Group #31780181 — subgroups

Subgroups — #31780181

Order 17536.

156 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22577806	C₂	1	1	yes	maximal, characteristic
2	#22577613	C₂	1	1	yes	maximal, characteristic
3	Q₈×#11764040	C₂	1	1	yes	maximal, characteristic
4	#22577155	C₂	1	1	yes	maximal, characteristic
5	#22577150	C₂	1	1	yes	maximal, characteristic
6	#22577149	C₂	1	1	yes	maximal, characteristic
7	#462×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555500	—	2	1	no	—
9	#17984088	C₂²	1	1	yes	characteristic
10	#17984021	—	2	1	no	—
11	#17984021	—	2	1	no	—
12	C₄⋊#11764040	—	2	1	no	—
13	C₄⋊#11764040	C₂²	1	1	yes	characteristic
14	C₄⋊#11764040	C₂²	1	1	yes	characteristic
15	#17984011	—	2	1	no	—
16	C₄×#11764040	—	2	1	no	—
17	C₄×#11764040	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₃₇	—	2	1	no	—
19	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	#103×C₁₃₇	—	2	1	no	—
23	C₂×Dic₂₇₄	—	2	1	no	—
24	#14555500	—	4	2	no	—
25	C₂×#11764040	—	2	1	no	—
26	C₂×#11764040	—	2	1	no	—
27	C₂×#11764040	D₄	1	1	yes	characteristic
28	C₂×#11764040	D₄	1	1	yes	characteristic
29	C₄⋊Dic₁₃₇	—	2	1	no	—
30	C₄⋊Dic₁₃₇	—	2	1	no	—
31	C₂×Q₈×C₁₃₇	—	2	1	no	—
32	C₂×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
33	Q₁₆×C₁₃₇	—	4	2	no	—
34	C₂×C₈×C₁₃₇	—	2	1	no	—
35	C₄⋊C₄×C₁₃₇	—	2	1	no	—
36	C₄⋊C₄×C₁₃₇	—	2	1	no	—
37	C₄⋊C₄×C₁₃₇	—	2	1	no	—
38	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
39	C₄²×C₁₃₇	—	2	1	no	—
40	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
41	Dic₂₇₄	—	4	1	no	—
42	Dic₂₇₄	—	4	1	no	—
43	C₂×Dic₁₃₇	—	4	1	no	—
44	#11764040	—	2	1	no	—
45	#11764040	—	4	1	no	—
46	#11764040	—	4	1	no	—
47	#11764040	Q₁₆	1	2	yes	complemented
48	Q₈×C₁₃₇	—	2	1	no	—
49	Q₈×C₁₃₇	—	4	1	no	—
50	Q₈×C₁₃₇	—	4	1	no	—
51	Q₈×C₁₃₇	D₈	1	2	yes	—
52	C₂×C₄×C₁₃₇	—	2	1	no	—
53	C₂×C₄×C₁₃₇	—	2	1	no	—
54	C₂×C₄×C₁₃₇	—	4	1	no	—
55	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
58	C₈×C₁₃₇	—	4	1	no	—
59	C₂²×C₁₃₇	#121	1	1	yes	characteristic
60	C₄×C₁₃₇	—	2	2	no	—
61	C₄×C₁₃₇	—	2	1	no	—
62	C₄×C₁₃₇	—	4	1	no	—
63	C₄×C₁₃₇	—	8	1	no	—
64	C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
65	C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₄×C₁₃₇	#135	1	1	yes	characteristic
67	C₄×C₁₃₇	#137	1	1	yes	characteristic
68	Dic₁₃₇	—	8	1	no	—
69	C₂×C₁₃₇	#462	1	1	yes	characteristic
70	C₂×C₁₃₇	#466	1	1	yes	characteristic
71	C₂×C₁₃₇	#473	1	1	yes	characteristic
72	C₁₃₇	#1674	1	1	yes	characteristic, complemented, Sylow
73	#1674	—	137	1	no	maximal, Sylow
74	#500	—	137	1	no	—
75	#462	—	137	1	no	—
76	#462	D₁₃₇	1	1	yes	characteristic
77	C₈×Q₈	—	137	1	no	—
78	#335	—	137	1	no	—
79	#332	—	137	1	no	—
80	#332	—	137	1	no	—
81	C₂×Q₁₆	—	2	1	no	—
82	C₂×Q₁₆	—	274	1	no	—
83	C₄⋊Q₈	—	137	1	no	—
84	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
85	C₄×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
86	#107	—	274	1	no	—
87	#107	—	274	1	no	—
88	C₄⋊C₈	—	274	1	no	—
89	C₄⋊C₈	—	137	1	no	—
90	C₄⋊C₈	—	137	1	no	—
91	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
92	#103	—	2	1	no	—
93	#103	—	274	1	no	—
94	C₄×C₈	—	274	1	no	—
95	C₄×C₈	—	137	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	274	1	no	—
98	C₂×Q₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
99	Q₁₆	—	4	2	no	—
100	Q₁₆	—	548	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	274	1	no	—
103	C₂×C₈	—	274	1	no	—
104	C₂×C₈	—	137	1	no	—
105	C₂×C₈	—	137	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	274	1	no	—
110	C₄⋊C₄	—	274	1	no	—
111	C₄⋊C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	548	1	no	—
118	Q₈	—	548	1	no	—
119	Q₈	#14555495	1	2	yes	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	548	1	no	—
124	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
126	C₂×C₄	#14555509	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	274	1	no	—
129	C₈	—	548	1	no	—
130	C₈	—	548	1	no	—
131	C₈	—	137	2	no	—
132	C₂²	#17984060	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	2	no	—
135	C₄	—	4	1	no	—
136	C₄	—	8	1	no	—
137	C₄	—	1096	1	no	—
138	C₄	C₂×#14555495	1	1	yes	characteristic
139	C₄	Q₁₆×D₁₃₇	1	1	yes	characteristic
140	C₄	#17984108	1	1	yes	characteristic
141	C₄	#17984120	1	1	yes	characteristic
142	C₂	#22577611	1	1	yes	characteristic
143	C₂	#22577637	1	1	yes	characteristic
144	C₂	#22577676	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577806

