Subgroups — C₇×#21275 #13198491

Order 1792.

974 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2293	C₂	1	2	yes	maximal, complemented
3	C₇×#1970	C₂	1	2	yes	maximal
4	C₇×#1942	C₂	1	2	yes	maximal, complemented
5	C₇×#1934	C₂	1	4	yes	maximal, complemented
6	C₄×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₇×#102	C₂	1	2	yes	maximal, complemented
8	C₄²×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
11	C₂×C₇×#117	—	2	2	no	—
12	C₄×C₇×D₈	C₄	1	8	yes	complemented
13	C₂×C₇×#107	C₂²	1	2	yes	—
14	C₂×C₇×#107	C₂²	1	1	yes	characteristic
15	C₂×C₇×#102	C₂²	1	4	yes	—
16	C₂×C₇×#102	C₂²	1	2	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₇×#390	C₂²	1	4	yes	—
22	C₇×#389	C₂²	1	4	yes	—
23	C₇×#388	—	2	4	no	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
26	C₄³×C₇	C₂²	1	1	yes	characteristic
27	C₇×#340	C₂²	1	4	yes	—
28	C₇×#340	C₂²	1	2	yes	—
29	#21275	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×D₄	C₂³	1	2	yes	—
31	C₂³×C₄×C₇	—	2	2	no	—
32	C₂×C₇×D₈	—	2	4	no	—
33	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
34	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₇×C₈	C₂³	1	2	yes	—
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₇×D₄	—	2	4	no	—
38	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
39	C₇×#117	—	4	8	no	—
40	C₂×C₇×C₄⋊C₄	—	2	4	no	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
46	C₂×C₄²×C₇	—	2	2	no	—
47	C₂×C₄²×C₇	D₄	1	2	yes	—
48	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
52	C₇×#107	—	2	4	no	—
53	C₇×#107	C₂×C₄	1	4	yes	—
54	C₇×#102	—	2	8	no	—
55	C₇×#102	C₂×C₄	1	8	yes	—
56	C₄×C₇×C₈	—	2	4	no	—
57	C₄×C₇×C₈	—	2	4	no	—
58	C₄×C₇×C₈	—	2	2	no	—
59	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
60	C₇×#95	—	2	4	no	—
61	C₇×#95	—	2	2	no	—
62	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
63	#2293	C₂×C₇	1	2	yes	complemented
64	#1970	C₂×C₇	1	2	yes	—
65	#1942	C₂×C₇	1	2	yes	complemented
66	#1934	C₂×C₇	1	4	yes	complemented
67	C₄×#107	C₂×C₇	1	1	yes	characteristic, complemented
68	C₄×#102	C₂×C₇	1	2	yes	complemented
69	C₄²×C₈	C₂×C₇	1	1	yes	characteristic, complemented
70	C₂⁴×C₇	—	2	2	no	—
71	C₂×C₇×D₄	—	2	4	no	—
72	C₂×C₇×D₄	—	2	4	no	—
73	C₂×C₇×D₄	—	2	2	no	—
74	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₇	—	2	2	no	—
76	C₂²×C₄×C₇	—	2	8	no	—
77	C₂²×C₄×C₇	—	2	4	no	—
78	C₂²×C₄×C₇	—	4	4	no	—
79	C₂²×C₄×C₇	—	4	4	no	—
80	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
85	C₇×D₈	—	2	8	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
90	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₇×C₈	#41	1	4	yes	—
92	C₂×C₇×C₈	#41	1	4	yes	—
93	C₇×C₄⋊C₄	—	2	2	no	—
94	C₇×C₄⋊C₄	—	2	4	no	—
95	C₇×C₄⋊C₄	—	2	4	no	—
96	C₇×C₄⋊C₄	—	4	8	no	—
97	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
98	C₇×C₂²⋊C₄	—	2	8	no	—
99	C₇×C₂²⋊C₄	—	4	8	no	—
100	C₇×C₂²⋊C₄	—	4	8	no	—
101	C₄²×C₇	—	2	4	no	—
102	C₄²×C₇	—	2	2	no	—
103	C₄²×C₇	—	2	2	no	—
104	C₄²×C₇	—	2	2	no	—
105	C₄²×C₇	—	4	4	no	—
106	C₄²×C₇	D₈	1	4	yes	complemented
107	C₄²×C₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₇	#41	1	2	yes	—
109	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
110	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
