Group #26130991 — subgroups

Subgroups — #26130991

Order 12032.

310 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20245641	C₂	1	1	yes	maximal, characteristic
2	#20245640	C₂	1	2	yes	maximal
3	#20243004	C₂	1	2	yes	maximal
4	C₁₆×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2247	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₇₆	—	2	2	no	—
7	C₂×#13093970	—	2	4	no	—
8	#15689526	—	2	4	no	—
9	#15689503	C₂²	1	2	yes	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₄₇	—	2	2	no	—
13	C₄₇×#494	C₂²	1	2	yes	—
14	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₈₈	—	2	2	no	—
16	C₂×Dic₁₈₈	—	2	2	no	—
17	#13094024	—	2	2	no	—
18	Dic₃₇₆	—	4	4	no	—
19	#13093970	—	4	8	no	—
20	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
21	#13093957	—	4	8	no	—
22	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
23	C₂×Q₁₆×C₄₇	—	2	2	no	—
24	C₂×Q₁₆×C₄₇	—	2	2	no	—
25	C₄⋊Q₈×C₄₇	—	2	2	no	—
26	Q₃₂×C₄₇	—	4	4	no	—
27	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₇	—	4	2	no	—
30	C₂×Dic₉₄	—	4	2	no	—
31	Q₁₆×C₄₇	—	4	4	no	—
32	Q₁₆×C₄₇	—	4	4	no	—
33	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₄₇	—	4	4	no	—
36	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₄₇	—	4	4	no	—
38	C₂×C₄₇⋊C₈	D₈	1	4	yes	—
39	Dic₁₈₈	—	4	4	no	—
40	Dic₁₈₈	—	4	4	no	—
41	C₁₆×C₄₇	D₈	1	4	yes	—
42	C₄₇⋊C₁₆	D₈	1	4	yes	—
43	Q₈×C₄₇	—	8	2	no	—
44	Q₈×C₄₇	—	8	2	no	—
45	C₂×C₄×C₄₇	—	8	2	no	—
46	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
48	C₂×Dic₄₇	—	8	2	no	—
49	Dic₉₄	—	8	2	no	—
50	Dic₉₄	—	8	2	no	—
51	C₈×C₄₇	C₂×D₈	1	2	yes	—
52	C₈×C₄₇	C₂×D₈	1	2	yes	—
53	C₄₇⋊C₈	Q₃₂	1	8	yes	complemented
54	#12516	—	47	1	no	maximal, Sylow
55	C₂²×C₄₇	#493	1	1	yes	characteristic
56	C₄×C₄₇	—	16	2	no	—
57	C₄×C₄₇	#493	1	1	yes	characteristic
58	C₄×C₄₇	#493	1	1	yes	characteristic
59	C₄×C₄₇	C₂×Q₃₂	1	4	yes	—
60	Dic₄₇	—	16	2	no	—
61	#2247	—	47	2	no	—
62	#2247	—	47	1	no	—
63	#2247	D₄₇	1	1	yes	characteristic
64	#1713	—	47	2	no	—
65	C₈×C₁₆	—	47	1	no	—
66	C₂×C₄₇	#1712	1	1	yes	characteristic
67	C₂×C₄₇	#2247	1	2	yes	—
68	C₂×Q₃₂	—	2	2	no	—
69	C₂×Q₃₂	—	94	4	no	—
70	C₂×Q₃₂	—	94	2	no	—
71	#500	—	94	4	no	—
72	#494	—	47	2	no	—
73	#494	C₂×D₄₇	1	2	yes	—
74	C₄×C₁₆	—	47	1	no	—
75	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
76	C₈²	—	47	1	no	—
77	C₄₇	#12516	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	—	2	2	no	—
80	C₂×Q₁₆	—	94	2	no	—
81	C₂×Q₁₆	—	94	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	94	2	no	—
84	Q₃₂	—	4	4	no	—
85	Q₃₂	—	188	8	no	—
86	Q₃₂	—	188	4	no	—
87	C₂×C₁₆	—	47	2	no	—
88	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
89	#107	—	188	8	no	—
90	C₄×C₈	—	47	2	no	—
91	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	188	2	no	—
94	Q₁₆	—	4	4	no	—
95	Q₁₆	—	4	4	no	—
96	Q₁₆	—	188	4	no	—
97	Q₁₆	—	188	4	no	—
98	C₂×C₈	—	47	4	no	—
99	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	188	4	no	—
103	C₄²	D₄×D₄₇	1	1	yes	characteristic
104	C₁₆	—	47	4	no	—
105	C₁₆	#10625334	1	4	yes	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	Q₈	—	376	2	no	—
109	Q₈	—	376	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	376	2	no	—
112	C₂×C₄	#13094020	1	1	yes	characteristic
113	C₂×C₄	D₈×D₄₇	1	2	yes	—
114	C₈	—	47	8	no	—
115	C₈	D₈×D₄₇	1	2	yes	—
116	C₈	C₂×#10625334	1	2	yes	—
117	C₂²	#15689498	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	752	2	no	—
120	C₄	#15689498	1	1	yes	characteristic
121	C₄	#15689499	1	1	yes	characteristic
122	C₄	Q₃₂×D₄₇	1	4	yes	—
123	C₂	#20243002	1	1	yes	characteristic
124	C₂	#20245639	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245641

