Group #26131002 — subgroups

Subgroups — #26131002

Order 12032.

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

normal subgroups only

This group has 330 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	#20245645	C₂	1	2	yes	maximal, complemented
2	#20245638	C₂	1	1	yes	maximal, characteristic
3	#20242995	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13093967	—	2	4	no	—
7	C₂×D₇₅₂	—	2	2	no	—
8	#15689500	C₂²	1	2	yes	—
9	#15689471	—	2	4	no	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₄₇	—	2	2	no	—
13	C₄₇×#494	C₂²	1	2	yes	—
14	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×D₃₇₆	—	2	2	no	—
16	C₂×D₃₇₆	—	2	2	no	—
17	#13094021	—	2	2	no	—
18	#13093967	—	4	8	no	—
19	D₇₅₂	—	4	4	no	—
20	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
21	#13093944	—	4	8	no	—
22	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
23	C₂×Q₁₆×C₄₇	—	2	2	no	—
24	C₂×Q₁₆×C₄₇	—	2	2	no	—
25	C₄⋊Q₈×C₄₇	—	2	2	no	—
26	Q₃₂×C₄₇	—	4	4	no	—
27	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₇	—	4	2	no	—
30	C₂×D₁₈₈	—	4	2	no	—
31	C₂×D₁₈₈	—	4	4	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	4	no	—
37	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄₇⋊C₈	D₈	1	4	yes	—
39	D₃₇₆	—	4	4	no	—
40	D₃₇₆	—	4	4	no	—
41	C₁₆×C₄₇	D₈	1	4	yes	—
42	C₄₇⋊C₁₆	D₈	1	4	yes	—
43	C₂²×D₄₇	—	8	2	no	—
44	Q₈×C₄₇	—	8	2	no	—
45	Q₈×C₄₇	—	8	2	no	—
46	C₂×C₄×C₄₇	—	8	2	no	—
47	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
49	D₁₈₈	—	8	2	no	—
50	D₁₈₈	—	8	2	no	—
51	D₁₈₈	—	8	8	no	—
52	C₈×C₄₇	C₂×D₈	1	2	yes	—
53	C₈×C₄₇	C₂×D₈	1	2	yes	—
54	C₄₇⋊C₈	—	2	4	no	—
55	#12518	—	47	1	no	maximal, Sylow
56	C₂²×C₄₇	#493	1	1	yes	characteristic
57	C₂×D₄₇	—	16	2	no	—
58	C₂×D₄₇	—	16	4	no	—
59	C₄×C₄₇	—	16	2	no	—
60	C₄×C₄₇	#493	1	1	yes	characteristic
61	C₄×C₄₇	#493	1	1	yes	characteristic
62	C₄×C₄₇	#508	1	4	yes	—
63	#2248	—	47	2	no	—
64	#2247	D₄₇	1	1	yes	characteristic, complemented
65	#2246	—	47	1	no	—
66	#1709	—	47	2	no	—
67	C₈×C₁₆	—	47	1	no	—
68	C₂×C₄₇	#1712	1	1	yes	characteristic
69	C₂×C₄₇	#2249	1	2	yes	—
70	D₄₇	—	32	2	no	—
71	C₂×Q₃₂	—	2	2	no	—
72	C₂×SD₃₂	—	94	4	no	—
73	C₂×D₁₆	—	94	2	no	—
74	#494	C₂×D₄₇	1	2	yes	—
75	#493	—	47	2	no	—
76	#486	—	94	4	no	—
77	C₄×C₁₆	—	47	1	no	—
78	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
79	C₈²	—	47	1	no	—
80	C₄₇	#12518	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×D₈	—	94	2	no	—
84	C₂×D₈	—	94	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	#127	—	94	2	no	—
87	Q₃₂	—	4	4	no	—
88	SD₃₂	—	188	8	no	—
89	D₁₆	—	188	4	no	—
90	C₂×C₁₆	—	47	2	no	—
91	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
92	#102	—	188	8	no	—
93	C₄×C₈	—	47	2	no	—
94	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	188	2	no	—
97	C₂×D₄	—	188	4	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	D₈	—	188	4	no	—
101	D₈	—	188	4	no	—
102	C₂×C₈	—	47	4	no	—
103	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
104	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
105	C₄⋊C₄	—	4	4	no	—
106	C₄²	D₄×D₄₇	1	1	yes	characteristic
107	C₁₆	—	47	4	no	—
108	C₁₆	#10625334	1	4	yes	—
109	C₂³	—	376	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	376	2	no	—
113	D₄	—	376	2	no	—
114	D₄	—	376	8	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	#13094020	1	1	yes	characteristic
117	C₂×C₄	D₈×D₄₇	1	2	yes	—
118	C₈	—	94	4	no	—
119	C₈	D₈×D₄₇	1	2	yes	—
120	C₈	C₂×#10625334	1	2	yes	—
121	C₂²	—	752	2	no	—
122	C₂²	—	752	4	no	—
123	C₂²	#15689498	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	#15689498	1	1	yes	characteristic
126	C₄	#15689499	1	1	yes	characteristic
127	C₄	#15689556	1	4	yes	—
128	C₂	—	1504	2	no	—
129	C₂	#20243002	1	1	yes	characteristic
130	C₂	#20245650	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245645

