Group #31796352 — subgroups

Subgroups — #31796352

Order 17536.

168 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 66 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	#22578425	C₂	1	2	yes	maximal
2	#22578267	C₂	1	2	yes	maximal, complemented
3	#22578265	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11764047	C₂²	1	4	yes	—
6	#17984145	—	2	2	no	—
7	#17984143	—	2	2	no	—
8	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic
9	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic
10	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
11	C₄×#11764040	C₄	1	1	yes	characteristic
12	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
15	C₂×#11764047	—	2	4	no	—
16	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
18	C₂×C₄×D₁₃₇	Q₈	1	1	yes	characteristic
19	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
20	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
21	C₈×D₁₃₇	—	2	2	no	—
22	C₈×D₁₃₇	—	2	2	no	—
23	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
24	C₄×Dic₁₃₇	Q₈	1	1	yes	characteristic
25	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	#11764047	—	4	4	no	—
30	C₂²×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₁₃₇	C₂×Q₈	1	1	yes	characteristic
34	C₄×D₁₃₇	—	2	2	no	—
35	C₄×D₁₃₇	—	2	1	no	—
36	C₄×D₁₃₇	—	2	1	no	—
37	C₄×D₁₃₇	—	2	2	no	—
38	#11764040	M₁₆	1	2	yes	—
39	#11764040	M₁₆	1	2	yes	—
40	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₁₃₇	—	2	1	no	—
44	C₈×C₁₃₇	—	2	1	no	—
45	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×D₁₃₇	—	2	1	no	—
47	C₂×D₁₃₇	—	2	1	no	—
48	C₂×D₁₃₇	#108	1	2	yes	—
49	C₄×C₁₃₇	—	2	1	no	—
50	C₄×C₁₃₇	—	2	1	no	—
51	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
52	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
53	Dic₁₃₇	—	2	1	no	—
54	Dic₁₃₇	—	2	1	no	—
55	Dic₁₃₇	#135	1	2	yes	—
56	C₂×C₁₃₇	#423	1	1	yes	characteristic
57	C₂×C₁₃₇	#427	1	1	yes	characteristic
58	C₂×C₁₃₇	C₂×#108	1	1	yes	characteristic
59	D₁₃₇	—	2	2	no	—
60	C₁₃₇	#1568	1	1	yes	characteristic, complemented, Sylow
61	#1568	—	137	1	no	maximal, Sylow
62	#432	—	137	2	no	—
63	C₂×C₄×C₈	—	137	1	no	—
64	#335	—	137	2	no	—
65	#334	—	137	2	no	—
66	C₂²×C₈	—	137	1	no	—
67	C₂²×C₈	—	137	1	no	—
68	C₂×C₄²	—	137	1	no	—
69	C₄⋊C₈	—	137	4	no	—
70	C₂²⋊C₈	—	274	2	no	—
71	#97	—	274	2	no	—
72	C₄×C₈	—	137	1	no	—
73	C₄×C₈	—	137	1	no	—
74	C₄×C₈	—	137	1	no	—
75	C₄×C₈	#10587739	1	1	yes	characteristic
76	C₂²×C₄	—	137	1	no	—
77	C₂²×C₄	—	137	1	no	—
78	C₂²×C₄	—	137	1	no	—
79	C₂×C₈	—	274	2	no	—
80	C₂×C₈	—	274	2	no	—
81	C₂×C₈	—	274	4	no	—
82	C₂×C₈	—	137	1	no	—
83	C₂×C₈	—	137	1	no	—
84	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
85	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
86	C₄²	—	137	1	no	—
87	C₄²	—	137	1	no	—
88	C₄²	—	137	1	no	—
89	C₄²	C₂×#10587739	1	1	yes	characteristic
90	C₂³	—	137	1	no	—
91	C₂×C₄	—	274	2	no	—
92	C₂×C₄	—	274	2	no	—
93	C₂×C₄	—	274	1	no	—
94	C₂×C₄	—	274	1	no	—
95	C₂×C₄	—	137	1	no	—
96	C₂×C₄	—	137	1	no	—
97	C₂×C₄	—	137	1	no	—
98	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
99	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
100	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₈	—	548	4	no	—
104	C₈	—	137	2	no	—
105	C₈	—	137	2	no	—
106	C₂²	—	274	1	no	—
107	C₂²	—	274	1	no	—
108	C₂²	—	137	2	no	—
109	C₂²	C₂×C₄⋊#10587739	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	274	1	no	—
113	C₄	—	274	1	no	—
114	C₄	—	137	2	no	—
115	C₄	D₁₃₇⋊M₁₆	1	1	yes	characteristic
116	C₄	D₁₃₇⋊M₁₆	1	1	yes	characteristic
117	C₂	—	274	2	no	—
118	C₂	#22578408	1	1	yes	characteristic
119	C₂	#22578413	1	1	yes	characteristic
120	C₂	D₁₃₇⋊#108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578425

