—	2	1	no	—
11	C₇×#532	C₂²	1	2	yes	—
12	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₇×#495	—	2	1	no	—
14	C₇×#495	—	2	1	no	—
15	C₇×#494	C₂²	1	1	yes	characteristic
16	C₇×#493	C₂²	1	1	yes	characteristic
17	C₇×#492	C₂²	1	2	yes	—
18	C₇×#481	—	2	2	no	—
19	C₇×#475	C₂²	1	4	yes	—
20	C₇×#460	C₂²	1	4	yes	—
21	C₇×#448	—	2	2	no	—
22	C₇×#446	—	2	1	no	—
23	C₇×#446	—	2	1	no	—
24	C₇×#446	C₂²	1	2	yes	—
25	C₇×#446	C₂²	1	2	yes	—
26	C₄×C₇×SD₁₆	—	2	4	no	—
27	C₇×#431	C₂²	1	2	yes	—
28	C₂×C₇×C₄⋊C₈	—	2	1	no	—
29	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₇×#97	—	2	1	no	—
31	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	C₇×#335	C₂²	1	2	yes	—
33	C₇×#334	C₂²	1	2	yes	—
34	C₇×#332	C₂²	1	4	yes	—
35	C₇×#331	C₂²	1	4	yes	—
36	#19146	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₇×#41	—	2	2	no	—
38	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
39	C₇×#135	—	4	4	no	—
40	C₇×#135	—	4	4	no	—
41	C₂×C₇×Q₁₆	—	2	1	no	—
42	C₂×C₇×Q₁₆	—	2	1	no	—
43	C₂×C₇×SD₁₆	—	2	4	no	—
44	C₂×C₇×SD₁₆	—	2	2	no	—
45	C₂×C₇×SD₁₆	—	2	2	no	—
46	C₂×C₇×D₈	—	2	1	no	—
47	C₂×C₇×D₈	—	2	1	no	—
48	C₂²×C₇×C₈	—	2	1	no	—
49	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
51	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₇×#127	C₂³	1	2	yes	—
53	C₇×#124	—	2	2	no	—
54	C₇×#121	—	2	2	no	—
55	C₇×#121	—	2	2	no	—
56	C₄×C₇×Q₈	—	2	1	no	—
57	C₄×C₇×Q₈	—	2	1	no	—
58	C₄×C₇×Q₈	D₄	1	2	yes	—
59	C₄×C₇×Q₈	C₂³	1	2	yes	—
60	C₄×C₇×D₄	—	2	2	no	—
61	C₄×C₇×D₄	—	2	2	no	—
62	C₂×C₇×C₄⋊C₄	—	2	1	no	—
63	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₇	—	2	1	no	—
65	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
66	C₇×#106	—	2	4	no	—
67	C₇×C₄⋊C₈	—	2	2	no	—
68	C₇×C₄⋊C₈	—	2	2	no	—
69	C₇×C₄⋊C₈	—	2	1	no	—
70	C₇×C₄⋊C₈	C₂³	1	4	yes	—
71	C₇×C₄⋊C₈	C₂³	1	2	yes	—
72	C₇×#103	—	2	4	no	—
73	C₇×#102	—	2	4	no	—
74	C₇×C₂²⋊C₈	—	2	2	no	—
75	C₇×#97	—	2	2	no	—
76	C₇×#97	—	2	2	no	—
77	C₄×C₇×C₈	—	2	2	no	—
78	C₄×C₇×C₈	D₄	1	1	yes	characteristic
79	C₄×C₇×C₈	D₄	1	1	yes	characteristic
80	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
82	#3147	C₂×C₇	1	1	yes	characteristic
83	#3103	C₂×C₇	1	2	yes	complemented
84	C₂×#446	C₂×C₇	1	1	yes	characteristic, complemented
85	#1690	C₂×C₇	1	2	yes	complemented
86	#1688	C₂×C₇	1	2	yes	complemented
87	#1686	C₂×C₇	1	4	yes	complemented
88	#1564	C₂×C₇	1	1	yes	characteristic, complemented
89	#1540	C₂×C₇	1	2	yes	complemented
90	C₇×#41	—	4	4	no	—
91	C₇×#41	—	4	4	no	—
92	C₂×C₇×Q₈	—	2	1	no	—
93	C₂×C₇×Q₈	—	2	2	no	—
94	C₂×C₇×Q₈	—	4	2	no	—
95	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
96	C₂×C₇×D₄	—	2	2	no	—
97	C₂×C₇×D₄	—	2	2	no	—
98	C₂×C₇×D₄	—	2	2	no	—
99	C₂×C₇×D₄	—	4	2	no	—
100	C₂²×C₄×C₇	—	2	1	no	—
101	C₂²×C₄×C₇	—	2	1	no	—
102	C₂²×C₄×C₇	—	4	2	no	—
103	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₇	#41	1	1	yes	characteristic
106	C₇×Q₁₆	—	4	2	no	—
107	C₇×Q₁₆	—	4	2	no	—
108	C₇×SD₁₆	—	4	4	no	—
109	C₇×SD₁₆	—	4	8	no	—
110	C₇×SD₁₆	—	4	4	no	—
111	C₇×D₈	—	4	2	no	—
112	C₇×D₈	—	4	2	no	—
113	C₂×C₇×C₈	—	2	2	no	—
114	C₂×C₇×C₈	—	2	4	no	—
115	C₂×C₇×C₈	—	2	2	no	—
116	C₂×C₇×C₈	—	4	2	no	—
117	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
118	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
122	C₇×C₄⋊C₄	—	2	2	no	—
