Group #31780907 — subgroups

Subgroups — #31780907

Order 17536.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×#17984017	C₂	1	2	yes	maximal
2	#22577249	C₂	1	1	yes	maximal, characteristic
3	#22577158	C₂	1	1	yes	maximal, characteristic
4	#22577157	C₂	1	2	yes	maximal
5	#336×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11764040	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
9	#17984017	—	2	4	no	—
10	#17983995	C₄	1	2	yes	—
11	#17983995	C₄	1	2	yes	—
12	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₃₇	C₂×C₄	1	2	yes	—
17	C₂×#11764040	—	2	2	no	—
18	C₂×#11764040	—	2	2	no	—
19	C₂×#11764040	—	2	2	no	—
20	C₄×Dic₁₃₇	—	2	2	no	—
21	C₄×Dic₁₃₇	D₄	1	2	yes	—
22	C₄×Dic₁₃₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₃₇	—	2	2	no	—
26	C₂×C₈×C₁₃₇	—	2	2	no	—
27	C₂×C₈×C₁₃₇	—	2	2	no	—
28	C₄²×C₁₃₇	—	2	2	no	—
29	C₂×Dic₁₃₇	—	2	2	no	—
30	C₂×Dic₁₃₇	—	2	2	no	—
31	C₂×Dic₁₃₇	—	2	1	no	—
32	C₂×Dic₁₃₇	—	2	1	no	—
33	C₂×Dic₁₃₇	—	2	2	no	—
34	C₂×Dic₁₃₇	—	2	2	no	—
35	C₂×Dic₁₃₇	C₄⋊C₄	1	4	yes	—
36	#11764040	—	4	2	no	—
37	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₃₇	#41	1	2	yes	—
44	C₂×C₄×C₁₃₇	#41	1	2	yes	—
45	C₈×C₁₃₇	—	4	2	no	—
46	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₃₇	#117	1	1	yes	characteristic
48	C₂²×C₁₃₇	#117	1	1	yes	characteristic
49	C₂²×C₁₃₇	#131	1	2	yes	—
50	C₂²×C₁₃₇	#131	1	2	yes	—
51	C₄×C₁₃₇	—	4	2	no	—
52	C₄×C₁₃₇	#124	1	2	yes	—
53	C₄×C₁₃₇	#125	1	2	yes	—
54	Dic₁₃₇	—	2	4	no	—
55	Dic₁₃₇	—	4	1	no	—
56	Dic₁₃₇	—	4	1	no	—
57	C₂×C₁₃₇	#382	1	1	yes	characteristic
58	C₂×C₁₃₇	#424	1	2	yes	—
59	C₂×C₁₃₇	#433	1	2	yes	—
60	C₂×C₁₃₇	#433	1	1	yes	characteristic
61	C₂×C₁₃₇	#433	1	1	yes	characteristic
62	C₁₃₇	#1832	1	1	yes	characteristic, complemented, Sylow
63	#1832	—	137	1	no	maximal, Sylow
64	C₂×C₄⋊C₈	—	137	2	no	—
65	#376	—	137	1	no	—
66	#336	—	137	2	no	—
67	#336	—	137	1	no	—
68	#336	D₁₃₇	1	1	yes	characteristic
69	C₂²×C₈	—	137	2	no	—
70	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
71	C₂×C₄²	—	137	1	no	—
72	C₂×C₄²	—	137	1	no	—
73	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
74	C₄⋊C₈	—	274	4	no	—
75	#95	—	137	2	no	—
76	#95	—	137	2	no	—
77	C₂²×C₄	—	137	1	no	—
78	C₂²×C₄	—	137	1	no	—
79	C₂²×C₄	—	137	2	no	—
80	C₂²×C₄	C₄×D₁₃₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	274	2	no	—
86	C₂×C₈	—	274	2	no	—
87	C₂×C₈	—	274	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	274	2	no	—
90	C₄²	—	137	2	no	—
91	C₄²	—	137	2	no	—
92	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	274	2	no	—
97	C₂×C₄	—	274	2	no	—
98	C₂×C₄	—	274	1	no	—
99	C₂×C₄	—	274	1	no	—
100	C₂×C₄	—	274	2	no	—
101	C₂×C₄	—	274	2	no	—
102	C₂×C₄	—	137	4	no	—
103	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
104	C₂×C₄	Q₈×D₁₃₇	1	1	yes	characteristic
105	C₂×C₄	#14555509	1	2	yes	—
106	C₂×C₄	#14555510	1	2	yes	—
107	C₈	—	4	2	no	—
108	C₈	—	548	2	no	—
109	C₂²	C₄⋊C₄×D₁₃₇	1	1	yes	characteristic
110	C₂²	#17984044	1	1	yes	characteristic
111	C₂²	#17984045	1	1	yes	characteristic
112	C₂²	#17984094	1	2	yes	—
113	C₂²	#17984096	1	2	yes	—
114	C₄	—	4	2	no	—
115	C₄	—	274	4	no	—
116	C₄	—	548	1	no	—
117	C₄	—	548	1	no	—
118	C₄	#17984074	1	2	yes	—
119	C₄	#17984078	1	2	yes	—
120	C₂	#22577267	1	1	yes	characteristic
121	C₂	#22577428	1	2	yes	—
122	C₂	#22577461	1	2	yes	—
123	C₂	#22577462	1	1	yes	characteristic
124	C₂	#22577465	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984017

