Subgroups — C₇×#2497 #11741806

Order 896.

428 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#400	C₂	1	2	yes	maximal, complemented
2	C₇×#396	C₂	1	2	yes	maximal
3	C₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#388	C₂	1	2	yes	maximal
6	C₇×#387	C₂	1	2	yes	maximal, complemented
7	C₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#380	C₂	1	1	yes	maximal, characteristic
9	C₇×#380	C₂	1	1	yes	maximal, characteristic
10	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₇	C₂²	1	2	yes	—
28	C₇×#95	C₂²	1	1	yes	characteristic
29	C₇×#95	C₂²	1	1	yes	characteristic
30	C₇×#95	C₂²	1	2	yes	—
31	C₇×#95	C₂²	1	2	yes	—
32	C₇×#95	C₂²	1	2	yes	—
33	C₇×#95	C₂²	1	2	yes	—
34	C₇×#95	C₂²	1	2	yes	—
35	#2497	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂⁴×C₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₇	—	2	2	no	—
38	C₂²×C₄×C₇	—	2	2	no	—
39	C₂²×C₄×C₇	—	2	2	no	—
40	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₇×C₄⋊C₄	—	2	4	no	—
51	C₇×C₄⋊C₄	—	2	2	no	—
52	C₇×C₄⋊C₄	—	2	2	no	—
53	C₇×C₄⋊C₄	—	2	2	no	—
54	C₇×C₄⋊C₄	—	2	2	no	—
55	C₇×C₄⋊C₄	—	4	2	no	—
56	C₇×C₄⋊C₄	—	4	2	no	—
57	C₇×C₄⋊C₄	—	4	2	no	—
58	C₇×C₂²⋊C₄	—	2	2	no	—
59	C₇×C₂²⋊C₄	—	4	2	no	—
60	C₇×C₂²⋊C₄	—	4	2	no	—
61	C₇×C₂²⋊C₄	Q₈	1	4	yes	—
62	C₄²×C₇	—	2	2	no	—
63	C₄²×C₇	—	2	2	no	—
64	C₄²×C₇	—	4	2	no	—
65	#400	C₂×C₇	1	2	yes	complemented
66	#396	C₂×C₇	1	2	yes	—
67	#389	C₂×C₇	1	1	yes	characteristic, complemented
68	#389	C₂×C₇	1	1	yes	characteristic, complemented
69	#388	C₂×C₇	1	2	yes	—
70	#387	C₂×C₇	1	2	yes	complemented
71	#384	C₂×C₇	1	1	yes	characteristic, complemented
72	#380	C₂×C₇	1	1	yes	characteristic
73	#380	C₂×C₇	1	1	yes	characteristic
74	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
75	C₄×C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic
76	C₂³×C₇	—	2	1	no	—
77	C₂³×C₇	—	2	1	no	—
78	C₂³×C₇	—	2	1	no	—
79	C₂³×C₇	—	2	1	no	—
80	C₂³×C₇	—	2	1	no	—
81	C₂³×C₇	—	2	1	no	—
82	C₂³×C₇	C₂×Q₈	1	2	yes	—
83	C₂³×C₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	1	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	4	4	no	—
109	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
110	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₇	#41	1	2	yes	—
112	C₂×C₄×C₇	#41	1	2	yes	—
113	C₂×C₄×C₇	#41	1	2	yes	—
114	C₂×C₄×C₇	#41	1	2	yes	—
115	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
121	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
125	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
126	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
128	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
129	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
130	C₂×C₄²	C₂²×C₇	1	2	yes	—
131	#95	C₂²×C₇	1	1	yes	characteristic
132	#95	C₂²×C₇	1	1	yes	characteristic
133	#95	C₂²×C₇	1	2	yes	—
134	#95	C₂²×C₇	1	2	yes	—
135	#95	C₂²×C₇	1	2	yes	—
136	#95	C₂²×C₇	1	2	yes	—
137	#95	C₂²×C₇	1	2	yes	—
138	C₂²×C₇	—	2	2	no	—
139	C₂²×C₇	—	2	2	no	—
140	C₂²×C₇	—	2	2	no	—
141	C₂²×C₇	—	4	1	no	—
