Group #31796729 — subgroups

Subgroups — #31796729

Order 17536.

220 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 90 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×#17984159	C₂	1	2	yes	maximal
2	C₂×#17984158	C₂	1	2	yes	maximal
3	#22578408	C₂	1	2	yes	maximal
4	C₄×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555520	C₂²	1	4	yes	—
6	#17984159	—	2	4	no	—
7	#17984158	—	2	4	no	—
8	#17984155	—	2	4	no	—
9	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
11	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
12	C₄×#11764040	C₄	1	1	yes	characteristic
13	C₈×Dic₁₃₇	C₄	1	1	yes	characteristic
14	C₈×Dic₁₃₇	C₄	1	1	yes	characteristic
15	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
16	#14555520	—	2	4	no	—
17	#14555520	—	2	4	no	—
18	C₂×#11764047	—	2	4	no	—
19	C₂×C₄×D₁₃₇	Q₈	1	2	yes	—
20	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
21	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
22	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
23	C₈×D₁₃₇	D₄	1	2	yes	—
24	C₈×D₁₃₇	D₄	1	2	yes	—
25	C₈×D₁₃₇	Q₈	1	2	yes	—
26	C₈×D₁₃₇	Q₈	1	2	yes	—
27	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₃₇	D₄	1	2	yes	—
29	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
32	#11764047	—	4	4	no	—
33	C₂²×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
36	C₄×D₁₃₇	—	2	4	no	—
37	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
38	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
39	C₄×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
40	C₄×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
41	#11764040	C₄⋊C₄	1	2	yes	—
42	#11764040	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₁₃₇	—	2	2	no	—
49	C₂×D₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×D₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₃₇	—	2	2	no	—
52	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	Dic₁₃₇	#108	1	4	yes	complemented
55	Dic₁₃₇	#127	1	1	yes	characteristic
56	Dic₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₁₃₇	#384	1	1	yes	characteristic
58	C₂×C₁₃₇	C₂×#108	1	2	yes	—
59	D₁₃₇	—	2	1	no	—
60	D₁₃₇	—	2	1	no	—
61	C₁₃₇	#1848	1	1	yes	characteristic, complemented, Sylow
62	#1848	—	137	1	no	maximal, Sylow
63	C₂×#108	—	137	2	no	—
64	C₂×#108	—	137	2	no	—
65	#423	—	137	2	no	—
66	C₂×C₄×C₈	—	137	1	no	—
67	C₂×M₁₆	—	137	4	no	—
68	C₂²×C₈	—	137	1	no	—
69	C₂²×C₈	—	137	1	no	—
70	C₂×C₄²	—	137	1	no	—
71	#108	—	274	4	no	—
72	#108	—	274	4	no	—
73	C₄⋊C₈	—	274	4	no	—
74	C₄×C₈	—	137	1	no	—
75	C₄×C₈	—	137	1	no	—
76	C₄×C₈	—	137	1	no	—
77	C₄×C₈	#10587739	1	1	yes	characteristic
78	C₂²×C₄	—	137	2	no	—
79	C₂²×C₄	—	137	1	no	—
80	M₁₆	—	274	4	no	—
81	M₁₆	—	274	4	no	—
82	C₂×C₈	—	274	4	no	—
83	C₂×C₈	—	137	1	no	—
84	C₂×C₈	—	137	1	no	—
85	C₂×C₈	—	137	2	no	—
86	C₂×C₈	—	137	2	no	—
87	C₂×C₈	—	137	2	no	—
88	C₂×C₈	—	137	2	no	—
89	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
90	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
91	C₄²	—	137	2	no	—
92	C₄²	—	137	1	no	—
93	C₄²	C₂×#10587739	1	1	yes	characteristic
94	C₂³	—	137	1	no	—
95	C₂×C₄	—	274	4	no	—
96	C₂×C₄	—	137	1	no	—
97	C₂×C₄	—	137	2	no	—
98	C₂×C₄	—	137	1	no	—
99	C₂×C₄	—	137	1	no	—
100	C₂×C₄	—	137	1	no	—
101	C₂×C₄	—	137	1	no	—
102	C₂×C₄	C₄⋊#10587739	1	2	yes	—
103	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
104	C₈	—	548	4	no	—
105	C₈	—	137	2	no	—
106	C₈	—	137	2	no	—
107	C₈	C₄⋊#10587739	1	2	yes	—
108	C₈	C₄⋊#10587739	1	2	yes	—
109	C₂²	—	274	2	no	—
110	C₂²	—	137	1	no	—
111	C₂²	—	137	1	no	—
112	C₂²	C₂×C₄⋊#10587739	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	137	1	no	—
115	C₄	—	137	1	no	—
116	C₄	—	137	4	no	—
117	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
118	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
119	C₂	—	274	1	no	—
120	C₂	—	274	1	no	—
121	C₂	#22578349	1	1	yes	characteristic
122	C₂	D₁₃₇⋊#108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984159

