Subgroups — C₇×#21282 #13198498

Order 1792.

862 conjugacy classes of subgroups.

The subgroup lattice has 308 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×#2319	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×#1936	C₂	1	4	yes	maximal, complemented
6	C₄×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1762	C₂	1	2	yes	maximal, complemented
8	C₇×#1726	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	C₄×C₇×D₈	C₄	1	8	yes	complemented
12	C₂×C₇×#107	C₂²	1	2	yes	—
13	C₂×C₇×#107	C₂²	1	1	yes	characteristic
14	C₂×C₇×#102	C₂²	1	4	yes	—
15	C₂×C₇×#102	C₂²	1	2	yes	—
16	C₂×C₇×#97	C₂²	1	1	yes	characteristic
17	C₂×C₇×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#390	C₂²	1	4	yes	—
21	C₇×#389	C₂²	1	4	yes	—
22	C₇×#388	—	2	4	no	—
23	C₇×#381	—	2	2	no	—
24	C₇×#380	—	2	2	no	—
25	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
26	C₇×#376	C₂²	1	1	yes	characteristic
27	C₇×#340	C₂²	1	4	yes	—
28	C₇×#340	C₂²	1	2	yes	—
29	#21282	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	C₂×C₇×C₄⋊C₄	—	2	4	no	—
40	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
44	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₄²×C₇	—	2	2	no	—
46	C₂×C₄²×C₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₇×#107	—	2	4	no	—
49	C₇×#107	C₂×C₄	1	4	yes	—
50	C₇×#102	—	2	8	no	—
51	C₇×#102	C₂×C₄	1	8	yes	—
52	C₇×#97	—	2	4	no	—
53	C₇×#97	—	2	2	no	—
54	C₄×C₇×C₈	—	2	4	no	—
55	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
56	C₇×#95	—	2	2	no	—
57	C₇×#95	—	2	4	no	—
58	C₇×#95	D₄	1	1	yes	characteristic
59	C₇×#95	D₄	1	1	yes	characteristic
60	C₇×#95	D₄	1	2	yes	—
61	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
62	#2319	C₂×C₇	1	2	yes	complemented
63	#1970	C₂×C₇	1	2	yes	—
64	#1942	C₂×C₇	1	2	yes	complemented
65	#1936	C₂×C₇	1	4	yes	complemented
66	C₄×#107	C₂×C₇	1	1	yes	characteristic, complemented
67	#1762	C₂×C₇	1	2	yes	complemented
68	#1726	C₂×C₇	1	1	yes	characteristic, complemented
69	C₂⁴×C₇	—	2	2	no	—
70	C₂×C₇×D₄	—	2	4	no	—
71	C₂×C₇×D₄	—	2	4	no	—
72	C₂×C₇×D₄	—	2	2	no	—
73	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₇	—	2	2	no	—
75	C₂²×C₄×C₇	—	2	8	no	—
76	C₂²×C₄×C₇	—	2	4	no	—
77	C₂²×C₄×C₇	—	8	4	no	—
78	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
83	C₇×D₈	—	2	8	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₇×C₈	#41	1	4	yes	—
90	C₂×C₇×C₈	#41	1	4	yes	—
91	C₇×C₄⋊C₄	—	2	2	no	—
92	C₇×C₄⋊C₄	—	2	4	no	—
93	C₇×C₄⋊C₄	—	2	4	no	—
94	C₇×C₄⋊C₄	—	8	4	no	—
95	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
96	C₇×C₂²⋊C₄	—	2	8	no	—
97	C₇×C₂²⋊C₄	—	8	4	no	—
98	C₇×C₂²⋊C₄	—	8	4	no	—
99	C₄²×C₇	—	2	2	no	—
100	C₄²×C₇	—	2	2	no	—
101	C₄²×C₇	—	8	2	no	—
102	C₄²×C₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₇	#41	1	2	yes	—
104	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
105	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
106	C₄×D₈	C₄×C₇	1	8	yes	complemented
107	C₂×#107	C₂²×C₇	1	2	yes	—
108	C₂×#107	C₂²×C₇	1	1	yes	characteristic
109	C₂×#102	C₂²×C₇	1	4	yes	—
110	C₂×#102	C₂²×C₇	1	2	yes	—
111	C₂×#97	C₂²×C₇	1	2	yes	—
112	C₂×#97	C₂²×C₇	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
115	#390	C₂²×C₇	1	4	yes	—
116	#389	C₂²×C₇	1	4	yes	—
117	#388	—	2	4	no	—
118	#381	—	2	2	no	—
119	#380	—	2	2	no	—
120	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
121	#376	C₂²×C₇	1	1	yes	characteristic
122	#340	C₂²×C₇	1	2	yes	—
123	#340	C₂²×C₇	1	4	yes	—
124	C₂³×C₇	—	2	4	no	—
125	C₂³×C₇	—	4	4	no	—
126	C₂³×C₇	—	4	2	no	—
127	C₂³×C₇	—	4	4	no	—
128	C₂³×C₇	—	4	2	no	—
129	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
130	C₇×D₄	—	2	4	no	—
131	C₇×D₄	—	2	4	no	—
132	C₇×D₄	—	4	4	no	—
133	C₂×C₄×C₇	—	2	1	no	—
134	C₂×C₄×C₇	—	2	2	no	—
135	C₂×C₄×C₇	—	2	1	no	—
136	C₂×C₄×C₇	—	2	4	no	—
137	C₂×C₄×C₇	—	2	2	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	—	4	4	no	—
144	C₂×C₄×C₇	—	4	4	no	—
145	C₂×C₄×C₇	—	4	4	no	—
146	C₂×C₄×C₇	—	4	4	no	—
147	C₂×C₄×C₇	—	8	8	no	—
148	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
150	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂×#41	1	2	yes	—
152	C₂×C₄×C₇	C₂×#41	1	2	yes	—
153	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
154	C₇×C₈	—	2	2	no	—
155	C₇×C₈	—	2	4	no	—
156	C₇×C₈	#117	1	4	yes	—
157	C₂²×D₄	C₂³×C₇	1	2	yes	—
158	C₂³×C₄	—	2	2	no	—
159	C₂×D₈	—	2	4	no	—
160	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
