Subgroups — C₇×#2523 #11741832

Order 896.

420 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#397	C₂	1	2	yes	maximal
2	C₇×#397	C₂	1	1	yes	maximal, characteristic
3	C₇×#393	C₂	1	1	yes	maximal, characteristic
4	C₇×#391	C₂	1	2	yes	maximal, complemented
5	C₇×#387	C₂	1	2	yes	maximal, complemented
6	C₇×#387	C₂	1	2	yes	maximal, complemented
7	C₇×#385	C₂	1	1	yes	maximal, characteristic
8	C₇×#383	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×C₂²⋊C₄	C₂	1	2	yes	maximal
10	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
16	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₇	C₂²	1	2	yes	—
21	C₇×#95	C₂²	1	2	yes	—
22	C₇×#95	C₂²	1	2	yes	—
23	C₇×#95	C₂²	1	1	yes	characteristic
24	C₇×#95	C₂²	1	2	yes	—
25	C₇×#95	C₂²	1	2	yes	—
26	C₇×#95	C₂²	1	2	yes	—
27	C₇×#95	C₂²	1	2	yes	—
28	C₇×#95	C₂²	1	2	yes	—
29	C₇×#95	C₂²	1	1	yes	characteristic
30	C₇×#95	C₂²	1	2	yes	—
31	#2523	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₇×Q₈	—	2	4	no	—
34	C₂×C₇×Q₈	—	4	1	no	—
35	C₂²×C₄×C₇	—	2	4	no	—
36	C₂²×C₄×C₇	—	4	1	no	—
37	C₂²×C₄×C₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₇×C₄⋊C₄	—	2	4	no	—
46	C₇×C₄⋊C₄	—	2	4	no	—
47	C₇×C₄⋊C₄	—	4	1	no	—
48	C₇×C₂²⋊C₄	—	2	4	no	—
49	C₇×C₂²⋊C₄	—	2	2	no	—
50	C₇×C₂²⋊C₄	—	2	4	no	—
51	C₇×C₂²⋊C₄	—	4	1	no	—
52	C₄²×C₇	—	2	4	no	—
53	C₄²×C₇	—	4	2	no	—
54	#397	C₂×C₇	1	2	yes	—
55	#397	C₂×C₇	1	1	yes	characteristic
56	#393	C₂×C₇	1	1	yes	characteristic
57	#391	C₂×C₇	1	2	yes	complemented
58	#387	C₂×C₇	1	2	yes	complemented
59	#387	C₂×C₇	1	2	yes	complemented
60	#385	C₂×C₇	1	1	yes	characteristic
61	#383	C₂×C₇	1	2	yes	complemented
62	C₄×C₂²⋊C₄	C₂×C₇	1	2	yes	—
63	C₂³×C₇	—	2	2	no	—
64	C₂³×C₇	—	2	2	no	—
65	C₂³×C₇	—	2	1	no	—
66	C₂³×C₇	—	2	1	no	—
67	C₂³×C₇	#41	1	2	yes	—
68	C₂³×C₇	C₂⁴	1	1	yes	characteristic
69	C₇×Q₈	—	4	4	no	—
70	C₂×C₄×C₇	—	2	2	no	—
71	C₂×C₄×C₇	—	2	2	no	—
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	—	2	2	no	—
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	4	4	no	—
91	C₂×C₄×C₇	#41	1	4	yes	—
92	C₂×C₄×C₇	#41	1	4	yes	—
93	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
94	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
96	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
97	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
98	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
99	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
100	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
101	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
102	C₂×C₄²	C₂²×C₇	1	2	yes	—
103	C₂×C₄²	C₂²×C₇	1	2	yes	—
104	#95	C₂²×C₇	1	2	yes	—
105	#95	C₂²×C₇	1	2	yes	—
106	#95	C₂²×C₇	1	1	yes	characteristic
107	#95	C₂²×C₇	1	2	yes	—
108	#95	C₂²×C₇	1	2	yes	—
109	#95	C₂²×C₇	1	2	yes	—