111	C₂×#117	—	2	2	no	—
112	C₄×D₈	C₄×C₇	1	8	yes	complemented
113	C₂×#107	C₂²×C₇	1	2	yes	—
114	C₂×#107	C₂²×C₇	1	1	yes	characteristic
115	C₂×#102	C₂²×C₇	1	4	yes	—
116	C₂×#102	C₂²×C₇	1	2	yes	—
117	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
121	#390	C₂²×C₇	1	4	yes	—
122	#389	C₂²×C₇	1	4	yes	—
123	#388	—	2	4	no	—
124	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
125	C₄×C₂²⋊C₄	—	2	2	no	—
126	C₄³	C₂²×C₇	1	1	yes	characteristic
127	#340	C₂²×C₇	1	2	yes	—
128	#340	C₂²×C₇	1	4	yes	—
129	C₂³×C₇	—	2	4	no	—
130	C₂³×C₇	—	4	4	no	—
131	C₂³×C₇	—	4	2	no	—
132	C₂³×C₇	—	4	4	no	—
133	C₂³×C₇	—	4	2	no	—
134	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
135	C₇×D₄	—	2	4	no	—
136	C₇×D₄	—	2	4	no	—
137	C₇×D₄	—	4	4	no	—
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	2	2	no	—
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	—	2	4	no	—
144	C₂×C₄×C₇	—	4	2	no	—
145	C₂×C₄×C₇	—	4	2	no	—
146	C₂×C₄×C₇	—	4	16	no	—
147	C₂×C₄×C₇	—	4	4	no	—
148	C₂×C₄×C₇	—	4	4	no	—
149	C₂×C₄×C₇	—	4	4	no	—
150	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
151	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
153	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₇	#135	1	4	yes	—
155	C₂×C₄×C₇	#135	1	2	yes	—
156	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂×#41	1	2	yes	—
158	C₂×C₄×C₇	C₂×#41	1	2	yes	—
159	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
160	C₇×C₈	—	2	2	no	—
161	C₇×C₈	—	2	4	no	—
162	C₇×C₈	#117	1	4	yes	complemented
163	C₂²×D₄	C₂³×C₇	1	2	yes	—
164	C₂³×C₄	—	2	2	no	—
165	C₂×D₈	—	2	4	no	—
166	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
167	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₇	1	2	yes	—
169	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
170	C₄×D₄	—	2	4	no	—
171	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
172	#117	—	4	8	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
175	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
176	C₂×C₂²⋊C₄	—	2	2	no	—
177	C₂×C₂²⋊C₄	—	2	4	no	—
178	C₂×C₂²⋊C₄	—	2	4	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
181	C₂×C₄²	C₇×D₄	1	2	yes	—
182	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₇	1	2	yes	—
184	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
185	#107	—	2	4	no	—
186	#107	C₂×C₄×C₇	1	4	yes	—
187	#102	—	2	8	no	—
188	#102	C₂×C₄×C₇	1	8	yes	—
189	C₄×C₈	—	2	4	no	—
190	C₄×C₈	—	2	4	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
193	#95	—	2	4	no	—
194	#95	—	2	2	no	—
195	C₂²×C₇	—	4	4	no	—
196	C₂²×C₇	—	4	4	no	—
197	C₂²×C₇	—	4	4	no	—
198	C₂²×C₇	—	8	4	no	—
199	C₂²×C₇	—	8	4	no	—
200	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂²×C₇	#532	1	2	yes	—
202	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
203	C₂²×C₇	C₂×#135	1	1	yes	characteristic
204	C₂²×C₇	C₂×#135	1	2	yes	—
205	C₄×C₇	—	2	4	no	—
206	C₄×C₇	—	2	4	no	—
207	C₄×C₇	—	2	2	no	—
208	C₄×C₇	—	4	4	no	—
209	C₄×C₇	—	8	4	no	—
210	C₄×C₇	C₄×D₈	1	4	yes	—
211	C₄×C₇	C₂×#117	1	1	yes	characteristic
212	C₄×C₇	C₄×#41	1	1	yes	characteristic
213	C₄×C₇	#529	1	2	yes	—
214	C₂⁴	—	2	2	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
226	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
