Group #26130976 — subgroups

Subgroups — #26130976

Order 12032.

268 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 79 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	#20245615	C₂	1	2	yes	maximal, complemented
2	#20245611	C₂	1	2	yes	maximal
3	#20243006	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
6	#15689528	—	2	1	no	—
7	#15689528	C₂²	1	2	yes	—
8	#15689503	—	2	1	no	—
9	#15688927	—	2	4	no	—
10	#15688924	—	2	4	no	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
12	C₄₇×#494	C₂²	1	1	yes	characteristic
13	C₄₇×#493	C₂²	1	1	yes	characteristic
14	C₄₇×#486	—	2	1	no	—
15	C₄₇×#486	—	2	1	no	—
16	C₈²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	—	4	2	no	—
18	#13094024	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
20	#13093958	—	2	2	no	—
21	#13093958	—	2	2	no	—
22	#13093958	—	4	2	no	—
23	C₂×Q₁₆×C₄₇	—	2	2	no	—
24	C₂×D₈×C₄₇	—	2	2	no	—
25	C₄⋊Q₈×C₄₇	—	2	1	no	—
26	#127×C₄₇	—	2	1	no	—
27	#102×C₄₇	—	4	2	no	—
28	#102×C₄₇	—	4	2	no	—
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×Q₈×C₄₇	—	4	1	no	—
33	C₂×D₄×C₄₇	—	4	1	no	—
34	C₂×D₄×C₄₇	—	4	1	no	—
35	C₂×D₄×C₄₇	—	4	1	no	—
36	C₂×Dic₉₄	—	4	2	no	—
37	Q₁₆×C₄₇	—	4	4	no	—
38	D₈×C₄₇	—	4	4	no	—
39	C₂×C₈×C₄₇	D₈	1	2	yes	—
40	C₂×C₈×C₄₇	D₈	1	2	yes	—
41	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₄₇	—	4	1	no	—
43	C₄⋊C₄×C₄₇	—	4	1	no	—
44	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Dic₄₇	—	4	2	no	—
46	C₄⋊Dic₄₇	—	4	2	no	—
47	Dic₁₈₈	—	8	4	no	—
48	C₄₇⋊C₁₆	—	2	4	no	—
49	C₂³×C₄₇	—	8	1	no	—
50	Q₈×C₄₇	—	8	2	no	—
51	D₄×C₄₇	—	8	2	no	—
52	D₄×C₄₇	—	8	2	no	—
53	D₄×C₄₇	—	8	2	no	—
54	C₂×C₄×C₄₇	—	8	1	no	—
55	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₄₇	—	8	2	no	—
59	Dic₉₄	—	8	4	no	—
60	C₈×C₄₇	—	2	2	no	—
61	C₈×C₄₇	—	2	2	no	—
62	C₈×C₄₇	#135	1	4	yes	—
63	#12512	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	—	16	1	no	—
65	C₂²×C₄₇	—	16	1	no	—
66	C₂²×C₄₇	—	16	1	no	—
67	C₂²×C₄₇	#493	1	1	yes	characteristic
68	C₄×C₄₇	—	16	1	no	—
69	C₄×C₄₇	#495	1	2	yes	—
70	C₄×C₄₇	#509	1	2	yes	—
71	C₄×C₄₇	#510	1	2	yes	—
72	Dic₄₇	—	16	2	no	—
73	#2241	—	47	2	no	—
74	#2240	—	47	2	no	—
75	#1713	—	47	1	no	—
76	#1709	D₄₇	1	1	yes	characteristic
77	#1311	—	47	1	no	—
78	C₂×C₄₇	—	32	1	no	—
79	C₂×C₄₇	#1709	1	1	yes	characteristic
80	C₂×C₄₇	#2250	1	1	yes	characteristic
81	C₂×C₄₇	#2251	1	1	yes	characteristic
82	#500	—	47	2	no	—
83	#500	—	94	1	no	—
84	#494	—	94	1	no	—
85	#494	C₂×D₄₇	1	1	yes	characteristic
86	#493	C₂×D₄₇	1	1	yes	characteristic
87	#486	—	2	1	no	—
88	#486	—	2	1	no	—
89	#358	—	94	4	no	—
90	#357	—	94	4	no	—
91	C₄×C₁₆	—	47	2	no	—
92	C₈²	C₂×D₄₇	1	1	yes	characteristic
93	C₄₇	#12512	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	2	2	no	—
95	C₂×Q₁₆	—	188	2	no	—
96	C₂×D₈	—	2	2	no	—
97	C₄⋊Q₈	—	2	1	no	—
98	C₄⋊Q₈	—	94	2	no	—
99	#127	—	2	1	no	—
100	C₂×C₁₆	—	47	4	no	—
101	#107	—	94	2	no	—
102	#107	—	94	2	no	—
103	#107	—	188	2	no	—
104	#102	—	4	2	no	—
105	#102	—	4	2	no	—
106	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
109	C₂×Q₈	—	4	1	no	—
110	C₂×Q₈	—	188	2	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	4	1	no	—
113	C₂×D₄	—	4	1	no	—
114	Q₁₆	—	4	4	no	—
115	Q₁₆	—	376	4	no	—
116	D₈	—	4	4	no	—
117	C₂×C₈	#10625334	1	2	yes	—
118	C₂×C₈	#10625334	1	2	yes	—
119	C₂×C₈	D₄×D₄₇	1	2	yes	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	188	2	no	—
123	C₄⋊C₄	—	188	2	no	—
124	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
125	C₁₆	—	94	4	no	—
126	C₂³	—	8	1	no	—
127	Q₈	—	8	2	no	—
128	Q₈	—	376	4	no	—
129	D₄	—	8	2	no	—
130	D₄	—	8	2	no	—
131	D₄	—	8	2	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	376	2	no	—
134	C₂×C₄	#13094020	1	1	yes	characteristic
135	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
136	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	#13094043	1	4	yes	—
140	C₂²	—	16	1	no	—
141	C₂²	—	16	1	no	—
142	C₂²	—	16	1	no	—
143	C₂²	#15689499	1	1	yes	characteristic
144	C₄	—	16	1	no	—
145	C₄	—	752	2	no	—
146	C₄	#15689506	1	2	yes	—
147	C₄	#15689558	1	2	yes	—
148	C₄	#15689563	1	2	yes	—
149	C₂	—	32	1	no	—
150	C₂	#20242993	1	1	yes	characteristic
151	C₂	#20245655	1	1	yes	characteristic
152	C₂	#20245660	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245615