C₂

yes

maximal, characteristic

#22577613

C₂

yes

maximal, characteristic

Q₈×#11764040

C₂

yes

maximal, characteristic

#22577155

C₂

yes

maximal, characteristic

#22577150

C₂

yes

maximal, characteristic

#22577149

C₂

yes

maximal, characteristic

#462×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555500

—

#17984088

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊#11764040

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×Q₁₆×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

characteristic

C₄×Q₈×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

C₂²

yes

characteristic

#103×C₁₃₇

—

C₂×Dic₂₇₄

—

—

—

—

D₄

yes

characteristic

C₂×#11764040

D₄

yes

characteristic

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

D₄

yes

characteristic

Q₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

D₄

yes

characteristic

C₄²×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

—

—

—

Q₁₆

yes

complemented

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

D₈

yes

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

#121

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×D₈

yes

characteristic

C₄×C₁₃₇

C₂×Q₁₆

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

Dic₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1674

—

137

maximal, Sylow

#500

—

137

—

#462

—

137

—

#462

D₁₃₇

yes

characteristic

C₈×Q₈

—

137

—

#335

—

137

—

#332

—

137

—

#332

—

137

—

C₂×Q₁₆

—

C₂×Q₁₆

—

274

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

C₂×D₁₃₇

yes

characteristic

C₄×Q₈

C₂×D₁₃₇

yes

characteristic

#107

—

274

—

#107

—

274

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

C₂×D₁₃₇

yes

characteristic

#103

—

#103

—

274

—

C₄×C₈

—

274

—

C₄×C₈

—

137

—

C₂×Q₈

—

C₂×Q₈

—

274

—

C₂×Q₈

C₂²⋊D₁₃₇

yes

characteristic

Q₁₆

—

100

Q₁₆

—

548

—

101

C₂×C₈

—

102

C₂×C₈

—

274

—

103

C₂×C₈

—

274

—

104

C₂×C₈

—

137

—

105

C₂×C₈

—

137

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

274

—

110

C₄⋊C₄

—

274

—

111

C₄⋊C₄

C₂²⋊D₁₃₇

yes

characteristic

112

C₄²

—

113

C₄²

C₂²×D₁₃₇

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

548

—

118

Q₈

—

548

—

119

Q₈

#14555495

yes

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

548

—

124

C₂×C₄

D₄×D₁₃₇

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

126

C₂×C₄

#14555509

yes

characteristic

127

C₈

—

128

C₈

—

274

—

129

C₈

—

548

—

130

C₈

—

548

—

131

C₈

—

137

—

132

C₂²

#17984060

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

1096

—

138

C₄

C₂×#14555495

yes

characteristic

139

C₄

Q₁₆×D₁₃₇

yes

characteristic

140

C₄

#17984108

yes

characteristic

141

C₄

#17984120

yes

characteristic

142

C₂

#22577611

yes

characteristic

143

C₂

#22577637

yes

characteristic

144

C₂

#22577676

yes

characteristic

	subgroup	G/H
center	C₂²	#17984060
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#462×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	#121
hypercenter	#462	D₁₃₇
hypocenter	C₁₃₇	#1674
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#17984060

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#462×C₁₃₇

C₂

C₂²×C₁₃₇

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
47. #11764040	Q₁₆	semidirect
72. C₁₃₇	#1674	semidirect

H ≅ G/N

type

47. #11764040

Q₁₆

semidirect

72. C₁₃₇

#1674

semidirect

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

subgroup	index
#22577806	2
#22577613	2
Q₈×#11764040	2
#22577155	2
#22577150	2
#22577149	2
#462×C₁₃₇	2
#1674	137

subgroup

index

#462×C₁₃₇

137

Subgroups — #31780181

Conjugacy classes of subgroups

Complements

Maximal subgroups