Group #36321982 — subgroups

Subgroups — #36321982

Order 21376.

152 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 77 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	#24923322	C₂	1	1	yes	maximal, characteristic
2	C₂×#19128743	C₂	1	1	yes	maximal, characteristic
3	#431×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆₇⋊C₈	C₄	1	2	yes	—
5	C₄⋊(C₁₆₇⋊C₈)	—	2	1	no	—
6	C₂²⋊(C₁₆₇⋊C₈)	C₄	1	2	yes	—
7	#19128743	—	2	1	no	—
8	#19128743	—	2	1	no	—
9	C₄×C₁₆₇⋊C₈	—	2	1	no	—
10	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₆₇	—	2	1	no	—
12	C₂²⋊C₈×C₁₆₇	C₄	1	2	yes	—
13	C₄×C₈×C₁₆₇	—	2	1	no	—
14	C₂×C₁₆₇⋊C₈	—	2	2	no	—
15	C₂×C₁₆₇⋊C₈	—	2	2	no	—
16	C₂×C₁₆₇⋊C₈	—	2	2	no	—
17	C₂×C₁₆₇⋊C₈	C₈	1	4	yes	complemented
18	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₆₇	—	2	2	no	—
22	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
23	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
24	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
25	C₄²×C₁₆₇	Q₈	1	1	yes	characteristic
26	C₁₆₇⋊C₈	—	2	4	no	—
27	C₁₆₇⋊C₈	—	4	2	no	—
28	C₂³×C₁₆₇	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₁₆₇	—	4	1	no	—
30	C₂×C₄×C₁₆₇	—	4	1	no	—
31	C₂×C₄×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₆₇	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₁₆₇	M₁₆	1	2	yes	—
39	C₈×C₁₆₇	—	4	2	no	—
40	C₂²×C₁₆₇	—	2	1	no	—
41	C₂²×C₁₆₇	—	2	1	no	—
42	C₂²×C₁₆₇	#95	1	1	yes	characteristic
43	C₂²×C₁₆₇	C₄×C₈	1	1	yes	characteristic
44	C₂²×C₁₆₇	#97	1	1	yes	characteristic
45	C₄×C₁₆₇	—	2	1	no	—
46	C₄×C₁₆₇	—	2	1	no	—
47	C₄×C₁₆₇	—	2	1	no	—
48	C₄×C₁₆₇	—	2	1	no	—
49	C₄×C₁₆₇	C₂²⋊C₈	1	2	yes	—
50	C₄×C₁₆₇	C₄⋊C₈	1	2	yes	—
51	C₂×C₁₆₇	—	2	1	no	—
52	C₂×C₁₆₇	—	2	1	no	—
53	C₂×C₁₆₇	#336	1	1	yes	characteristic
54	C₂×C₁₆₇	#344	1	1	yes	characteristic
55	C₂×C₁₆₇	#344	1	1	yes	characteristic
56	C₁₆₇	#1281	1	1	yes	characteristic, complemented, Sylow
57	#1281	—	167	1	no	maximal, Sylow
58	#431	—	167	1	no	—
59	#431	D₁₆₇	1	1	yes	characteristic
60	C₂×#97	—	167	1	no	—
61	C₂²×C₈	—	167	2	no	—
62	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	—	334	1	no	—
65	C₂²⋊C₈	—	167	2	no	—
66	C₂²⋊C₈	Dic₁₆₇	1	2	yes	—
67	#97	—	334	1	no	—
68	#97	—	334	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	334	1	no	—
71	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
73	C₂²×C₄	C₂×Dic₁₆₇	1	1	yes	characteristic
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	334	2	no	—
76	C₂×C₈	—	334	2	no	—
77	C₂×C₈	—	334	2	no	—
78	C₂×C₈	—	167	4	no	—
79	C₄²	C₂²⋊D₁₆₇	1	1	yes	characteristic
80	C₄²	C₂²⋊D₁₆₇	1	1	yes	characteristic
81	C₄²	D₆₆₈	1	1	yes	characteristic
82	C₄²	Dic₃₃₄	1	1	yes	characteristic
83	C₂³	C₄×Dic₁₆₇	1	1	yes	characteristic
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	(C₂×D₁₆₇)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	(C₂×D₁₆₇)⋊C₄	1	1	yes	characteristic
88	C₂×C₄	C₂²⋊Dic₁₆₇	1	1	yes	characteristic
89	C₂×C₄	Dic₁₆₇⋊C₄	1	1	yes	characteristic
90	C₂×C₄	Dic₁₆₇⋊C₄	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₁₆₇	1	1	yes	characteristic
92	C₂×C₄	C₈×D₁₆₇	1	2	yes	—
93	C₂×C₄	#14581933	1	2	yes	—
94	C₈	—	4	2	no	—
95	C₈	—	334	4	no	—
96	C₈	—	668	2	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	C₁₆₇⋊#95	1	1	yes	characteristic
100	C₂²	C₈×Dic₁₆₇	1	1	yes	characteristic
101	C₂²	#19128742	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	(C₂×D₁₆₇)⋊C₈	1	2	yes	—
107	C₄	Dic₁₆₇⋊C₈	1	2	yes	—
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	#24923023	1	1	yes	characteristic
111	C₂	#24923039	1	1	yes	characteristic
112	C₂	#24923041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24923322

