Group #26130933 — subgroups

Subgroups — #26130933

Order 12032.

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

normal subgroups only

This group has 262 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	#20245663	C₂	1	1	yes	maximal, characteristic
2	#20245615	C₂	1	2	yes	maximal
3	#20242982	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	#15689528	C₂²	1	2	yes	—
7	#15689519	—	2	2	no	—
8	#15689475	—	2	2	no	—
9	#15688949	—	2	2	no	—
10	#15688927	—	2	4	no	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₄₇	—	2	2	no	—
14	C₄₇×#494	C₂²	1	2	yes	—
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	#13093955	—	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	C₂×Q₁₆×C₄₇	—	2	2	no	—
25	C₂×Q₁₆×C₄₇	—	2	2	no	—
26	C₄⋊Q₈×C₄₇	—	2	2	no	—
27	Q₃₂×C₄₇	—	4	4	no	—
28	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
29	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₄₇	—	4	2	no	—
31	C₂×Dic₉₄	—	4	2	no	—
32	Q₁₆×C₄₇	—	4	4	no	—
33	Q₁₆×C₄₇	—	4	4	no	—
34	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₄₇	—	4	2	no	—
37	C₄⋊C₄×C₄₇	—	4	2	no	—
38	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₄₇	—	4	2	no	—
40	C₄⋊Dic₄₇	—	4	2	no	—
41	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
42	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
43	C₁₆×C₄₇	SD₁₆	1	4	yes	—
44	C₄₇⋊C₁₆	—	2	2	no	—
45	Q₈×C₄₇	—	8	2	no	—
46	Q₈×C₄₇	—	8	2	no	—
47	C₂×C₄×C₄₇	—	8	2	no	—
48	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₄₇	—	8	2	no	—
52	Dic₉₄	—	8	4	no	—
53	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
54	C₈×C₄₇	#135	1	2	yes	—
55	C₄₇⋊C₈	—	2	2	no	—
56	C₄₇⋊C₈	Q₃₂	1	4	yes	complemented
57	#12504	—	47	1	no	maximal, Sylow
58	C₂²×C₄₇	#493	1	1	yes	characteristic
59	C₄×C₄₇	—	16	2	no	—
60	C₄×C₄₇	#492	1	1	yes	characteristic
61	C₄×C₄₇	#495	1	1	yes	characteristic
62	C₄×C₄₇	C₂×Q₃₂	1	2	yes	—
63	C₄×C₄₇	#508	1	2	yes	—
64	Dic₄₇	—	16	2	no	—
65	#2252	—	47	1	no	—
66	#2247	D₄₇	1	1	yes	characteristic
67	#2241	—	47	2	no	—
68	#1707	—	47	2	no	—
69	C₈×C₁₆	—	47	1	no	—
70	C₂×C₄₇	#1706	1	1	yes	characteristic
71	C₂×C₄₇	#2247	1	1	yes	characteristic
72	C₂×C₄₇	#2249	1	1	yes	characteristic
73	C₂×Q₃₂	—	2	2	no	—
74	#500	—	47	2	no	—
75	#498	—	94	2	no	—
76	#494	C₂×D₄₇	1	2	yes	—
77	#487	—	94	2	no	—
78	#366	—	94	2	no	—
79	#358	—	94	4	no	—
80	C₄×C₁₆	—	47	1	no	—
81	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
82	C₈²	—	47	1	no	—
83	C₄₇	#12504	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₁₆	—	2	2	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	—	94	2	no	—
88	Q₃₂	—	4	4	no	—
89	C₂×C₁₆	—	47	2	no	—
90	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
91	#107	—	94	2	no	—
92	#107	—	94	2	no	—
93	#106	—	188	4	no	—
94	#103	—	188	4	no	—
95	C₄×C₈	—	47	1	no	—
96	C₄×C₈	—	47	1	no	—
97	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	188	2	no	—
100	Q₁₆	—	4	4	no	—
101	Q₁₆	—	4	4	no	—
102	C₂×C₈	—	47	2	no	—
103	C₂×C₈	—	47	2	no	—
104	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
105	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	188	2	no	—
109	C₄⋊C₄	—	188	2	no	—
110	C₄²	D₄×D₄₇	1	1	yes	characteristic
111	C₁₆	—	94	2	no	—
112	C₁₆	#10625335	1	4	yes	—
113	Q₈	—	8	2	no	—
114	Q₈	—	8	2	no	—
115	Q₈	—	376	4	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	376	2	no	—
118	C₂×C₄	#13094020	1	1	yes	characteristic
119	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
120	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
121	C₈	—	47	4	no	—
122	C₈	—	94	2	no	—
123	C₈	C₂×#10625335	1	2	yes	—
124	C₈	#13094043	1	2	yes	—
125	C₂²	#15689498	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	752	2	no	—
128	C₄	#15689495	1	1	yes	characteristic
129	C₄	#15689506	1	1	yes	characteristic
130	C₄	Q₃₂×D₄₇	1	2	yes	—
131	C₄	#15689556	1	2	yes	—
132	C₂	#20242977	1	1	yes	characteristic
133	C₂	#20245639	1	1	yes	characteristic
134	C₂	#20245650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245663

