Group #13244251 — subgroups

Subgroups — #13244251

Order 1792.

294 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1450	C₂	1	1	yes	maximal, characteristic
2	#11727505	C₂	1	1	yes	maximal, characteristic
3	#11727480	C₂	1	1	yes	maximal, characteristic
4	#11725546	C₂	1	2	yes	maximal
5	#11725543	C₂	1	2	yes	maximal
6	C₇×#431	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	#89704	C₂²	1	1	yes	characteristic
9	#89697	C₂²	1	2	yes	—
10	C₂×#6377	C₂²	1	1	yes	characteristic
11	C₇×#322	C₄	1	2	yes	complemented
12	C₇×C₈²	C₄	1	2	yes	complemented
13	#89345	C₄	1	4	yes	—
14	#89249	C₄	1	4	yes	complemented
15	#8141	—	7	1	no	maximal, Sylow
16	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
18	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
19	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₇×C₄⋊C₈	—	2	2	no	—
21	C₇×C₂²⋊C₈	—	2	2	no	—
22	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
24	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₇⋊#95	—	2	2	no	—
27	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₇⋊C₈)	—	2	2	no	—
29	#6377	—	2	2	no	—
30	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
31	#1746	—	7	1	no	—
32	#1452	—	7	1	no	—
33	#1450	D₇	1	1	yes	characteristic
34	#1275	—	7	2	no	—
35	#1274	—	7	2	no	—
36	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇	#41	1	2	yes	—
38	C₂²×Dic₇	—	2	2	no	—
39	C₂×C₇×C₈	—	2	4	no	—
40	C₂×C₇×C₈	—	2	4	no	—
41	C₂×C₇×C₈	C₄²	1	2	yes	—
42	C₂×C₇×C₈	C₄²	1	2	yes	—
43	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₇	#41	1	2	yes	—
46	C₄×Dic₇	—	2	4	no	—
47	C₂×C₇⋊C₈	—	2	4	no	—
48	C₂×C₇⋊C₈	—	2	2	no	—
49	C₂×C₇⋊C₈	—	2	4	no	—
50	#431	—	7	2	no	—
51	#431	C₂×D₇	1	2	yes	—
52	C₂×#97	—	7	1	no	—
53	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
54	#376	—	7	1	no	—
55	#336	—	7	4	no	—
56	#322	—	7	4	no	—
57	#322	Dic₇	1	2	yes	complemented
58	C₈²	Dic₇	1	2	yes	complemented
59	C₂³×C₇	#117	1	1	yes	characteristic
60	C₂×C₄×C₇	—	4	2	no	—
61	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₇	#117	1	2	yes	—
63	C₂×C₄×C₇	#117	1	1	yes	characteristic
64	C₂×C₄×C₇	#117	1	2	yes	—
65	C₂×C₄×C₇	#131	1	4	yes	—
66	C₂×Dic₇	—	4	4	no	—
67	C₂×Dic₇	—	4	2	no	—
68	C₇×C₈	—	2	2	no	—
69	C₇×C₈	—	2	2	no	—
70	C₇×C₈	—	4	4	no	—
71	C₇⋊C₈	—	4	4	no	—
72	C₇⋊C₈	—	4	2	no	—
73	C₂²×C₈	—	7	2	no	—
74	C₂²×C₈	C₄×D₇	1	2	yes	—
75	C₂×C₄²	—	7	2	no	—
76	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	—	14	2	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	—	14	2	no	—
81	#97	—	14	2	no	—
82	C₄×C₈	—	7	4	no	—
83	C₄×C₈	C₄×D₇	1	2	yes	—
84	C₄×C₈	C₂×Dic₇	1	4	yes	—
85	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
87	#95	—	14	2	no	—
88	C₂²×C₇	—	2	2	no	—
89	C₂²×C₇	#376	1	1	yes	characteristic
90	C₂²×C₇	#433	1	2	yes	—
91	C₄×C₇	—	2	2	no	—
92	C₄×C₇	—	2	2	no	—
93	C₄×C₇	#405	1	2	yes	—
94	C₄×C₇	#433	1	2	yes	—
95	Dic₇	—	8	2	no	—
96	C₂²×C₄	—	14	2	no	—
97	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
98	C₂²×C₄	#1157	1	2	yes	—
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	14	4	no	—
102	C₂×C₈	—	14	2	no	—
103	C₂×C₈	—	14	4	no	—
104	C₂×C₈	C₄×Dic₇	1	2	yes	—
105	C₂×C₈	C₄×Dic₇	1	2	yes	—
106	C₄²	—	14	4	no	—
107	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
108	C₄²	#1157	1	2	yes	—
109	C₄²	C₂²×Dic₇	1	1	yes	characteristic
110	C₂×C₇	—	2	2	no	—
111	C₂×C₇	#1746	1	1	yes	characteristic
112	C₂×C₇	#1747	1	2	yes	—
113	C₂³	#6435	1	1	yes	characteristic
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	28	4	no	—
116	C₂×C₄	—	28	2	no	—
117	C₂×C₄	#6435	1	2	yes	—
118	C₂×C₄	#6435	1	1	yes	characteristic
119	C₂×C₄	#6464	1	4	yes	—
120	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
121	C₂×C₄	#6489	1	2	yes	—
122	C₈	—	2	2	no	—
123	C₈	—	2	2	no	—
124	C₈	—	4	4	no	—
125	C₈	—	28	4	no	—
126	C₈	—	28	2	no	—
127	C₇	#8141	1	1	yes	characteristic, complemented, Sylow
128	C₂²	—	2	2	no	—
129	C₂²	#89462	1	2	yes	—
130	C₂²	#89704	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	56	2	no	—
134	C₄	#89462	1	2	yes	—
135	C₄	#89873	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#11727505	1	1	yes	characteristic
138	C₂	#11727649	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1450