C₂

yes

maximal, characteristic

C₂×#19128743

C₂

yes

maximal, characteristic

#431×C₁₆₇

C₂

yes

maximal, characteristic

C₂²×C₁₆₇⋊C₈

C₄

yes

—

C₄⋊(C₁₆₇⋊C₈)

—

C₂²⋊(C₁₆₇⋊C₈)

C₄

yes

—

#19128743

—

#19128743

—

C₄×C₁₆₇⋊C₈

—

C₂×C₄²×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₇

—

C₂²⋊C₈×C₁₆₇

C₄

yes

—

C₄×C₈×C₁₆₇

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

C₈

yes

complemented

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₇

—

C₄²×C₁₆₇

D₄

yes

characteristic

C₄²×C₁₆₇

D₄

yes

characteristic

C₄²×C₁₆₇

D₄

yes

characteristic

C₄²×C₁₆₇

Q₈

yes

characteristic

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

C₄²

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×C₈

yes

—

C₂×C₄×C₁₆₇

M₁₆

yes

—

C₈×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

#95

yes

characteristic

C₂²×C₁₆₇

C₄×C₈

yes

characteristic

C₂²×C₁₆₇

#97

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂²⋊C₈

yes

—

C₄×C₁₆₇

C₄⋊C₈

yes

—

C₂×C₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#1281

—

167

maximal, Sylow

#431

—

167

—

#431

D₁₆₇

yes

characteristic

C₂×#97

—

167

—

C₂²×C₈

—

167

—

C₂×C₄²

C₂×D₁₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

334

—

C₂²⋊C₈

—

167

—

C₂²⋊C₈

Dic₁₆₇

yes

—

#97

—

334

—

#97

—

334

—

C₄×C₈

—

C₄×C₈

—

334

—

C₂²×C₄

C₄×D₁₆₇

yes

characteristic

C₂²×C₄

C₄×D₁₆₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

334

—

C₂×C₈

—

334

—

C₂×C₈

—

334

—

C₂×C₈

—

167

—

C₄²

C₂²⋊D₁₆₇

yes

characteristic

C₄²

C₂²⋊D₁₆₇

yes

characteristic

C₄²

D₆₆₈

yes

characteristic

C₄²

Dic₃₃₄

yes

characteristic

C₂³

C₄×Dic₁₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₁₆₇)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₁₆₇)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₁₆₇

yes

characteristic

C₂×C₄

Dic₁₆₇⋊C₄

yes

characteristic

C₂×C₄

Dic₁₆₇⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₁₆₇

yes

characteristic

C₂×C₄

C₈×D₁₆₇

yes

—

C₂×C₄

#14581933

yes

—

C₈

—

C₈

—

334

—

C₈

—

668

—

C₂²

—

C₂²

—

C₂²

C₁₆₇⋊#95

yes

characteristic

100

C₂²

C₈×Dic₁₆₇

yes

characteristic

101

C₂²

#19128742

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

(C₂×D₁₆₇)⋊C₈

yes

—

107

C₄

Dic₁₆₇⋊C₈

yes

—

108

C₂

—

109

C₂

—

110

C₂

#24923023

yes

characteristic

111

C₂

#24923039

yes

characteristic

112

C₂

#24923041

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₁₆₇)⋊C₄
commutator	C₂²×C₁₆₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₆₇
Fitting	#431×C₁₆₇	C₂
socle	C₂²×C₁₆₇	#95
prime radical	C₂³×C₁₆₇	C₄²
cosocle	C₂×C₄²×C₁₆₇	C₂²
hypercenter	#431	D₁₆₇
hypocenter	C₁₆₇	#1281
LCS	G ▷ C₂²×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₁₆₇)⋊C₄

commutator

C₂²×C₁₆₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₆₇

Fitting

#431×C₁₆₇

C₂

socle

C₂²×C₁₆₇

#95

prime radical

C₂³×C₁₆₇

C₄²

cosocle

C₂×C₄²×C₁₆₇

C₂²

D₁₆₇

C₁₆₇

G ▷ C₂²×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
17. C₂×C₁₆₇⋊C₈	C₈	semidirect
56. C₁₆₇	#1281	semidirect

H ≅ G/N

type

17. C₂×C₁₆₇⋊C₈

C₈

semidirect

56. C₁₆₇

#1281

semidirect

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

subgroup	index
#24923322	2
C₂×#19128743	2
#431×C₁₆₇	2
#1281	167

subgroup

index

#24923322

C₂×#19128743

#431×C₁₆₇

#1281

167

Subgroups — #36321982

Conjugacy classes of subgroups

Complements

Maximal subgroups