Group #26131007 — subgroups

Subgroups — #26131007

Order 12032.

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

normal subgroups only

This group has 360 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	#20245637	C₂	1	2	yes	maximal, complemented
3	#20243003	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241305	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1709	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13093967	—	2	4	no	—
7	C₂×#13093965	—	2	4	no	—
8	#15689500	—	2	2	no	—
9	#15689500	C₂²	1	2	yes	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
11	C₄₇×#494	C₂²	1	1	yes	characteristic
12	C₄₇×#493	C₂²	1	1	yes	characteristic
13	C₄₇×#486	—	2	2	no	—
14	C₈²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×D₃₇₆	—	2	2	no	—
16	C₂×D₃₇₆	—	2	2	no	—
17	C₂×D₃₇₆	—	4	4	no	—
18	#13094021	—	2	2	no	—
19	#13093967	—	4	8	no	—
20	#13093965	—	4	8	no	—
21	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
22	C₂×Q₁₆×C₄₇	—	2	2	no	—
23	C₂×D₈×C₄₇	—	2	2	no	—
24	C₄⋊Q₈×C₄₇	—	2	1	no	—
25	#127×C₄₇	—	2	1	no	—
26	#102×C₄₇	—	4	4	no	—
27	C₄×C₈×C₄₇	D₄	1	2	yes	—
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₇	—	4	1	no	—
30	C₂×D₄×C₄₇	—	4	1	no	—
31	C₂×D₄×C₄₇	—	4	2	no	—
32	C₂×D₁₈₈	—	4	2	no	—
33	C₂×D₁₈₈	—	4	4	no	—
34	Q₁₆×C₄₇	—	4	4	no	—
35	D₈×C₄₇	—	4	4	no	—
36	C₂×C₈×C₄₇	D₈	1	4	yes	—
37	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₄₇	—	4	2	no	—
39	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
40	D₃₇₆	—	4	4	no	—
41	D₃₇₆	—	4	4	no	—
42	D₃₇₆	—	8	8	no	—
43	C₄₇⋊C₁₆	D₈	1	8	yes	complemented
44	C₂³×C₄₇	—	8	1	no	—
45	C₂²×D₄₇	—	8	2	no	—
46	Q₈×C₄₇	—	8	2	no	—
47	D₄×C₄₇	—	8	2	no	—
48	D₄×C₄₇	—	8	4	no	—
49	C₂×C₄×C₄₇	—	8	1	no	—
50	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
52	D₁₈₈	—	8	2	no	—
53	D₁₈₈	—	8	2	no	—
54	D₁₈₈	—	8	8	no	—
55	C₈×C₄₇	—	2	4	no	—
56	C₈×C₄₇	C₂×D₈	1	4	yes	—
57	#12519	—	47	1	no	maximal, Sylow
58	C₂²×C₄₇	—	16	1	no	—
59	C₂²×C₄₇	—	16	2	no	—
60	C₂²×C₄₇	#493	1	1	yes	characteristic
61	C₂×D₄₇	—	16	2	no	—
62	C₂×D₄₇	—	16	4	no	—
63	C₄×C₄₇	—	16	1	no	—
64	C₄×C₄₇	#493	1	2	yes	—
65	C₄×C₄₇	#509	1	4	yes	—
66	#2248	—	47	2	no	—
67	#2246	—	47	2	no	—
68	#1712	—	47	1	no	—
69	#1709	D₄₇	1	1	yes	characteristic, complemented
70	#1311	—	47	1	no	—
71	C₂×C₄₇	—	32	1	no	—
72	C₂×C₄₇	#1712	1	1	yes	characteristic
73	C₂×C₄₇	#2250	1	2	yes	—
74	D₄₇	—	32	2	no	—
75	C₂×SD₃₂	—	94	4	no	—
76	C₂×D₁₆	—	94	4	no	—
77	#494	C₂×D₄₇	1	1	yes	characteristic
78	#493	—	47	2	no	—
79	#493	—	94	2	no	—
80	#493	C₂×D₄₇	1	1	yes	characteristic
81	#486	—	2	2	no	—
82	C₄×C₁₆	—	47	2	no	—
83	C₈²	C₂×D₄₇	1	1	yes	characteristic
84	C₄₇	#12519	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₂×D₈	—	94	2	no	—
88	C₂×D₈	—	94	2	no	—
89	C₂×D₈	—	188	4	no	—
90	C₄⋊Q₈	—	2	1	no	—
91	#127	—	2	1	no	—
92	#127	—	94	2	no	—
93	SD₃₂	—	188	8	no	—
94	D₁₆	—	188	8	no	—
95	C₂×C₁₆	—	47	4	no	—
96	#102	—	4	4	no	—
97	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
98	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×Q₈	—	4	1	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	188	2	no	—
103	C₂×D₄	—	188	4	no	—
104	Q₁₆	—	4	4	no	—
105	D₈	—	4	4	no	—
106	D₈	—	376	8	no	—
107	D₈	—	188	4	no	—
108	D₈	—	188	4	no	—
109	C₂×C₈	#10625334	1	4	yes	—
110	C₂×C₈	D₄×D₄₇	1	2	yes	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
113	C₁₆	—	47	8	no	—
114	C₂³	—	8	1	no	—
115	C₂³	—	376	2	no	—
116	Q₈	—	8	2	no	—
117	D₄	—	8	2	no	—
118	D₄	—	8	4	no	—
119	D₄	—	376	2	no	—
120	D₄	—	376	2	no	—
121	D₄	—	376	8	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	#13094020	1	1	yes	characteristic
124	C₂×C₄	C₂×#10625334	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	D₈×D₄₇	1	4	yes	—
127	C₂²	—	16	1	no	—
128	C₂²	—	16	2	no	—
129	C₂²	—	752	2	no	—
130	C₂²	—	752	4	no	—
131	C₂²	#15689499	1	1	yes	characteristic
132	C₄	—	16	1	no	—
133	C₄	#15689498	1	2	yes	—
134	C₄	#15689558	1	4	yes	—
135	C₂	—	32	1	no	—
136	C₂	—	1504	2	no	—
137	C₂	#20243002	1	1	yes	characteristic
138	C₂	#20245655	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245645