C₂

yes

maximal, complemented

#20245638

C₂

yes

maximal, characteristic

#20242995

C₂

yes

maximal, complemented

C₁₆×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄₇×#2247

C₂

yes

maximal, characteristic, complemented

C₂×#13093967

—

C₂×D₇₅₂

—

#15689500

C₂²

yes

—

#15689471

—

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

C₂×D₃₇₆

—

C₂×D₃₇₆

—

#13094021

—

#13093967

—

D₇₅₂

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093944

—

C₄×C₄₇⋊C₈

D₄

yes

—

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×D₁₈₈

—

C₂×D₁₈₈

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₈

yes

—

D₃₇₆

—

D₃₇₆

—

C₁₆×C₄₇

D₈

yes

—

C₄₇⋊C₁₆

D₈

yes

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

C₂×D₈

yes

—

C₈×C₄₇

C₂×D₈

yes

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

#2248

—

#2247

D₄₇

yes

characteristic, complemented

#2246

—

#1709

—

C₈×C₁₆

—

C₂×C₄₇

#1712

yes

characteristic

C₂×C₄₇

#2249

yes

—

D₄₇

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

C₂×D₄₇

yes

—

#493

—

#486

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₈²

—

C₄₇

#12518

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

188

—

D₁₆

—

188

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

—

#102

—

188

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

Q₁₆

—

Q₁₆

—

100

D₈

—

188

—

101

D₈

—

188

—

102

C₂×C₈

—

103

C₂×C₈

D₄×D₄₇

yes

characteristic

104

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

D₄×D₄₇

yes

characteristic

107

C₁₆

—

108

C₁₆

#10625334

yes

—

109

C₂³

—

376

—

110

Q₈

—

111

Q₈

—

112

D₄

—

376

—

113

D₄

—

376

—

114

D₄

—

376

—

115

C₂×C₄

—

116

C₂×C₄

#13094020

yes

characteristic

117

C₂×C₄

D₈×D₄₇

yes

—

118

C₈

—

119

C₈

D₈×D₄₇

yes

—

120

C₈

C₂×#10625334

yes

—

121

C₂²

—

752

—

122

C₂²

—

752

—

123

C₂²

#15689498

yes

characteristic

124

C₄

—

125

C₄

#15689498

yes

characteristic

126

C₄

#15689499

yes

characteristic

127

C₄

#15689556

yes

—

128

C₂

—

1504

—

129

C₂

#20243002

yes

characteristic

130

C₂

#20245650

yes

—

	subgroup	G/H
center	C₂²	#15689498
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2247	C₂
socle	C₂²×C₄₇	#493
prime radical	#20245638	C₂
hypercenter	#2247	D₄₇
hypocenter	C₄₇	#12518
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

C₂

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
1. #20245645	C₂	semidirect
3. #20242995	C₂	semidirect
4. C₁₆×C₄₇⋊C₈	C₂	semidirect
5. C₄₇×#2247	C₂	semidirect
64. #2247	D₄₇	semidirect
80. C₄₇	#12518	semidirect

H ≅ G/N

type

1. #20245645

C₂

semidirect

3. #20242995

C₂

semidirect

4. C₁₆×C₄₇⋊C₈

C₂

semidirect

5. C₄₇×#2247

C₂

semidirect

64. #2247

D₄₇

semidirect

80. C₄₇

#12518

semidirect

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

subgroup	index
#20245645	2
#20245638	2
#20242995	2
C₁₆×C₄₇⋊C₈	2
C₄₇×#2247	2
#12518	47

subgroup

index

#20245645

#20245638

#20242995

C₁₆×C₄₇⋊C₈

C₄₇×#2247

#12518

Subgroups — #26131002

Conjugacy classes of subgroups

Complements

Maximal subgroups