Group #36334652 — subgroups

Subgroups — #36334652

Order 21376.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 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	#24923826	C₂	1	2	yes	maximal
2	#24923674	C₂	1	2	yes	maximal, complemented
3	#24923671	C₂	1	4	yes	maximal, complemented
4	#24923667	C₂	1	2	yes	maximal, complemented
5	#24923307	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
7	#19128832	C₂²	1	4	yes	—
8	#19128824	C₂²	1	4	yes	—
9	#19128811	—	2	4	no	—
10	C₄×Dic₃₃₄	—	2	4	no	—
11	#19128791	C₂²	1	4	yes	—
12	#19128766	C₂²	1	8	yes	—
13	#19128764	C₂²	1	8	yes	—
14	C₂²×C₈×C₁₆₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₁₆₇	C₂²	1	2	yes	—
18	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₃₄	—	2	4	no	—
20	C₄⋊Dic₁₆₇	—	2	4	no	—
21	C₄⋊Dic₁₆₇	C₂³	1	8	yes	—
22	Dic₁₆₇⋊C₄	—	2	8	no	—
23	C₂²⋊Dic₁₆₇	—	2	4	no	—
24	C₄×Dic₁₆₇	—	2	4	no	—
25	C₂²×C₄×C₁₆₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₆₇	Q₈	1	8	yes	complemented
28	C₂×C₈×C₁₆₇	C₂³	1	2	yes	—
29	C₂×C₈×C₁₆₇	C₂³	1	2	yes	—
30	C₄²×C₁₆₇	D₄	1	2	yes	—
31	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
32	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
33	Dic₃₃₄	—	4	8	no	—
34	C₂×Dic₁₆₇	—	2	8	no	—
35	C₂³×C₁₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₆₇	—	2	4	no	—
37	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₆₇	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
42	C₈×C₁₆₇	C₂×Q₈	1	4	yes	—
43	C₈×C₁₆₇	C₂×Q₈	1	4	yes	—
44	C₂²×C₁₆₇	—	2	2	no	—
45	C₂²×C₁₆₇	C₄⋊Q₈	1	2	yes	—
46	C₂²×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₁₆₇	—	2	2	no	—
48	C₄×C₁₆₇	C₄⋊Q₈	1	2	yes	—
49	C₄×C₁₆₇	#135	1	4	yes	—
50	C₄×C₁₆₇	C₂²×Q₈	1	2	yes	—
51	Dic₁₆₇	—	4	8	no	—
52	C₂×C₁₆₇	—	2	2	no	—
53	C₂×C₁₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₁₆₇	C₂×#135	1	2	yes	—
55	C₁₆₇	#3160	1	1	yes	characteristic, complemented, Sylow
56	#3160	—	167	1	no	maximal, Sylow
57	#533	—	167	2	no	—
58	#500	—	167	2	no	—
59	#499	—	167	4	no	—
60	#498	—	167	2	no	—
61	#427	—	167	4	no	—
62	C₂×C₄×C₈	D₁₆₇	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₁₆₇	1	2	yes	—
64	C₄⋊Q₈	—	167	4	no	—
65	#125	—	167	4	no	—
66	C₂²⋊Q₈	—	334	4	no	—
67	C₄×Q₈	—	334	4	no	—
68	#117	—	167	4	no	—
69	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
70	#107	—	167	8	no	—
71	#106	—	167	8	no	—
72	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
73	C₄×C₈	C₂×D₁₆₇	1	2	yes	—
74	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
75	C₂×Q₈	—	334	4	no	—
76	C₂²×C₄	D₆₆₈	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
78	C₂×C₈	Dic₃₃₄	1	8	yes	complemented
79	C₂×C₈	C₂²×D₁₆₇	1	2	yes	—
80	C₂×C₈	C₂²×D₁₆₇	1	2	yes	—
81	C₄⋊C₄	—	334	8	no	—
82	C₄⋊C₄	—	334	4	no	—
83	C₄⋊C₄	—	167	8	no	—
84	C₂²⋊C₄	—	334	4	no	—
85	C₄²	—	334	4	no	—
86	C₄²	D₆₆₈	1	2	yes	—
87	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
88	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
89	C₂³	C₂×D₆₆₈	1	1	yes	characteristic
90	Q₈	—	668	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	334	8	no	—
93	C₂×C₄	C₂×D₆₆₈	1	2	yes	—
94	C₂×C₄	C₂×D₆₆₈	1	2	yes	—
95	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
96	C₂×C₄	C₂×Dic₃₃₄	1	4	yes	—
97	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
98	C₈	C₂×Dic₃₃₄	1	4	yes	—
99	C₈	C₂×Dic₃₃₄	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#19128832	1	2	yes	—
102	C₂²	C₂²×D₆₆₈	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	668	8	no	—
105	C₄	#19128832	1	2	yes	—
106	C₄	#19128854	1	4	yes	—
107	C₄	C₂²×Dic₃₃₄	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#19128832	1	1	yes	characteristic
110	C₂	C₂×#19128854	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24923826