C₂

yes

maximal

#22577249

C₂

yes

maximal, characteristic

#22577158

C₂

yes

maximal, characteristic

#22577157

C₂

yes

maximal

#336×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×#11764040

C₂²

yes

—

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17984017

—

#17983995

C₄

yes

—

#17983995

C₄

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₂²×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₇

C₂×C₄

yes

—

C₂×#11764040

—

C₂×#11764040

—

C₂×#11764040

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

D₄

yes

—

C₄×Dic₁₃₇

Q₈

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄²×C₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

#11764040

—

C₂³×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

—

C₂²×C₁₃₇

#131

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#124

yes

—

C₄×C₁₃₇

#125

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

#382

yes

characteristic

C₂×C₁₃₇

#424

yes

—

C₂×C₁₃₇

#433

yes

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1832

—

137

maximal, Sylow

C₂×C₄⋊C₈

—

137

—

#376

—

137

—

#336

—

137

—

#336

—

137

—

#336

D₁₃₇

yes

characteristic

C₂²×C₈

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

C₄⋊C₈

—

274

—

#95

—

137

—

#95

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₄×D₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₄²

—

C₄²

—

274

—

C₄²

—

137

—

C₄²

—

137

—

C₂³

C₂×C₄×D₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

100

C₂×C₄

—

274

—

101

C₂×C₄

—

274

—

102

C₂×C₄

—

137

—

103

C₂×C₄

D₄×D₁₃₇

yes

characteristic

104

C₂×C₄

Q₈×D₁₃₇

yes

characteristic

105

C₂×C₄

#14555509

yes

—

106

C₂×C₄

#14555510

yes

—

107

C₈

—

108

C₈

—

548

—

109

C₂²

C₄⋊C₄×D₁₃₇

yes

characteristic

110

C₂²

#17984044

yes

characteristic

111

C₂²

#17984045

yes

characteristic

112

C₂²

#17984094

yes

—

113

C₂²

#17984096

yes

—

114

C₄

—

115

C₄

—

274

—

116

C₄

—

548

—

117

C₄

—

548

—

118

C₄

#17984074

yes

—

119

C₄

#17984078

yes

—

120

C₂

#22577267

yes

characteristic

121

C₂

#22577428

yes

—

122

C₂

#22577461

yes

—

123

C₂

#22577462

yes

characteristic

124

C₂

#22577465

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₇
commutator, socle, prime radical	C₂³×C₁₃₇	C₂²×C₄
Fitting	#336×C₁₃₇	C₂
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#336	D₁₃₇
hypocenter	C₁₃₇	#1832
LCS	G ▷ C₂³×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃₇

commutator, socle, prime radical

C₂³×C₁₃₇

C₂²×C₄

Fitting

#336×C₁₃₇

C₂

cosocle

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂³×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
62. C₁₃₇	#1832	semidirect

H ≅ G/N

type

62. C₁₃₇

#1832

semidirect

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

subgroup	index
C₂×#17984017	2
#22577249	2
#22577158	2
#22577157	2
#336×C₁₃₇	2
#1832	137

subgroup

index

#336×C₁₃₇

137

Subgroups — #31780907

Conjugacy classes of subgroups

Complements

Maximal subgroups