Group #26119031 — subgroups

Subgroups — #26119031

Order 12032.

592 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20254769	C₂	1	3	yes	maximal, complemented
2	#20244215	C₂	1	3	yes	maximal, complemented
3	C₄₇×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689673	C₂²	1	3	yes	—
5	#15689414	—	2	6	no	—
6	#15689376	—	2	6	no	—
7	#15689319	—	2	6	no	—
8	C₂×#13093950	—	2	6	no	—
9	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×#423	C₂²	1	3	yes	—
11	C₂×#10625350	—	2	3	no	—
12	C₂×D₃₇₆	—	2	6	no	—
13	#13094024	D₄	1	3	yes	—
14	#13094021	D₄	1	3	yes	—
15	#13094012	—	2	3	no	—
16	#13093998	—	2	6	no	—
17	C₄×D₁₈₈	—	2	6	no	—
18	C₂×C₄×Dic₄₇	—	2	3	no	—
19	#13093956	—	2	6	no	—
20	#13093950	—	4	12	no	—
21	#13093946	—	2	12	no	—
22	C₂×M₁₆×C₄₇	D₄	1	6	yes	—
23	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₄₇	—	2	6	no	—
25	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
26	C₂×C₂²⋊D₄₇	—	4	3	no	—
27	C₂²×Dic₄₇	—	4	1	no	—
28	#10625350	—	4	6	no	—
29	#10625350	—	4	6	no	—
30	C₂×D₁₈₈	—	2	6	no	—
31	C₂×D₁₈₈	—	2	3	no	—
32	C₂×C₄×D₄₇	—	4	3	no	—
33	C₂×Dic₉₄	—	2	3	no	—
34	M₁₆×C₄₇	—	4	6	no	—
35	C₂×C₈×C₄₇	—	2	6	no	—
36	C₄²×C₄₇	C₂×D₄	1	3	yes	—
37	C₄²×C₄₇	#41	1	1	yes	characteristic
38	(C₂×D₄₇)⋊C₄	—	4	6	no	—
39	C₄⋊Dic₄₇	—	2	6	no	—
40	C₄×Dic₄₇	—	2	6	no	—
41	C₄×Dic₄₇	—	4	3	no	—
42	D₃₇₆	—	8	6	no	—
43	C₂³×C₄₇	#127	1	1	yes	characteristic
44	C₂²×D₄₇	—	4	3	no	—
45	C₂×C₄×C₄₇	—	2	3	no	—
46	C₂×C₄×C₄₇	—	2	3	no	—
47	C₂×C₄×C₄₇	#120	1	3	yes	—
48	C₂×C₄×C₄₇	#121	1	3	yes	—
49	C₂²⋊D₄₇	—	8	6	no	—
50	C₂×Dic₄₇	—	4	6	no	—
51	C₂×Dic₄₇	—	4	3	no	—
52	D₁₈₈	—	4	6	no	—
53	D₁₈₈	—	4	6	no	—
54	D₁₈₈	—	4	6	no	—
55	C₄×D₄₇	—	8	3	no	—
56	C₄×D₄₇	—	8	3	no	—
57	Dic₉₄	—	4	6	no	—
58	C₈×C₄₇	—	4	6	no	—
59	#10300	—	47	1	no	maximal, Sylow
60	C₂²×C₄₇	—	2	3	no	—
61	C₂²×C₄₇	#392	1	1	yes	characteristic
62	C₂×D₄₇	—	8	3	no	—
63	C₂×D₄₇	—	8	6	no	—
64	C₄×C₄₇	—	2	3	no	—
65	C₄×C₄₇	#453	1	3	yes	—
66	C₄×C₄₇	#454	1	3	yes	—
67	Dic₄₇	—	4	4	no	—
68	Dic₄₇	—	8	3	no	—
69	#3253	—	47	3	no	—
70	#1962	—	47	3	no	—
71	#1309	D₄₇	1	1	yes	characteristic, complemented
72	C₂×C₄₇	—	4	1	no	—
73	C₂×C₄₇	#2006	1	3	yes	—
74	D₄₇	—	16	3	no	—
75	#532	—	47	3	no	—
76	#476	—	94	6	no	—
77	#469	—	94	6	no	—
78	#459	—	94	6	no	—
79	#423	C₂×D₄₇	1	3	yes	—
80	C₂×#104	—	94	6	no	—
81	C₄³	—	47	1	no	—
82	C₄₇	#10300	1	1	yes	characteristic, complemented, Sylow
83	C₂×#41	—	94	3	no	—
84	C₂×D₈	—	94	6	no	—
85	C₂×M₁₆	D₁₈₈	1	6	yes	—
86	C₄⋊Q₈	—	47	3	no	—
87	#127	—	47	3	no	—
88	#124	—	94	3	no	—
89	#121	—	94	6	no	—
90	C₄×D₄	—	94	6	no	—
91	C₂×C₄²	—	94	3	no	—
92	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
93	#106	—	94	6	no	—
94	C₄⋊C₈	—	2	6	no	—
95	#104	—	188	12	no	—
96	#102	—	94	12	no	—
97	#41	—	188	6	no	—
98	#41	—	188	6	no	—
99	C₂×Q₈	—	94	3	no	—
100	C₂×D₄	—	94	6	no	—
101	C₂×D₄	—	94	3	no	—
102	C₂×D₄	—	188	3	no	—
103	C₂²×C₄	—	188	3	no	—
104	C₂²×C₄	—	188	1	no	—
105	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
106	D₈	—	376	6	no	—
107	M₁₆	—	4	6	no	—
108	C₂×C₈	—	2	6	no	—
109	C₄⋊C₄	—	94	6	no	—
110	C₂²⋊C₄	—	188	6	no	—
111	C₄²	—	94	6	no	—
112	C₄²	—	188	3	no	—
113	C₄²	D₄×D₄₇	1	3	yes	—
114	C₄²	#10625354	1	1	yes	characteristic
115	C₂³	—	188	3	no	—
116	C₂³	#13094021	1	1	yes	characteristic
117	Q₈	—	188	6	no	—
118	D₄	—	376	6	no	—
119	D₄	—	188	6	no	—
120	D₄	—	188	6	no	—
121	D₄	—	188	6	no	—
122	C₂×C₄	—	2	3	no	—
123	C₂×C₄	—	2	3	no	—
124	C₂×C₄	—	376	3	no	—
125	C₂×C₄	—	376	3	no	—
126	C₂×C₄	—	188	3	no	—
127	C₂×C₄	—	188	6	no	—
128	C₂×C₄	#13093994	1	3	yes	—
129	C₂×C₄	#13093999	1	3	yes	—
130	C₈	—	4	6	no	—
131	C₂²	—	2	3	no	—
132	C₂²	—	376	3	no	—
133	C₂²	—	376	6	no	—
134	C₂²	#15689033	1	1	yes	characteristic
135	C₄	—	2	3	no	—
136	C₄	—	376	3	no	—
137	C₄	—	188	4	no	—
138	C₄	#15689293	1	3	yes	—
139	C₄	#15689298	1	3	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	752	3	no	—
142	C₂	#20244432	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20254769

