Group #36322909 — subgroups

Subgroups — #36322909

Order 21376.

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	#24923293	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#24923020	C₂	1	4	yes	maximal, complemented
4	#431×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581934	—	2	1	no	—
6	C₂²×C₁₆₇⋊C₈	—	2	1	no	—
7	C₄⋊(C₁₆₇⋊C₈)	—	2	1	no	—
8	C₄⋊(C₁₆₇⋊C₈)	—	2	1	no	—
9	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	2	yes	—
10	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₆₇	C₄	1	2	yes	—
13	C₂²⋊C₈×C₁₆₇	C₄	1	2	yes	—
14	C₄×C₈×C₁₆₇	C₂²	1	2	yes	—
15	#14581934	—	4	2	no	—
16	C₂×C₁₆₇⋊C₈	—	2	2	no	—
17	C₂×C₁₆₇⋊C₈	—	2	2	no	—
18	C₂×C₁₆₇⋊C₈	—	4	2	no	—
19	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₆₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₆₇	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₁₆₇	C₂×C₄	1	4	yes	—
23	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
25	C₄²×C₁₆₇	Q₈	1	1	yes	characteristic
26	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
27	C₁₆₇⋊C₈	—	4	2	no	—
28	C₁₆₇⋊C₈	—	4	2	no	—
29	C₂³×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₆₇	—	2	2	no	—
31	C₂×C₄×C₁₆₇	—	4	1	no	—
32	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₆₇	SD₁₆	1	2	yes	—
36	C₂×C₄×C₁₆₇	SD₁₆	1	2	yes	—
37	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₁₆₇	—	2	4	no	—
41	C₂²×C₁₆₇	—	2	1	no	—
42	C₂²×C₁₆₇	—	2	1	no	—
43	C₂²×C₁₆₇	#106	1	2	yes	—
44	C₂²×C₁₆₇	C₂×C₄⋊C₄	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	#106	1	2	yes	—
49	C₄×C₁₆₇	#131	1	2	yes	—
50	C₄×C₁₆₇	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₁₆₇	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×C₁₆₇	—	2	2	no	—
53	C₂×C₁₆₇	#424	1	1	yes	characteristic
54	C₂×C₁₆₇	C₂×#106	1	1	yes	characteristic
55	C₂×C₁₆₇	#430	1	1	yes	characteristic
56	C₁₆₇	#1570	1	1	yes	characteristic, complemented, Sylow
57	#1570	—	167	1	no	maximal, Sylow
58	#431	D₁₆₇	1	1	yes	characteristic
59	#423	—	167	1	no	—
60	C₂×C₄⋊C₈	—	167	1	no	—
61	#334	—	167	4	no	—
62	C₂×M₁₆	—	334	1	no	—
63	C₂²×C₈	—	334	1	no	—
64	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
65	C₄⋊C₈	—	334	1	no	—
66	C₄⋊C₈	—	334	1	no	—
67	C₄⋊C₈	—	167	2	no	—
68	C₄⋊C₈	—	167	2	no	—
69	C₄⋊C₈	Dic₁₆₇	1	2	yes	—
70	C₂²⋊C₈	Dic₁₆₇	1	2	yes	—
71	C₄×C₈	C₂×D₁₆₇	1	2	yes	—
72	C₂²×C₄	C₂×Dic₁₆₇	1	1	yes	characteristic
73	C₂²×C₄	D₆₆₈	1	1	yes	characteristic
74	C₂²×C₄	Dic₃₃₄	1	1	yes	characteristic
75	M₁₆	—	668	2	no	—
76	C₂×C₈	—	334	2	no	—
77	C₂×C₈	—	334	2	no	—
78	C₂×C₈	—	668	2	no	—
79	C₂×C₈	C₂×Dic₁₆₇	1	4	yes	—
80	C₄²	C₂×Dic₁₆₇	1	1	yes	characteristic
81	C₄²	D₆₆₈	1	1	yes	characteristic
82	C₄²	Dic₃₃₄	1	1	yes	characteristic
83	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₁₆₇	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	C₄⋊Dic₁₆₇	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Dic₁₆₇	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₁₆₇	1	1	yes	characteristic
90	C₂×C₄	#14581939	1	2	yes	—
91	C₂×C₄	#14581939	1	2	yes	—
92	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
93	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₃₃₄	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	668	2	no	—
97	C₈	—	668	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#19128764	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₁₆₇	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#19128764	1	2	yes	—
106	C₄	#19128839	1	2	yes	—
107	C₄	C₂×#14581939	1	1	yes	characteristic
108	C₄	C₂×#14581939	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#24923296	1	1	yes	characteristic
111	C₂	C₂×#19128764	1	1	yes	characteristic
112	C₂	#24923318	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24923293

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₆₇⋊C₈)

