Group #26204083 — subgroups

Subgroups — #26204083

Order 12032.

690 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 227 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	#20253974	C₂	1	4	yes	maximal
2	#20253902	C₂	1	2	yes	maximal
3	#20243065	C₂	1	8	yes	maximal, complemented
4	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
5	#15689680	C₂²	1	4	yes	—
6	#15689528	C₂²	1	4	yes	—
7	#15689528	C₂²	1	4	yes	—
8	#15689525	—	2	4	no	—
9	#15689521	C₂²	1	4	yes	—
10	#15689503	C₂²	1	4	yes	—
11	#15689474	C₂²	1	8	yes	—
12	#15689364	—	2	8	no	—
13	C₄×Dic₁₈₈	—	2	8	no	—
14	#15689161	—	2	4	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#322	C₂²	1	4	yes	—
18	C₂×Dic₁₈₈	—	2	8	no	—
19	#13094024	C₂³	1	8	yes	—
20	#13094016	—	2	4	no	—
21	#13094003	—	2	8	no	—
22	C₄×Dic₉₄	—	2	8	no	—
23	#13093983	—	2	4	no	—
24	#13093958	—	2	8	no	—
25	#13093958	—	2	8	no	—
26	#13093956	—	2	8	no	—
27	#13093949	—	2	16	no	—
28	C₂²×C₈×C₄₇	D₄	1	4	yes	—
29	C₂²×C₈×C₄₇	D₄	1	2	yes	—
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₄₇	D₄	1	4	yes	—
32	C₄×C₈×C₄₇	D₄	1	2	yes	—
33	C₄×C₈×C₄₇	C₂³	1	4	yes	—
34	C₄×C₈×C₄₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₉₄	—	2	8	no	—
38	C₂×C₈×C₄₇	—	4	4	no	—
39	C₂×C₈×C₄₇	Q₁₆	1	8	yes	complemented
40	C₂×C₈×C₄₇	C₂×D₄	1	8	yes	—
41	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
42	C₄²×C₄₇	C₂×D₄	1	2	yes	—
43	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
45	C₂²⋊Dic₄₇	—	4	4	no	—
46	C₄⋊Dic₄₇	—	2	8	no	—
47	C₄⋊Dic₄₇	—	2	8	no	—
48	Dic₄₇⋊C₄	—	4	8	no	—
49	C₄×Dic₄₇	—	4	4	no	—
50	Dic₁₈₈	—	4	16	no	—
51	C₂³×C₄₇	#127	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	4	no	—
53	C₂×C₄×C₄₇	#127	1	2	yes	—
54	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
56	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
58	C₂×Dic₄₇	—	4	8	no	—
59	Dic₉₄	—	4	16	no	—
60	C₈×C₄₇	—	2	8	no	—
61	C₈×C₄₇	C₂×Q₁₆	1	8	yes	—
62	#19618	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	—	2	2	no	—
64	C₂²×C₄₇	#494	1	2	yes	—
65	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
66	C₄×C₄₇	—	2	2	no	—
67	C₄×C₄₇	#494	1	2	yes	—
68	C₄×C₄₇	C₂²×Q₁₆	1	2	yes	—
69	C₄×C₄₇	#575	1	4	yes	—
70	Dic₄₇	—	8	8	no	—
71	#3170	—	47	4	no	—
72	#3160	—	47	2	no	—
73	#1723	—	47	8	no	—
74	C₂×#322	D₄₇	1	1	yes	characteristic
75	C₂×C₄₇	—	2	2	no	—
76	C₂×C₄₇	C₂×#494	1	1	yes	characteristic
77	C₂×C₄₇	#3150	1	2	yes	—
78	#533	—	47	4	no	—
79	#500	—	47	4	no	—
80	#500	—	47	4	no	—
81	#499	—	94	4	no	—
82	#498	—	47	4	no	—
83	#494	—	47	4	no	—
84	#487	—	47	8	no	—
85	#467	—	94	8	no	—
86	C₄×Q₁₆	—	94	8	no	—
87	#427	—	94	4	no	—
88	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
90	#322	C₂×D₄₇	1	4	yes	—
91	C₄₇	#19618	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	94	8	no	—
93	C₂²×C₈	D₁₈₈	1	4	yes	—
94	C₂²×C₈	D₁₈₈	1	2	yes	—
95	C₄⋊Q₈	—	47	8	no	—
96	#125	—	94	4	no	—
97	C₂²⋊Q₈	—	94	8	no	—
98	C₄×Q₈	—	94	8	no	—
99	#117	—	94	4	no	—
100	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
101	#107	—	94	8	no	—
102	#107	—	94	8	no	—
103	#106	—	94	8	no	—
104	#103	—	94	16	no	—
105	C₄×C₈	D₁₈₈	1	4	yes	—
106	C₄×C₈	D₁₈₈	1	2	yes	—
107	C₄×C₈	C₂²×D₄₇	1	4	yes	—
108	C₄×C₈	C₂²×D₄₇	1	2	yes	—
109	C₂×Q₈	—	94	8	no	—
110	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
111	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
112	Q₁₆	—	188	16	no	—
113	C₂×C₈	—	4	4	no	—
114	C₂×C₈	Dic₁₈₈	1	8	yes	complemented
115	C₂×C₈	C₂×D₁₈₈	1	8	yes	—
116	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
117	C₄⋊C₄	—	94	8	no	—
118	C₄⋊C₄	—	94	8	no	—
119	C₄⋊C₄	—	188	8	no	—
120	C₂²⋊C₄	—	188	4	no	—
121	C₄²	—	188	4	no	—
122	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
123	C₄²	C₂×D₁₈₈	1	2	yes	—
124	C₄²	C₂³×D₄₇	1	1	yes	characteristic
125	C₂³	#13094021	1	1	yes	characteristic
126	Q₈	—	188	16	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	188	8	no	—
129	C₂×C₄	#13094021	1	2	yes	—
130	C₂×C₄	#13094021	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₁₈₈	1	4	yes	—
132	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
134	C₈	—	2	8	no	—
135	C₈	C₂×Dic₁₈₈	1	8	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	#15689503	1	2	yes	—
138	C₂²	C₂×#13094021	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	376	8	no	—
141	C₄	#15689503	1	2	yes	—
142	C₄	C₂²×Dic₁₈₈	1	2	yes	—
143	C₄	#15689923	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	C₂×#15689503	1	1	yes	characteristic
146	C₂	#20253834	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20253974