227	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
228	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
229	D₈	—	2	8	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
234	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
235	C₂×C₈	C₇×#41	1	4	yes	—
236	C₂×C₈	C₇×#41	1	4	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	8	no	—
241	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
242	C₂²⋊C₄	—	2	8	no	—
243	C₂²⋊C₄	—	4	8	no	—
244	C₂²⋊C₄	—	4	8	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	4	4	no	—
250	C₄²	C₇×D₈	1	4	yes	complemented
251	C₄²	C₂²×C₄×C₇	1	2	yes	—
252	C₄²	C₇×#41	1	2	yes	—
253	C₂×C₇	—	8	4	no	—
254	C₂×C₇	#2293	1	1	yes	characteristic
255	C₂×C₇	C₂×C₄×D₈	1	1	yes	characteristic
256	C₂×C₇	C₄×#135	1	1	yes	characteristic
257	C₂×C₇	#3168	1	2	yes	—
258	C₂×C₇	#3169	1	2	yes	—
259	C₂³	—	2	4	no	—
260	C₂³	—	4	4	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
265	D₄	—	2	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	16	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
281	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
282	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
283	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
284	C₂×C₄	C₇×#135	1	4	yes	—
285	C₂×C₄	C₇×#135	1	2	yes	—
286	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
287	C₂×C₄	C₂×C₇×#41	1	2	yes	—
288	C₂×C₄	C₂×C₇×#41	1	2	yes	—
289	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	2	4	no	—
292	C₈	C₇×#117	1	4	yes	complemented
293	C₇	#21275	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²	—	4	4	no	—
295	C₂²	—	4	4	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
300	C₂²	C₇×#532	1	2	yes	—
301	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
302	C₂²	C₂×C₇×#135	1	1	yes	characteristic
303	C₂²	C₂×C₇×#135	1	2	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	4	no	—
308	C₄	—	8	4	no	—
309	C₄	C₄×C₇×D₈	1	4	yes	—
310	C₄	C₂×C₇×#117	1	1	yes	characteristic
311	C₄	C₄×C₇×#41	1	1	yes	characteristic
312	C₄	C₇×#529	1	2	yes	—
313	C₂	—	8	4	no	—
314	C₂	C₇×#2293	1	1	yes	characteristic
315	C₂	C₂×C₄×C₇×D₈	1	1	yes	characteristic
316	C₂	C₄×C₇×#135	1	1	yes	characteristic
317	C₂	C₇×#3168	1	2	yes	—
318	C₂	C₇×#3169	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂²×D₄
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2293	C₂	semidirect
4. C₇×#1942	C₂	semidirect
5. C₇×#1934	C₂	semidirect
6. C₄×C₇×#107	C₂	semidirect
7. C₄×C₇×#102	C₂	semidirect
8. C₄²×C₇×C₈	C₂	semidirect
12. C₄×C₇×D₈	C₄	semidirect
29. #21275	C₇	direct (N × H)
63. #2293	C₂×C₇	semidirect
65. #1942	C₂×C₇	semidirect
66. #1934	C₂×C₇	semidirect
67. C₄×#107	C₂×C₇	semidirect
68. C₄×#102	C₂×C₇	semidirect
69. C₄²×C₈	C₂×C₇	semidirect
106. C₄²×C₇	D₈	semidirect
112. C₄×D₈	C₄×C₇	semidirect
162. C₇×C₈	#117	semidirect
250. C₄²	C₇×D₈	semidirect
292. C₈	C₇×#117	semidirect
293. C₇	#21275	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×C₄×C₇×D₈	2
C₇×#2293	2
C₇×#1970	2
C₇×#1942	2
C₇×#1934	2
C₄×C₇×#107	2
C₄×C₇×#102	2
C₄²×C₇×C₈	2
#21275	7

Subgroups — C7×#21275 #13198491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21275 #13198491