Group #26131006 — subgroups

Subgroups — #26131006

Order 12032.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20245644	C₂	1	2	yes	maximal, complemented
2	#20245638	C₂	1	2	yes	maximal, complemented
3	#20243003	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13093969	—	2	4	no	—
7	C₂×D₇₅₂	—	2	4	no	—
8	#15689503	C₂²	1	1	yes	characteristic
9	#15689500	—	2	2	no	—
10	#15689500	C₂²	1	1	yes	characteristic
11	#15689500	C₂²	1	2	yes	—
12	#15689470	—	2	2	no	—
13	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
14	C₈²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₈₈	—	2	2	no	—
16	C₂×D₃₇₆	—	2	2	no	—
17	C₂×D₃₇₆	—	2	2	no	—
18	C₂×D₃₇₆	—	2	2	no	—
19	C₂×D₃₇₆	—	4	4	no	—
20	#13094024	—	2	1	no	—
21	#13094021	—	2	2	no	—
22	#13094021	—	2	1	no	—
23	#13093969	—	4	8	no	—
24	D₇₅₂	—	4	8	no	—
25	#13093946	—	4	4	no	—
26	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
27	C₄×C₈×C₄₇	D₄	1	2	yes	—
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×D₁₈₈	—	4	2	no	—
30	C₂×D₁₈₈	—	4	1	no	—
31	C₂×D₁₈₈	—	4	2	no	—
32	C₂×D₁₈₈	—	4	4	no	—
33	C₂×Dic₉₄	—	4	1	no	—
34	C₂×C₈×C₄₇	D₈	1	4	yes	—
35	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
36	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₄₇	—	4	2	no	—
38	Dic₁₈₈	—	4	4	no	—
39	D₃₇₆	—	4	4	no	—
40	D₃₇₆	—	4	4	no	—
41	D₃₇₆	—	4	4	no	—
42	D₃₇₆	—	8	8	no	—
43	C₁₆×C₄₇	D₈	1	8	yes	complemented
44	C₂²×D₄₇	—	8	1	no	—
45	C₂²×D₄₇	—	8	2	no	—
46	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
48	C₂×Dic₄₇	—	8	1	no	—
49	D₁₈₈	—	8	2	no	—
50	D₁₈₈	—	8	2	no	—
51	D₁₈₈	—	8	2	no	—
52	D₁₈₈	—	8	8	no	—
53	D₁₈₈	—	8	4	no	—
54	Dic₉₄	—	8	2	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	#493	1	1	yes	characteristic
59	C₂×D₄₇	—	16	1	no	—
60	C₂×D₄₇	—	16	2	no	—
61	C₂×D₄₇	—	16	4	no	—
62	C₂×D₄₇	—	16	2	no	—
63	C₄×C₄₇	#493	1	2	yes	—
64	C₄×C₄₇	#509	1	4	yes	—
65	Dic₄₇	—	16	1	no	—
66	#2248	—	47	2	no	—
67	#2246	—	47	2	no	—
68	#1712	—	47	1	no	—
69	#1709	—	47	1	no	—
70	#1311	D₄₇	1	1	yes	characteristic, complemented
71	C₂×C₄₇	#1712	1	1	yes	characteristic
72	C₂×C₄₇	#2250	1	2	yes	—
73	D₄₇	—	32	2	no	—
74	D₄₇	—	32	1	no	—
75	C₂×SD₃₂	—	94	4	no	—
76	C₂×D₁₆	—	94	4	no	—
77	#494	—	47	1	no	—
78	#493	—	47	1	no	—
79	#493	—	47	2	no	—
80	#493	—	94	2	no	—
81	#486	—	94	2	no	—
82	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
83	C₈²	C₂×D₄₇	1	1	yes	characteristic
84	C₄₇	#12519	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	94	2	no	—
86	C₂×D₈	—	94	2	no	—
87	C₂×D₈	—	94	2	no	—
88	C₂×D₈	—	94	2	no	—
89	C₂×D₈	—	188	4	no	—
90	C₄⋊Q₈	—	94	1	no	—
91	#127	—	94	1	no	—
92	#127	—	94	2	no	—
93	SD₃₂	—	188	8	no	—
94	D₁₆	—	188	8	no	—
95	C₂×C₁₆	D₁₈₈	1	4	yes	—
96	#102	—	188	4	no	—
97	C₄×C₈	D₁₈₈	1	2	yes	—
98	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×Q₈	—	188	1	no	—
100	C₂×D₄	—	188	1	no	—
101	C₂×D₄	—	188	2	no	—
102	C₂×D₄	—	188	2	no	—
103	C₂×D₄	—	188	4	no	—
104	Q₁₆	—	188	4	no	—
105	D₈	—	376	8	no	—
106	D₈	—	188	4	no	—
107	D₈	—	188	4	no	—
108	D₈	—	188	4	no	—
109	C₂×C₈	D₃₇₆	1	4	yes	—
110	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
111	C₄⋊C₄	—	188	2	no	—
112	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
113	C₁₆	D₃₇₆	1	8	yes	complemented
114	C₂³	—	376	1	no	—
115	C₂³	—	376	2	no	—
116	Q₈	—	376	2	no	—
117	D₄	—	376	2	no	—
118	D₄	—	376	2	no	—
119	D₄	—	376	2	no	—
120	D₄	—	376	8	no	—
121	D₄	—	376	4	no	—
122	C₂×C₄	—	376	1	no	—
123	C₂×C₄	#13094021	1	1	yes	characteristic
124	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	C₂×D₃₇₆	1	4	yes	—
127	C₂²	—	752	1	no	—
128	C₂²	—	752	2	no	—
129	C₂²	—	752	4	no	—
130	C₂²	—	752	2	no	—
131	C₂²	#15689500	1	1	yes	characteristic
132	C₄	—	752	1	no	—
133	C₄	#15689500	1	2	yes	—
134	C₄	#15689561	1	4	yes	—
135	C₂	—	1504	2	no	—
136	C₂	—	1504	1	no	—
137	C₂	#20243003	1	1	yes	characteristic
138	C₂	#20245654	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245644