C₂

yes

maximal

#20253902

C₂

yes

maximal

#20243065

C₂

yes

maximal, complemented

C₂×C₄₇×#322

C₂

yes

maximal, characteristic

#15689680

C₂²

yes

—

#15689528

C₂²

yes

—

#15689528

C₂²

yes

—

#15689525

—

#15689521

C₂²

yes

—

#15689503

C₂²

yes

—

#15689474

C₂²

yes

—

#15689364

—

C₄×Dic₁₈₈

—

#15689161

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

C₂²

yes

—

C₂×Dic₁₈₈

—

#13094024

C₂³

yes

—

#13094016

—

#13094003

—

C₄×Dic₉₄

—

—

—

—

—

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

—

C₄×C₈×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

Q₁₆

yes

complemented

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂⁴

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

Dic₁₈₈

—

C₂³×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×D₄

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂×Q₁₆

yes

—

#19618

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#494

yes

—

C₂²×C₄₇

C₂×#127

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#494

yes

—

C₄×C₄₇

C₂²×Q₁₆

yes

—

C₄×C₄₇

#575

yes

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

C₂×#494

yes

characteristic

C₂×C₄₇

#3150

yes

—

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

#427

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

—

#322

C₂×D₄₇

yes

—

C₄₇

#19618

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

D₁₈₈

yes

—

C₂²×C₈

D₁₈₈

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

100

C₂×C₄²

C₂²×D₄₇

yes

characteristic

101

#107

—

102

#107

—

103

#106

—

104

#103

—

105

C₄×C₈

D₁₈₈

yes

—

106

C₄×C₈

D₁₈₈

yes

—

107

C₄×C₈

C₂²×D₄₇

yes

—

108

C₄×C₈

C₂²×D₄₇

yes

—

109

C₂×Q₈

—

110

C₂²×C₄

C₂×D₁₈₈

yes

—

111

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

112

Q₁₆

—

188

—

113

C₂×C₈

—

114

C₂×C₈

Dic₁₈₈

yes

complemented

115

C₂×C₈

C₂×D₁₈₈

yes

—

116

C₂×C₈

C₂×D₁₈₈

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

188

—

120

C₂²⋊C₄

—

188

—

121

C₄²

—

188

—

122

C₄²

C₂×D₁₈₈

yes

characteristic

123

C₄²

C₂×D₁₈₈

yes

—

124

C₄²

C₂³×D₄₇

yes

characteristic

125

C₂³

#13094021

yes

characteristic

126

Q₈

—

188

—

127

C₂×C₄

—

128

C₂×C₄

—

188

—

129

C₂×C₄

#13094021

yes

—

130

C₂×C₄

#13094021

yes

characteristic

131

C₂×C₄

C₂×Dic₁₈₈

yes

—

132

C₂×C₄

C₂²×D₁₈₈

yes

characteristic

133

C₂×C₄

C₂²×D₁₈₈

yes

—

134

C₈

—

135

C₈

C₂×Dic₁₈₈

yes

—

136

C₂²

—

137

C₂²

#15689503

yes

—

138

C₂²

C₂×#13094021

yes

characteristic

139

C₄

—

140

C₄

—

376

—

141

C₄

#15689503

yes

—

142

C₄

C₂²×Dic₁₈₈

yes

—

143

C₄

#15689923

yes

—

144

C₂

—

145

C₂

C₂×#15689503

yes

characteristic

146

C₂

#20253834

yes

—

	subgroup	G/H
center	C₂×C₄	#13094021
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	C₂³×C₄₇	#127
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#19618
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#13094021

commutator, cosocle

C₄²×C₄₇

C₂⁴

Frattini

C₄²

C₂³×D₄₇

Fitting

C₂×C₄₇×#322

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
3. #20243065	C₂	semidirect
39. C₂×C₈×C₄₇	Q₁₆	semidirect
91. C₄₇	#19618	semidirect
114. C₂×C₈	Dic₁₈₈	semidirect

H ≅ G/N

type

3. #20243065

C₂

semidirect

39. C₂×C₈×C₄₇

Q₁₆

semidirect

91. C₄₇

#19618

semidirect

114. C₂×C₈

Dic₁₈₈

semidirect

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

subgroup	index
#20253974	2
#20253902	2
#20243065	2
C₂×C₄₇×#322	2
#19618	47

subgroup

index

C₂×C₄₇×#322

Subgroups — #26204083

Conjugacy classes of subgroups

Complements

Maximal subgroups