Subgroups — C₇×#20821 #13198037

Order 1792.

810 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
2	C₇×#2314	C₂	1	2	yes	maximal
3	C₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1836	C₂	1	2	yes	maximal, complemented
6	C₇×#1809	C₂	1	2	yes	maximal
7	C₇×#1808	C₂	1	2	yes	maximal
8	C₇×#1776	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×#107	C₂	1	2	yes	maximal, complemented
10	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₇×#117	C₂²	1	1	yes	characteristic
13	C₂×C₇×#107	C₂²	1	2	yes	—
14	C₂×C₇×#107	C₂²	1	2	yes	—
15	C₂×C₇×#107	C₂²	1	2	yes	—
16	C₂×C₇×#106	C₂²	1	2	yes	—
17	C₇×#405	C₄	1	8	yes	complemented
18	C₂×C₇×#97	C₂²	1	1	yes	characteristic
19	C₂×C₇×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
22	C₇×#388	—	2	4	no	—
23	C₇×#387	—	2	4	no	—
24	C₇×#384	C₂²	1	2	yes	—
25	C₇×#382	C₂²	1	2	yes	—
26	C₇×#380	C₂²	1	4	yes	—
27	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
28	C₇×#340	C₂²	1	4	yes	—
29	C₇×#340	C₂²	1	4	yes	—
30	#20821	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂×C₇×M₁₆	—	2	2	no	—
33	C₂×C₇×M₁₆	—	2	2	no	—
34	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
35	C₂²×C₇×C₈	—	2	2	no	—
36	C₂²×C₇×C₈	—	2	2	no	—
37	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
42	C₇×#117	C₂×C₄	1	4	yes	—
43	C₇×#117	Q₈	1	4	yes	—
44	C₂×C₇×C₄⋊C₄	—	2	4	no	—
45	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
46	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
48	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
49	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
50	C₂×C₄²×C₇	—	2	4	no	—
51	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₇×#107	—	2	4	no	—
54	C₇×#107	—	2	4	no	—
55	C₇×#107	—	2	4	no	—
56	C₇×#106	—	2	4	no	—
57	C₇×#97	C₂×C₄	1	4	yes	—
58	C₇×#97	C₂×C₄	1	4	yes	—
59	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
60	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
61	C₇×#95	—	2	4	no	—
62	C₇×#95	—	2	4	no	—
63	C₇×#95	—	2	4	no	—
64	C₂×#405	C₂×C₇	1	1	yes	characteristic
65	#2314	C₂×C₇	1	2	yes	—
66	#1846	C₂×C₇	1	1	yes	characteristic, complemented
67	#1845	C₂×C₇	1	1	yes	characteristic, complemented
68	#1836	C₂×C₇	1	2	yes	complemented
69	#1809	C₂×C₇	1	2	yes	—
70	#1808	C₂×C₇	1	2	yes	—
71	#1776	C₂×C₇	1	2	yes	complemented
72	C₄×#107	C₂×C₇	1	2	yes	complemented
73	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	—	2	1	no	—
75	C₂²×C₄×C₇	—	2	1	no	—
76	C₂²×C₄×C₇	—	2	1	no	—
77	C₂²×C₄×C₇	—	2	1	no	—
78	C₂²×C₄×C₇	—	2	4	no	—
79	C₂²×C₄×C₇	—	2	4	no	—
80	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
86	C₇×M₁₆	—	2	8	no	—
87	C₂×C₇×C₈	—	2	1	no	—
88	C₂×C₇×C₈	—	2	1	no	—
89	C₂×C₇×C₈	—	2	1	no	—
90	C₂×C₇×C₈	—	2	1	no	—
91	C₂×C₇×C₈	—	2	8	no	—
92	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₇×C₈	#41	1	2	yes	—
97	C₂×C₇×C₈	#41	1	2	yes	—
98	C₂×C₇×C₈	#41	1	2	yes	—
99	C₂×C₇×C₈	#41	1	2	yes	—
100	C₇×C₄⋊C₄	—	2	4	no	—
101	C₇×C₄⋊C₄	—	2	4	no	—
102	C₇×C₄⋊C₄	—	2	4	no	—
103	C₇×C₄⋊C₄	—	2	4	no	—
104	C₇×C₄⋊C₄	—	8	4	no	—
105	C₇×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
