Group #26130953 — subgroups

Subgroups — #26130953

Order 12032.

258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20245670	C₂	1	1	yes	maximal, characteristic
2	#20245641	C₂	1	1	yes	maximal, characteristic
3	#20245616	C₂	1	2	yes	maximal
4	#20242985	C₂	1	2	yes	maximal
5	C₄₇×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₇₆	—	2	2	no	—
7	#15689528	C₂²	1	2	yes	—
8	#15689521	—	2	2	no	—
9	#15689503	C₂²	1	2	yes	—
10	#15689474	—	2	2	no	—
11	#15688951	—	2	2	no	—
12	#15688928	—	2	4	no	—
13	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₈²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₈₈	—	2	2	no	—
17	C₂×Dic₁₈₈	—	2	2	no	—
18	#13094024	—	2	2	no	—
19	#13094024	—	2	2	no	—
20	Dic₃₇₆	—	4	4	no	—
21	#13093958	—	2	2	no	—
22	#13093958	—	2	2	no	—
23	#13093956	—	4	4	no	—
24	#13093949	—	4	4	no	—
25	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
26	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
27	C₄×C₈×C₄₇	D₄	1	2	yes	—
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₉₄	—	4	2	no	—
30	C₂×Dic₉₄	—	4	2	no	—
31	C₂×C₈×C₄₇	D₈	1	4	yes	—
32	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₄₇	—	4	2	no	—
36	C₄⋊Dic₄₇	—	4	2	no	—
37	C₄⋊Dic₄₇	—	4	2	no	—
38	C₄⋊Dic₄₇	—	4	2	no	—
39	Dic₁₈₈	—	4	4	no	—
40	Dic₁₈₈	—	4	4	no	—
41	C₁₆×C₄₇	—	2	2	no	—
42	C₁₆×C₄₇	SD₁₆	1	4	yes	—
43	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
45	C₂×Dic₄₇	—	8	2	no	—
46	C₂×Dic₄₇	—	8	2	no	—
47	Dic₉₄	—	8	2	no	—
48	Dic₉₄	—	8	2	no	—
49	Dic₉₄	—	8	4	no	—
50	C₈×C₄₇	—	2	4	no	—
51	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
52	C₈×C₄₇	#135	1	2	yes	—
53	#12507	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	#493	1	1	yes	characteristic
55	C₄×C₄₇	#492	1	1	yes	characteristic
56	C₄×C₄₇	#495	1	1	yes	characteristic
57	C₄×C₄₇	#510	1	4	yes	—
58	Dic₄₇	—	16	2	no	—
59	Dic₄₇	—	16	2	no	—
60	#2254	—	47	1	no	—
61	#2247	—	47	1	no	—
62	#2241	—	47	2	no	—
63	#1707	—	47	2	no	—
64	#1311	D₄₇	1	1	yes	characteristic
65	C₂×C₄₇	#1706	1	1	yes	characteristic
66	C₂×C₄₇	#2251	1	2	yes	—
67	C₂×Q₃₂	—	94	2	no	—
68	#500	—	47	2	no	—
69	#498	—	94	2	no	—
70	#494	—	47	2	no	—
71	#487	—	94	2	no	—
72	#367	—	94	2	no	—
73	#358	—	94	4	no	—
74	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
75	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
76	C₈²	C₂×D₄₇	1	1	yes	characteristic
77	C₄₇	#12507	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	94	2	no	—
79	C₂×Q₁₆	—	94	2	no	—
80	C₄⋊Q₈	—	94	2	no	—
81	C₄⋊Q₈	—	94	2	no	—
82	Q₃₂	—	188	4	no	—
83	C₂×C₁₆	D₁₈₈	1	2	yes	—
84	C₂×C₁₆	D₁₈₈	1	2	yes	—
85	#107	—	94	2	no	—
86	#107	—	94	2	no	—
87	#106	—	188	4	no	—
88	#103	—	188	4	no	—
89	C₄×C₈	D₁₈₈	1	2	yes	—
90	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
91	C₂×Q₈	—	188	2	no	—
92	C₂×Q₈	—	188	2	no	—
93	Q₁₆	—	188	4	no	—
94	Q₁₆	—	188	4	no	—
95	C₂×C₈	D₃₇₆	1	4	yes	—
96	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
97	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
98	C₄⋊C₄	—	188	2	no	—
99	C₄⋊C₄	—	188	2	no	—
100	C₄⋊C₄	—	188	2	no	—
101	C₄⋊C₄	—	188	2	no	—
102	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	#10625325	1	4	yes	—
105	Q₈	—	376	2	no	—
106	Q₈	—	376	2	no	—
107	Q₈	—	376	4	no	—
108	C₂×C₄	—	376	2	no	—
109	C₂×C₄	—	376	2	no	—
110	C₂×C₄	#13094021	1	1	yes	characteristic
111	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	C₂×#10625325	1	2	yes	—
114	C₈	#13094046	1	2	yes	—
115	C₂²	#15689500	1	1	yes	characteristic
116	C₄	—	752	2	no	—
117	C₄	—	752	2	no	—
118	C₄	#15689496	1	1	yes	characteristic
119	C₄	#15689507	1	1	yes	characteristic
120	C₄	#15689566	1	4	yes	—
121	C₂	#20242980	1	1	yes	characteristic
122	C₂	#20245659	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245670

