Group #13244241 — subgroups

Subgroups — #13244241

Order 1792.

354 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 195 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×#1726	C₂	1	1	yes	maximal, characteristic
2	#11727478	C₂	1	2	yes	maximal
3	#11725466	C₂	1	2	yes	maximal
4	#11725420	C₂	1	2	yes	maximal
5	C₂×C₇×#97	C₄	1	2	yes	—
6	C₂×C₇×#97	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#376	C₂²	1	1	yes	characteristic
10	#89697	C₂²	1	4	yes	—
11	#89250	C₄	1	4	yes	complemented
12	C₈×C₇⋊C₈	C₄	1	4	yes	complemented
13	#8138	—	7	1	no	maximal, Sylow
14	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
15	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
19	C₇×#97	—	2	2	no	—
20	C₇×#97	C₈	1	8	yes	complemented
21	C₄×C₇×C₈	—	2	4	no	—
22	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
24	C₇×#95	—	2	2	no	—
25	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
26	C₄⋊(C₇⋊C₈)	—	2	4	no	—
27	C₄×C₇⋊C₈	C₂×C₄	1	8	yes	—
28	#1726	D₇	1	1	yes	characteristic
29	#1450	—	7	2	no	—
30	#1275	—	7	2	no	—
31	#1274	—	7	2	no	—
32	C₂²×C₄×C₇	—	2	2	no	—
33	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇	#41	1	2	yes	—
35	C₂×C₇×C₈	—	2	2	no	—
36	C₂×C₇×C₈	—	2	2	no	—
37	C₂×C₇×C₈	C₄²	1	2	yes	—
38	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
39	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
40	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
41	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
42	C₂×C₇×C₈	M₁₆	1	4	yes	—
43	C₄²×C₇	—	2	2	no	—
44	C₄²×C₇	C₂×C₈	1	4	yes	—
45	C₄²×C₇	C₂²×C₄	1	2	yes	—
46	C₄²×C₇	#41	1	2	yes	—
47	C₂×C₇⋊C₈	—	2	8	no	—
48	#431	—	7	4	no	—
49	C₂×#97	Dic₇	1	2	yes	—
50	C₂×#97	C₂×D₇	1	1	yes	characteristic
51	C₂×C₄×C₈	Dic₇	1	2	yes	—
52	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
53	#376	C₂×D₇	1	1	yes	characteristic
54	#322	—	7	4	no	—
55	C₈²	—	7	4	no	—
56	C₂³×C₇	#117	1	1	yes	characteristic
57	C₂×C₄×C₇	—	2	4	no	—
58	C₂×C₄×C₇	—	4	2	no	—
59	C₂×C₄×C₇	—	4	4	no	—
60	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₇	#117	1	4	yes	—
62	C₂×C₄×C₇	#117	1	1	yes	characteristic
63	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
64	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
65	C₇×C₈	—	2	4	no	—
66	C₇×C₈	C₄×C₈	1	4	yes	—
67	C₇×C₈	#97	1	2	yes	—
68	C₇×C₈	#97	1	2	yes	—
69	C₇⋊C₈	—	4	8	no	—
70	C₂²×C₈	C₂×Dic₇	1	2	yes	—
71	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
72	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
73	C₂×C₄²	C₂×Dic₇	1	2	yes	—
74	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
75	C₄⋊C₈	—	14	4	no	—
76	C₂²⋊C₈	—	14	4	no	—
77	#97	—	2	2	no	—
78	#97	C₇⋊C₈	1	8	yes	complemented
79	C₄×C₈	—	2	4	no	—
80	C₄×C₈	—	7	8	no	—
81	C₄×C₈	C₄×D₇	1	2	yes	—
82	C₄×C₈	C₄×D₇	1	2	yes	—
83	#95	—	2	2	no	—
84	C₂²×C₇	—	2	2	no	—
85	C₂²×C₇	#376	1	1	yes	characteristic
86	C₂²×C₇	#431	1	2	yes	—
87	C₄×C₇	—	2	4	no	—
88	C₄×C₇	—	4	4	no	—
89	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₇	C₂×#97	1	1	yes	characteristic
91	C₄×C₇	#431	1	2	yes	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	#1157	1	2	yes	—
94	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	14	8	no	—
98	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
99	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
100	C₂×C₈	#1136	1	4	yes	—
101	C₂×C₈	C₄×Dic₇	1	2	yes	—
102	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
103	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
104	C₄²	—	2	2	no	—
105	C₄²	C₂×C₇⋊C₈	1	4	yes	—
106	C₄²	C₂×C₄×D₇	1	2	yes	—
107	C₄²	#1157	1	2	yes	—
108	C₂×C₇	—	2	2	no	—
109	C₂×C₇	#1744	1	1	yes	characteristic
110	C₂×C₇	#1747	1	2	yes	—
111	C₂³	#6489	1	1	yes	characteristic
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	#6435	1	4	yes	—
116	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
117	C₂×C₄	C₂×#1136	1	2	yes	—
118	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
119	C₂×C₄	#6489	1	1	yes	characteristic
120	C₈	—	2	4	no	—
121	C₈	—	28	8	no	—
122	C₈	C₄×C₇⋊C₈	1	4	yes	—
123	C₈	#6377	1	2	yes	—
124	C₈	#6377	1	2	yes	—
125	C₇	#8138	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	2	2	no	—
127	C₂²	#89697	1	2	yes	—
128	C₂²	#89704	1	1	yes	characteristic
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
132	C₄	C₂×#6377	1	1	yes	characteristic
133	C₄	#89697	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	#11727142	1	1	yes	characteristic
136	C₂	#11727649	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1726