C₂

yes

maximal, characteristic

#11727505

C₂

yes

maximal, characteristic

#11727480

C₂

yes

maximal, characteristic

#11725546

C₂

yes

maximal

#11725543

C₂

yes

maximal

C₇×#431

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

#89704

C₂²

yes

characteristic

#89697

C₂²

yes

—

C₂×#6377

C₂²

yes

characteristic

C₇×#322

C₄

yes

complemented

C₇×C₈²

C₄

yes

complemented

#89345

C₄

yes

—

#89249

C₄

yes

complemented

#8141

—

maximal, Sylow

C₂²×C₇×C₈

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄×Dic₇

C₂×C₄

yes

—

C₂²×C₇⋊C₈

C₂×C₄

yes

—

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₇⋊#95

—

C₂²⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

#6377

—

C₄×C₇⋊C₈

C₂×C₄

yes

—

#1746

—

#1452

—

#1450

D₇

yes

characteristic

#1275

—

#1274

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂²×Dic₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄²

yes

—

C₂×C₇×C₈

C₄²

yes

—

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

#41

yes

—

C₄×Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

#431

—

#431

C₂×D₇

yes

—

C₂×#97

—

C₂×C₄×C₈

C₂×D₇

yes

—

—

—

Dic₇

yes

complemented

C₈²

Dic₇

yes

complemented

C₂³×C₇

#117

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#117

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#131

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₄×D₇

yes

—

C₄×C₈

C₂×Dic₇

yes

—

C₄×C₈

C₂×Dic₇

yes

characteristic

C₄×C₈

C₂×Dic₇

yes

characteristic

#95

—

C₂²×C₇

—

C₂²×C₇

#376

yes

characteristic

C₂²×C₇

#433

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#405

yes

—

C₄×C₇

#433

yes

—

Dic₇

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

C₂²×C₄

#1157

yes

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₄×Dic₇

yes

—

105

C₂×C₈

C₄×Dic₇

yes

—

106

C₄²

—

107

C₄²

C₂×C₄×D₇

yes

characteristic

108

C₄²

#1157

yes

—

109

C₄²

C₂²×Dic₇

yes

characteristic

110

C₂×C₇

—

111

C₂×C₇

#1746

yes

characteristic

112

C₂×C₇

#1747

yes

—

113

C₂³

#6435

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#6435

yes

—

118

C₂×C₄

#6435

yes

characteristic

119

C₂×C₄

#6464

yes

—

120

C₂×C₄

C₂×C₄×Dic₇

yes

characteristic

121

C₂×C₄

#6489

yes

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₇

#8141

yes

characteristic, complemented, Sylow

128

C₂²

—

129

C₂²

#89462

yes

—

130

C₂²

#89704

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#89462

yes

—

135

C₄

#89873

yes

—

136

C₂

—

137

C₂

#11727505

yes

characteristic

138

C₂

#11727649

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1450	C₂
socle	C₂²×C₇	#376
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1450	D₇
hypocenter	C₇	#8141
LCS	G ▷ C₂×C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #1450

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₇

commutator

C₂×C₄×C₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₇

C₁ ◁ C₂×C₄ ◁ #1450

N	H ≅ G/N	type
11. C₇×#322	C₄	semidirect
12. C₇×C₈²	C₄	semidirect
14. #89249	C₄	semidirect
57. #322	Dic₇	semidirect
58. C₈²	Dic₇	semidirect
127. C₇	#8141	semidirect

H ≅ G/N

type

11. C₇×#322

C₄

semidirect

12. C₇×C₈²

C₄

semidirect

14. #89249

C₄

semidirect

57. #322

Dic₇

semidirect

58. C₈²

Dic₇

semidirect

127. C₇

#8141

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#1450	2
#11727505	2
#11727480	2
#11725546	2
#11725543	2
#8141	7

subgroup

index

Subgroups — #13244251

Conjugacy classes of subgroups

Complements

Maximal subgroups