106	C₇×C₂²⋊C₄	—	8	4	no	—
107	C₇×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
108	C₄²×C₇	—	8	4	no	—
109	C₄²×C₇	C₂²×C₄	1	2	yes	—
110	C₄²×C₇	C₂²×C₄	1	2	yes	—
111	C₄²×C₇	C₂×Q₈	1	2	yes	—
112	C₄²×C₇	C₂×Q₈	1	2	yes	—
113	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
114	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
115	C₂×#117	C₂²×C₇	1	1	yes	characteristic
116	C₂×#107	C₂²×C₇	1	2	yes	—
117	C₂×#107	C₂²×C₇	1	2	yes	—
118	C₂×#107	C₂²×C₇	1	2	yes	—
119	C₂×#106	C₂²×C₇	1	2	yes	—
120	#405	C₄×C₇	1	8	yes	complemented
121	C₂×#97	C₂²×C₇	1	1	yes	characteristic
122	C₂×#97	C₂²×C₇	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
125	#388	—	2	4	no	—
126	#387	—	2	4	no	—
127	#384	C₂²×C₇	1	2	yes	—
128	#382	C₂²×C₇	1	2	yes	—
129	#380	C₂²×C₇	1	4	yes	—
130	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
131	#340	C₂²×C₇	1	4	yes	—
132	#340	C₂²×C₇	1	4	yes	—
133	C₂³×C₇	—	2	1	no	—
134	C₂³×C₇	—	2	1	no	—
135	C₂³×C₇	—	2	2	no	—
136	C₂³×C₇	—	2	2	no	—
137	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
138	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇	—	2	1	no	—
140	C₂×C₄×C₇	—	2	1	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	2	no	—
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	4	4	no	—
150	C₂×C₄×C₇	—	4	4	no	—
151	C₂×C₄×C₇	—	4	4	no	—
152	C₂×C₄×C₇	—	4	4	no	—
153	C₂×C₄×C₇	—	4	4	no	—
154	C₂×C₄×C₇	—	4	4	no	—
155	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
156	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
157	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
158	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
164	C₇×C₈	—	2	2	no	—
165	C₇×C₈	—	2	2	no	—
166	C₇×C₈	#117	1	4	yes	—
167	C₇×C₈	#117	1	4	yes	—
168	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
175	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
179	#117	C₂×C₄×C₇	1	4	yes	—
180	#117	C₇×Q₈	1	4	yes	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
184	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
185	C₂×C₂²⋊C₄	—	2	4	no	—
186	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
187	C₂×C₄²	—	2	4	no	—
188	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
190	#107	—	2	4	no	—
191	#107	—	2	4	no	—
192	#107	—	2	4	no	—
193	#106	—	2	4	no	—
194	#97	C₂×C₄×C₇	1	4	yes	—
195	#97	C₂×C₄×C₇	1	4	yes	—
196	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
197	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
198	#95	—	2	4	no	—
199	#95	—	2	4	no	—
200	#95	—	2	4	no	—
201	C₂²×C₇	—	2	4	no	—
202	C₂²×C₇	—	2	2	no	—
203	C₂²×C₇	—	4	2	no	—
204	C₂²×C₇	—	4	1	no	—
205	C₂²×C₇	—	4	1	no	—
206	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
207	C₂²×C₇	#532	1	1	yes	characteristic
208	C₂²×C₇	#532	1	1	yes	characteristic
209	C₂²×C₇	#576	1	2	yes	—
210	C₂²×C₇	#578	1	2	yes	—
211	C₄×C₇	—	2	4	no	—
212	C₄×C₇	—	2	4	no	—
213	C₄×C₇	—	2	2	no	—
214	C₄×C₇	—	8	4	no	—
215	C₄×C₇	—	8	4	no	—
216	C₄×C₇	C₂×#117	1	1	yes	characteristic
217	C₄×C₇	C₂×#117	1	1	yes	characteristic
218	C₄×C₇	#533	1	1	yes	characteristic
219	C₄×C₇	#533	1	1	yes	characteristic