161	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₇	1	2	yes	—
163	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
164	C₄×D₄	—	2	4	no	—
165	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
168	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
169	C₂×C₂²⋊C₄	—	2	2	no	—
170	C₂×C₂²⋊C₄	—	2	4	no	—
171	C₂×C₂²⋊C₄	—	2	4	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	C₂³×C₇	1	2	yes	—
174	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
175	#107	—	2	4	no	—
176	#107	C₂×C₄×C₇	1	4	yes	—
177	#102	—	2	8	no	—
178	#102	C₂×C₄×C₇	1	8	yes	—
179	#97	—	2	4	no	—
180	#97	—	2	2	no	—
181	C₄×C₈	—	2	4	no	—
182	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
183	#95	—	2	2	no	—
184	#95	—	2	4	no	—
185	#95	C₇×D₄	1	1	yes	characteristic
186	#95	C₇×D₄	1	1	yes	characteristic
187	#95	C₇×D₄	1	2	yes	—
188	C₂²×C₇	—	4	4	no	—
189	C₂²×C₇	—	4	4	no	—
190	C₂²×C₇	—	4	4	no	—
191	C₂²×C₇	—	8	4	no	—
192	C₂²×C₇	—	8	4	no	—
193	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂²×C₇	#532	1	2	yes	—
195	C₂²×C₇	#576	1	2	yes	—
196	C₂²×C₇	#576	1	1	yes	characteristic
197	C₂²×C₇	#578	1	1	yes	characteristic
198	C₄×C₇	—	2	2	no	—
199	C₄×C₇	—	2	2	no	—
200	C₄×C₇	—	4	2	no	—
201	C₄×C₇	—	4	2	no	—
202	C₄×C₇	—	4	4	no	—
203	C₄×C₇	—	8	4	no	—
204	C₄×C₇	C₂×#117	1	1	yes	characteristic
205	C₄×C₇	C₄×#41	1	1	yes	characteristic
206	C₄×C₇	#529	1	2	yes	—
207	C₂⁴	—	2	2	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	8	4	no	—
216	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
217	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
218	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
220	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
221	D₈	—	2	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
227	C₂×C₈	C₇×#41	1	4	yes	—
228	C₂×C₈	C₇×#41	1	4	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	8	4	no	—
233	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
234	C₂²⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	8	4	no	—
236	C₂²⋊C₄	—	8	4	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	8	2	no	—
240	C₄²	C₂²×C₄×C₇	1	2	yes	—
241	C₄²	C₇×#41	1	2	yes	—
242	C₂×C₇	—	8	4	no	—
243	C₂×C₇	#2293	1	1	yes	characteristic
244	C₂×C₇	#2948	1	1	yes	characteristic
245	C₂×C₇	#2951	1	1	yes	characteristic
246	C₂×C₇	#3175	1	2	yes	—
247	C₂×C₇	#3176	1	2	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	—	4	4	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	4	no	—
252	C₂³	—	4	2	no	—
253	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
254	D₄	—	2	4	no	—
255	D₄	—	2	4	no	—
256	D₄	—	4	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	8	no	—
272	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
273	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
274	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₇×#41	1	2	yes	—
276	C₂×C₄	C₂×C₇×#41	1	2	yes	—
277	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	2	4	no	—
280	C₈	C₇×#117	1	4	yes	—
281	C₇	#21282	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
288	C₂²	C₇×#532	1	2	yes	—
289	C₂²	C₇×#576	1	2	yes	—
290	C₂²	C₇×#576	1	1	yes	characteristic
291	C₂²	C₇×#578	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	4	no	—
297	C₄	—	8	4	no	—
298	C₄	C₂×C₇×#117	1	1	yes	characteristic
299	C₄	C₄×C₇×#41	1	1	yes	characteristic
300	C₄	C₇×#529	1	2	yes	—
301	C₂	—	8	4	no	—
302	C₂	C₇×#2293	1	1	yes	characteristic
303	C₂	C₇×#2948	1	1	yes	characteristic
304	C₂	C₇×#2951	1	1	yes	characteristic
305	C₂	C₇×#3175	1	2	yes	—
306	C₂	C₇×#3176	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₇×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2319	C₂	semidirect
4. C₇×#1942	C₂	semidirect
5. C₇×#1936	C₂	semidirect
6. C₄×C₇×#107	C₂	semidirect
7. C₇×#1762	C₂	semidirect
8. C₇×#1726	C₂	semidirect
11. C₄×C₇×D₈	C₄	semidirect
29. #21282	C₇	direct (N × H)
62. #2319	C₂×C₇	semidirect
64. #1942	C₂×C₇	semidirect
65. #1936	C₂×C₇	semidirect
66. C₄×#107	C₂×C₇	semidirect
67. #1762	C₂×C₇	semidirect
68. #1726	C₂×C₇	semidirect
106. C₄×D₈	C₄×C₇	semidirect
281. C₇	#21282	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×#2319	2
C₇×#1970	2
C₇×#1942	2
C₇×#1936	2
C₄×C₇×#107	2
C₇×#1762	2
C₇×#1726	2
#21282	7

Subgroups — C7×#21282 #13198498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21282 #13198498