Subgroups — C₇×#2490 #11741799

Order 896.

560 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#121	C₂	1	1	yes	maximal, characteristic
3	C₇×#397	C₂	1	2	yes	maximal, complemented
4	C₇×#394	C₂	1	2	yes	maximal, complemented
5	C₇×#390	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#390	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#390	C₂	1	2	yes	maximal, complemented
8	C₇×#388	C₂	1	2	yes	maximal, complemented
9	C₇×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₇×D₄	C₂²	1	2	yes	—
14	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₇×#125	—	2	4	no	—
16	C₇×#121	—	2	4	no	—
17	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂²	1	2	yes	—
33	C₇×#95	C₂²	1	1	yes	characteristic
34	C₇×#95	C₂²	1	1	yes	characteristic
35	C₇×#95	C₂²	1	2	yes	—
36	C₇×#95	C₂²	1	2	yes	—
37	#2490	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂⁴×C₇	C₂³	1	2	yes	—
39	C₂⁴×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₇×D₄	—	2	4	no	—
41	C₂×C₇×D₄	—	2	4	no	—
42	C₂×C₇×D₄	—	2	2	no	—
43	C₂×C₇×D₄	—	2	2	no	—
44	C₂×C₇×D₄	—	2	2	no	—
45	C₂×C₇×D₄	—	4	2	no	—
46	C₂²×C₄×C₇	—	2	2	no	—
47	C₂²×C₄×C₇	—	2	2	no	—
48	C₂²×C₄×C₇	—	2	2	no	—
49	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
58	C₇×C₄⋊C₄	—	2	4	no	—
59	C₇×C₄⋊C₄	—	2	4	no	—
60	C₇×C₄⋊C₄	—	2	2	no	—
61	C₇×C₄⋊C₄	—	2	2	no	—
62	C₇×C₄⋊C₄	D₄	1	4	yes	—
63	C₇×C₂²⋊C₄	—	2	2	no	—
64	C₇×C₂²⋊C₄	—	2	4	no	—
65	C₇×C₂²⋊C₄	—	2	2	no	—
66	C₇×C₂²⋊C₄	—	2	4	no	—
67	C₇×C₂²⋊C₄	—	4	2	no	—
68	C₇×C₂²⋊C₄	—	4	2	no	—
69	C₇×C₂²⋊C₄	—	4	2	no	—
70	C₇×C₂²⋊C₄	—	4	2	no	—
71	C₄²×C₇	—	2	2	no	—
72	C₄²×C₇	—	2	2	no	—
73	C₄²×C₇	—	4	2	no	—
74	C₂×#125	C₂×C₇	1	1	yes	characteristic, complemented
75	C₂×#121	C₂×C₇	1	1	yes	characteristic
76	#397	C₂×C₇	1	2	yes	complemented
77	#394	C₂×C₇	1	2	yes	complemented
78	#390	C₂×C₇	1	1	yes	characteristic, complemented
79	#390	C₂×C₇	1	1	yes	characteristic, complemented
80	#390	C₂×C₇	1	2	yes	complemented
81	#388	C₂×C₇	1	2	yes	complemented
82	#380	C₂×C₇	1	1	yes	characteristic, complemented
83	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
84	C₄×C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
85	C₂³×C₇	—	2	1	no	—
86	C₂³×C₇	—	2	1	no	—
87	C₂³×C₇	—	2	1	no	—
88	C₂³×C₇	—	2	1	no	—
89	C₂³×C₇	—	2	1	no	—
90	C₂³×C₇	—	2	1	no	—
91	C₂³×C₇	—	2	2	no	—
92	C₂³×C₇	—	2	2	no	—
93	C₂³×C₇	—	2	2	no	—
94	C₂³×C₇	—	2	2	no	—
95	C₂³×C₇	—	2	2	no	—
96	C₂³×C₇	—	2	2	no	—
97	C₂³×C₇	—	2	2	no	—
98	C₂³×C₇	#41	1	2	yes	—
99	C₂³×C₇	C₂⁴	1	1	yes	characteristic
100	C₇×D₄	—	4	8	no	—
101	C₇×D₄	—	4	4	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	—	2	1	no	—
115	C₂×C₄×C₇	—	2	2	no	—
116	C₂×C₄×C₇	—	2	1	no	—
117	C₂×C₄×C₇	—	2	1	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	1	no	—
121	C₂×C₄×C₇	—	2	1	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	1	no	—
124	C₂×C₄×C₇	—	4	4	no	—
125	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
126	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
127	C₂×C₄×C₇	#41	1	2	yes	—
128	C₂×C₄×C₇	#41	1	2	yes	—
129	C₂×C₄×C₇	#41	1	2	yes	—
130	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
131	C₂²×D₄	C₂²×C₇	1	2	yes	—
132	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
133	#125	—	2	4	no	—
134	#121	—	2	4	no	—
135	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
137	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
142	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
144	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
148	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
149	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
150	C₂×C₄²	C₂²×C₇	1	2	yes	—
151	#95	C₂²×C₇	1	1	yes	characteristic
152	#95	C₂²×C₇	1	1	yes	characteristic
153	#95	C₂²×C₇	1	2	yes	—
154	#95	C₂²×C₇	1	2	yes	—