220	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
228	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
230	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
231	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
233	M₁₆	—	2	8	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	8	no	—
239	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
240	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
243	C₂×C₈	C₇×#41	1	2	yes	—
244	C₂×C₈	C₇×#41	1	2	yes	—
245	C₂×C₈	C₇×#41	1	2	yes	—
246	C₂×C₈	C₇×#41	1	2	yes	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	8	4	no	—
252	C₄⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
253	C₂²⋊C₄	—	8	4	no	—
254	C₂²⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
255	C₄²	—	8	4	no	—
256	C₄²	C₂²×C₄×C₇	1	2	yes	—
257	C₄²	C₂²×C₄×C₇	1	2	yes	—
258	C₄²	C₂×C₇×Q₈	1	2	yes	—
259	C₄²	C₂×C₇×Q₈	1	2	yes	—
260	C₂×C₇	—	4	2	no	—
261	C₂×C₇	#2285	1	1	yes	characteristic
262	C₂×C₇	#2913	1	2	yes	—
263	C₂×C₇	#3175	1	2	yes	—
264	C₂×C₇	#3176	1	1	yes	characteristic
265	C₂×C₇	#3177	1	1	yes	characteristic
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
271	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
289	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
290	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
291	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
293	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
294	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
295	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
296	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	C₇×#117	1	4	yes	—
300	C₈	C₇×#117	1	4	yes	—
301	C₇	#20821	1	1	yes	characteristic, direct product factor, Sylow
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
308	C₂²	C₇×#532	1	1	yes	characteristic
309	C₂²	C₇×#532	1	1	yes	characteristic
310	C₂²	C₇×#576	1	2	yes	—
311	C₂²	C₇×#578	1	2	yes	—
312	C₄	—	2	4	no	—
313	C₄	—	2	4	no	—
314	C₄	—	2	2	no	—
315	C₄	—	8	4	no	—
316	C₄	—	8	4	no	—
317	C₄	C₂×C₇×#117	1	1	yes	characteristic
318	C₄	C₂×C₇×#117	1	1	yes	characteristic
319	C₄	C₇×#533	1	1	yes	characteristic
320	C₄	C₇×#533	1	1	yes	characteristic
321	C₂	—	4	2	no	—
322	C₂	C₇×#2285	1	1	yes	characteristic
323	C₂	C₇×#2913	1	2	yes	—
324	C₂	C₇×#3175	1	2	yes	—
325	C₂	C₇×#3176	1	1	yes	characteristic
326	C₂	C₇×#3177	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1846	C₂	semidirect
4. C₇×#1845	C₂	semidirect
5. C₇×#1836	C₂	semidirect
8. C₇×#1776	C₂	semidirect
9. C₄×C₇×#107	C₂	semidirect
17. C₇×#405	C₄	semidirect
30. #20821	C₇	direct (N × H)
66. #1846	C₂×C₇	semidirect
67. #1845	C₂×C₇	semidirect
68. #1836	C₂×C₇	semidirect
71. #1776	C₂×C₇	semidirect
72. C₄×#107	C₂×C₇	semidirect
120. #405	C₄×C₇	semidirect
301. C₇	#20821	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#405	2
C₇×#2314	2
C₇×#1846	2
C₇×#1845	2
C₇×#1836	2
C₇×#1809	2
C₇×#1808	2
C₇×#1776	2
C₄×C₇×#107	2
#20821	7

Subgroups — C7×#20821 #13198037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20821 #13198037