C₂

yes

maximal, characteristic

#11727478

C₂

yes

maximal

#11725466

C₂

yes

maximal

#11725420

C₂

yes

maximal

C₂×C₇×#97

C₄

yes

—

C₂×C₇×#97

C₂²

yes

characteristic

C₂×C₄×C₇×C₈

C₄

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#376

C₂²

yes

characteristic

#89697

C₂²

yes

—

#89250

C₄

yes

complemented

C₈×C₇⋊C₈

C₄

yes

complemented

#8138

—

maximal, Sylow

C₂²×C₇×C₈

C₂×C₄

yes

—

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×#97

—

C₇×#97

C₈

yes

complemented

C₄×C₇×C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₇×#95

—

C₂²⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄×C₇⋊C₈

C₂×C₄

yes

—

#1726

D₇

yes

—

—

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄²

yes

—

C₂×C₇×C₈

C₄²

yes

characteristic

C₂×C₇×C₈

C₄²

yes

characteristic

C₂×C₇×C₈

C₂×C₈

yes

—

C₂×C₇×C₈

C₂×C₈

yes

—

C₂×C₇×C₈

M₁₆

yes

—

C₄²×C₇

—

C₄²×C₇

C₂×C₈

yes

—

C₄²×C₇

C₂²×C₄

yes

—

C₄²×C₇

#41

yes

—

C₂×C₇⋊C₈

—

#431

—

C₂×#97

Dic₇

yes

—

C₂×#97

C₂×D₇

yes

characteristic

C₂×C₄×C₈

Dic₇

yes

—

C₂×C₄×C₈

C₂×D₇

yes

characteristic

#376

C₂×D₇

yes

characteristic

#322

—

C₈²

—

C₂³×C₇

#117

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#117

yes

characteristic

C₂×C₄×C₇

C₂²×C₈

yes

—

C₂×C₄×C₇

C₂×M₁₆

yes

—

C₇×C₈

—

C₇×C₈

C₄×C₈

yes

—

C₇×C₈

#97

yes

—

C₇×C₈

#97

yes

—

C₇⋊C₈

—

C₂²×C₈

C₂×Dic₇

yes

—

C₂²×C₈

C₂×Dic₇

yes

characteristic

C₂²×C₈

C₂×Dic₇

yes

characteristic

C₂×C₄²

C₂×Dic₇

yes

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

C₇⋊C₈

yes

complemented

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇

yes

—

C₄×C₈

C₄×D₇

yes

—

#95

—

C₂²×C₇

—

C₂²×C₇

#376

yes

characteristic

C₂²×C₇

#431

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×C₄×C₈

yes

characteristic

C₄×C₇

C₂×#97

yes

characteristic

C₄×C₇

#431

yes

—

C₂²×C₄

—

C₂²×C₄

#1157

yes

—

C₂²×C₄

C₂²×Dic₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₇⋊C₈

yes

—

C₂×C₈

C₂×C₇⋊C₈

yes

—

100

C₂×C₈

#1136

yes

—

101

C₂×C₈

C₄×Dic₇

yes

—

102

C₂×C₈

C₄×Dic₇

yes

characteristic

103

C₂×C₈

C₄×Dic₇

yes

characteristic

104

C₄²

—

105

C₄²

C₂×C₇⋊C₈

yes

—

106

C₄²

C₂×C₄×D₇

yes

—

107

C₄²

#1157

yes

—

108

C₂×C₇

—

109

C₂×C₇

#1744

yes

characteristic

110

C₂×C₇

#1747

yes

—

111

C₂³

#6489

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#6435

yes

—

116

C₂×C₄

C₂²×C₇⋊C₈

yes

—

117

C₂×C₄

C₂×#1136

yes

—

118

C₂×C₄

C₂×C₄×Dic₇

yes

characteristic

119

C₂×C₄

#6489

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

C₄×C₇⋊C₈

yes

—

123

C₈

#6377

yes

—

124

C₈

#6377

yes

—

125

C₇

#8138

yes

characteristic, complemented, Sylow

126

C₂²

—

127

C₂²

#89697

yes

—

128

C₂²

#89704

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

C₂×C₄×C₇⋊C₈

yes

characteristic

132

C₄

C₂×#6377

yes

characteristic

133

C₄

#89697

yes

—

134

C₂

—

135

C₂

#11727142

yes

characteristic

136

C₂

#11727649

yes

—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1726	C₂
socle	C₂²×C₇	#376
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1726	D₇
hypocenter	C₇	#8138
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₄×Dic₇

commutator

C₄×C₇

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₄×C₇ ▷ C₇

C₁ ◁ C₂×C₈ ◁ #1726

N	H ≅ G/N	type
11. #89250	C₄	semidirect
12. C₈×C₇⋊C₈	C₄	semidirect
20. C₇×#97	C₈	semidirect
78. #97	C₇⋊C₈	semidirect
125. C₇	#8138	semidirect

H ≅ G/N

type

11. #89250

C₄

semidirect

12. C₈×C₇⋊C₈

C₄

semidirect

20. C₇×#97

C₈

semidirect

78. #97

C₇⋊C₈

semidirect

125. C₇

#8138

semidirect

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

subgroup	index
C₇×#1726	2
#11727478	2
#11725466	2
#11725420	2
#8138	7

subgroup

index

Subgroups — #13244241

Conjugacy classes of subgroups

Complements

Maximal subgroups