Group #26130964 — subgroups

Subgroups — #26130964

Order 12032.

306 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20245612	C₂	1	4	yes	maximal, complemented
2	#20243006	C₂	1	1	yes	maximal, characteristic, complemented
3	#20243003	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689528	—	2	1	no	—
6	#15689528	C₂²	1	2	yes	—
7	#15689503	—	2	1	no	—
8	#15689500	—	2	1	no	—
9	#15689500	—	2	1	no	—
10	#15689500	C₂²	1	2	yes	—
11	#15688925	—	2	8	no	—
12	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
13	C₈²×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₈₈	—	4	2	no	—
15	C₂×D₃₇₆	—	2	4	no	—
16	C₂×D₃₇₆	—	4	2	no	—
17	C₂×D₃₇₆	—	4	2	no	—
18	#13094024	—	2	2	no	—
19	#13094021	—	2	2	no	—
20	#13093958	—	2	4	no	—
21	#13093958	—	4	2	no	—
22	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
23	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×D₁₈₈	—	4	2	no	—
27	C₂×D₁₈₈	—	4	2	no	—
28	C₂×D₁₈₈	—	4	2	no	—
29	C₂×Dic₉₄	—	4	2	no	—
30	C₂×C₈×C₄₇	D₈	1	2	yes	—
31	C₂×C₈×C₄₇	D₈	1	2	yes	—
32	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
33	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₄⋊Dic₄₇	—	4	2	no	—
35	C₄⋊Dic₄₇	—	4	2	no	—
36	Dic₁₈₈	—	8	4	no	—
37	D₃₇₆	—	4	8	no	—
38	D₃₇₆	—	8	4	no	—
39	D₃₇₆	—	8	4	no	—
40	C₁₆×C₄₇	—	2	4	no	—
41	C₂²×D₄₇	—	8	2	no	—
42	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
45	C₂×Dic₄₇	—	8	2	no	—
46	D₁₈₈	—	8	4	no	—
47	D₁₈₈	—	8	4	no	—
48	D₁₈₈	—	8	4	no	—
49	Dic₉₄	—	8	4	no	—
50	C₈×C₄₇	D₁₆	1	4	yes	—
51	C₈×C₄₇	SD₃₂	1	4	yes	—
52	C₈×C₄₇	#135	1	4	yes	—
53	#12509	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	#493	1	1	yes	characteristic
55	C₂×D₄₇	—	16	2	no	—
56	C₂×D₄₇	—	16	2	no	—
57	C₂×D₄₇	—	16	2	no	—
58	C₄×C₄₇	#495	1	2	yes	—
59	C₄×C₄₇	C₂×D₁₆	1	2	yes	—
60	C₄×C₄₇	C₂×SD₃₂	1	2	yes	—
61	Dic₄₇	—	16	2	no	—
62	#2240	—	47	4	no	—
63	#1713	—	47	1	no	—
64	#1712	—	47	1	no	—
65	C₈×C₁₆	D₄₇	1	1	yes	characteristic, complemented
66	C₂×C₄₇	#1709	1	1	yes	characteristic
67	C₂×C₄₇	#2246	1	1	yes	characteristic
68	C₂×C₄₇	#2248	1	1	yes	characteristic
69	D₄₇	—	32	2	no	—
70	#500	—	47	2	no	—
71	#500	—	94	1	no	—
72	#494	—	94	1	no	—
73	#493	—	47	2	no	—
74	#493	—	94	1	no	—
75	#493	—	94	1	no	—
76	#357	—	94	8	no	—
77	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
78	C₈²	C₂×D₄₇	1	1	yes	characteristic
79	C₄₇	#12509	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	188	2	no	—
81	C₂×D₈	—	94	4	no	—
82	C₂×D₈	—	188	2	no	—
83	C₂×D₈	—	188	2	no	—
84	C₄⋊Q₈	—	94	2	no	—
85	#127	—	94	2	no	—
86	C₂×C₁₆	D₁₈₈	1	4	yes	—
87	#107	—	94	4	no	—
88	#107	—	188	2	no	—
89	C₄×C₈	D₁₈₈	1	1	yes	characteristic
90	C₄×C₈	D₁₈₈	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
92	C₂×Q₈	—	188	2	no	—
93	C₂×D₄	—	188	2	no	—
94	C₂×D₄	—	188	2	no	—
95	C₂×D₄	—	188	2	no	—
96	Q₁₆	—	376	4	no	—
97	D₈	—	376	4	no	—
98	D₈	—	376	4	no	—
99	D₈	—	188	8	no	—
100	C₂×C₈	D₃₇₆	1	2	yes	—
101	C₂×C₈	D₃₇₆	1	2	yes	—
102	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
103	C₄⋊C₄	—	188	2	no	—
104	C₄⋊C₄	—	188	2	no	—
105	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
106	C₁₆	—	2	4	no	—
107	C₂³	—	376	2	no	—
108	Q₈	—	376	4	no	—
109	D₄	—	376	4	no	—
110	D₄	—	376	4	no	—
111	D₄	—	376	4	no	—
112	C₂×C₄	—	376	2	no	—
113	C₂×C₄	#13094021	1	1	yes	characteristic
114	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
115	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
116	C₈	D₇₅₂	1	4	yes	—
117	C₈	#13093969	1	4	yes	—
118	C₈	#13094046	1	4	yes	—
119	C₂²	—	752	2	no	—
120	C₂²	—	752	2	no	—
121	C₂²	—	752	2	no	—
122	C₂²	#15689500	1	1	yes	characteristic
123	C₄	—	752	2	no	—
124	C₄	#15689507	1	2	yes	—
125	C₄	C₂×D₇₅₂	1	2	yes	—
126	C₄	C₂×#13093969	1	2	yes	—
127	C₂	—	1504	2	no	—
128	C₂	#20242994	1	1	yes	characteristic
129	C₂	#20245638	1	1	yes	characteristic
130	C₂	#20245644	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245612

