Group #31797018 — subgroups

Subgroups — #31797018

Order 17536.

260 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₂×#17984162	C₂	1	2	yes	maximal, complemented
2	#22578298	C₂	1	4	yes	maximal, complemented
3	C₂²×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#17984162	—	2	4	no	—
5	C₂×#14555513	C₂²	1	4	yes	complemented
6	C₂²×C₄×D₁₃₇	C₄	1	1	yes	characteristic
7	C₂²×#11764040	C₂²	1	1	yes	characteristic
8	C₂×C₈×D₁₃₇	—	2	4	no	—
9	C₂×C₈×D₁₃₇	C₄	1	2	yes	—
10	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic
12	#14555513	—	2	4	no	—
13	C₂³×D₁₃₇	C₈	1	1	yes	characteristic
14	C₂²×Dic₁₃₇	C₈	1	1	yes	characteristic
15	C₂×C₄×D₁₃₇	—	2	2	no	—
16	C₂×C₄×D₁₃₇	—	2	2	no	—
17	C₂×C₄×D₁₃₇	C₈	1	2	yes	—
18	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
19	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
20	C₂×#11764040	D₄	1	4	yes	—
21	C₂×#11764040	C₂³	1	1	yes	characteristic
22	C₈×D₁₃₇	—	2	4	no	—
23	C₈×D₁₃₇	—	2	4	no	—
24	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₃₇	—	2	2	no	—
26	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂²×D₁₃₇	—	2	2	no	—
28	C₂²×D₁₃₇	—	2	4	no	—
29	C₂²×D₁₃₇	C₂×C₈	1	2	yes	—
30	C₂×Dic₁₃₇	—	2	2	no	—
31	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
32	C₄×D₁₃₇	—	2	4	no	—
33	C₄×D₁₃₇	—	2	2	no	—
34	C₄×D₁₃₇	—	2	2	no	—
35	#11764040	C₂²⋊C₄	1	2	yes	—
36	#11764040	C₂×D₄	1	2	yes	—
37	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	M₁₆	1	2	yes	—
41	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₈×C₁₃₇	—	2	2	no	—
43	C₂²×C₁₃₇	—	2	2	no	—
44	C₂²×C₁₃₇	C₂²⋊C₈	1	2	yes	—
45	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
46	C₂×D₁₃₇	—	2	8	no	—
47	C₂×D₁₃₇	—	2	2	no	—
48	C₂×D₁₃₇	—	2	2	no	—
49	C₂×D₁₃₇	M₃₂	1	4	yes	complemented
50	C₄×C₁₃₇	C₂²⋊C₈	1	2	yes	—
51	C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
53	Dic₁₃₇	—	2	2	no	—
54	C₂×C₁₃₇	—	2	2	no	—
55	C₂×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₁₃₇	C₂×M₃₂	1	2	yes	—
57	D₁₃₇	—	2	4	no	—
58	C₁₃₇	#2112	1	1	yes	characteristic, complemented, Sylow
59	#2112	—	137	1	no	maximal, Sylow
60	C₂³×C₈	—	137	1	no	—
61	C₂×M₃₂	—	137	2	no	—
62	C₂²⋊C₁₆	—	137	4	no	—
63	C₂³×C₄	—	137	1	no	—
64	C₂²×C₈	—	274	4	no	—
65	C₂²×C₈	—	137	2	no	—
66	C₂²×C₈	—	137	2	no	—
67	C₂²×C₈	—	137	1	no	—
68	C₂²×C₈	#10587739	1	1	yes	characteristic
69	M₃₂	—	274	4	no	—
70	C₂×C₁₆	—	137	4	no	—
71	C₂⁴	—	137	1	no	—
72	C₂²×C₄	—	274	2	no	—
73	C₂²×C₄	—	274	2	no	—
74	C₂²×C₄	—	137	1	no	—
75	C₂²×C₄	—	137	2	no	—
76	C₂²×C₄	—	137	2	no	—
77	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	274	4	no	—
80	C₂×C₈	—	274	4	no	—
81	C₂×C₈	—	137	4	no	—
82	C₂×C₈	—	137	1	no	—
83	C₂×C₈	—	137	1	no	—
84	C₂×C₈	C₂×#10587739	1	2	yes	—
85	C₁₆	—	274	4	no	—
86	C₂³	—	274	4	no	—
87	C₂³	—	274	2	no	—
88	C₂³	—	137	2	no	—
89	C₂³	#14555518	1	1	yes	characteristic
90	C₂×C₄	—	274	2	no	—
91	C₂×C₄	—	274	4	no	—
92	C₂×C₄	—	274	2	no	—
93	C₂×C₄	—	274	2	no	—
94	C₂×C₄	—	137	2	no	—
95	C₂×C₄	C₂²⋊#10587739	1	2	yes	—
96	C₂×C₄	#14555518	1	1	yes	characteristic
97	C₂×C₄	#14555520	1	2	yes	—
98	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	137	2	no	—
101	C₈	—	137	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	274	8	no	—
104	C₂²	—	274	2	no	—
105	C₂²	—	274	2	no	—
106	C₂²	—	137	4	no	—
107	C₂²	#17984145	1	2	yes	—
108	C₂²	C₂×#14555518	1	1	yes	characteristic
109	C₄	—	274	2	no	—
110	C₄	#17984145	1	2	yes	—
111	C₄	C₂×C₂²⋊#10587739	1	1	yes	characteristic
112	C₄	C₂×#14555520	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	—	274	4	no	—
115	C₂	C₂×#17984145	1	1	yes	characteristic
116	C₂	#22578459	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984162

