Group #26137309 — subgroups

Subgroups — #26137309

Order 12032.

150 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 75 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	#20245667	C₂	1	1	yes	maximal, characteristic
2	#20245663	C₂	1	1	yes	maximal, characteristic
3	#20241590	C₂	1	2	yes	maximal
4	#20241588	C₂	1	2	yes	maximal
5	C₄₇×#1397	C₂	1	1	yes	maximal, characteristic
6	#15689528	—	2	1	no	—
7	#15689525	—	2	1	no	—
8	#15688951	—	2	1	no	—
9	#15688949	—	2	1	no	—
10	#15688949	C₂²	1	2	yes	—
11	#15688949	C₂²	1	1	yes	characteristic
12	#15688949	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₄₇	C₂²	1	2	yes	—
14	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	#13094024	—	4	1	no	—
16	#13094016	—	4	1	no	—
17	#13093958	—	2	1	no	—
18	#13093958	—	2	1	no	—
19	#13093958	—	2	2	no	—
20	#13093958	—	4	1	no	—
21	#13093956	—	4	1	no	—
22	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₃₂×C₄₇	Q₈	1	4	yes	complemented
26	C₂×Dic₉₄	—	8	1	no	—
27	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
28	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
29	C₄²×C₄₇	D₈	1	1	yes	characteristic
30	C₄⋊Dic₄₇	—	4	1	no	—
31	C₄⋊Dic₄₇	—	4	1	no	—
32	C₄⋊Dic₄₇	—	4	2	no	—
33	C₄⋊Dic₄₇	—	8	1	no	—
34	Dic₄₇⋊C₄	—	8	1	no	—
35	Dic₄₇⋊C₄	—	8	1	no	—
36	C₁₆×C₄₇	Q₁₆	1	2	yes	—
37	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
39	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
41	C₂×Dic₄₇	—	8	1	no	—
42	C₂×Dic₄₇	—	8	1	no	—
43	C₂×Dic₄₇	—	8	2	no	—
44	Dic₉₄	—	16	2	no	—
45	C₈×C₄₇	Q₃₂	1	2	yes	—
46	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
47	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
48	#13734	—	47	1	no	maximal, Sylow
49	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
50	C₄×C₄₇	—	2	1	no	—
51	C₄×C₄₇	—	2	1	no	—
52	C₄×C₄₇	#500	1	1	yes	characteristic
53	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
54	Dic₄₇	—	16	2	no	—
55	Dic₄₇	—	16	1	no	—
56	Dic₄₇	—	16	1	no	—
57	#2253	—	47	1	no	—
58	#2252	—	47	1	no	—
59	#1424	—	47	2	no	—
60	#1423	—	47	2	no	—
61	#1397	D₄₇	1	1	yes	characteristic
62	C₂×C₄₇	#2252	1	1	yes	characteristic
63	C₂×C₄₇	#2263	1	1	yes	characteristic
64	C₂×C₄₇	#2264	1	1	yes	characteristic
65	#500	—	94	1	no	—
66	#499	—	94	1	no	—
67	C₂×C₃₂	C₂×D₄₇	1	2	yes	—
68	#367	—	94	1	no	—
69	#366	—	47	1	no	—
70	#366	—	47	2	no	—
71	#366	—	47	1	no	—
72	#366	—	94	1	no	—
73	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
74	C₄₇	#13734	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	—	188	1	no	—
76	#125	—	188	1	no	—
77	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
78	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
79	#107	—	94	2	no	—
80	#107	—	94	1	no	—
81	#107	—	94	1	no	—
82	#107	—	188	1	no	—
83	#106	—	188	1	no	—
84	C₄×C₈	D₁₈₈	1	1	yes	characteristic
85	C₃₂	Dic₉₄	1	4	yes	complemented
86	C₂×Q₈	—	376	1	no	—
87	C₂×C₈	D₃₇₆	1	1	yes	characteristic
88	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
89	C₄⋊C₄	—	376	1	no	—
90	C₄⋊C₄	—	376	1	no	—
91	C₄⋊C₄	—	376	1	no	—
92	C₄⋊C₄	—	188	1	no	—
93	C₄⋊C₄	—	188	1	no	—
94	C₄⋊C₄	—	188	2	no	—
95	C₄²	D₃₇₆	1	1	yes	characteristic
96	C₁₆	Dic₁₈₈	1	2	yes	—
97	C₁₆	C₂×Dic₉₄	1	2	yes	—
98	Q₈	—	752	2	no	—
99	C₂×C₄	—	376	1	no	—
100	C₂×C₄	—	376	1	no	—
101	C₂×C₄	—	376	2	no	—
102	C₂×C₄	D₇₅₂	1	1	yes	characteristic
103	C₂×C₄	D₇₅₂	1	1	yes	characteristic
104	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
105	C₈	Dic₃₇₆	1	2	yes	—
106	C₈	#13094024	1	1	yes	characteristic
107	C₈	C₂×Dic₁₈₈	1	1	yes	characteristic
108	C₂²	C₂×D₇₅₂	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	752	2	no	—
112	C₄	—	752	1	no	—
113	C₄	—	752	1	no	—
114	C₄	#15689528	1	1	yes	characteristic
115	C₄	C₂×Dic₃₇₆	1	1	yes	characteristic
116	C₂	#20245663	1	1	yes	characteristic
117	C₂	#20245703	1	1	yes	characteristic
118	C₂	#20245708	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245667

