Group #31779613 — subgroups

Subgroups — #31779613

Order 17536.

192 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 87 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	D₄×#11764040	C₂	1	2	yes	maximal, complemented
2	#22577154	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#11764040	C₂	1	2	yes	maximal, complemented
4	#22577124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11764040	—	2	2	no	—
7	C₄⋊#11764040	C₂²	1	2	yes	—
8	C₂²⋊#11764040	—	2	2	no	—
9	C₄×#11764040	C₂²	1	1	yes	characteristic
10	C₄×#11764040	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₁₃₇	C₄	1	1	yes	characteristic
12	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
13	#107×C₁₃₇	C₄	1	1	yes	characteristic
14	#102×C₁₃₇	C₄	1	2	yes	—
15	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#11764040	—	2	2	no	—
17	C₂×#11764040	—	2	1	no	—
18	C₂×#11764040	—	4	4	no	—
19	C₂×#11764040	D₄	1	2	yes	—
20	C₂×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₃₇	—	2	2	no	—
22	D₈×C₁₃₇	C₈	1	4	yes	complemented
23	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
27	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
28	#11764040	—	2	2	no	—
29	#11764040	—	4	1	no	—
30	#11764040	—	4	2	no	—
31	C₂³×C₁₃₇	—	2	2	no	—
32	D₄×C₁₃₇	—	2	2	no	—
33	D₄×C₁₃₇	C₂×C₈	1	4	yes	complemented
34	C₂×C₄×C₁₃₇	—	2	2	no	—
35	C₂×C₄×C₁₃₇	—	4	2	no	—
36	C₂×C₄×C₁₃₇	—	4	2	no	—
37	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
40	C₈×C₁₃₇	—	2	1	no	—
41	C₈×C₁₃₇	C₂×C₈	1	2	yes	complemented
42	C₂²×C₁₃₇	—	2	4	no	—
43	C₂²×C₁₃₇	—	4	2	no	—
44	C₂²×C₁₃₇	—	4	2	no	—
45	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
46	C₄×C₁₃₇	—	2	1	no	—
47	C₄×C₁₃₇	—	4	2	no	—
48	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₁₃₇	#131	1	1	yes	characteristic
50	C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
51	C₂×C₁₃₇	—	4	4	no	—
52	C₂×C₁₃₇	C₈×D₄	1	1	yes	characteristic
53	C₂×C₁₃₇	#442	1	1	yes	characteristic
54	C₂×C₁₃₇	#444	1	1	yes	characteristic
55	C₁₃₇	#1580	1	1	yes	characteristic, complemented, Sylow
56	#1580	—	137	1	no	maximal, Sylow
57	C₄×D₈	D₁₃₇	1	1	yes	characteristic
58	C₈×D₄	—	137	2	no	—
59	#334	—	137	1	no	—
60	D₄⋊C₈	—	137	2	no	—
61	#322	—	137	1	no	—
62	C₂×D₈	Dic₁₃₇	1	1	yes	characteristic
63	C₂²×C₈	—	274	2	no	—
64	C₄×D₄	C₂×D₁₃₇	1	2	yes	—
65	#107	Dic₁₃₇	1	1	yes	characteristic
66	C₄⋊C₈	—	137	2	no	—
67	#102	Dic₁₃₇	1	2	yes	—
68	C₂²⋊C₈	—	274	2	no	—
69	C₄×C₈	—	137	1	no	—
70	C₄×C₈	—	137	1	no	—
71	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
72	C₂×D₄	C₂×Dic₁₃₇	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	D₈	#11764040	1	4	yes	complemented
75	C₂×C₈	—	274	2	no	—
76	C₂×C₈	—	274	1	no	—
77	C₂×C₈	—	548	4	no	—
78	C₂×C₈	—	137	2	no	—
79	C₂×C₈	C₂×Dic₁₃₇	1	1	yes	characteristic
80	C₂×C₈	C₂×Dic₁₃₇	1	1	yes	characteristic
81	C₄⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	C₂×#11764040	1	4	yes	complemented
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
91	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
92	C₂×C₄	#14555509	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	274	2	no	—
95	C₈	—	548	1	no	—
96	C₈	—	548	2	no	—
97	C₈	C₂×#11764040	1	2	yes	complemented
98	C₂²	—	2	4	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×Dic₁₃₇	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	C₂²×#11764040	1	1	yes	characteristic
105	C₄	#17984095	1	1	yes	characteristic
106	C₄	#17984112	1	2	yes	—
107	C₂	—	4	4	no	—
108	C₂	D₄×#11764040	1	1	yes	characteristic
109	C₂	#22577515	1	1	yes	characteristic
110	C₂	#22577525	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×#11764040