C₂

yes

maximal

#24923674

C₂

yes

maximal, complemented

#24923671

C₂

yes

maximal, complemented

#24923667

C₂

yes

maximal, complemented

#24923307

C₂

yes

maximal

C₂×C₄×C₈×C₁₆₇

C₂

yes

maximal, characteristic

#19128832

C₂²

yes

—

#19128824

C₂²

yes

—

#19128811

—

C₄×Dic₃₃₄

—

#19128791

C₂²

yes

—

#19128766

C₂²

yes

—

#19128764

C₂²

yes

—

C₂²×C₈×C₁₆₇

C₂²

yes

—

C₂×C₄²×C₁₆₇

C₂²

yes

characteristic

C₄×C₈×C₁₆₇

C₂²

yes

characteristic

C₄×C₈×C₁₆₇

C₂²

yes

—

C₄×C₈×C₁₆₇

C₂²

yes

characteristic

C₂×Dic₃₃₄

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

C₂³

yes

—

Dic₁₆₇⋊C₄

—

C₂²⋊Dic₁₆₇

—

C₄×Dic₁₆₇

—

C₂²×C₄×C₁₆₇

D₄

yes

—

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₈×C₁₆₇

Q₈

yes

complemented

C₂×C₈×C₁₆₇

C₂³

yes

—

C₂×C₈×C₁₆₇

C₂³

yes

—

C₄²×C₁₆₇

D₄

yes

—

C₄²×C₁₆₇

C₂³

yes

characteristic

C₄²×C₁₆₇

C₂³

yes

characteristic

Dic₃₃₄

—

C₂×Dic₁₆₇

—

C₂³×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×Q₈

yes

—

C₂×C₄×C₁₆₇

C₂⁴

yes

characteristic

C₈×C₁₆₇

C₂×Q₈

yes

—

C₈×C₁₆₇

C₂×Q₈

yes

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₄⋊Q₈

yes

—

C₂²×C₁₆₇

C₂²×D₄

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

C₄⋊Q₈

yes

—

C₄×C₁₆₇

#135

yes

—

C₄×C₁₆₇

C₂²×Q₈

yes

—

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₆₇

C₂×#135

yes

—

C₁₆₇

#3160

yes

characteristic, complemented, Sylow

#3160

—

167

maximal, Sylow

#533

—

167

—

#500

—

167

—

#499

—

167

—

#498

—

167

—

#427

—

167

—

C₂×C₄×C₈

D₁₆₇

yes

characteristic

C₂²×C₈

C₂×D₁₆₇

yes

—

C₄⋊Q₈

—

167

—

#125

—

167

—

C₂²⋊Q₈

—

334

—

C₄×Q₈

—

334

—

#117

—

167

—

C₂×C₄²

C₂×D₁₆₇

yes

characteristic

#107

—

167

—

#106

—

167

—

C₄×C₈

C₂×D₁₆₇

yes

characteristic

C₄×C₈

C₂×D₁₆₇

yes

—

C₄×C₈

C₂×D₁₆₇

yes

characteristic

C₂×Q₈

—

334

—

C₂²×C₄

D₆₆₈

yes

—

C₂²×C₄

C₂²×D₁₆₇

yes

characteristic

C₂×C₈

Dic₃₃₄

yes

complemented

C₂×C₈

C₂²×D₁₆₇

yes

—

C₂×C₈

C₂²×D₁₆₇

yes

—

C₄⋊C₄

—

334

—

C₄⋊C₄

—

334

—

C₄⋊C₄

—

167

—

C₂²⋊C₄

—

334

—

C₄²

—

334

—

C₄²

D₆₆₈

yes

—

C₄²

C₂²×D₁₆₇

yes

characteristic

C₄²

C₂²×D₁₆₇

yes

characteristic

C₂³

C₂×D₆₆₈

yes

characteristic

Q₈

—

668

—

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

C₂×D₆₆₈

yes

—

C₂×C₄

C₂×D₆₆₈

yes

—

C₂×C₄

C₂×D₆₆₈

yes

characteristic

C₂×C₄

C₂×Dic₃₃₄

yes

—

C₂×C₄

C₂³×D₁₆₇

yes

characteristic

C₈

C₂×Dic₃₃₄

yes

—

C₈

C₂×Dic₃₃₄

yes

—

100

C₂²

—

101

C₂²

#19128832

yes

—

102

C₂²

C₂²×D₆₆₈

yes

characteristic

103

C₄

—

104

C₄

—

668

—

105

C₄

#19128832

yes

—

106

C₄

#19128854

yes

—

107

C₄

C₂²×Dic₃₃₄

yes

—

108

C₂

—

109

C₂

C₂×#19128832

yes

characteristic

110

C₂

C₂×#19128854

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₆₈
commutator, cosocle	C₂×C₄×C₁₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₇
Fitting	C₂×C₄×C₈×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂²×D₄
prime radical	C₂³×C₁₆₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₆₇
hypocenter	C₁₆₇	#3160
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₆₆₈

commutator, cosocle

C₂×C₄×C₁₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₆₇

Fitting

C₂×C₄×C₈×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₂²×D₄

prime radical

C₂³×C₁₆₇

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₆₇

hypocenter

C₁₆₇

#3160

LCS

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #24923674	C₂	semidirect
3. #24923671	C₂	semidirect
4. #24923667	C₂	semidirect
27. C₂×C₈×C₁₆₇	Q₈	semidirect
55. C₁₆₇	#3160	semidirect
78. C₂×C₈	Dic₃₃₄	semidirect

H ≅ G/N

type

2. #24923674

C₂

semidirect

3. #24923671

C₂

semidirect

4. #24923667

C₂

semidirect

27. C₂×C₈×C₁₆₇

Q₈

semidirect

55. C₁₆₇

#3160

semidirect

78. C₂×C₈

Dic₃₃₄

semidirect

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

subgroup	index
#24923826	2
#24923674	2
#24923671	2
#24923667	2
#24923307	2
C₂×C₄×C₈×C₁₆₇	2
#3160	167

subgroup

index

C₂×C₄×C₈×C₁₆₇

167

Subgroups — #36334652

Conjugacy classes of subgroups

Complements

Maximal subgroups