Group #13247493 — subgroups

Subgroups — #13247493

Order 1792.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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×#1835	C₂	1	1	yes	maximal, characteristic
2	#11727546	C₂	1	1	yes	maximal, characteristic
3	#11727480	C₂	1	1	yes	maximal, characteristic
4	#11725437	C₂	1	4	yes	maximal
5	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₇×#382	C₂²	1	2	yes	—
7	C₂×#6393	—	2	2	no	—
8	#89697	C₂²	1	2	yes	—
9	#89696	C₂²	1	2	yes	—
10	C₇×#340	—	2	2	no	—
11	#89249	C₄	1	4	yes	complemented
12	#8794	—	7	1	no	maximal, Sylow
13	C₂²×C₇×C₈	D₄	1	2	yes	—
14	C₂×C₇×C₄⋊C₄	—	2	2	no	—
15	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
16	C₂×#1136	—	2	2	no	—
17	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
18	C₄×C₇×C₈	Q₈	1	2	yes	—
19	C₇×#95	—	2	4	no	—
20	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
21	#6393	—	4	4	no	—
22	C₄⋊(C₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇⋊C₈)	—	2	4	no	—
24	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
25	#1848	—	7	1	no	—
26	#1835	D₇	1	1	yes	characteristic
27	#1452	—	7	1	no	—
28	#1286	—	7	4	no	—
29	C₂²×C₄×C₇	—	4	2	no	—
30	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇	#41	1	2	yes	—
32	C₂×C₇×C₈	—	2	2	no	—
33	C₂×C₇×C₈	—	2	2	no	—
34	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
35	C₇×C₄⋊C₄	—	4	2	no	—
36	C₇×C₄⋊C₄	—	4	2	no	—
37	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₇	#41	1	2	yes	—
40	#1136	—	4	4	no	—
41	C₂×C₇⋊C₈	—	2	4	no	—
42	C₂×C₇⋊C₈	—	4	2	no	—
43	#431	—	7	2	no	—
44	C₂×#108	—	14	2	no	—
45	#423	—	7	2	no	—
46	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
47	#382	C₂×D₇	1	2	yes	—
48	#340	—	2	2	no	—
49	#322	—	7	4	no	—
50	C₂³×C₇	#124	1	1	yes	characteristic
51	C₂×C₄×C₇	—	4	2	no	—
52	C₂×C₄×C₇	—	8	4	no	—
53	C₂×C₄×C₇	—	8	2	no	—
54	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇	#117	1	2	yes	—
56	C₂×C₄×C₇	#124	1	1	yes	characteristic
57	C₂×C₄×C₇	#125	1	2	yes	—
58	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
59	C₂×C₄×C₇	#137	1	2	yes	—
60	C₇×C₈	—	2	4	no	—
61	C₇⋊C₈	—	4	4	no	—
62	C₇⋊C₈	—	8	2	no	—
63	C₂×M₁₆	—	14	2	no	—
64	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
65	C₂×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
67	#108	—	28	4	no	—
68	C₄⋊C₈	—	14	2	no	—
69	C₄⋊C₈	—	14	4	no	—
70	C₂²⋊C₈	—	14	2	no	—
71	C₄×C₈	—	7	4	no	—
72	C₄×C₈	Dic₁₄	1	2	yes	—
73	C₄×C₈	C₄×D₇	1	2	yes	—
74	#95	—	2	4	no	—
75	C₂²×C₇	—	2	2	no	—
76	C₂²×C₇	#382	1	1	yes	characteristic
77	C₂²×C₇	#489	1	2	yes	—
78	C₄×C₇	—	2	2	no	—
79	C₄×C₇	—	2	2	no	—
80	C₄×C₇	—	16	2	no	—
81	C₄×C₇	#428	1	2	yes	—
82	C₄×C₇	#490	1	1	yes	characteristic
83	C₄×C₇	#491	1	1	yes	characteristic
84	C₂²×C₄	—	4	2	no	—
85	C₂²×C₄	#1159	1	2	yes	—
86	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
87	M₁₆	—	28	4	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	14	4	no	—
91	C₂×C₈	—	28	2	no	—
92	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
96	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
97	C₄²	#1161	1	2	yes	—
98	C₂×C₇	—	2	2	no	—
99	C₂×C₇	#1833	1	2	yes	—
100	C₂×C₇	#1836	1	1	yes	characteristic
101	C₂³	#6499	1	1	yes	characteristic
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	8	4	no	—
104	C₂×C₄	—	8	2	no	—
105	C₂×C₄	#6453	1	2	yes	—
106	C₂×C₄	#6455	1	2	yes	—
107	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
108	C₂×C₄	#6510	1	1	yes	characteristic
109	C₂×C₄	#6512	1	2	yes	—
110	C₂×C₄	#6514	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	—	28	4	no	—
113	C₈	—	56	2	no	—
114	C₇	#8794	1	1	yes	characteristic, complemented, Sylow
115	C₂²	—	2	2	no	—
116	C₂²	#89749	1	1	yes	characteristic
117	C₂²	#89827	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	#89767	1	2	yes	—
122	C₄	#89839	1	1	yes	characteristic
123	C₄	#89841	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#11727302	1	1	yes	characteristic
126	C₂	#11728411	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1835