C₂

yes

maximal, complemented

#22578298

C₂

yes

maximal, complemented

C₂²×C₈×D₁₃₇

C₂

yes

maximal, characteristic

#17984162

—

C₂×#14555513

C₂²

yes

complemented

C₂²×C₄×D₁₃₇

C₄

yes

characteristic

C₂²×#11764040

C₂²

yes

characteristic

C₂×C₈×D₁₃₇

—

C₂×C₈×D₁₃₇

C₄

yes

—

C₂×C₈×D₁₃₇

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₄

yes

characteristic

#14555513

—

C₂³×D₁₃₇

C₈

yes

characteristic

C₂²×Dic₁₃₇

C₈

yes

characteristic

C₂×C₄×D₁₃₇

—

C₂×C₄×D₁₃₇

—

C₂×C₄×D₁₃₇

C₈

yes

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

—

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

D₄

yes

—

C₂×#11764040

C₂³

yes

characteristic

C₈×D₁₃₇

—

C₈×D₁₃₇

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₂²×D₁₃₇

—

C₂²×D₁₃₇

—

C₂²×D₁₃₇

C₂×C₈

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

C₂²⋊C₄

yes

—

#11764040

C₂×D₄

yes

—

C₂³×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

M₁₆

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²⋊C₈

yes

—

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

M₃₂

yes

complemented

C₄×C₁₃₇

C₂²⋊C₈

yes

—

C₄×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₃₇

C₂×M₃₂

yes

—

D₁₃₇

—

C₁₃₇

#2112

yes

characteristic, complemented, Sylow

#2112

—

137

maximal, Sylow

C₂³×C₈

—

137

—

C₂×M₃₂

—

137

—

C₂²⋊C₁₆

—

137

—

C₂³×C₄

—

137

—

C₂²×C₈

—

274

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

#10587739

yes

characteristic

M₃₂

—

274

—

C₂×C₁₆

—

137

—

C₂⁴

—

137

—

C₂²×C₄

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×#10587739

yes

characteristic

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

C₂×#10587739

yes

—

C₁₆

—

274

—

C₂³

—

274

—

C₂³

—

274

—

C₂³

—

137

—

C₂³

#14555518

yes

characteristic

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

C₂²⋊#10587739

yes

—

C₂×C₄

#14555518

yes

characteristic

C₂×C₄

#14555520

yes

—

C₂×C₄

C₂²×#10587739

yes

characteristic

C₈

—

100

C₈

—

137

—

101

C₈

—

137

—

102

C₂²

—

103

C₂²

—

274

—

104

C₂²

—

274

—

105

C₂²

—

274

—

106

C₂²

—

137

—

107

C₂²

#17984145

yes

—

108

C₂²

C₂×#14555518

yes

characteristic

109

C₄

—

274

—

110

C₄

#17984145

yes

—

111

C₄

C₂×C₂²⋊#10587739

yes

characteristic

112

C₄

C₂×#14555520

yes

characteristic

113

C₂

—

114

C₂

—

274

—

115

C₂

C₂×#17984145

yes

characteristic

116

C₂

#22578459

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₂²×C₈×C₁₃₇	C₄
prime radical	C₂³×D₁₃₇	C₈
cosocle	C₂×#11764040	C₂³
hypercenter	C₂²×C₈	#10587739
hypocenter	C₁₃₇	#2112
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10587739

commutator, socle

C₂²×C₁₃₇

C₂²×C₈

Fitting

C₂²×C₈×C₁₃₇

C₄

prime radical

C₂³×D₁₃₇

C₈

C₂³

C₂²×C₈

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
1. C₂×#17984162	C₂	semidirect
2. #22578298	C₂	semidirect
5. C₂×#14555513	C₂²	semidirect
49. C₂×D₁₃₇	M₃₂	semidirect
58. C₁₃₇	#2112	semidirect

H ≅ G/N

type

1. C₂×#17984162

C₂

semidirect

2. #22578298

C₂

semidirect

5. C₂×#14555513

C₂²

semidirect

49. C₂×D₁₃₇

M₃₂

semidirect

58. C₁₃₇

#2112

semidirect

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

subgroup	index
C₂×#17984162	2
#22578298	2
C₂²×C₈×D₁₃₇	2
#2112	137

subgroup

index

C₂×#17984162

#22578298

C₂²×C₈×D₁₃₇

#2112

137

Subgroups — #31797018

Conjugacy classes of subgroups

Complements

Maximal subgroups