Group #31791299 — subgroups

Subgroups — #31791299

Order 17536.

356 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 157 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	#22577960	C₂	1	1	yes	maximal, characteristic
2	#22577806	C₂	1	2	yes	maximal, complemented
3	#22577802	C₂	1	4	yes	maximal, complemented
4	#22577799	C₂	1	2	yes	maximal, complemented
5	#22577439	C₂	1	4	yes	maximal
6	C₂×C₄×#11764040	C₂	1	1	yes	maximal, characteristic
7	#533×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11764040	C₂²	1	2	yes	—
9	#17984088	C₂²	1	2	yes	—
10	#17984080	C₂²	1	2	yes	—
11	#17984067	—	2	2	no	—
12	C₄×Dic₂₇₄	—	2	2	no	—
13	#17984047	C₂²	1	2	yes	—
14	#17984021	C₂²	1	8	yes	—
15	#17984019	C₂²	1	8	yes	—
16	C₄×#11764040	C₂²	1	1	yes	characteristic
17	C₄×#11764040	C₂²	1	2	yes	—
18	C₄×#11764040	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
20	#125×C₁₃₇	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
22	C₄×Q₈×C₁₃₇	—	2	2	no	—
23	#117×C₁₃₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×Dic₂₇₄	—	2	2	no	—
26	C₂×#11764040	Q₈	1	8	yes	complemented
27	C₂×#11764040	C₂³	1	2	yes	—
28	C₂×#11764040	C₂³	1	2	yes	—
29	C₄⋊Dic₁₃₇	—	2	2	no	—
30	C₄⋊Dic₁₃₇	C₂³	1	4	yes	—
31	#14555489	—	2	4	no	—
32	C₂²⋊Dic₁₃₇	—	2	2	no	—
33	C₄×Dic₁₃₇	—	2	2	no	—
34	C₂×Q₈×C₁₃₇	—	2	2	no	—
35	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
36	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₃₇	—	2	4	no	—
38	C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
40	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
41	C₄²×C₁₃₇	—	2	2	no	—
42	C₄²×C₁₃₇	D₄	1	2	yes	—
43	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
45	Dic₂₇₄	—	4	4	no	—
46	C₂×Dic₁₃₇	—	2	4	no	—
47	#11764040	C₂×Q₈	1	4	yes	—
48	#11764040	C₂×Q₈	1	4	yes	—
49	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₁₃₇	—	4	4	no	—
51	C₂×C₄×C₁₃₇	—	2	4	no	—
52	C₂×C₄×C₁₃₇	—	2	4	no	—
53	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
57	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
58	C₂²×C₁₃₇	—	2	2	no	—
59	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
60	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₁₃₇	—	2	2	no	—
62	C₄×C₁₃₇	—	4	4	no	—
63	C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
64	C₄×C₁₃₇	#135	1	4	yes	—
65	C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
66	Dic₁₃₇	—	4	4	no	—
67	C₂×C₁₃₇	—	2	2	no	—
68	C₂×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₁₃₇	C₂×#135	1	2	yes	—
70	C₁₃₇	#3160	1	1	yes	characteristic, complemented, Sylow
71	#3160	—	137	1	no	maximal, Sylow
72	#533	—	137	1	no	—
73	#533	D₁₃₇	1	1	yes	characteristic
74	#500	—	137	2	no	—
75	#499	—	137	4	no	—
76	#498	—	137	2	no	—
77	#427	—	137	4	no	—
78	C₂×C₄×C₈	—	137	1	no	—
79	C₂²×C₈	—	137	2	no	—
80	C₄⋊Q₈	—	137	2	no	—
81	C₄⋊Q₈	C₂×D₁₃₇	1	2	yes	—
82	#125	—	137	2	no	—
83	#125	C₂×D₁₃₇	1	2	yes	—
84	C₂²⋊Q₈	—	2	2	no	—
85	C₂²⋊Q₈	—	274	2	no	—
86	C₄×Q₈	—	2	2	no	—
87	C₄×Q₈	—	274	2	no	—
88	#117	—	137	2	no	—
89	#117	C₂×D₁₃₇	1	2	yes	—
90	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
91	#107	—	137	8	no	—
92	#106	—	137	8	no	—
93	C₄×C₈	—	137	1	no	—
94	C₄×C₈	—	137	2	no	—
95	C₄×C₈	—	137	1	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	274	2	no	—
98	C₂²×C₄	C₂²⋊D₁₃₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
100	C₂×C₈	—	137	8	no	—
101	C₂×C₈	—	137	2	no	—
102	C₂×C₈	—	137	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	274	4	no	—
106	C₄⋊C₄	—	274	2	no	—
107	C₄⋊C₄	—	137	4	no	—
108	C₄⋊C₄	C₂²×D₁₃₇	1	4	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	274	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	274	2	no	—
113	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
114	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
115	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
116	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	548	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	274	4	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₁₃₇	1	4	yes	—
126	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
127	C₈	—	137	4	no	—
128	C₈	—	137	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	#17984086	1	2	yes	—
131	C₂²	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	4	4	no	—
134	C₄	—	548	4	no	—
135	C₄	#17984086	1	2	yes	—
136	C₄	#17984108	1	4	yes	—
137	C₄	C₂×Q₈×D₁₃₇	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	C₂×#17984086	1	1	yes	characteristic
140	C₂	C₂×#17984108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22577960