C₂

yes

maximal, characteristic

#20245640

C₂

yes

maximal

#20243004

C₂

yes

maximal

C₁₆×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#2247

C₂

yes

maximal, characteristic

C₂×Dic₃₇₆

—

C₂×#13093970

—

#15689526

—

#15689503

C₂²

yes

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₄₇

—

C₄₇×#494

C₂²

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

—

Dic₃₇₆

—

#13093970

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093957

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

—

Q₃₂×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

D₈

yes

—

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

D₈

yes

—

C₄₇⋊C₁₆

D₈

yes

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

C₂×D₈

yes

—

C₈×C₄₇

C₂×D₈

yes

—

C₄₇⋊C₈

Q₃₂

yes

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×Q₃₂

yes

—

Dic₄₇

—

#2247

—

#2247

—

#2247

D₄₇

yes

characteristic

#1713

—

C₈×C₁₆

—

C₂×C₄₇

#1712

yes

characteristic

C₂×C₄₇

#2247

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#500

—

#494

—

#494

C₂×D₄₇

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₈²

—

C₄₇

#12516

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

188

—

Q₃₂

—

188

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

—

#107

—

188

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

188

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

188

—

Q₁₆

—

188

—

C₂×C₈

—

C₂×C₈

D₄×D₄₇

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

188

—

103

C₄²

D₄×D₄₇

yes

characteristic

104

C₁₆

—

105

C₁₆

#10625334

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

376

—

109

Q₈

—

376

—

110

C₂×C₄

—

111

C₂×C₄

—

376

—

112

C₂×C₄

#13094020

yes

characteristic

113

C₂×C₄

D₈×D₄₇

yes

—

114

C₈

—

115

C₈

D₈×D₄₇

yes

—

116

C₈

C₂×#10625334

yes

—

117

C₂²

#15689498

yes

characteristic

118

C₄

—

119

C₄

—

752

—

120

C₄

#15689498

yes

characteristic

121

C₄

#15689499

yes

characteristic

122

C₄

Q₃₂×D₄₇

yes

—

123

C₂

#20243002

yes

characteristic

124

C₂

#20245639

yes

—

	subgroup	G/H
center	C₂²	#15689498
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2247	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	#2247	D₄₇
hypocenter	C₄₇	#12516
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#15689498

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2247

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
53. C₄₇⋊C₈	Q₃₂	semidirect
77. C₄₇	#12516	semidirect

H ≅ G/N

type

53. C₄₇⋊C₈

Q₃₂

semidirect

77. C₄₇

#12516

semidirect

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

subgroup	index
#20245641	2
#20245640	2
#20243004	2
C₁₆×C₄₇⋊C₈	2
C₄₇×#2247	2
#12516	47

subgroup

index

#20245641

#20245640

#20243004

C₁₆×C₄₇⋊C₈

C₄₇×#2247

#12516

Subgroups — #26130991

Conjugacy classes of subgroups

Complements

Maximal subgroups