Group #26102935 — subgroups

Subgroups — #26102935

Order 12032.

140 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₈×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#20241306	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1367	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₉₄	—	2	1	no	—
5	C₄×Dic₁₈₈	C₄	1	1	yes	characteristic
6	C₄×C₄₇⋊C₁₆	—	2	1	no	—
7	#15688876	C₄	1	1	yes	characteristic
8	#15688855	—	2	1	no	—
9	C₄₇×C₄⋊C₁₆	—	2	1	no	—
10	C₈²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₈₈	C₈	1	1	yes	characteristic
12	C₄×Dic₉₄	—	2	1	no	—
13	C₂×C₄₇⋊C₁₆	—	2	2	no	—
14	#13093958	C₈	1	1	yes	characteristic
15	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
16	Dic₄₇⋊C₈	—	4	1	no	—
17	#13093949	—	2	1	no	—
18	C₈×Dic₄₇	—	4	1	no	—
19	C₂×C₁₆×C₄₇	—	4	1	no	—
20	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
22	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
23	C₂×Dic₉₄	—	2	1	no	—
24	C₂×C₈×C₄₇	—	2	1	no	—
25	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
28	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
29	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₄⋊Dic₄₇	—	2	1	no	—
31	Dic₄₇⋊C₄	—	4	1	no	—
32	C₄×Dic₄₇	—	4	1	no	—
33	C₂×C₄₇⋊C₈	—	4	1	no	—
34	Dic₁₈₈	—	2	1	no	—
35	Dic₁₈₈	C₁₆	1	2	yes	complemented
36	C₁₆×C₄₇	—	8	1	no	—
37	C₄₇⋊C₁₆	—	4	2	no	—
38	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
40	C₂×C₄×C₄₇	#104	1	1	yes	characteristic
41	C₂×Dic₄₇	—	4	1	no	—
42	C₂×Dic₄₇	—	4	1	no	—
43	Dic₉₄	—	2	2	no	—
44	Dic₉₄	—	4	1	no	—
45	C₈×C₄₇	—	2	2	no	—
46	C₈×C₄₇	—	2	1	no	—
47	C₈×C₄₇	—	2	1	no	—
48	C₈×C₄₇	—	2	1	no	—
49	C₈×C₄₇	C₂×C₁₆	1	1	yes	characteristic
50	C₈×C₄₇	M₃₂	1	1	yes	characteristic
51	C₄₇⋊C₈	—	8	1	no	—
52	#7071	—	47	1	no	maximal, Sylow
53	C₂²×C₄₇	D₄⋊C₈	1	1	yes	characteristic
54	C₄×C₄₇	—	2	1	no	—
55	C₄×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
56	C₄×C₄₇	#350	1	1	yes	characteristic
57	C₄×C₄₇	#360	1	1	yes	characteristic
58	C₄×C₄₇	#362	1	1	yes	characteristic
59	Dic₄₇	—	4	2	no	—
60	Dic₄₇	—	8	1	no	—
61	C₈×Q₁₆	—	47	1	no	—
62	#1367	D₄₇	1	1	yes	characteristic
63	#1312	—	47	1	no	—
64	C₂×C₄₇	D₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₄₇	#1334	1	1	yes	characteristic
66	C₂×C₄₇	#1336	1	1	yes	characteristic
67	C₈×Q₈	—	94	1	no	—
68	C₄×Q₁₆	—	47	1	no	—
69	C₄⋊C₁₆	—	2	1	no	—
70	C₄×C₁₆	—	94	1	no	—
71	#335	—	47	1	no	—
72	Q₈⋊C₈	—	94	1	no	—
73	C₈²	C₂×D₄₇	1	1	yes	characteristic
74	C₄₇	#7071	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₁₆	—	47	1	no	—
76	C₄×Q₈	—	94	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	C₂×C₁₆	—	94	2	no	—
79	#107	—	47	1	no	—
80	C₄⋊C₈	—	94	1	no	—
81	C₄⋊C₈	—	188	1	no	—
82	#103	—	94	1	no	—
83	C₄×C₈	—	188	1	no	—
84	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
85	C₄×C₈	D₁₈₈	1	1	yes	characteristic
86	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
87	C₂×Q₈	—	94	1	no	—
88	Q₁₆	—	47	2	no	—
89	Q₁₆	—	94	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	188	1	no	—
92	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
93	C₂×C₈	#10625324	1	1	yes	characteristic
94	C₂×C₈	#10625334	1	1	yes	characteristic
95	C₂×C₈	#10625336	1	1	yes	characteristic
96	C₄⋊C₄	—	94	1	no	—
97	C₄⋊C₄	—	188	1	no	—
98	C₄²	—	188	1	no	—
99	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
100	C₁₆	—	8	1	no	—
101	C₁₆	—	188	2	no	—
102	Q₈	—	94	2	no	—
103	Q₈	—	188	1	no	—
104	C₂×C₄	—	188	1	no	—
105	C₂×C₄	—	188	1	no	—
106	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
107	C₂×C₄	#13093944	1	1	yes	characteristic
108	C₂×C₄	#13093950	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	376	1	no	—
114	C₈	C₁₆×D₄₇	1	1	yes	characteristic
115	C₈	#13093963	1	1	yes	characteristic
116	C₂²	#15688850	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	376	1	no	—
119	C₄	—	188	2	no	—
120	C₄	(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
121	C₄	#15688908	1	1	yes	characteristic
122	C₄	#15688932	1	1	yes	characteristic
123	C₄	#15688938	1	1	yes	characteristic
124	C₂	#20241342	1	1	yes	characteristic
125	C₂	#20241357	1	1	yes	characteristic
126	C₂	#20241363	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₁₈₈