C₂

yes

maximal, characteristic

#20245641

C₂

yes

maximal, characteristic

#20245616

C₂

yes

maximal

#20242985

C₂

yes

maximal

C₄₇×#1311

C₂

yes

maximal, characteristic

C₂×Dic₃₇₆

—

#15689528

C₂²

yes

—

#15689521

—

#15689503

C₂²

yes

—

#15689474

—

#15688951

—

#15688928

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

—

#13094024

—

Dic₃₇₆

—

—

—

—

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂×SD₁₆

yes

—

C₈×C₄₇

#135

yes

—

—

C₂²×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×Q₃₂

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12507

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

188

—

C₂×C₁₆

D₁₈₈

yes

—

C₂×C₁₆

D₁₈₈

yes

—

#107

—

#107

—

#106

—

188

—

#103

—

188

—

C₄×C₈

D₁₈₈

yes

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

188

—

C₂×Q₈

—

188

—

Q₁₆

—

188

—

Q₁₆

—

188

—

C₂×C₈

D₃₇₆

yes

—

C₂×C₈

C₂×D₁₈₈

yes

characteristic

C₂×C₈

C₂×D₁₈₈

yes

characteristic

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

100

C₄⋊C₄

—

188

—

101

C₄⋊C₄

—

188

—

102

C₄²

C₂×D₁₈₈

yes

characteristic

103

C₁₆

—

104

C₁₆

#10625325

yes

—

105

Q₈

—

376

—

106

Q₈

—

376

—

107

Q₈

—

376

—

108

C₂×C₄

—

376

—

109

C₂×C₄

—

376

—

110

C₂×C₄

#13094021

yes

characteristic

111

C₂×C₄

C₂×D₃₇₆

yes

—

112

C₈

—

113

C₈

C₂×#10625325

yes

—

114

C₈

#13094046

yes

—

115

C₂²

#15689500

yes

characteristic

116

C₄

—

752

—

117

C₄

—

752

—

118

C₄

#15689496

yes

characteristic

119

C₄

#15689507

yes

characteristic

120

C₄

#15689566

yes

—

121

C₂

#20242980

yes

characteristic

122

C₂

#20245659

yes

—

	subgroup	G/H
center	C₂²	#15689500
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1311	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	#1311	D₄₇
hypocenter	C₄₇	#12507
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#15689500

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1311

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
77. C₄₇	#12507	semidirect

H ≅ G/N

type

77. C₄₇

#12507

semidirect

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

subgroup	index
#20245670	2
#20245641	2
#20245616	2
#20242985	2
C₄₇×#1311	2
#12507	47

subgroup

index

C₄₇×#1311

Subgroups — #26130953

Conjugacy classes of subgroups

Complements

Maximal subgroups