C₂

yes

maximal

#22578267

C₂

yes

maximal, complemented

#22578265

C₂

yes

maximal, complemented

C₄×C₈×D₁₃₇

C₂

yes

maximal, characteristic

C₄⋊#11764047

C₂²

yes

—

#17984145

—

#17984143

—

C₂×C₈×D₁₃₇

C₄

yes

characteristic

C₂×C₈×D₁₃₇

C₄

yes

characteristic

C₄²×D₁₃₇

C₂²

yes

characteristic

C₄×#11764040

C₄

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₄

yes

characteristic

C₂×#11764047

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₇

D₄

yes

characteristic

C₂×C₄×D₁₃₇

Q₈

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₈×D₁₃₇

—

C₈×D₁₃₇

—

C₄×Dic₁₃₇

D₄

yes

characteristic

C₄×Dic₁₃₇

Q₈

yes

characteristic

C₄×Dic₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

#11764047

—

C₂²×D₁₃₇

C₄⋊C₄

yes

characteristic

C₂×Dic₁₃₇

C₄⋊C₄

yes

characteristic

C₂×Dic₁₃₇

C₂×D₄

yes

characteristic

C₂×Dic₁₃₇

C₂×Q₈

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

M₁₆

yes

—

#11764040

M₁₆

yes

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

#108

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

#135

yes

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

D₁₃₇

—

C₁₃₇

#1568

yes

characteristic, complemented, Sylow

#1568

—

137

maximal, Sylow

#432

—

137

—

C₂×C₄×C₈

—

137

—

#335

—

137

—

#334

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

274

—

#97

—

274

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

#10587739

yes

characteristic

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

yes

C₂×C₈

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

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₂²×#10587739

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

548

—

104

C₈

—

137

—

105

C₈

—

137

—

106

C₂²

—

274

—

107

C₂²

—

274

—

108

C₂²

—

137

—

109

C₂²

C₂×C₄⋊#10587739

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

274

—

113

C₄

—

274

—

114

C₄

—

137

—

115

C₄

D₁₃₇⋊M₁₆

yes

characteristic

116

C₄

D₁₃₇⋊M₁₆

yes

characteristic

117

C₂

—

274

—

118

C₂

#22578408

yes

characteristic

119

C₂

#22578413

yes

characteristic

120

C₂

D₁₃₇⋊#108

yes

characteristic

	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	C₄⋊C₄
cosocle	C₄×Dic₁₃₇	C₂³
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	#1568
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

C₄⋊C₄

cosocle

C₄×Dic₁₃₇

C₂³

C₄×C₈

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
2. #22578267	C₂	semidirect
3. #22578265	C₂	semidirect
60. C₁₃₇	#1568	semidirect

H ≅ G/N

type

2. #22578267

C₂

semidirect

3. #22578265

C₂

semidirect

60. C₁₃₇

#1568

semidirect

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

subgroup	index
#22578425	2
#22578267	2
#22578265	2
C₄×C₈×D₁₃₇	2
#1568	137

subgroup

index

C₄×C₈×D₁₃₇

137

Subgroups — #31796352

Conjugacy classes of subgroups

Complements

Maximal subgroups