Group #26131026 — subgroups

Subgroups — #26131026

Order 12032.

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

normal subgroups only

This group has 246 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	#20245670	C₂	1	2	yes	maximal
2	#20245663	C₂	1	2	yes	maximal
3	#20243006	C₂	1	1	yes	maximal, characteristic
4	#20242998	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1311	C₂	1	1	yes	maximal, characteristic
6	#15689528	—	2	1	no	—
7	#15689528	C₂²	1	1	yes	characteristic
8	#15689528	C₂²	1	1	yes	characteristic
9	#15689528	C₂²	1	2	yes	—
10	#15689503	—	2	1	no	—
11	#15689474	—	2	1	no	—
12	#15689474	—	2	1	no	—
13	#15688951	—	2	4	no	—
14	#15688949	—	2	4	no	—
15	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
16	C₈²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	—	4	2	no	—
18	#13094024	—	2	1	no	—
19	#13094024	—	2	1	no	—
20	#13094024	—	2	2	no	—
21	#13093958	—	2	2	no	—
22	#13093958	—	2	2	no	—
23	#13093958	—	2	2	no	—
24	#13093958	—	2	2	no	—
25	#13093958	—	4	2	no	—
26	#13093949	—	4	2	no	—
27	#13093949	—	4	2	no	—
28	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
29	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₉₄	—	4	1	no	—
33	C₂×Dic₉₄	—	4	1	no	—
34	C₂×Dic₉₄	—	4	2	no	—
35	C₂×C₈×C₄₇	D₈	1	2	yes	—
36	C₂×C₈×C₄₇	D₈	1	2	yes	—
37	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
38	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₄₇	—	4	1	no	—
40	C₄⋊Dic₄₇	—	4	2	no	—
41	C₄⋊Dic₄₇	—	4	1	no	—
42	C₄⋊Dic₄₇	—	4	1	no	—
43	C₄⋊Dic₄₇	—	4	1	no	—
44	C₄⋊Dic₄₇	—	4	2	no	—
45	Dic₁₈₈	—	8	4	no	—
46	C₁₆×C₄₇	Q₁₆	1	8	yes	complemented
47	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₄₇	—	8	1	no	—
51	C₂×Dic₄₇	—	8	2	no	—
52	C₂×Dic₄₇	—	8	1	no	—
53	Dic₉₄	—	8	4	no	—
54	Dic₉₄	—	8	2	no	—
55	Dic₉₄	—	8	2	no	—
56	C₈×C₄₇	—	2	2	no	—
57	C₈×C₄₇	—	2	2	no	—
58	C₈×C₄₇	C₂×Q₁₆	1	4	yes	—
59	#12524	—	47	1	no	maximal, Sylow
60	C₂²×C₄₇	#493	1	1	yes	characteristic
61	C₄×C₄₇	#494	1	2	yes	—
62	C₄×C₄₇	#509	1	2	yes	—
63	C₄×C₄₇	#510	1	2	yes	—
64	Dic₄₇	—	16	2	no	—
65	Dic₄₇	—	16	1	no	—
66	Dic₄₇	—	16	1	no	—
67	#2254	—	47	2	no	—
68	#2252	—	47	2	no	—
69	#1713	—	47	1	no	—
70	#1710	—	47	1	no	—
71	#1311	D₄₇	1	1	yes	characteristic
72	C₂×C₄₇	#1713	1	1	yes	characteristic
73	C₂×C₄₇	#2250	1	1	yes	characteristic
74	C₂×C₄₇	#2251	1	1	yes	characteristic
75	#500	—	47	1	no	—
76	#500	—	47	1	no	—
77	#500	—	47	2	no	—
78	#500	—	94	1	no	—
79	#494	—	94	1	no	—
80	#487	—	94	1	no	—
81	#487	—	94	1	no	—
82	#367	—	94	4	no	—
83	#366	—	94	4	no	—
84	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
85	C₈²	C₂×D₄₇	1	1	yes	characteristic
86	C₄₇	#12524	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	—	188	2	no	—
88	C₄⋊Q₈	—	94	1	no	—
89	C₄⋊Q₈	—	94	1	no	—
90	C₄⋊Q₈	—	94	2	no	—
91	C₂×C₁₆	D₁₈₈	1	4	yes	—
92	#107	—	94	2	no	—
93	#107	—	94	2	no	—
94	#107	—	94	2	no	—
95	#107	—	94	2	no	—
96	#107	—	188	2	no	—
97	#103	—	188	2	no	—
98	#103	—	188	2	no	—
99	C₄×C₈	D₁₈₈	1	1	yes	characteristic
100	C₄×C₈	D₁₈₈	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
102	C₂×Q₈	—	188	1	no	—
103	C₂×Q₈	—	188	1	no	—
104	C₂×Q₈	—	188	2	no	—
105	Q₁₆	—	376	4	no	—
106	C₂×C₈	D₃₇₆	1	2	yes	—
107	C₂×C₈	D₃₇₆	1	2	yes	—
108	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
109	C₄⋊C₄	—	188	1	no	—
110	C₄⋊C₄	—	188	2	no	—
111	C₄⋊C₄	—	188	1	no	—
112	C₄⋊C₄	—	188	1	no	—
113	C₄⋊C₄	—	188	1	no	—
114	C₄⋊C₄	—	188	2	no	—
115	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
116	C₁₆	Dic₁₈₈	1	8	yes	complemented
117	Q₈	—	376	4	no	—
118	Q₈	—	376	2	no	—
119	Q₈	—	376	2	no	—
120	C₂×C₄	—	376	1	no	—
121	C₂×C₄	—	376	2	no	—
122	C₂×C₄	—	376	1	no	—
123	C₂×C₄	#13094021	1	1	yes	characteristic
124	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
125	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	C₂×Dic₁₈₈	1	4	yes	—
129	C₂²	#15689500	1	1	yes	characteristic
130	C₄	—	752	2	no	—
131	C₄	—	752	1	no	—
132	C₄	—	752	1	no	—
133	C₄	#15689503	1	2	yes	—
134	C₄	#15689561	1	2	yes	—
135	C₄	#15689566	1	2	yes	—
136	C₂	#20243006	1	1	yes	characteristic
137	C₂	#20245654	1	1	yes	characteristic
138	C₂	#20245659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245670