C₂

yes

maximal, complemented

#20243006

C₂

yes

maximal, characteristic, complemented

#20243003

C₂

yes

maximal, characteristic

C₈×C₁₆×C₄₇

C₂

yes

maximal, characteristic, complemented

#15689528

—

#15689528

C₂²

yes

—

—

—

—

C₂²

yes

—

#15688925

—

C₄×C₁₆×C₄₇

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

—

—

—

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

D₃₇₆

—

D₃₇₆

—

D₃₇₆

—

C₁₆×C₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

C₈×C₄₇

D₁₆

yes

—

C₈×C₄₇

SD₃₂

yes

—

C₈×C₄₇

#135

yes

—

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#495

yes

—

C₄×C₄₇

C₂×D₁₆

yes

—

C₄×C₄₇

C₂×SD₃₂

yes

—

Dic₄₇

—

#2240

—

#1713

—

#1712

—

C₈×C₁₆

D₄₇

yes

characteristic, complemented

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₇

yes

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12509

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

188

—

C₂×D₈

—

C₂×D₈

—

188

—

C₂×D₈

—

188

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₁₈₈

yes

—

#107

—

#107

—

188

—

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

188

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

Q₁₆

—

376

—

D₈

—

376

—

D₈

—

376

—

D₈

—

188

—

100

C₂×C₈

D₃₇₆

yes

—

101

C₂×C₈

D₃₇₆

yes

—

102

C₂×C₈

C₂×D₁₈₈

yes

—

103

C₄⋊C₄

—

188

—

104

C₄⋊C₄

—

188

—

105

C₄²

C₂×D₁₈₈

yes

characteristic

106

C₁₆

—

107

C₂³

—

376

—

108

Q₈

—

376

—

109

D₄

—

376

—

110

D₄

—

376

—

111

D₄

—

376

—

112

C₂×C₄

—

376

—

113

C₂×C₄

#13094021

yes

characteristic

114

C₂×C₄

C₂×D₃₇₆

yes

characteristic

115

C₂×C₄

C₂×D₃₇₆

yes

characteristic

116

C₈

D₇₅₂

yes

—

117

C₈

#13093969

yes

—

118

C₈

#13094046

yes

—

119

C₂²

—

752

—

120

C₂²

—

752

—

121

C₂²

—

752

—

122

C₂²

#15689500

yes

characteristic

123

C₄

—

752

—

124

C₄

#15689507

yes

—

125

C₄

C₂×D₇₅₂

yes

—

126

C₄

C₂×#13093969

yes

—

127

C₂

—

1504

—

128

C₂

#20242994

yes

characteristic

129

C₂

#20245638

yes

characteristic

130

C₂

#20245644

yes

characteristic

	subgroup	G/H
center	C₂²	#15689500
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₈×C₁₆×C₄₇	C₂
socle	C₂²×C₄₇	#493
prime radical	#20243003	C₂
hypercenter	C₈×C₁₆	D₄₇
hypocenter	C₄₇	#12509
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#15689500

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₈×C₁₆×C₄₇

C₂

C₂²×C₄₇

C₂

C₈×C₁₆

D₄₇

hypocenter

C₄₇

#12509

LCS

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #20245612	C₂	semidirect
2. #20243006	C₂	semidirect
4. C₈×C₁₆×C₄₇	C₂	semidirect
65. C₈×C₁₆	D₄₇	semidirect
79. C₄₇	#12509	semidirect

H ≅ G/N

type

1. #20245612

C₂

semidirect

2. #20243006

C₂

semidirect

4. C₈×C₁₆×C₄₇

C₂

semidirect

65. C₈×C₁₆

D₄₇

semidirect

79. C₄₇

#12509

semidirect

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

subgroup	index
#20245612	2
#20243006	2
#20243003	2
C₈×C₁₆×C₄₇	2
#12509	47

subgroup

index

C₈×C₁₆×C₄₇

Subgroups — #26130964

Conjugacy classes of subgroups

Complements

Maximal subgroups