C₂

yes

maximal, complemented

#20245638

C₂

yes

maximal, complemented

#20243003

C₂

yes

maximal, characteristic, complemented

#20242994

C₂

yes

maximal, characteristic, complemented

C₄₇×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#13093969

—

C₂×D₇₅₂

—

#15689503

C₂²

yes

characteristic

#15689500

—

#15689500

C₂²

yes

characteristic

#15689500

C₂²

yes

—

#15689470

—

C₄×C₁₆×C₄₇

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

—

—

—

—

D₇₅₂

—

#13093946

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

Dic₁₈₈

—

D₃₇₆

—

D₃₇₆

—

D₃₇₆

—

D₃₇₆

—

C₁₆×C₄₇

D₈

yes

complemented

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂×D₈

yes

—

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#493

yes

—

C₄×C₄₇

#509

yes

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×C₄₇

#1712

yes

characteristic

C₂×C₄₇

#2250

yes

—

D₄₇

—

D₄₇

—

C₂×SD₃₂

—

C₂×D₁₆

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₇

yes

—

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

D₁₈₈

yes

—

#102

—

188

—

C₄×C₈

D₁₈₈

yes

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

188

—

100

C₂×D₄

—

188

—

101

C₂×D₄

—

188

—

102

C₂×D₄

—

188

—

103

C₂×D₄

—

188

—

104

Q₁₆

—

188

—

105

D₈

—

376

—

106

D₈

—

188

—

107

D₈

—

188

—

108

D₈

—

188

—

109

C₂×C₈

D₃₇₆

yes

—

110

C₂×C₈

C₂×D₁₈₈

yes

—

111

C₄⋊C₄

—

188

—

112

C₄²

C₂×D₁₈₈

yes

characteristic

113

C₁₆

D₃₇₆

yes

complemented

114

C₂³

—

376

—

115

C₂³

—

376

—

116

Q₈

—

376

—

117

D₄

—

376

—

118

D₄

—

376

—

119

D₄

—

376

—

120

D₄

—

376

—

121

D₄

—

376

—

122

C₂×C₄

—

376

—

123

C₂×C₄

#13094021

yes

characteristic

124

C₂×C₄

C₂×D₃₇₆

yes

—

125

C₈

—

126

C₈

C₂×D₃₇₆

yes

—

127

C₂²

—

752

—

128

C₂²

—

752

—

129

C₂²

—

752

—

130

C₂²

—

752

—

131

C₂²

#15689500

yes

characteristic

132

C₄

—

752

—

133

C₄

#15689500

yes

—

134

C₄

#15689561

yes

—

135

C₂

—

1504

—

136

C₂

—

1504

—

137

C₂

#20243003

yes

characteristic

138

C₂

#20245654

yes

—

	subgroup	G/H
center	C₂²	#15689500
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1311	C₂
socle	C₂²×C₄₇	#493
hypercenter	#1311	D₄₇
hypocenter	C₄₇	#12519
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
1. #20245644	C₂	semidirect
2. #20245638	C₂	semidirect
3. #20243003	C₂	semidirect
4. #20242994	C₂	semidirect
5. C₄₇×#1311	C₂	semidirect
43. C₁₆×C₄₇	D₈	semidirect
70. #1311	D₄₇	semidirect
84. C₄₇	#12519	semidirect
113. C₁₆	D₃₇₆	semidirect

H ≅ G/N

type

1. #20245644

C₂

semidirect

2. #20245638

C₂

semidirect

3. #20243003

C₂

semidirect

4. #20242994

C₂

semidirect

5. C₄₇×#1311

C₂

semidirect

43. C₁₆×C₄₇

D₈

semidirect

70. #1311

D₄₇

semidirect

84. C₄₇

#12519

semidirect

113. C₁₆

D₃₇₆

semidirect

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

subgroup	index
#20245644	2
#20245638	2
#20243003	2
#20242994	2
C₄₇×#1311	2
#12519	47

subgroup

index

C₄₇×#1311

Subgroups — #26131006

Conjugacy classes of subgroups

Complements

Maximal subgroups