Group #31795995 — subgroups

Subgroups — #31795995

Order 17536.

212 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 104 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	#22578422	C₂	1	1	yes	maximal, characteristic
2	C₄×#14555518	C₂	1	1	yes	maximal, characteristic
3	#22578245	C₂	1	2	yes	maximal, complemented
4	C₈×#11764047	C₂	1	2	yes	maximal, complemented
5	#22577455	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555518	C₄	1	2	yes	—
7	C₄⋊#11764047	C₄	1	2	yes	—
8	#17984145	C₄	1	2	yes	—
9	C₄×#11764047	C₄	1	2	yes	—
10	C₄×#11764047	C₂²	1	1	yes	characteristic
11	C₄×#11764047	C₂²	1	2	yes	—
12	C₄×#11764047	C₂²	1	1	yes	characteristic
13	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
14	C₄⋊#11764040	C₄	1	1	yes	characteristic
15	#17984000	C₄	1	2	yes	—
16	C₈×Dic₁₃₇	C₂²	1	2	yes	—
17	C₄⋊C₈×C₁₃₇	C₄	1	1	yes	characteristic
18	#14555518	C₈	1	8	yes	complemented
19	C₂×#11764047	C₂×C₄	1	4	yes	—
20	C₂×#11764047	C₂×C₄	1	2	yes	—
21	C₂×#11764047	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×#11764040	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
31	#11764047	—	2	4	no	—
32	#11764047	C₂×C₈	1	4	yes	—
33	#11764047	C₂×C₈	1	4	yes	—
34	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
35	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
36	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₄×D₁₃₇	—	2	1	no	—
39	C₄×D₁₃₇	—	2	1	no	—
40	C₄×D₁₃₇	—	2	2	no	—
41	C₄×D₁₃₇	C₂×C₈	1	4	yes	—
42	#11764040	—	2	2	no	—
43	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
45	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₁₃₇	—	2	2	no	—
47	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
48	C₂×D₁₃₇	—	2	1	no	—
49	C₂×D₁₃₇	—	2	1	no	—
50	C₂×D₁₃₇	C₄×C₈	1	2	yes	—
51	C₄×C₁₃₇	—	2	1	no	—
52	C₄×C₁₃₇	C₄×C₈	1	2	yes	—
53	C₄×C₁₃₇	#131	1	1	yes	characteristic
54	C₄×C₁₃₇	#131	1	1	yes	characteristic
55	Dic₁₃₇	—	2	1	no	—
56	Dic₁₃₇	C₂²×C₈	1	2	yes	—
57	Dic₁₃₇	#131	1	2	yes	—
58	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
59	C₂×C₁₃₇	#405	1	1	yes	characteristic
60	C₂×C₁₃₇	#405	1	1	yes	characteristic
61	D₁₃₇	—	2	2	no	—
62	C₁₃₇	#1450	1	1	yes	characteristic, complemented, Sylow
63	#1450	—	137	1	no	maximal, Sylow
64	#431	—	137	1	no	—
65	#431	—	137	1	no	—
66	C₂×C₄×C₈	—	137	1	no	—
67	#322	—	137	2	no	—
68	C₈²	—	137	2	no	—
69	C₂²×C₈	—	137	2	no	—
70	C₂×C₄²	—	137	1	no	—
71	C₄⋊C₈	—	137	2	no	—
72	C₄⋊C₈	—	137	1	no	—
73	C₄⋊C₈	#10587739	1	1	yes	characteristic
74	C₂²⋊C₈	—	137	2	no	—
75	C₂²⋊C₈	—	137	2	no	—
76	C₄×C₈	—	137	1	no	—
77	C₄×C₈	—	137	2	no	—
78	C₄×C₈	—	137	1	no	—
79	C₄×C₈	—	137	2	no	—
80	C₄×C₈	—	137	2	no	—
81	C₂²×C₄	—	137	1	no	—
82	C₂²×C₄	—	137	1	no	—
83	C₂²×C₄	—	137	1	no	—
84	C₂×C₈	—	137	4	no	—
85	C₂×C₈	—	137	2	no	—
86	C₂×C₈	—	137	2	no	—
87	C₂×C₈	—	137	2	no	—
88	C₂×C₈	—	137	8	no	—
89	C₂×C₈	C₂×#10587739	1	2	yes	—
90	C₄²	—	137	1	no	—
91	C₄²	—	137	1	no	—
92	C₄²	—	137	1	no	—
93	C₄²	C₂×#10587739	1	1	yes	characteristic
94	C₂³	—	137	1	no	—
95	C₂×C₄	—	274	2	no	—
96	C₂×C₄	—	274	1	no	—
97	C₂×C₄	—	274	1	no	—
98	C₂×C₄	—	137	1	no	—
99	C₂×C₄	—	137	1	no	—
100	C₂×C₄	—	137	1	no	—
101	C₂×C₄	—	137	4	no	—
102	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
103	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
104	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	274	4	no	—
107	C₈	—	274	2	no	—
108	C₈	—	137	4	no	—
109	C₈	—	137	4	no	—
110	C₂²	—	274	1	no	—
111	C₂²	—	274	1	no	—
112	C₂²	—	137	2	no	—
113	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	274	1	no	—
116	C₄	—	137	2	no	—
117	C₄	—	137	2	no	—
118	C₄	C₈×#10587739	1	2	yes	—
119	C₄	#17984175	1	1	yes	characteristic
120	C₄	#17984176	1	1	yes	characteristic
121	C₂	—	274	2	no	—
122	C₂	C₂×C₈×#10587739	1	1	yes	characteristic
123	C₂	#22578368	1	1	yes	characteristic
124	C₂	#22578369	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578422