C₂

yes

maximal

#20245663

C₂

yes

maximal

#20243006

C₂

yes

maximal, characteristic

#20242998

C₂

yes

maximal, characteristic

C₄₇×#1311

C₂

yes

maximal, characteristic

#15689528

—

#15689528

C₂²

yes

characteristic

#15689528

C₂²

yes

characteristic

#15689528

C₂²

yes

—

—

—

—

—

—

C₄×C₁₆×C₄₇

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

—

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

C₁₆×C₄₇

Q₁₆

yes

complemented

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×Q₁₆

yes

—

—

C₂²×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#509

yes

—

C₄×C₄₇

#510

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₇

yes

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12524

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

188

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

D₁₈₈

yes

—

—

—

—

—

—

188

—

#103

—

188

—

#103

—

188

—

C₄×C₈

D₁₈₈

yes

characteristic

100

C₄×C₈

D₁₈₈

yes

characteristic

101

C₄×C₈

C₂²×D₄₇

yes

characteristic

102

C₂×Q₈

—

188

—

103

C₂×Q₈

—

188

—

104

C₂×Q₈

—

188

—

105

Q₁₆

—

376

—

106

C₂×C₈

D₃₇₆

yes

—

107

C₂×C₈

D₃₇₆

yes

—

108

C₂×C₈

C₂×D₁₈₈

yes

—

109

C₄⋊C₄

—

188

—

110

C₄⋊C₄

—

188

—

111

C₄⋊C₄

—

188

—

112

C₄⋊C₄

—

188

—

113

C₄⋊C₄

—

188

—

114

C₄⋊C₄

—

188

—

115

C₄²

C₂×D₁₈₈

yes

characteristic

116

C₁₆

Dic₁₈₈

yes

complemented

117

Q₈

—

376

—

118

Q₈

—

376

—

119

Q₈

—

376

—

120

C₂×C₄

—

376

—

121

C₂×C₄

—

376

—

122

C₂×C₄

—

376

—

123

C₂×C₄

#13094021

yes

characteristic

124

C₂×C₄

C₂×D₃₇₆

yes

characteristic

125

C₂×C₄

C₂×D₃₇₆

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

C₂×Dic₁₈₈

yes

—

129

C₂²

#15689500

yes

characteristic

130

C₄

—

752

—

131

C₄

—

752

—

132

C₄

—

752

—

133

C₄

#15689503

yes

—

134

C₄

#15689561

yes

—

135

C₄

#15689566

yes

—

136

C₂

#20243006

yes

characteristic

137

C₂

#20245654

yes

characteristic

138

C₂

#20245659

yes

characteristic

	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	#12524
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
46. C₁₆×C₄₇	Q₁₆	semidirect
86. C₄₇	#12524	semidirect
116. C₁₆	Dic₁₈₈	semidirect

H ≅ G/N

type

46. C₁₆×C₄₇

Q₁₆

semidirect

86. C₄₇

#12524

semidirect

116. C₁₆

Dic₁₈₈

semidirect

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

subgroup	index
#20245670	2
#20245663	2
#20243006	2
#20242998	2
C₄₇×#1311	2
#12524	47

subgroup

index

C₄₇×#1311

Subgroups — #26131026

Conjugacy classes of subgroups

Complements

Maximal subgroups