Group #26131017 — subgroups

Subgroups — #26131017

Order 12032.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20245668	C₂	1	2	yes	maximal
2	#20245663	C₂	1	1	yes	maximal, characteristic
3	#20242997	C₂	1	2	yes	maximal
4	C₁₆×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2252	C₂	1	1	yes	maximal, characteristic
6	#15689528	C₂²	1	2	yes	—
7	#15689475	—	2	2	no	—
8	#15689473	—	2	2	no	—
9	#15688950	—	2	4	no	—
10	#15688949	—	2	2	no	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄₇×#500	C₂²	1	2	yes	—
14	C₄₇×#366	—	2	2	no	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	#13094024	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
18	#13093958	—	2	2	no	—
19	#13093958	—	2	2	no	—
20	#13093948	—	4	4	no	—
21	#13093947	—	4	4	no	—
22	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Q₈×C₄₇	—	2	2	no	—
25	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
26	#107×C₄₇	—	2	2	no	—
27	#107×C₄₇	—	2	2	no	—
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₇	—	4	2	no	—
30	C₂×Dic₉₄	—	4	2	no	—
31	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₄₇	—	4	2	no	—
34	C₄⋊C₄×C₄₇	—	4	2	no	—
35	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₄₇	—	4	2	no	—
37	C₄⋊Dic₄₇	—	4	2	no	—
38	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
39	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
40	C₁₆×C₄₇	Q₁₆	1	4	yes	—
41	C₄₇⋊C₁₆	Q₁₆	1	4	yes	—
42	Q₈×C₄₇	—	8	4	no	—
43	C₂×C₄×C₄₇	—	8	2	no	—
44	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₄₇	—	8	2	no	—
48	Dic₉₄	—	8	4	no	—
49	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
50	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
51	C₄₇⋊C₈	—	2	2	no	—
52	C₄₇⋊C₈	—	2	2	no	—
53	#12522	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	#493	1	1	yes	characteristic
55	C₄×C₄₇	—	16	2	no	—
56	C₄×C₄₇	#494	1	1	yes	characteristic
57	C₄×C₄₇	#494	1	1	yes	characteristic
58	C₄×C₄₇	#508	1	2	yes	—
59	C₄×C₄₇	#508	1	2	yes	—
60	Dic₄₇	—	16	2	no	—
61	#2254	—	47	2	no	—
62	#2252	—	47	1	no	—
63	#2252	D₄₇	1	1	yes	characteristic
64	#1710	—	47	2	no	—
65	C₈×C₁₆	—	47	1	no	—
66	C₂×C₄₇	#1713	1	1	yes	characteristic
67	C₂×C₄₇	#2249	1	1	yes	characteristic
68	C₂×C₄₇	#2249	1	1	yes	characteristic
69	#500	—	47	2	no	—
70	#500	C₂×D₄₇	1	2	yes	—
71	#487	—	94	2	no	—
72	#487	—	94	2	no	—
73	#367	—	94	4	no	—
74	#366	—	2	2	no	—
75	#366	—	94	2	no	—
76	C₄×C₁₆	—	47	1	no	—
77	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
78	C₈²	—	47	1	no	—
79	C₄₇	#12522	1	1	yes	characteristic, complemented, Sylow
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	94	2	no	—
82	C₂×C₁₆	—	47	2	no	—
83	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
84	#107	—	2	2	no	—
85	#107	—	2	2	no	—
86	#107	—	94	2	no	—
87	#107	—	94	2	no	—
88	#103	—	188	4	no	—
89	#103	—	188	4	no	—
90	C₄×C₈	—	47	1	no	—
91	C₄×C₈	—	47	1	no	—
92	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
93	C₂×Q₈	—	4	2	no	—
94	C₂×Q₈	—	188	2	no	—
95	C₂×C₈	—	47	2	no	—
96	C₂×C₈	—	47	2	no	—
97	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
98	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	188	2	no	—
102	C₄⋊C₄	—	188	2	no	—
103	C₄²	D₄×D₄₇	1	1	yes	characteristic
104	C₁₆	—	47	4	no	—
105	C₁₆	#10625337	1	4	yes	—
106	Q₈	—	8	4	no	—
107	Q₈	—	376	4	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	376	2	no	—
110	C₂×C₄	#13094020	1	1	yes	characteristic
111	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
112	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
113	C₈	—	94	2	no	—
114	C₈	—	94	2	no	—
115	C₈	Q₁₆×D₄₇	1	2	yes	—
116	C₈	C₂×#10625337	1	2	yes	—
117	C₂²	#15689498	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	752	2	no	—
120	C₄	#15689501	1	1	yes	characteristic
121	C₄	#15689502	1	1	yes	characteristic
122	C₄	#15689552	1	2	yes	—
123	C₄	#15689556	1	2	yes	—
124	C₂	#20243005	1	1	yes	characteristic
125	C₂	#20245646	1	1	yes	characteristic
126	C₂	#20245650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245668

C₂

yes

maximal

#20245663

C₂

yes

maximal, characteristic

#20242997

C₂

yes

maximal

C₁₆×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#2252

C₂

yes

maximal, characteristic

#15689528

C₂²

yes

—

—

—

—

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₄₇×#500

C₂²

yes

—

C₄₇×#366

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

#13094024

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

—

—

—

—

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄⋊Q₈×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

—

#107×C₄₇

—

#107×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₁₆×C₄₇

Q₁₆

yes

—

C₄₇⋊C₁₆

Q₁₆

yes

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

C₂×Q₁₆

yes

—

C₈×C₄₇

C₂×Q₁₆

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#508

yes

—

Dic₄₇

—

#2254

—

#2252

—

#2252

D₄₇

yes

characteristic

#1710

—

C₈×C₁₆

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×D₄₇

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₈²

—

C₄₇

#12522

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

—

—

—

—

—

—

188

—

#103

—

188

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₄₇

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

188

—

102

C₄⋊C₄

—

188

—

103

C₄²

D₄×D₄₇

yes

characteristic

104

C₁₆

—

105

C₁₆

#10625337

yes

—

106

Q₈

—

107

Q₈

—

376

—

108

C₂×C₄

—

109

C₂×C₄

—

376

—

110

C₂×C₄

#13094020

yes

characteristic

111

C₂×C₄

D₈×D₄₇

yes

characteristic

112

C₂×C₄

D₈×D₄₇

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Q₁₆×D₄₇

yes

—

116

C₈

C₂×#10625337

yes

—

117

C₂²

#15689498

yes

characteristic

118

C₄

—

119

C₄

—

752

—

120

C₄

#15689501

yes

characteristic

121

C₄

#15689502

yes

characteristic

122

C₄

#15689552

yes

—

123

C₄

#15689556

yes

—

124

C₂

#20243005

yes

characteristic

125

C₂

#20245646

yes

characteristic

126

C₂

#20245650

yes

characteristic

	subgroup	G/H
center	C₂²	#15689498
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2252	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	#2252	D₄₇
hypocenter	C₄₇	#12522
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#15689498

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2252

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
79. C₄₇	#12522	semidirect

H ≅ G/N

type

79. C₄₇

#12522

semidirect

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

subgroup	index
#20245668	2
#20245663	2
#20242997	2
C₁₆×C₄₇⋊C₈	2
C₄₇×#2252	2
#12522	47

subgroup

index

#20245668

#20245663

#20242997

C₁₆×C₄₇⋊C₈

C₄₇×#2252

#12522

Subgroups — #26131017

Conjugacy classes of subgroups

Complements

Maximal subgroups