C₂

yes

maximal, complemented

#22577154

C₂

yes

maximal, characteristic, complemented

D₄⋊#11764040

C₂

yes

maximal, complemented

#22577124

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×#11764040

—

C₄⋊#11764040

C₂²

yes

—

C₂²⋊#11764040

—

C₄×#11764040

C₂²

yes

characteristic

C₄×#11764040

C₂²

yes

characteristic

C₂×D₈×C₁₃₇

C₄

yes

characteristic

C₄×D₄×C₁₃₇

C₂²

yes

—

#107×C₁₃₇

C₄

yes

characteristic

#102×C₁₃₇

C₄

yes

—

C₄×C₈×C₁₃₇

C₂²

yes

—

—

—

D₄

yes

—

C₂×D₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

—

D₈×C₁₃₇

C₈

yes

complemented

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

—

—

—

C₂³×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

C₂×C₈

yes

complemented

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

C₂×C₈

yes

complemented

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂²×C₈

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₈×D₄

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1580

—

137

maximal, Sylow

C₄×D₈

D₁₃₇

yes

characteristic

C₈×D₄

—

137

—

#334

—

137

—

D₄⋊C₈

—

137

—

#322

—

137

—

C₂×D₈

Dic₁₃₇

yes

characteristic

C₂²×C₈

—

274

—

C₄×D₄

C₂×D₁₃₇

yes

—

#107

Dic₁₃₇

yes

characteristic

C₄⋊C₈

—

137

—

#102

Dic₁₃₇

yes

—

C₂²⋊C₈

—

274

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₂×D₄

C₂×Dic₁₃₇

yes

—

C₂²×C₄

—

D₈

#11764040

yes

complemented

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

548

—

C₂×C₈

—

137

—

C₂×C₈

C₂×Dic₁₃₇

yes

characteristic

C₂×C₈

C₂×Dic₁₃₇

yes

characteristic

C₄⋊C₄

C₂×Dic₁₃₇

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₁₃₇

yes

characteristic

C₂³

—

D₄

—

D₄

C₂×#11764040

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₁₃₇

yes

characteristic

C₂×C₄

D₄×D₁₃₇

yes

characteristic

C₂×C₄

#14555509

yes

characteristic

C₈

—

C₈

—

274

—

C₈

—

548

—

C₈

—

548

—

C₈

C₂×#11764040

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×Dic₁₃₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂²×#11764040

yes

characteristic

105

C₄

#17984095

yes

characteristic

106

C₄

#17984112

yes

—

107

C₂

—

108

C₂

D₄×#11764040

yes

characteristic

109

C₂

#22577515

yes

characteristic

110

C₂

#22577525

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₇
commutator	C₄×C₁₃₇	C₂²×C₈
Frattini	C₄²	C₂²×D₁₃₇
Fitting	C₄×D₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₄×D₄
prime radical	C₂×D₈×C₁₃₇	C₄
cosocle	C₄²×C₁₃₇	C₂³
hypercenter	C₄×D₈	D₁₃₇
hypocenter	C₁₃₇	#1580
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₇

commutator

C₄×C₁₃₇

C₂²×C₈

Frattini

C₄²

C₂²×D₁₃₇

Fitting

C₄×D₈×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₄×D₄

prime radical

C₂×D₈×C₁₃₇

C₄

cosocle

C₄²×C₁₃₇

C₂³

hypercenter

C₄×D₈

D₁₃₇

hypocenter

C₁₃₇

#1580

LCS

G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. D₄×#11764040	C₂	semidirect
2. #22577154	C₂	semidirect
3. D₄⋊#11764040	C₂	semidirect
4. #22577124	C₂	semidirect
22. D₈×C₁₃₇	C₈	semidirect
33. D₄×C₁₃₇	C₂×C₈	semidirect
41. C₈×C₁₃₇	C₂×C₈	semidirect
55. C₁₃₇	#1580	semidirect
74. D₈	#11764040	semidirect
86. D₄	C₂×#11764040	semidirect
97. C₈	C₂×#11764040	semidirect

H ≅ G/N

type

1. D₄×#11764040

C₂

semidirect

2. #22577154

C₂

semidirect

3. D₄⋊#11764040

C₂

semidirect

4. #22577124

C₂

semidirect

22. D₈×C₁₃₇

C₈

semidirect

33. D₄×C₁₃₇

C₂×C₈

semidirect

41. C₈×C₁₃₇

C₂×C₈

semidirect

55. C₁₃₇

#1580

semidirect

74. D₈

#11764040

semidirect

86. D₄

C₂×#11764040

semidirect

97. C₈

C₂×#11764040

semidirect

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

subgroup	index
D₄×#11764040	2
#22577154	2
D₄⋊#11764040	2
#22577124	2
C₄×D₈×C₁₃₇	2
#1580	137

subgroup

index

C₄×D₈×C₁₃₇

137

Subgroups — #31779613

Conjugacy classes of subgroups

Complements

Maximal subgroups