110	#95	C₂²×C₇	1	2	yes	—
111	#95	C₂²×C₇	1	2	yes	—
112	#95	C₂²×C₇	1	1	yes	characteristic
113	#95	C₂²×C₇	1	2	yes	—
114	C₂²×C₇	—	2	4	no	—
115	C₂²×C₇	—	2	2	no	—
116	C₂²×C₇	—	4	1	no	—
117	C₂²×C₇	—	4	1	no	—
118	C₂²×C₇	—	4	2	no	—
119	C₂²×C₇	C₂×#41	1	1	yes	characteristic
120	C₂²×C₇	C₂×#41	1	2	yes	—
121	C₂²×C₇	C₂×#41	1	2	yes	—
122	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
124	C₄×C₇	—	2	4	no	—
125	C₄×C₇	—	2	4	no	—
126	C₄×C₇	—	4	2	no	—
127	C₄×C₇	—	4	2	no	—
128	C₄×C₇	—	4	2	no	—
129	C₄×C₇	—	4	1	no	—
130	C₄×C₇	—	4	1	no	—
131	C₄×C₇	—	4	2	no	—
132	C₂⁴	C₂³×C₇	1	1	yes	characteristic
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	4	1	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	C₂³×C₇	1	2	yes	—
138	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₇	1	2	yes	—
140	C₂²×C₄	C₂³×C₇	1	2	yes	—
141	C₂²×C₄	C₂³×C₇	1	2	yes	—
142	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₇	1	2	yes	—
144	C₂²×C₄	C₂³×C₇	1	2	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₂²⋊C₄	—	2	2	no	—
150	C₂²⋊C₄	—	2	4	no	—
151	C₂²⋊C₄	—	4	1	no	—
152	C₄²	—	2	4	no	—
153	C₄²	—	4	2	no	—
154	C₂×C₇	—	4	2	no	—
155	C₂×C₇	#529	1	2	yes	—
156	C₂×C₇	#539	1	1	yes	characteristic
157	C₂×C₇	#551	1	1	yes	characteristic
158	C₂×C₇	#551	1	1	yes	characteristic
159	C₂×C₇	#556	1	2	yes	—
160	C₂³	—	2	2	no	—
161	C₂³	—	2	2	no	—
162	C₂³	—	2	1	no	—
163	C₂³	—	2	1	no	—
164	C₂³	C₇×#41	1	2	yes	—
165	C₂³	C₂⁴×C₇	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	C₇×#41	1	4	yes	—
189	C₂×C₄	C₇×#41	1	4	yes	—
190	C₇	#2523	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²	—	2	4	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	4	1	no	—
194	C₂²	—	4	1	no	—
195	C₂²	—	4	2	no	—
196	C₂²	C₂×C₇×#41	1	1	yes	characteristic
197	C₂²	C₂×C₇×#41	1	2	yes	—
198	C₂²	C₂×C₇×#41	1	2	yes	—
199	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
200	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	—	2	4	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	1	no	—
207	C₄	—	4	1	no	—
208	C₄	—	4	2	no	—
209	C₂	—	4	2	no	—
210	C₂	C₇×#529	1	2	yes	—
211	C₂	C₇×#539	1	1	yes	characteristic
212	C₂	C₇×#551	1	1	yes	characteristic
213	C₂	C₇×#551	1	1	yes	characteristic
214	C₂	C₇×#556	1	2	yes	—

	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
4. C₇×#391	C₂	semidirect
5. C₇×#387	C₂	semidirect
6. C₇×#387	C₂	semidirect
8. C₇×#383	C₂	semidirect
31. #2523	C₇	direct (N × H)
57. #391	C₂×C₇	semidirect
58. #387	C₂×C₇	semidirect
59. #387	C₂×C₇	semidirect
61. #383	C₂×C₇	semidirect
190. C₇	#2523	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×#397	2
C₇×#393	2
C₇×#391	2
C₇×#387	2
C₇×#385	2
C₇×#383	2
C₄×C₇×C₂²⋊C₄	2
#2523	7

Subgroups — C7×#2523 #11741832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2523 #11741832