C₂

yes

maximal, characteristic

C₄×#14555518

C₂

yes

maximal, characteristic

#22578245

C₂

yes

maximal, complemented

C₈×#11764047

C₂

yes

maximal, complemented

#22577455

C₂

yes

maximal, characteristic

C₂×#14555518

C₄

yes

—

C₄⋊#11764047

C₄

yes

—

#17984145

C₄

yes

—

C₄×#11764047

C₄

yes

—

C₄×#11764047

C₂²

yes

characteristic

C₄×#11764047

C₂²

yes

—

C₄×#11764047

C₂²

yes

characteristic

C₄²×D₁₃₇

C₂²

yes

characteristic

C₄⋊#11764040

C₄

yes

characteristic

#17984000

C₄

yes

—

C₈×Dic₁₃₇

C₂²

yes

—

C₄⋊C₈×C₁₃₇

C₄

yes

characteristic

#14555518

C₈

yes

complemented

C₂×#11764047

C₂×C₄

yes

—

C₂×#11764047

C₂×C₄

yes

—

C₂×#11764047

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×#11764040

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

#11764047

—

#11764047

C₂×C₈

yes

—

#11764047

C₂×C₈

yes

—

C₂²×D₁₃₇

C₄²

yes

characteristic

C₂×Dic₁₃₇

C₄²

yes

characteristic

C₂×Dic₁₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₃₇

C₂²×C₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

C₂×C₈

yes

—

#11764040

—

C₂×C₄×C₁₃₇

C₄²

yes

characteristic

C₂×C₄×C₁₃₇

C₄²

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄²

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

C₄×C₈

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄×C₈

yes

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

Dic₁₃₇

—

Dic₁₃₇

C₂²×C₈

yes

—

Dic₁₃₇

#131

yes

—

C₂×C₁₃₇

C₂×C₄×C₈

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

D₁₃₇

—

C₁₃₇

#1450

yes

characteristic, complemented, Sylow

#1450

—

137

maximal, Sylow

#431

—

137

—

#431

—

137

—

C₂×C₄×C₈

—

137

—

#322

—

137

—

C₈²

—

137

—

C₂²×C₈

—

137

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

#10587739

yes

characteristic

C₂²⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

C₂×#10587739

yes

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

100

C₂×C₄

—

137

—

101

C₂×C₄

—

137

—

102

C₂×C₄

C₄×#10587739

yes

characteristic

103

C₂×C₄

C₄×#10587739

yes

characteristic

104

C₂×C₄

C₂²×#10587739

yes

characteristic

105

C₈

—

106

C₈

—

274

—

107

C₈

—

274

—

108

C₈

—

137

—

109

C₈

—

137

—

110

C₂²

—

274

—

111

C₂²

—

274

—

112

C₂²

—

137

—

113

C₂²

C₂×C₄×#10587739

yes

characteristic

114

C₄

—

115

C₄

—

274

—

116

C₄

—

137

—

117

C₄

—

137

—

118

C₄

C₈×#10587739

yes

—

119

C₄

#17984175

yes

characteristic

120

C₄

#17984176

yes

characteristic

121

C₂

—

274

—

122

C₂

C₂×C₈×#10587739

yes

characteristic

123

C₂

#22578368

yes

characteristic

124

C₂

#22578369

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator	C₂×C₁₃₇	C₂×C₄×C₈
Fitting	C₄⋊C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄²
prime radical	C₂²×D₁₃₇	C₄²
cosocle	C₄×Dic₁₃₇	C₂³
hypercenter	C₄⋊C₈	#10587739
hypocenter	C₁₃₇	#1450
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10587739

commutator

C₂×C₁₃₇

C₂×C₄×C₈

Fitting

C₄⋊C₈×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂×C₄²

prime radical

C₂²×D₁₃₇

C₄²

cosocle

C₄×Dic₁₃₇

C₂³

C₄⋊C₈

C₁₃₇

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
3. #22578245	C₂	semidirect
4. C₈×#11764047	C₂	semidirect
18. #14555518	C₈	semidirect
62. C₁₃₇	#1450	semidirect

H ≅ G/N

type

3. #22578245

C₂

semidirect

4. C₈×#11764047

C₂

semidirect

18. #14555518

C₈

semidirect

62. C₁₃₇

#1450

semidirect

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

subgroup	index
#22578422	2
C₄×#14555518	2
#22578245	2
C₈×#11764047	2
#22577455	2
#1450	137

subgroup

index

137

Subgroups — #31795995

Conjugacy classes of subgroups

Complements

Maximal subgroups