C₂

yes

maximal, complemented

#20245637

C₂

yes

maximal, complemented

#20243003

C₂

yes

maximal, characteristic, complemented

#20241305

C₂

yes

maximal, characteristic, complemented

C₄₇×#1709

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#494

C₂²

yes

characteristic

C₄₇×#493

C₂²

yes

characteristic

C₄₇×#486

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

#13094021

—

#13093967

—

#13093965

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₂×Q₁₆×C₄₇

—

C₂×D₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

#127×C₄₇

—

#102×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

Q₁₆×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

D₃₇₆

—

D₃₇₆

—

D₃₇₆

—

C₄₇⋊C₁₆

D₈

yes

complemented

C₂³×C₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₈×C₄₇

C₂×D₈

yes

—

#12519

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#493

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#493

yes

—

C₄×C₄₇

#509

yes

—

—

—

—

D₄₇

yes

characteristic, complemented

#1311

—

C₂×C₄₇

—

C₂×C₄₇

#1712

yes

characteristic

C₂×C₄₇

#2250

yes

—

D₄₇

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12519

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

188

—

C₄⋊Q₈

—

#127

—

#127

—

SD₃₂

—

188

—

D₁₆

—

188

—

C₂×C₁₆

—

#102

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

188

—

103

C₂×D₄

—

188

—

104

Q₁₆

—

105

D₈

—

106

D₈

—

376

—

107

D₈

—

188

—

108

D₈

—

188

—

109

C₂×C₈

#10625334

yes

—

110

C₂×C₈

D₄×D₄₇

yes

—

111

C₄⋊C₄

—

112

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

113

C₁₆

—

114

C₂³

—

115

C₂³

—

376

—

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

376

—

120

D₄

—

376

—

121

D₄

—

376

—

122

C₂×C₄

—

123

C₂×C₄

#13094020

yes

characteristic

124

C₂×C₄

C₂×#10625334

yes

—

125

C₈

—

126

C₈

D₈×D₄₇

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

752

—

130

C₂²

—

752

—

131

C₂²

#15689499

yes

characteristic

132

C₄

—

133

C₄

#15689498

yes

—

134

C₄

#15689558

yes

—

135

C₂

—

136

C₂

—

1504

—

137

C₂

#20243002

yes

characteristic

138

C₂

#20245655

yes

—

	subgroup	G/H
center	C₂²	#15689499
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1709	C₂
socle	C₂²×C₄₇	#493
hypercenter	#1709	D₄₇
hypocenter	C₄₇	#12519
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#15689499

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1709

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #20245645	C₂	semidirect
2. #20245637	C₂	semidirect
3. #20243003	C₂	semidirect
4. #20241305	C₂	semidirect
5. C₄₇×#1709	C₂	semidirect
43. C₄₇⋊C₁₆	D₈	semidirect
69. #1709	D₄₇	semidirect
84. C₄₇	#12519	semidirect

H ≅ G/N

type

1. #20245645

C₂

semidirect

2. #20245637

C₂

semidirect

3. #20243003

C₂

semidirect

4. #20241305

C₂

semidirect

5. C₄₇×#1709

C₂

semidirect

43. C₄₇⋊C₁₆

D₈

semidirect

69. #1709

D₄₇

semidirect

84. C₄₇

#12519

semidirect

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

subgroup	index
#20245645	2
#20245637	2
#20243003	2
#20241305	2
C₄₇×#1709	2
#12519	47

subgroup

index

C₄₇×#1709

Subgroups — #26131007

Conjugacy classes of subgroups

Complements

Maximal subgroups