C₂

yes

maximal, complemented

#20245611

C₂

yes

maximal

#20243006

C₂

yes

maximal, characteristic, complemented

#20241305

C₂

yes

maximal, characteristic, complemented

C₄₇×#1709

C₂

yes

maximal, characteristic

#15689528

—

#15689528

C₂²

yes

—

#15689503

—

#15688927

—

#15688924

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#494

C₂²

yes

characteristic

C₄₇×#493

C₂²

yes

characteristic

C₄₇×#486

—

C₄₇×#486

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

#13094024

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

—

#13093958

—

#13093958

—

C₂×Q₁₆×C₄₇

—

C₂×D₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

#127×C₄₇

—

#102×C₄₇

—

#102×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×Dic₉₄

—

Q₁₆×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#135

yes

—

#12512

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#493

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#495

yes

—

C₄×C₄₇

#509

yes

—

C₄×C₄₇

#510

yes

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

#1311

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

C₂×D₄₇

yes

characteristic

#493

C₂×D₄₇

yes

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12512

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

188

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

100

C₂×C₁₆

—

101

#107

—

102

#107

—

103

#107

—

188

—

104

#102

—

105

#102

—

106

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

107

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

108

C₄×C₈

C₂²×D₄₇

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

188

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

Q₁₆

—

115

Q₁₆

—

376

—

116

D₈

—

117

C₂×C₈

#10625334

yes

—

118

C₂×C₈

#10625334

yes

—

119

C₂×C₈

D₄×D₄₇

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

188

—

123

C₄⋊C₄

—

188

—

124

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

125

C₁₆

—

126

C₂³

—

127

Q₈

—

128

Q₈

—

376

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

376

—

134

C₂×C₄

#13094020

yes

characteristic

135

C₂×C₄

C₂×#10625334

yes

characteristic

136

C₂×C₄

C₂×#10625334

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#13094043

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#15689499

yes

characteristic

144

C₄

—

145

C₄

—

752

—

146

C₄

#15689506

yes

—

147

C₄

#15689558

yes

—

148

C₄

#15689563

yes

—

149

C₂

—

150

C₂

#20242993

yes

characteristic

151

C₂

#20245655

yes

characteristic

152

C₂

#20245660

yes

characteristic

	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
prime radical	C₄₇×#493	C₂²
hypercenter	#1709	D₄₇
hypocenter	C₄₇	#12512
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

socle

C₂²×C₄₇

#493

prime radical

C₄₇×#493

C₂²

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #20245615	C₂	semidirect
3. #20243006	C₂	semidirect
4. #20241305	C₂	semidirect
93. C₄₇	#12512	semidirect

H ≅ G/N

type

1. #20245615

C₂

semidirect

3. #20243006

C₂

semidirect

4. #20241305

C₂

semidirect

93. C₄₇

#12512

semidirect

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

subgroup	index
#20245615	2
#20245611	2
#20243006	2
#20241305	2
C₄₇×#1709	2
#12512	47

subgroup

index

C₄₇×#1709

Subgroups — #26130976

Conjugacy classes of subgroups

Complements

Maximal subgroups