C₂

yes

maximal

C₂×#17984158

C₂

yes

maximal

#22578408

C₂

yes

maximal

C₄×C₈×D₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555520

C₂²

yes

—

#17984159

—

#17984158

—

#17984155

—

C₂×C₈×D₁₃₇

C₂²

yes

characteristic

C₂×C₈×D₁₃₇

C₂²

yes

characteristic

C₄²×D₁₃₇

C₂²

yes

characteristic

C₄×#11764040

C₄

yes

characteristic

C₈×Dic₁₃₇

C₄

yes

characteristic

C₈×Dic₁₃₇

C₄

yes

characteristic

C₄×C₈×C₁₃₇

C₄

yes

—

—

—

C₂×C₄×D₁₃₇

Q₈

yes

—

C₂×C₄×D₁₃₇

C₂³

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₈×D₁₃₇

D₄

yes

—

C₈×D₁₃₇

D₄

yes

—

C₈×D₁₃₇

Q₈

yes

—

C₈×D₁₃₇

Q₈

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

D₄

yes

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

#11764047

—

C₂²×D₁₃₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

C₂×Dic₁₃₇

C₂×D₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

C₂×D₄

yes

characteristic

C₄×D₁₃₇

C₂×D₄

yes

characteristic

C₄×D₁₃₇

C₂×Q₈

yes

characteristic

C₄×D₁₃₇

C₂×Q₈

yes

characteristic

#11764040

C₄⋊C₄

yes

—

#11764040

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

C₄⋊C₄

yes

—

C₈×C₁₃₇

C₄⋊C₄

yes

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

C₄⋊Q₈

yes

characteristic

C₂×D₁₃₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

Dic₁₃₇

yes

Dic₁₃₇

yes

Dic₁₃₇

C₄⋊Q₈

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

—

D₁₃₇

—

D₁₃₇

—

C₁₃₇

#1848

yes

characteristic, complemented, Sylow

#1848

—

137

maximal, Sylow

C₂×#108

—

137

—

C₂×#108

—

137

—

#423

—

137

—

C₂×C₄×C₈

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

C₂×C₄²

—

137

—

#108

—

274

—

#108

—

274

—

C₄⋊C₈

—

274

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

#10587739

yes

characteristic

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

M₁₆

—

274

—

M₁₆

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

100

C₂×C₄

—

137

—

101

C₂×C₄

—

137

—

102

C₂×C₄

C₄⋊#10587739

yes

—

103

C₂×C₄

C₂²×#10587739

yes

characteristic

104

C₈

—

548

—

105

C₈

—

137

—

106

C₈

—

137

—

107

C₈

C₄⋊#10587739

yes

—

108

C₈

C₄⋊#10587739

yes

—

109

C₂²

—

274

—

110

C₂²

—

137

—

111

C₂²

—

137

—

112

C₂²

C₂×C₄⋊#10587739

yes

characteristic

113

C₄

—

114

C₄

—

137

—

115

C₄

—

137

—

116

C₄

—

137

—

117

C₄

C₂×C₄⋊#10587739

yes

characteristic

118

C₄

C₂×C₄⋊#10587739

yes

characteristic

119

C₂

—

274

—

120

C₂

—

274

—

121

C₂

#22578349

yes

characteristic

122

C₂

D₁₃₇⋊#108

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄×C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₃₇	C₂×Q₈
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	#1848
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#10587739

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10587739

Fitting

C₄×C₈×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂×C₄⋊C₄

prime radical

C₂²×D₁₃₇

C₂×Q₈

cosocle

C₂×C₄×D₁₃₇

C₂³

C₄×C₈

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
54. Dic₁₃₇	#108	semidirect
61. C₁₃₇	#1848	semidirect

H ≅ G/N

type

54. Dic₁₃₇

#108

semidirect

61. C₁₃₇

#1848

semidirect

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

subgroup	index
C₂×#17984159	2
C₂×#17984158	2
#22578408	2
C₄×C₈×D₁₃₇	2
#1848	137

subgroup

index

C₄×C₈×D₁₃₇

137

Subgroups — #31796729

Conjugacy classes of subgroups

Complements

Maximal subgroups