123	C₇×C₄⋊C₄	—	2	1	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	C₂×D₄	1	2	yes	—
128	C₇×C₂²⋊C₄	—	4	2	no	—
129	C₇×C₂²⋊C₄	—	4	2	no	—
130	C₄²×C₇	—	2	2	no	—
131	C₄²×C₇	—	2	2	no	—
132	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₇	#41	1	1	yes	characteristic
135	C₄²×C₇	C₂⁴	1	1	yes	characteristic
136	C₂×#135	—	2	1	no	—
137	C₂×#135	—	2	1	no	—
138	#532	C₂²×C₇	1	2	yes	—
139	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
140	#495	—	2	1	no	—
141	#495	—	2	1	no	—
142	#494	C₂²×C₇	1	1	yes	characteristic
143	#493	C₂²×C₇	1	1	yes	characteristic
144	#492	C₂²×C₇	1	2	yes	—
145	#481	—	2	2	no	—
146	#475	C₂²×C₇	1	4	yes	—
147	#460	C₂²×C₇	1	4	yes	—
148	#448	—	2	2	no	—
149	#446	—	2	1	no	—
150	#446	—	2	1	no	—
151	#446	C₂²×C₇	1	2	yes	—
152	#446	C₂²×C₇	1	2	yes	—
153	C₄×SD₁₆	—	2	4	no	—
154	#431	C₂²×C₇	1	2	yes	—
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
157	C₂×#97	—	2	1	no	—
158	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
159	#335	C₂²×C₇	1	2	yes	—
160	#334	C₂²×C₇	1	2	yes	—
161	#332	C₂²×C₇	1	4	yes	—
162	#331	C₂²×C₇	1	4	yes	—
163	C₂³×C₇	—	4	2	no	—
164	C₂³×C₇	#121	1	1	yes	characteristic
165	C₇×Q₈	—	2	4	no	—
166	C₇×Q₈	—	4	2	no	—
167	C₇×Q₈	—	4	2	no	—
168	C₇×Q₈	—	4	2	no	—
169	C₇×D₄	—	4	2	no	—
170	C₇×D₄	—	4	2	no	—
171	C₇×D₄	—	4	4	no	—
172	C₇×D₄	—	8	2	no	—
173	C₇×D₄	—	8	4	no	—
174	C₂×C₄×C₇	—	2	2	no	—
175	C₂×C₄×C₇	—	2	2	no	—
176	C₂×C₄×C₇	—	2	2	no	—
177	C₂×C₄×C₇	—	4	1	no	—
178	C₂×C₄×C₇	—	4	2	no	—
179	C₂×C₄×C₇	—	4	2	no	—
180	C₂×C₄×C₇	—	4	2	no	—
181	C₂×C₄×C₇	—	8	4	no	—
182	C₂×C₄×C₇	#121	1	1	yes	characteristic
183	C₂×C₄×C₇	#121	1	1	yes	characteristic
184	C₂×C₄×C₇	#121	1	1	yes	characteristic
185	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
186	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
187	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
190	C₇×C₈	—	2	1	no	—
191	C₇×C₈	—	2	1	no	—
192	C₇×C₈	—	4	2	no	—
193	C₇×C₈	—	4	4	no	—
194	C₇×C₈	C₂×SD₁₆	1	2	yes	—
195	C₇×C₈	C₂×SD₁₆	1	2	yes	—
196	C₂×#41	—	2	2	no	—
197	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
198	#135	—	4	4	no	—
199	#135	—	4	4	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×SD₁₆	—	2	4	no	—
203	C₂×SD₁₆	—	2	2	no	—
204	C₂×SD₁₆	—	2	2	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
209	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
210	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
211	#127	C₂³×C₇	1	2	yes	—
212	#124	—	2	2	no	—
213	#121	—	2	2	no	—
214	#121	—	2	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	C₇×D₄	1	2	yes	—
218	C₄×Q₈	C₂³×C₇	1	2	yes	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
225	#106	—	2	4	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	C₂³×C₇	1	2	yes	—
230	C₄⋊C₈	C₂³×C₇	1	4	yes	—
231	#103	—	2	4	no	—
232	#102	—	2	4	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	#97	—	2	2	no	—
235	#97	—	2	2	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	C₇×D₄	1	1	yes	characteristic
238	C₄×C₈	C₇×D₄	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
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	#460	1	2	yes	—
247	C₂²×C₇	C₂×#121	1	1	yes	characteristic
248	C₄×C₇	—	2	1	no	—
249	C₄×C₇	—	2	1	no	—
250	C₄×C₇	—	2	1	no	—