C₂

yes

maximal, characteristic

#20241306

C₂

yes

maximal, characteristic

C₄₇×#1367

C₂

yes

maximal, characteristic

C₈×Dic₉₄

—

C₄×Dic₁₈₈

C₄

yes

characteristic

C₄×C₄₇⋊C₁₆

—

#15688876

C₄

yes

characteristic

#15688855

—

C₄₇×C₄⋊C₁₆

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

C₈

yes

characteristic

C₄×Dic₉₄

—

C₂×C₄₇⋊C₁₆

—

#13093958

C₈

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

#13093949

—

C₈×Dic₄₇

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

M₁₆

yes

characteristic

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

SD₁₆

yes

characteristic

C₄²×C₄₇

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

Dic₁₈₈

C₁₆

yes

complemented

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₁₆

yes

characteristic

C₈×C₄₇

M₃₂

yes

characteristic

C₄₇⋊C₈

—

#7071

—

maximal, Sylow

C₂²×C₄₇

D₄⋊C₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊C₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

C₈×Q₁₆

—

#1367

D₄₇

yes

characteristic

#1312

—

C₂×C₄₇

D₄⋊C₁₆

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#7071

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

188

—

#103

—

C₄×C₈

—

188

—

C₄×C₈

C₄×D₄₇

yes

characteristic

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₈×D₄₇

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₄²

—

188

—

C₄²

(C₂×D₄₇)⋊C₄

yes

characteristic

100

C₁₆

—

101

C₁₆

—

188

—

102

Q₈

—

103

Q₈

—

188

—

104

C₂×C₄

—

188

—

105

C₂×C₄

—

188

—

106

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

characteristic

107

C₂×C₄

#13093944

yes

characteristic

108

C₂×C₄

#13093950

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

376

—

114

C₈

C₁₆×D₄₇

yes

characteristic

115

C₈

#13093963

yes

characteristic

116

C₂²

#15688850

yes

characteristic

117

C₄

—

118

C₄

—

376

—

119

C₄

—

188

—

120

C₄

(C₂×D₄₇)⋊C₁₆

yes

characteristic

121

C₄

#15688908

yes

characteristic

122

C₄

#15688932

yes

characteristic

123

C₄

#15688938

yes

characteristic

124

C₂

#20241342

yes

characteristic

125

C₂

#20241357

yes

characteristic

126

C₂

#20241363

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093944
commutator	C₈×C₄₇	C₂×C₁₆
Frattini	C₈²	C₂×D₄₇
Fitting	C₄₇×#1367	C₂
socle, prime radical	C₂²×C₄₇	D₄⋊C₈
cosocle	C₈²×C₄₇	C₂²
hypercenter	#1367	D₄₇
hypocenter	C₄₇	#7071
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂×C₄

#13093944

commutator

C₈×C₄₇

C₂×C₁₆

Frattini

C₈²

C₂×D₄₇

Fitting

C₄₇×#1367

C₂

socle, prime radical

C₂²×C₄₇

D₄⋊C₈

cosocle

C₈²×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

N	H ≅ G/N	type
35. Dic₁₈₈	C₁₆	semidirect
74. C₄₇	#7071	semidirect

H ≅ G/N

type

35. Dic₁₈₈

C₁₆

semidirect

74. C₄₇

#7071

semidirect

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

subgroup	index
C₈×Dic₁₈₈	2
#20241306	2
C₄₇×#1367	2
#7071	47

subgroup

index

C₈×Dic₁₈₈

#20241306

C₄₇×#1367

#7071

Subgroups — #26102935

Conjugacy classes of subgroups

Complements

Maximal subgroups