C₂

yes

maximal, characteristic

#11727546

C₂

yes

maximal, characteristic

#11727480

C₂

yes

maximal, characteristic

#11725437

C₂

yes

maximal

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#382

C₂²

yes

—

C₂×#6393

—

#89697

C₂²

yes

—

#89696

C₂²

yes

—

C₇×#340

—

#89249

C₄

yes

complemented

#8794

—

maximal, Sylow

C₂²×C₇×C₈

D₄

yes

—

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×#1136

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

Q₈

yes

—

C₇×#95

—

C₂²⋊(C₇⋊C₈)

—

#6393

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄×C₇⋊C₈

C₂×C₄

yes

—

#1848

—

#1835

D₇

yes

characteristic

#1452

—

#1286

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂×Q₈

yes

characteristic

C₄²×C₇

#41

yes

—

#1136

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

#431

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₇

yes

characteristic

#382

C₂×D₇

yes

—

#340

—

#322

—

C₂³×C₇

#124

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#124

yes

characteristic

C₂×C₄×C₇

#125

yes

—

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇

#137

yes

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₇

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₄

yes

—

C₄×C₈

C₄×D₇

yes

—

#95

—

C₂²×C₇

—

C₂²×C₇

#382

yes

characteristic

C₂²×C₇

#489

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#428

yes

—

C₄×C₇

yes

C₄×C₇

yes

C₂²×C₄

—

C₂²×C₄

#1159

yes

—

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Dic₁₄

yes

characteristic

C₄²

C₂×C₄×D₇

yes

characteristic

C₄²

#1161

yes

—

C₂×C₇

—

C₂×C₇

#1833

yes

—

100

C₂×C₇

#1836

yes

characteristic

101

C₂³

#6499

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

#6453

yes

—

106

C₂×C₄

#6455

yes

—

107

C₂×C₄

C₂×Dic₇⋊C₄

yes

characteristic

108

C₂×C₄

#6510

yes

characteristic

109

C₂×C₄

#6512

yes

—

110

C₂×C₄

#6514

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₇

#8794

yes

characteristic, complemented, Sylow

115

C₂²

—

116

C₂²

#89749

yes

characteristic

117

C₂²

#89827

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#89767

yes

—

122

C₄

#89839

yes

characteristic

123

C₄

#89841

yes

characteristic

124

C₂

—

125

C₂

#11727302

yes

characteristic

126

C₂

#11728411

yes

—

	subgroup	G/H
center	C₂²	#89749
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1835	C₂
socle	C₂²×C₇	#382
prime radical	C₂³×C₇	#124
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1835	D₇
hypocenter	C₇	#8794
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#89749

commutator

C₄²×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
11. #89249	C₄	semidirect
114. C₇	#8794	semidirect

H ≅ G/N

type

11. #89249

C₄

semidirect

114. C₇

#8794

semidirect

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

subgroup	index
C₇×#1835	2
#11727546	2
#11727480	2
#11725437	2
#8794	7

subgroup

index

Subgroups — #13247493

Conjugacy classes of subgroups

Complements

Maximal subgroups