251	C₄×C₇	—	4	2	no	—
252	C₄×C₇	—	4	2	no	—
253	C₄×C₇	—	8	2	no	—
254	C₄×C₇	#460	1	2	yes	—
255	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
256	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₄×C₇	#575	1	2	yes	—
258	#41	—	4	4	no	—
259	#41	—	4	4	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	4	2	no	—
263	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
273	C₂²×C₄	C₇×#41	1	1	yes	characteristic
274	Q₁₆	—	4	2	no	—
275	Q₁₆	—	4	2	no	—
276	SD₁₆	—	4	4	no	—
277	SD₁₆	—	4	8	no	—
278	SD₁₆	—	4	4	no	—
279	D₈	—	4	2	no	—
280	D₈	—	4	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	C₇×SD₁₆	1	4	yes	complemented
286	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
287	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
288	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
289	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
301	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
302	C₄²	C₇×#41	1	1	yes	characteristic
303	C₄²	C₂⁴×C₇	1	1	yes	characteristic
304	C₂×C₇	—	2	2	no	—
305	C₂×C₇	—	16	2	no	—
306	C₂×C₇	C₂×#460	1	1	yes	characteristic
307	C₂×C₇	#3054	1	1	yes	characteristic
308	C₂×C₇	#3067	1	1	yes	characteristic
309	C₂³	—	4	2	no	—
310	C₂³	C₇×#121	1	1	yes	characteristic
311	Q₈	—	2	4	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	4	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	8	4	no	—
328	C₂×C₄	C₇×#121	1	1	yes	characteristic
329	C₂×C₄	C₇×#121	1	1	yes	characteristic
330	C₂×C₄	C₇×#121	1	1	yes	characteristic
331	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
332	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
333	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
335	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	4	no	—
340	C₈	C₂×C₇×SD₁₆	1	2	yes	—
341	C₈	C₂×C₇×SD₁₆	1	2	yes	—
342	C₇	#19146	1	1	yes	characteristic, direct product factor, Sylow
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	C₇×#460	1	2	yes	—
349	C₂²	C₂×C₇×#121	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	8	2	no	—
356	C₄	C₇×#460	1	2	yes	—
357	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
358	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
359	C₄	C₇×#575	1	2	yes	—
360	C₂	—	2	2	no	—
361	C₂	—	16	2	no	—
362	C₂	C₂×C₇×#460	1	1	yes	characteristic
363	C₂	C₇×#3054	1	1	yes	characteristic
364	C₂	C₇×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×#121
prime radical	C₇×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#3103	C₂	semidirect
3. C₂×C₇×#446	C₂	semidirect
4. C₇×#1690	C₂	semidirect
5. C₇×#1688	C₂	semidirect
6. C₇×#1686	C₂	semidirect
7. C₇×#1564	C₂	semidirect
8. C₇×#1540	C₂	semidirect
36. #19146	C₇	direct (N × H)
83. #3103	C₂×C₇	semidirect
84. C₂×#446	C₂×C₇	semidirect
85. #1690	C₂×C₇	semidirect
86. #1688	C₂×C₇	semidirect
87. #1686	C₂×C₇	semidirect
88. #1564	C₂×C₇	semidirect
89. #1540	C₂×C₇	semidirect
117. C₂×C₇×C₈	SD₁₆	semidirect
285. C₂×C₈	C₇×SD₁₆	semidirect
342. C₇	#19146	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×#3147	2
C₇×#3103	2
C₂×C₇×#446	2
C₇×#1690	2
C₇×#1688	2
C₇×#1686	2
C₇×#1564	2
C₇×#1540	2
#19146	7

Subgroups — C7×#19146 #13196362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19146 #13196362