142	C₂²×C₇	—	4	1	no	—
143	C₂²×C₇	—	4	1	no	—
144	C₂²×C₇	—	4	1	no	—
145	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
146	C₂²×C₇	C₂×#41	1	1	yes	characteristic
147	C₂²×C₇	C₂×#41	1	1	yes	characteristic
148	C₂²×C₇	C₂×#41	1	1	yes	characteristic
149	C₂²×C₇	C₂×#41	1	1	yes	characteristic
150	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
152	C₄×C₇	—	2	2	no	—
153	C₄×C₇	—	2	2	no	—
154	C₄×C₇	—	2	2	no	—
155	C₄×C₇	—	4	1	no	—
156	C₄×C₇	—	4	2	no	—
157	C₄×C₇	—	4	2	no	—
158	C₄×C₇	—	4	2	no	—
159	C₄×C₇	—	4	1	no	—
160	C₄×C₇	—	4	2	no	—
161	C₄×C₇	#126	1	4	yes	—
162	C₂⁴	C₂³×C₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₇	1	2	yes	—
169	C₂²×C₄	C₂³×C₇	1	2	yes	—
170	C₂²×C₄	C₂³×C₇	1	2	yes	—
171	C₂²×C₄	C₂³×C₇	1	2	yes	—
172	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₂²⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	C₇×Q₈	1	4	yes	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	4	2	no	—
191	C₂×C₇	—	4	2	no	—
192	C₂×C₇	C₂×#126	1	1	yes	characteristic
193	C₂×C₇	#533	1	1	yes	characteristic
194	C₂×C₇	#540	1	1	yes	characteristic
195	C₂×C₇	#553	1	1	yes	characteristic
196	C₂×C₇	#554	1	1	yes	characteristic
197	C₂×C₇	#554	1	1	yes	characteristic
198	C₂×C₇	#555	1	1	yes	characteristic
199	C₂³	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	C₂×C₇×Q₈	1	2	yes	—
206	C₂³	C₂⁴×C₇	1	1	yes	characteristic
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
233	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
234	C₂×C₄	C₇×#41	1	2	yes	—
235	C₂×C₄	C₇×#41	1	2	yes	—
236	C₂×C₄	C₇×#41	1	2	yes	—
237	C₂×C₄	C₇×#41	1	2	yes	—
238	C₇	#2497	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
247	C₂²	C₂×C₇×#41	1	1	yes	characteristic
248	C₂²	C₂×C₇×#41	1	1	yes	characteristic
249	C₂²	C₂×C₇×#41	1	1	yes	characteristic
250	C₂²	C₂×C₇×#41	1	1	yes	characteristic
251	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
252	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	2	no	—
262	C₄	C₇×#126	1	4	yes	—
263	C₂	—	4	2	no	—
264	C₂	C₂×C₇×#126	1	1	yes	characteristic
265	C₂	C₇×#533	1	1	yes	characteristic
266	C₂	C₇×#540	1	1	yes	characteristic
267	C₂	C₇×#553	1	1	yes	characteristic
268	C₂	C₇×#554	1	1	yes	characteristic
269	C₂	C₇×#554	1	1	yes	characteristic
270	C₂	C₇×#555	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂³
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. C₇×#400	C₂	semidirect
3. C₇×#389	C₂	semidirect
4. C₇×#389	C₂	semidirect
6. C₇×#387	C₂	semidirect
7. C₇×#384	C₂	semidirect
10. C₄×C₇×C₄⋊C₄	C₂	semidirect
35. #2497	C₇	direct (N × H)
65. #400	C₂×C₇	semidirect
67. #389	C₂×C₇	semidirect
68. #389	C₂×C₇	semidirect
70. #387	C₂×C₇	semidirect
71. #384	C₂×C₇	semidirect
74. C₄×C₄⋊C₄	C₂×C₇	semidirect
238. C₇	#2497	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×#400	2
C₇×#396	2
C₇×#389	2
C₇×#388	2
C₇×#387	2
C₇×#384	2
C₇×#380	2
C₄×C₇×C₄⋊C₄	2
C₄×C₇×C₂²⋊C₄	2
#2497	7

Subgroups — C7×#2497 #11741806

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2497 #11741806