C₂

yes

maximal, characteristic

#24923020

C₂

yes

maximal, complemented

#431×C₁₆₇

C₂

yes

maximal, characteristic

C₂×#14581934

—

C₂²×C₁₆₇⋊C₈

—

C₄⋊(C₁₆₇⋊C₈)

—

C₄⋊(C₁₆₇⋊C₈)

—

C₄⋊(C₁₆₇⋊C₈)

C₂²

yes

—

C₄⋊(C₁₆₇⋊C₈)

C₂²

yes

—

C₂×C₄²×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₇

C₄

yes

—

C₂²⋊C₈×C₁₆₇

C₄

yes

—

C₄×C₈×C₁₆₇

C₂²

yes

—

#14581934

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₇

D₄

yes

characteristic

C₂²×C₄×C₁₆₇

Q₈

yes

characteristic

C₂×C₈×C₁₆₇

C₂×C₄

yes

—

C₄²×C₁₆₇

C₂×C₄

yes

characteristic

C₄²×C₁₆₇

D₄

yes

characteristic

C₄²×C₁₆₇

Q₈

yes

characteristic

C₄²×C₁₆₇

C₂³

yes

characteristic

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

SD₁₆

yes

—

C₂×C₄×C₁₆₇

SD₁₆

yes

—

C₂×C₄×C₁₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×Q₈

yes

characteristic

C₈×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

#106

yes

—

C₂²×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

#106

yes

—

C₄×C₁₆₇

#131

yes

—

C₄×C₁₆₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#1570

—

167

maximal, Sylow

#431

D₁₆₇

yes

characteristic

#423

—

167

—

C₂×C₄⋊C₈

—

167

—

#334

—

167

—

C₂×M₁₆

—

334

—

C₂²×C₈

—

334

—

C₂×C₄²

C₂×D₁₆₇

yes

characteristic

C₄⋊C₈

—

334

—

C₄⋊C₈

—

334

—

C₄⋊C₈

—

167

—

C₄⋊C₈

—

167

—

C₄⋊C₈

Dic₁₆₇

yes

—

C₂²⋊C₈

Dic₁₆₇

yes

—

C₄×C₈

C₂×D₁₆₇

yes

—

C₂²×C₄

C₂×Dic₁₆₇

yes

characteristic

C₂²×C₄

D₆₆₈

yes

characteristic

C₂²×C₄

Dic₃₃₄

yes

characteristic

M₁₆

—

668

—

C₂×C₈

—

334

—

C₂×C₈

—

334

—

C₂×C₈

—

668

—

C₂×C₈

C₂×Dic₁₆₇

yes

—

C₄²

C₂×Dic₁₆₇

yes

characteristic

C₄²

D₆₆₈

yes

characteristic

C₄²

Dic₃₃₄

yes

characteristic

C₄²

C₂²×D₁₆₇

yes

characteristic

C₂³

C₄⋊Dic₁₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₆₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₆₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₆₇

yes

characteristic

C₂×C₄

#14581939

yes

—

C₂×C₄

#14581939

yes

—

C₂×C₄

C₂²×Dic₁₆₇

yes

characteristic

C₂×C₄

C₂×D₆₆₈

yes

characteristic

C₂×C₄

C₂×Dic₃₃₄

yes

characteristic

C₈

—

C₈

—

668

—

C₈

—

668

—

C₂²

—

C₂²

—

100

C₂²

#19128764

yes

—

101

C₂²

C₂×C₄⋊Dic₁₆₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#19128764

yes

—

106

C₄

#19128839

yes

—

107

C₄

C₂×#14581939

yes

characteristic

108

C₄

C₂×#14581939

yes

characteristic

109

C₂

—

110

C₂

#24923296

yes

characteristic

111

C₂

C₂×#19128764

yes

characteristic

112

C₂

#24923318

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₆₈
commutator	C₂×C₄×C₁₆₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₆₇
Fitting	#431×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₆₇	C₄⋊C₄
cosocle	C₄²×C₁₆₇	C₂³
hypercenter	#431	D₁₆₇
hypocenter	C₁₆₇	#1570
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₆₆₈

commutator

C₂×C₄×C₁₆₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₆₇

Fitting

#431×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₆₇

C₄⋊C₄

cosocle

C₄²×C₁₆₇

C₂³

D₁₆₇

C₁₆₇

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #24923020	C₂	semidirect
56. C₁₆₇	#1570	semidirect

H ≅ G/N

type

3. #24923020

C₂

semidirect

56. C₁₆₇

#1570

semidirect

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

subgroup	index
#24923293	2
C₂×C₄⋊(C₁₆₇⋊C₈)	2
#24923020	2
#431×C₁₆₇	2
#1570	167

subgroup

index

#24923293

C₂×C₄⋊(C₁₆₇⋊C₈)

#24923020

#431×C₁₆₇

#1570

167

Subgroups — #36322909

Conjugacy classes of subgroups

Complements

Maximal subgroups