Group #36322425 — subgroups

Subgroups — #36322425

Order 21376.

208 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 121 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×C₄×C₁₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#24922991	C₂	1	2	yes	maximal, complemented
4	#24922991	C₂	1	2	yes	maximal, complemented
5	C₄×M₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₆₇⋊C₈	C₄	1	2	yes	—
7	C₄⋊(C₁₆₇⋊C₈)	C₄	1	2	yes	—
8	C₂²⋊(C₁₆₇⋊C₈)	C₄	1	2	yes	—
9	C₄×C₁₆₇⋊C₈	C₄	1	2	yes	—
10	C₄×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₁₆₇⋊C₈	C₂²	1	2	yes	—
13	C₂×M₁₆×C₁₆₇	C₄	1	2	yes	—
14	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
15	#97×C₁₆₇	C₄	1	2	yes	—
16	C₄×C₈×C₁₆₇	C₂²	1	2	yes	—
17	C₂×C₁₆₇⋊C₈	—	2	4	no	—
18	C₂×C₁₆₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₁₆₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₂×C₁₆₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
24	M₁₆×C₁₆₇	—	2	4	no	—
25	C₂×C₈×C₁₆₇	C₂×C₄	1	4	yes	—
26	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
30	C₁₆₇⋊C₈	—	2	2	no	—
31	C₁₆₇⋊C₈	—	2	4	no	—
32	C₁₆₇⋊C₈	M₁₆	1	4	yes	complemented
33	C₂³×C₁₆₇	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₁₆₇	—	2	2	no	—
35	C₂×C₄×C₁₆₇	—	2	2	no	—
36	C₂×C₄×C₁₆₇	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₁₆₇	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₁₆₇	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₁₆₇	M₁₆	1	4	yes	—
40	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₁₆₇	—	2	4	no	—
44	C₂²×C₁₆₇	—	2	1	no	—
45	C₂²×C₁₆₇	—	2	1	no	—
46	C₂²×C₁₆₇	#97	1	2	yes	—
47	C₂²×C₁₆₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₆₇	—	2	1	no	—
49	C₄×C₁₆₇	—	2	1	no	—
50	C₄×C₁₆₇	#97	1	2	yes	—
51	C₄×C₁₆₇	C₂×M₁₆	1	2	yes	—
52	C₄×C₁₆₇	C₂×M₁₆	1	2	yes	—
53	C₄×C₁₆₇	#131	1	2	yes	—
54	C₂×C₁₆₇	—	2	2	no	—
55	C₂×C₁₆₇	C₂×#97	1	1	yes	characteristic
56	C₂×C₁₆₇	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₁₆₇	#405	1	1	yes	characteristic
58	C₁₆₇	#1451	1	1	yes	characteristic, complemented, Sylow
59	#1451	—	167	1	no	maximal, Sylow
60	#431	—	167	1	no	—
61	C₄×M₁₆	D₁₆₇	1	1	yes	characteristic
62	C₂×C₄×C₈	—	167	1	no	—
63	#322	—	167	2	no	—
64	#322	—	167	2	no	—
65	C₂×M₁₆	Dic₁₆₇	1	2	yes	—
66	C₂²×C₈	—	167	2	no	—
67	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
68	C₄⋊C₈	—	167	2	no	—
69	C₂²⋊C₈	—	167	2	no	—
70	#97	Dic₁₆₇	1	2	yes	—
71	C₄×C₈	—	167	2	no	—
72	C₄×C₈	—	167	1	no	—
73	C₄×C₈	—	167	1	no	—
74	C₄×C₈	—	167	2	no	—
75	C₄×C₈	C₂×D₁₆₇	1	2	yes	—
76	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₁₆₇	1	1	yes	characteristic
79	M₁₆	—	2	4	no	—
80	C₂×C₈	—	334	4	no	—
81	C₂×C₈	—	167	2	no	—
82	C₂×C₈	—	167	4	no	—
83	C₂×C₈	—	167	2	no	—
84	C₂×C₈	C₂×Dic₁₆₇	1	4	yes	—
85	C₄²	C₄×D₁₆₇	1	1	yes	characteristic
86	C₄²	C₄×D₁₆₇	1	1	yes	characteristic
87	C₄²	C₂×Dic₁₆₇	1	1	yes	characteristic
88	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
89	C₂³	C₄×Dic₁₆₇	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	C₄×Dic₁₆₇	1	1	yes	characteristic
93	C₂×C₄	C₄×Dic₁₆₇	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₁₆₇	1	1	yes	characteristic
95	C₂×C₄	#14581934	1	4	yes	—
96	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
99	C₈	—	2	4	no	—
100	C₈	—	334	2	no	—
101	C₈	—	334	4	no	—
102	C₈	—	167	4	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#19128743	1	2	yes	—
106	C₂²	C₂×C₄×Dic₁₆₇	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	#19128743	1	2	yes	—
110	C₄	M₁₆×D₁₆₇	1	2	yes	—
111	C₄	C₂×#14581934	1	2	yes	—
112	C₄	#19128840	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×#19128743	1	1	yes	characteristic
115	C₂	M₁₆×Dic₁₆₇	1	1	yes	characteristic
116	C₂	#24923227	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24923322