C₂

yes

maximal, characteristic

#20245615

C₂

yes

maximal

#20242982

C₂

yes

maximal

C₁₆×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#2247

C₂

yes

maximal, characteristic

#15689528

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

#13094024

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

—

—

—

—

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

D₄

yes

characteristic

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

—

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

SD₁₆

yes

—

C₄₇⋊C₁₆

—

Q₈×C₄₇

—

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×SD₁₆

yes

—

C₈×C₄₇

#135

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

Q₃₂

yes

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×Q₃₂

yes

—

C₄×C₄₇

#508

yes

—

Dic₄₇

—

#2252

—

#2247

D₄₇

yes

characteristic

#2241

—

#1707

—

C₈×C₁₆

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×Q₃₂

—

#500

—

#498

—

#494

C₂×D₄₇

yes

—

#487

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₈²

—

C₄₇

#12504

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

—

#107

—

#107

—

#106

—

188

—

#103

—

188

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

188

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×D₄₇

yes

characteristic

105

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

188

—

109

C₄⋊C₄

—

188

—

110

C₄²

D₄×D₄₇

yes

characteristic

111

C₁₆

—

112

C₁₆

#10625335

yes

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

376

—

116

C₂×C₄

—

117

C₂×C₄

—

376

—

118

C₂×C₄

#13094020

yes

characteristic

119

C₂×C₄

D₈×D₄₇

yes

characteristic

120

C₂×C₄

D₈×D₄₇

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

C₂×#10625335

yes

—

124

C₈

#13094043

yes

—

125

C₂²

#15689498

yes

characteristic

126

C₄

—

127

C₄

—

752

—

128

C₄

#15689495

yes

characteristic

129

C₄

#15689506

yes

characteristic

130

C₄

Q₃₂×D₄₇

yes

—

131

C₄

#15689556

yes

—

132

C₂

#20242977

yes

characteristic

133

C₂

#20245639

yes

characteristic

134

C₂

#20245650

yes

characteristic

	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	#12504
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
56. C₄₇⋊C₈	Q₃₂	semidirect
83. C₄₇	#12504	semidirect

H ≅ G/N

type

56. C₄₇⋊C₈

Q₃₂

semidirect

83. C₄₇

#12504

semidirect

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

subgroup	index
#20245663	2
#20245615	2
#20242982	2
C₁₆×C₄₇⋊C₈	2
C₄₇×#2247	2
#12504	47

subgroup

index

#20245663

#20245615

#20242982

C₁₆×C₄₇⋊C₈

C₄₇×#2247

#12504

Subgroups — #26130933

Conjugacy classes of subgroups

Complements

Maximal subgroups