Group #26110034 — subgroups

Subgroups — #26110034

Order 12032.

402 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 211 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	#20243540	C₂	1	2	yes	maximal
2	C₂×#15688876	C₂	1	4	yes	maximal
3	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
6	#15688876	—	2	8	no	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
8	C₄³×C₄₇	C₂²	1	1	yes	characteristic
9	C₄₇×#336	C₄	1	4	yes	—
10	C₂²×C₄₇⋊C₈	—	2	4	no	—
11	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
12	C₄⋊(C₄₇⋊C₈)	—	4	8	no	—
13	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₄₇	D₄	1	2	yes	—
16	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
17	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₄₇	D₄	1	4	yes	—
19	C₄×C₈×C₄₇	Q₈	1	4	yes	—
20	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
21	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₈×C₄₇	—	2	8	no	—
25	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
26	C₄²×C₄₇	—	2	4	no	—
27	C₄²×C₄₇	—	4	2	no	—
28	C₄²×C₄₇	D₈	1	4	yes	—
29	C₄²×C₄₇	Q₁₆	1	4	yes	—
30	C₄²×C₄₇	C₂×D₄	1	2	yes	—
31	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
32	C₂×C₄₇⋊C₈	—	4	4	no	—
33	C₂×C₄₇⋊C₈	—	4	8	no	—
34	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	—	2	4	no	—
36	C₂×C₄×C₄₇	—	2	4	no	—
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	#107	1	8	yes	—
39	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
42	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₄×C₄₇	#131	1	4	yes	—
44	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
46	C₈×C₄₇	—	2	8	no	—
47	C₄₇⋊C₈	—	8	4	no	—
48	#8810	—	47	1	no	maximal, Sylow
49	C₂²×C₄₇	#384	1	1	yes	characteristic
50	C₂²×C₄₇	#424	1	2	yes	—
51	C₂²×C₄₇	C₂×#107	1	2	yes	—
52	C₂²×C₄₇	#430	1	2	yes	—
53	C₄×C₄₇	—	2	8	no	—
54	C₄×C₄₇	—	2	2	no	—
55	C₄×C₄₇	#424	1	4	yes	—
56	C₄×C₄₇	#493	1	1	yes	characteristic
57	C₄×C₄₇	#494	1	1	yes	characteristic
58	C₄×C₄₇	#500	1	2	yes	—
59	#1842	—	47	2	no	—
60	#1744	D₄₇	1	1	yes	characteristic
61	C₂×#335	—	47	4	no	—
62	C₂×C₄₇	#1571	1	4	yes	—
63	C₂×C₄₇	#1843	1	1	yes	characteristic
64	C₂×C₄₇	#1845	1	1	yes	characteristic
65	C₂×C₄₇	#1849	1	1	yes	characteristic
66	C₂×C₄⋊C₈	—	47	4	no	—
67	C₂×C₄⋊C₈	—	94	4	no	—
68	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
69	C₄³	C₂×D₄₇	1	1	yes	characteristic
70	#336	Dic₄₇	1	4	yes	—
71	#335	—	94	8	no	—
72	C₄₇	#8810	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	94	4	no	—
74	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
75	C₂×C₄²	Dic₉₄	1	2	yes	—
76	C₂×C₄²	D₁₈₈	1	2	yes	—
77	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
78	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
79	C₄⋊C₈	—	94	8	no	—
80	C₄⋊C₈	—	188	8	no	—
81	C₄×C₈	Dic₉₄	1	4	yes	—
82	C₄×C₈	D₁₈₈	1	4	yes	—
83	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
84	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
86	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
87	C₂×C₈	—	2	8	no	—
88	C₂×C₈	—	188	4	no	—
89	C₂×C₈	—	188	8	no	—
90	C₂×C₈	C₄⋊Dic₄₇	1	8	yes	—
91	C₄²	—	2	4	no	—
92	C₄²	—	4	2	no	—
93	C₄²	D₃₇₆	1	4	yes	—
94	C₄²	Dic₁₈₈	1	4	yes	—
95	C₄²	C₂×Dic₉₄	1	2	yes	—
96	C₄²	C₂×D₁₈₈	1	2	yes	—
97	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	#13093958	1	8	yes	—
102	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
103	C₂×C₄	#13094021	1	1	yes	characteristic
104	C₂×C₄	#13094024	1	2	yes	—
105	C₂×C₄	#13094024	1	1	yes	characteristic
106	C₂×C₄	#13094031	1	4	yes	—
107	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
108	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
109	C₈	—	2	8	no	—
110	C₈	—	376	4	no	—
111	C₂²	#15688993	1	1	yes	characteristic
112	C₂²	#15689150	1	2	yes	—
113	C₂²	C₂×#13093958	1	2	yes	—
114	C₂²	#15689172	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	—	2	8	no	—
117	C₄	#15689150	1	4	yes	—
118	C₄	#15689500	1	1	yes	characteristic
119	C₄	#15689503	1	1	yes	characteristic
120	C₄	#15689528	1	2	yes	—
121	C₂	#20242163	1	4	yes	—
122	C₂	#20243542	1	1	yes	characteristic
123	C₂	#20243548	1	1	yes	characteristic
124	C₂	#20243563	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243540

C₂

yes

maximal

C₂×#15688876

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#15688876

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

C₄

yes

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₈

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#107

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×#107

yes

—

C₂²×C₄₇

#430

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#424

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

#1842

—

#1744

D₄₇

yes

characteristic

C₂×#335

—

C₂×C₄₇

#1571

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₄³

C₂×D₄₇

yes

characteristic

#336

Dic₄₇

yes

—

#335

—

C₄₇

#8810

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂×C₄²

Dic₉₄

yes

—

C₂×C₄²

D₁₈₈

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

188

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

D₁₈₈

yes

—

C₂²×C₄

C₄⋊Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₄²

—

C₄²

—

C₄²

D₃₇₆

yes

—

C₄²

Dic₁₈₈

yes

—

C₄²

C₂×Dic₉₄

yes

—

C₄²

C₂×D₁₈₈

yes

—

C₂³

C₂×C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#13093958

yes

—

102

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

—

103

C₂×C₄

#13094021

yes

characteristic

104

C₂×C₄

#13094024

yes

—

105

C₂×C₄

#13094024

yes

characteristic

106

C₂×C₄

#13094031

yes

—

107

C₂×C₄

C₂×D₃₇₆

yes

—

108

C₂×C₄

C₂×Dic₁₈₈

yes

—

109

C₈

—

110

C₈

—

376

—

111

C₂²

#15688993

yes

characteristic

112

C₂²

#15689150

yes

—

113

C₂²

C₂×#13093958

yes

—

114

C₂²

#15689172

yes

—

115

C₄

—

116

C₄

—

117

C₄

#15689150

yes

—

118

C₄

#15689500

yes

characteristic

119

C₄

#15689503

yes

characteristic

120

C₄

#15689528

yes

—

121

C₂

#20242163

yes

—

122

C₂

#20243542

yes

characteristic

123

C₂

#20243548

yes

characteristic

124

C₂

#20243563

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#8810
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₁₈₈

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1744

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
72. C₄₇	#8810	semidirect

H ≅ G/N

type

72. C₄₇

#8810

semidirect

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

subgroup	index
#20243540	2
C₂×#15688876	2
C₄₇×#1744	2
#8810	47

subgroup

index

#20243540

C₂×#15688876

C₄₇×#1744

#8810

Subgroups — #26110034

Conjugacy classes of subgroups

Complements

Maximal subgroups