Group #31779564 — subgroups

Subgroups — #31779564

Order 17536.

168 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22577427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11764040	C₂	1	1	yes	maximal, characteristic
3	#22577156	C₂	1	4	yes	maximal, complemented
4	#431×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555494	—	2	1	no	—
6	C₂²×#11764040	—	2	1	no	—
7	C₄⋊#11764040	—	2	1	no	—
8	C₄⋊#11764040	—	2	1	no	—
9	C₄⋊#11764040	C₂²	1	2	yes	—
10	C₄⋊#11764040	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₃₇	C₄	1	2	yes	—
13	C₂²⋊C₈×C₁₃₇	C₄	1	2	yes	—
14	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
15	#14555494	—	4	2	no	—
16	C₂×#11764040	—	2	2	no	—
17	C₂×#11764040	—	2	2	no	—
18	C₂×#11764040	—	4	2	no	—
19	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₇	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
23	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
25	C₄²×C₁₃₇	Q₈	1	1	yes	characteristic
26	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
27	#11764040	—	4	2	no	—
28	#11764040	—	4	2	no	—
29	C₂³×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	—	2	2	no	—
31	C₂×C₄×C₁₃₇	—	4	1	no	—
32	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	D₈	1	2	yes	—
36	C₂×C₄×C₁₃₇	Q₁₆	1	2	yes	—
37	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₁₃₇	—	2	4	no	—
41	C₂²×C₁₃₇	—	2	1	no	—
42	C₂²×C₁₃₇	—	2	1	no	—
43	C₂²×C₁₃₇	#107	1	2	yes	—
44	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₁₃₇	—	2	1	no	—
46	C₄×C₁₃₇	—	2	1	no	—
47	C₄×C₁₃₇	—	2	1	no	—
48	C₄×C₁₃₇	#107	1	2	yes	—
49	C₄×C₁₃₇	#131	1	2	yes	—
50	C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
51	C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₁₃₇	—	2	2	no	—
53	C₂×C₁₃₇	#424	1	1	yes	characteristic
54	C₂×C₁₃₇	C₂×#107	1	1	yes	characteristic
55	C₂×C₁₃₇	#430	1	1	yes	characteristic
56	C₁₃₇	#1571	1	1	yes	characteristic, complemented, Sylow
57	#1571	—	137	1	no	maximal, Sylow
58	#431	D₁₃₇	1	1	yes	characteristic
59	#423	—	137	1	no	—
60	C₂×C₄⋊C₈	—	137	1	no	—
61	#335	—	137	4	no	—
62	C₂×M₁₆	—	274	1	no	—
63	C₂²×C₈	—	274	1	no	—
64	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
65	C₄⋊C₈	—	274	1	no	—
66	C₄⋊C₈	—	274	1	no	—
67	C₄⋊C₈	—	137	2	no	—
68	C₄⋊C₈	—	137	2	no	—
69	C₄⋊C₈	Dic₁₃₇	1	2	yes	—
70	C₂²⋊C₈	Dic₁₃₇	1	2	yes	—
71	C₄×C₈	C₂×D₁₃₇	1	2	yes	—
72	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
73	C₂²×C₄	D₅₄₈	1	1	yes	characteristic
74	C₂²×C₄	Dic₂₇₄	1	1	yes	characteristic
75	M₁₆	—	548	2	no	—
76	C₂×C₈	—	274	2	no	—
77	C₂×C₈	—	274	2	no	—
78	C₂×C₈	—	548	2	no	—
79	C₂×C₈	C₂×Dic₁₃₇	1	4	yes	—
80	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
81	C₄²	D₅₄₈	1	1	yes	characteristic
82	C₄²	Dic₂₇₄	1	1	yes	characteristic
83	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₁₃₇	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	C₄⋊Dic₁₃₇	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Dic₁₃₇	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₁₃₇	1	1	yes	characteristic
90	C₂×C₄	D₁₀₉₆	1	2	yes	—
91	C₂×C₄	Dic₅₄₈	1	2	yes	—
92	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
93	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	548	2	no	—
97	C₈	—	548	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#17984022	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#17984022	1	2	yes	—
106	C₄	#17984095	1	2	yes	—
107	C₄	C₂×D₁₀₉₆	1	1	yes	characteristic
108	C₄	C₂×Dic₅₄₈	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#22577430	1	1	yes	characteristic
111	C₂	C₂×#17984022	1	1	yes	characteristic
112	C₂	#22577452	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577427