C₂

yes

maximal, characteristic

#20245663

C₂

yes

maximal, characteristic

#20241590

C₂

yes

maximal

#20241588

C₂

yes

maximal

C₄₇×#1397

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#15688949

C₂²

yes

characteristic

#15688949

C₂²

yes

characteristic

C₂×C₃₂×C₄₇

C₂²

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

—

—

—

—

—

—

—

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₃₂×C₄₇

Q₈

yes

complemented

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₁₆×C₄₇

Q₁₆

yes

—

C₁₆×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

D₁₆

yes

characteristic

C₂×C₄×C₄₇

D₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

Q₃₂

yes

—

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

C₈×C₄₇

C₂×Q₁₆

yes

characteristic

#13734

—

maximal, Sylow

C₂²×C₄₇

C₂×D₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#500

yes

characteristic

C₄×C₄₇

C₂×Q₃₂

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₂×C₃₂

C₂×D₄₇

yes

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₄₇

#13734

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

188

—

#125

—

188

—

C₂×C₁₆

D₁₈₈

yes

characteristic

C₂×C₁₆

C₂²×D₄₇

yes

—

—

—

—

188

—

#106

—

188

—

C₄×C₈

D₁₈₈

yes

characteristic

C₃₂

Dic₉₄

yes

complemented

C₂×Q₈

—

376

—

C₂×C₈

D₃₇₆

yes

characteristic

C₂×C₈

C₂×D₁₈₈

yes

characteristic

C₄⋊C₄

—

376

—

C₄⋊C₄

—

376

—

C₄⋊C₄

—

376

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄²

D₃₇₆

yes

characteristic

C₁₆

Dic₁₈₈

yes

—

C₁₆

C₂×Dic₉₄

yes

—

Q₈

—

752

—

C₂×C₄

—

376

—

100

C₂×C₄

—

376

—

101

C₂×C₄

—

376

—

102

C₂×C₄

D₇₅₂

yes

characteristic

103

C₂×C₄

D₇₅₂

yes

characteristic

104

C₂×C₄

C₂×D₃₇₆

yes

characteristic

105

C₈

Dic₃₇₆

yes

—

106

C₈

#13094024

yes

characteristic

107

C₈

C₂×Dic₁₈₈

yes

characteristic

108

C₂²

C₂×D₇₅₂

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

752

—

112

C₄

—

752

—

113

C₄

—

752

—

114

C₄

#15689528

yes

characteristic

115

C₄

C₂×Dic₃₇₆

yes

characteristic

116

C₂

#20245663

yes

characteristic

117

C₂

#20245703

yes

characteristic

118

C₂

#20245708

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₇₅₂
commutator, cosocle	C₂×C₁₆×C₄₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄₇×#1397	C₂
socle, prime radical	C₂²×C₄₇	C₂×D₁₆
hypercenter	#1397	D₄₇
hypocenter	C₄₇	#13734
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₇₅₂

commutator, cosocle

C₂×C₁₆×C₄₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₇

Fitting

C₄₇×#1397

C₂

socle, prime radical

C₂²×C₄₇

C₂×D₁₆

D₄₇

C₄₇

G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
25. C₃₂×C₄₇	Q₈	semidirect
74. C₄₇	#13734	semidirect
85. C₃₂	Dic₉₄	semidirect

H ≅ G/N

type

25. C₃₂×C₄₇

Q₈

semidirect

74. C₄₇

#13734

semidirect

85. C₃₂

Dic₉₄

semidirect

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

subgroup	index
#20245667	2
#20245663	2
#20241590	2
#20241588	2
C₄₇×#1397	2
#13734	47

subgroup

index

C₄₇×#1397

Subgroups — #26137309

Conjugacy classes of subgroups

Complements

Maximal subgroups