155	C₂²×C₇	—	2	2	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	2	2	no	—
158	C₂²×C₇	—	4	1	no	—
159	C₂²×C₇	—	4	1	no	—
160	C₂²×C₇	—	4	1	no	—
161	C₂²×C₇	—	4	1	no	—
162	C₂²×C₇	—	4	2	no	—
163	C₂²×C₇	—	4	2	no	—
164	C₂²×C₇	—	4	2	no	—
165	C₂²×C₇	—	4	2	no	—
166	C₂²×C₇	—	4	2	no	—
167	C₂²×C₇	—	4	2	no	—
168	C₂²×C₇	—	4	2	no	—
169	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
170	C₂²×C₇	C₂×#41	1	1	yes	characteristic
171	C₂²×C₇	C₂×#41	1	1	yes	characteristic
172	C₂²×C₇	C₂×#41	1	1	yes	characteristic
173	C₂²×C₇	C₂×#41	1	1	yes	characteristic
174	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
176	C₄×C₇	—	2	4	no	—
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	4	1	no	—
179	C₄×C₇	—	4	2	no	—
180	C₄×C₇	—	4	1	no	—
181	C₄×C₇	—	4	1	no	—
182	C₄×C₇	—	4	2	no	—
183	C₄×C₇	—	4	1	no	—
184	C₄×C₇	#124	1	4	yes	—
185	C₂⁴	C₂³×C₇	1	2	yes	—
186	C₂⁴	C₂³×C₇	1	1	yes	characteristic
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₇	1	2	yes	—
199	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₇	1	2	yes	—
202	C₂²×C₄	C₂³×C₇	1	2	yes	—
203	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	C₇×D₄	1	4	yes	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	4	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₂²⋊C₄	—	4	2	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₂×C₇	—	4	2	no	—
222	C₂×C₇	—	8	2	no	—
223	C₂×C₇	C₂×#124	1	1	yes	characteristic
224	C₂×C₇	#529	1	1	yes	characteristic
225	C₂×C₇	#540	1	1	yes	characteristic
226	C₂×C₇	#546	1	1	yes	characteristic
227	C₂×C₇	#550	1	1	yes	characteristic
228	C₂×C₇	#553	1	1	yes	characteristic
229	C₂×C₇	#553	1	1	yes	characteristic
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₇×#41	1	2	yes	—
244	C₂³	C₂⁴×C₇	1	1	yes	characteristic
245	D₄	—	4	8	no	—
246	D₄	—	4	4	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	C₂×C₇×D₄	1	4	yes	—
271	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
272	C₂×C₄	C₇×#41	1	2	yes	—
273	C₂×C₄	C₇×#41	1	2	yes	—
274	C₂×C₄	C₇×#41	1	2	yes	—
275	C₇	#2490	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
291	C₂²	C₂×C₇×#41	1	1	yes	characteristic
292	C₂²	C₂×C₇×#41	1	1	yes	characteristic
293	C₂²	C₂×C₇×#41	1	1	yes	characteristic
294	C₂²	C₂×C₇×#41	1	1	yes	characteristic
295	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
296	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
297	C₄	—	2	4	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	1	no	—
305	C₄	C₇×#124	1	4	yes	—
306	C₂	—	4	2	no	—
307	C₂	—	8	2	no	—
308	C₂	C₂×C₇×#124	1	1	yes	characteristic
309	C₂	C₇×#529	1	1	yes	characteristic
310	C₂	C₇×#540	1	1	yes	characteristic
311	C₂	C₇×#546	1	1	yes	characteristic
312	C₂	C₇×#550	1	1	yes	characteristic
313	C₂	C₇×#553	1	1	yes	characteristic
314	C₂	C₇×#553	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂×C₇×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#125	C₂	semidirect
3. C₇×#397	C₂	semidirect
4. C₇×#394	C₂	semidirect
5. C₇×#390	C₂	semidirect
6. C₇×#390	C₂	semidirect
7. C₇×#390	C₂	semidirect
8. C₇×#388	C₂	semidirect
9. C₇×#380	C₂	semidirect
10. C₄×C₇×C₄⋊C₄	C₂	semidirect
11. C₄×C₇×C₂²⋊C₄	C₂	semidirect
37. #2490	C₇	direct (N × H)
74. C₂×#125	C₂×C₇	semidirect
76. #397	C₂×C₇	semidirect
77. #394	C₂×C₇	semidirect
78. #390	C₂×C₇	semidirect
79. #390	C₂×C₇	semidirect
80. #390	C₂×C₇	semidirect
81. #388	C₂×C₇	semidirect
82. #380	C₂×C₇	semidirect
83. C₄×C₄⋊C₄	C₂×C₇	semidirect
84. C₄×C₂²⋊C₄	C₂×C₇	semidirect
275. C₇	#2490	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×#125	2
C₂×C₇×#121	2
C₇×#397	2
C₇×#394	2
C₇×#390	2
C₇×#388	2
C₇×#380	2
C₄×C₇×C₄⋊C₄	2
C₄×C₇×C₂²⋊C₄	2
#2490	7

Subgroups — C7×#2490 #11741799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2490 #11741799