C₂

yes

maximal, characteristic

C₂×C₄⋊#11764040

C₂

yes

maximal, characteristic

#22577156

C₂

yes

maximal, complemented

#431×C₁₃₇

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

C₄⋊#11764040

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

C₄

yes

—

C₂²⋊C₈×C₁₃₇

C₄

yes

—

C₄×C₈×C₁₃₇

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

D₄

yes

characteristic

C₂²×C₄×C₁₃₇

Q₈

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

D₄

yes

characteristic

C₄²×C₁₃₇

Q₈

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

characteristic

#11764040

—

#11764040

—

C₂³×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

D₈

yes

—

C₂×C₄×C₁₃₇

Q₁₆

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#107

yes

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#107

yes

—

C₄×C₁₃₇

#131

yes

—

C₄×C₁₃₇

C₂×D₈

yes

characteristic

C₄×C₁₃₇

C₂×Q₁₆

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1571

—

137

maximal, Sylow

#431

D₁₃₇

yes

characteristic

#423

—

137

—

C₂×C₄⋊C₈

—

137

—

#335

—

137

—

C₂×M₁₆

—

274

—

C₂²×C₈

—

274

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

C₄⋊C₈

—

274

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

Dic₁₃₇

yes

—

C₂²⋊C₈

Dic₁₃₇

yes

—

C₄×C₈

C₂×D₁₃₇

yes

—

C₂²×C₄

C₂×Dic₁₃₇

yes

characteristic

C₂²×C₄

D₅₄₈

yes

characteristic

C₂²×C₄

Dic₂₇₄

yes

characteristic

M₁₆

—

548

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

548

—

C₂×C₈

C₂×Dic₁₃₇

yes

—

C₄²

C₂×Dic₁₃₇

yes

characteristic

C₄²

D₅₄₈

yes

characteristic

C₄²

Dic₂₇₄

yes

characteristic

C₄²

C₂²×D₁₃₇

yes

characteristic

C₂³

C₄⋊Dic₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₃₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₃₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₃₇

yes

characteristic

C₂×C₄

D₁₀₉₆

yes

—

C₂×C₄

Dic₅₄₈

yes

—

C₂×C₄

C₂²×Dic₁₃₇

yes

characteristic

C₂×C₄

C₂×D₅₄₈

yes

characteristic

C₂×C₄

C₂×Dic₂₇₄

yes

characteristic

C₈

—

C₈

—

548

—

C₈

—

548

—

C₂²

—

C₂²

—

100

C₂²

#17984022

yes

—

101

C₂²

C₂×C₄⋊Dic₁₃₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#17984022

yes

—

106

C₄

#17984095

yes

—

107

C₄

C₂×D₁₀₉₆

yes

characteristic

108

C₄

C₂×Dic₅₄₈

yes

characteristic

109

C₂

—

110

C₂

#22577430

yes

characteristic

111

C₂

C₂×#17984022

yes

characteristic

112

C₂

#22577452

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#431×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₇	C₄⋊C₄
cosocle	C₄²×C₁₃₇	C₂³
hypercenter	#431	D₁₃₇
hypocenter	C₁₃₇	#1571
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₅₄₈

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#431×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₃₇

C₄⋊C₄

cosocle

C₄²×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #22577156	C₂	semidirect
56. C₁₃₇	#1571	semidirect

H ≅ G/N

type

3. #22577156

C₂

semidirect

56. C₁₃₇

#1571

semidirect

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

subgroup	index
#22577427	2
C₂×C₄⋊#11764040	2
#22577156	2
#431×C₁₃₇	2
#1571	137

subgroup

index

#431×C₁₃₇

137

Subgroups — #31779564

Conjugacy classes of subgroups

Complements

Maximal subgroups