C₂

yes

maximal, complemented

#20244215

C₂

yes

maximal, complemented

C₄₇×#1309

C₂

yes

maximal, characteristic, complemented

#15689673

C₂²

yes

—

—

—

—

—

C₄²×Dic₄₇

C₂²

yes

characteristic

C₄₇×#423

C₂²

yes

—

C₂×#10625350

—

C₂×D₃₇₆

—

#13094024

D₄

yes

—

#13094021

D₄

yes

—

#13094012

—

#13093998

—

C₄×D₁₈₈

—

C₂×C₄×Dic₄₇

—

#13093956

—

#13093950

—

#13093946

—

C₂×M₁₆×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

characteristic

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

D₃₇₆

—

C₂³×C₄₇

#127

yes

characteristic

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

#10300

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#392

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#453

yes

—

C₄×C₄₇

#454

yes

—

Dic₄₇

—

Dic₄₇

—

#3253

—

#1962

—

#1309

D₄₇

yes

characteristic, complemented

C₂×C₄₇

—

C₂×C₄₇

#2006

yes

—

D₄₇

—

—

—

—

—

C₂×D₄₇

yes

—

C₂×#104

—

C₄³

—

C₄₇

#10300

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×D₈

—

C₂×M₁₆

D₁₈₈

yes

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#106

—

C₄⋊C₈

—

#104

—

188

—

#102

—

#41

—

188

—

#41

—

188

—

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

188

—

103

C₂²×C₄

—

188

—

104

C₂²×C₄

—

188

—

105

C₂²×C₄

C₂×D₁₈₈

yes

—

106

D₈

—

376

—

107

M₁₆

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

188

—

111

C₄²

—

112

C₄²

—

188

—

113

C₄²

D₄×D₄₇

yes

—

114

C₄²

#10625354

yes

characteristic

115

C₂³

—

188

—

116

C₂³

#13094021

yes

characteristic

117

Q₈

—

188

—

118

D₄

—

376

—

119

D₄

—

188

—

120

D₄

—

188

—

121

D₄

—

188

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

376

—

125

C₂×C₄

—

376

—

126

C₂×C₄

—

188

—

127

C₂×C₄

—

188

—

128

C₂×C₄

#13093994

yes

—

129

C₂×C₄

#13093999

yes

—

130

C₈

—

131

C₂²

—

132

C₂²

—

376

—

133

C₂²

—

376

—

134

C₂²

#15689033

yes

characteristic

135

C₄

—

136

C₄

—

376

—

137

C₄

—

188

—

138

C₄

#15689293

yes

—

139

C₄

#15689298

yes

—

140

C₂

—

141

C₂

—

752

—

142

C₂

#20244432

yes

—

	subgroup	G/H
center	C₂²	#15689033
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1309	C₂
socle	C₂²×C₄₇	#392
hypercenter	#1309	D₄₇
hypocenter	C₄₇	#10300
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#15689033

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1309

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #20254769	C₂	semidirect
2. #20244215	C₂	semidirect
3. C₄₇×#1309	C₂	semidirect
71. #1309	D₄₇	semidirect
82. C₄₇	#10300	semidirect

H ≅ G/N

type

1. #20254769

C₂

semidirect

2. #20244215

C₂

semidirect

3. C₄₇×#1309

C₂

semidirect

71. #1309

D₄₇

semidirect

82. C₄₇

#10300

semidirect

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

subgroup	index
#20254769	2
#20244215	2
C₄₇×#1309	2
#10300	47

subgroup

index

#20254769

#20244215

C₄₇×#1309

#10300

Subgroups — #26119031

Conjugacy classes of subgroups

Complements

Maximal subgroups