C₂

yes

maximal, characteristic

#22577806

C₂

yes

maximal, complemented

#22577802

C₂

yes

maximal, complemented

#22577799

C₂

yes

maximal, complemented

#22577439

C₂

yes

maximal

C₂×C₄×#11764040

C₂

yes

maximal, characteristic

#533×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×#11764040

C₂²

yes

—

#17984088

C₂²

yes

—

#17984080

C₂²

yes

—

#17984067

—

C₄×Dic₂₇₄

—

#17984047

C₂²

yes

—

#17984021

C₂²

yes

—

#17984019

C₂²

yes

—

C₄×#11764040

C₂²

yes

characteristic

C₄×#11764040

C₂²

yes

—

C₄×#11764040

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₇

C₂²

yes

—

#125×C₁₃₇

C₂²

yes

—

C₂²⋊Q₈×C₁₃₇

—

C₄×Q₈×C₁₃₇

—

#117×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

C₂×#11764040

Q₈

yes

complemented

C₂×#11764040

C₂³

yes

—

C₂×#11764040

C₂³

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂³

yes

—

#14555489

—

C₂²⋊Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

C₂³

yes

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

C₂×Dic₁₃₇

—

#11764040

C₂×Q₈

yes

—

#11764040

C₂×Q₈

yes

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄⋊Q₈

yes

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₄×C₁₃₇

#135

yes

—

C₄×C₁₃₇

C₂²×Q₈

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃₇

C₂×#135

yes

—

C₁₃₇

#3160

yes

characteristic, complemented, Sylow

#3160

—

137

maximal, Sylow

#533

—

137

—

#533

D₁₃₇

yes

characteristic

#500

—

137

—

#499

—

137

—

#498

—

137

—

#427

—

137

—

C₂×C₄×C₈

—

137

—

C₂²×C₈

—

137

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

C₂×D₁₃₇

yes

—

#125

—

137

—

#125

C₂×D₁₃₇

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

274

—

C₄×Q₈

—

C₄×Q₈

—

274

—

#117

—

137

—

#117

C₂×D₁₃₇

yes

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

#107

—

137

—

#106

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₂×Q₈

—

C₂×Q₈

—

274

—

C₂²×C₄

C₂²⋊D₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

100

C₂×C₈

—

137

—

101

C₂×C₈

—

137

—

102

C₂×C₈

—

137

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

274

—

106

C₄⋊C₄

—

274

—

107

C₄⋊C₄

—

137

—

108

C₄⋊C₄

C₂²×D₁₃₇

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

274

—

111

C₄²

—

112

C₄²

—

274

—

113

C₄²

C₂²⋊D₁₃₇

yes

—

114

C₄²

C₂²×D₁₃₇

yes

characteristic

115

C₄²

C₂²×D₁₃₇

yes

characteristic

116

C₂³

C₂×C₂²⋊D₁₃₇

yes

characteristic

117

Q₈

—

118

Q₈

—

548

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

274

—

122

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

124

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

125

C₂×C₄

Q₈×D₁₃₇

yes

—

126

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

127

C₈

—

137

—

128

C₈

—

137

—

129

C₂²

—

130

C₂²

#17984086

yes

—

131

C₂²

C₂²×C₂²⋊D₁₃₇

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

548

—

135

C₄

#17984086

yes

—

136

C₄

#17984108

yes

—

137

C₄

C₂×Q₈×D₁₃₇

yes

—

138

C₂

—

139

C₂

C₂×#17984086

yes

characteristic

140

C₂

C₂×#17984108

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₇
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#533×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂³×C₁₃₇	C₂×D₄
hypercenter	#533	D₁₃₇
hypocenter	C₁₃₇	#3160
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₃₇

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

#533×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂²×D₄

prime radical

C₂³×C₁₃₇

C₂×D₄

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #22577806	C₂	semidirect
3. #22577802	C₂	semidirect
4. #22577799	C₂	semidirect
26. C₂×#11764040	Q₈	semidirect
70. C₁₃₇	#3160	semidirect

H ≅ G/N

type

2. #22577806

C₂

semidirect

3. #22577802

C₂

semidirect

4. #22577799

C₂

semidirect

26. C₂×#11764040

Q₈

semidirect

70. C₁₃₇

#3160

semidirect

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

subgroup	index
#22577960	2
#22577806	2
#22577802	2
#22577799	2
#22577439	2
C₂×C₄×#11764040	2
#533×C₁₃₇	2
#3160	137

subgroup

index

#533×C₁₃₇

137

Subgroups — #31791299

Conjugacy classes of subgroups

Complements

Maximal subgroups