C₂

yes

maximal, characteristic

C₂×C₄×C₁₆₇⋊C₈

C₂

yes

maximal, characteristic

#24922991

C₂

yes

maximal, complemented

#24922991

C₂

yes

maximal, complemented

C₄×M₁₆×C₁₆₇

C₂

yes

maximal, characteristic

C₂²×C₁₆₇⋊C₈

C₄

yes

—

C₄⋊(C₁₆₇⋊C₈)

C₄

yes

—

C₂²⋊(C₁₆₇⋊C₈)

C₄

yes

—

C₄×C₁₆₇⋊C₈

C₄

yes

—

C₄×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₆₇⋊C₈

C₂²

yes

—

C₂×M₁₆×C₁₆₇

C₄

yes

—

C₂×C₄²×C₁₆₇

C₂²

yes

characteristic

#97×C₁₆₇

C₄

yes

—

C₄×C₈×C₁₆₇

C₂²

yes

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

M₁₆×C₁₆₇

—

C₂×C₈×C₁₆₇

C₂×C₄

yes

—

C₄²×C₁₆₇

C₂×C₄

yes

characteristic

C₄²×C₁₆₇

C₂×C₄

yes

characteristic

C₄²×C₁₆₇

C₂×C₄

yes

characteristic

C₄²×C₁₆₇

C₂³

yes

characteristic

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

M₁₆

yes

complemented

C₂³×C₁₆₇

C₄²

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₄²

yes

characteristic

C₂×C₄×C₁₆₇

C₄²

yes

characteristic

C₂×C₄×C₁₆₇

C₄²

yes

characteristic

C₂×C₄×C₁₆₇

M₁₆

yes

—

C₂×C₄×C₁₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂²×C₄

yes

characteristic

C₈×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

#97

yes

—

C₂²×C₁₆₇

C₂×C₄²

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

#97

yes

—

C₄×C₁₆₇

C₂×M₁₆

yes

—

C₄×C₁₆₇

C₂×M₁₆

yes

—

C₄×C₁₆₇

#131

yes

—

C₂×C₁₆₇

—

C₂×C₁₆₇

C₂×#97

yes

characteristic

C₂×C₁₆₇

C₄×M₁₆

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#1451

—

167

maximal, Sylow

#431

—

167

—

C₄×M₁₆

D₁₆₇

yes

characteristic

C₂×C₄×C₈

—

167

—

#322

—

167

—

#322

—

167

—

C₂×M₁₆

Dic₁₆₇

yes

—

C₂²×C₈

—

167

—

C₂×C₄²

C₂×D₁₆₇

yes

characteristic

C₄⋊C₈

—

167

—

C₂²⋊C₈

—

167

—

#97

Dic₁₆₇

yes

—

C₄×C₈

—

167

—

C₄×C₈

—

167

—

C₄×C₈

—

167

—

C₄×C₈

—

167

—

C₄×C₈

C₂×D₁₆₇

yes

—

C₂²×C₄

C₄×D₁₆₇

yes

characteristic

C₂²×C₄

C₄×D₁₆₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₆₇

yes

characteristic

M₁₆

—

C₂×C₈

—

334

—

C₂×C₈

—

167

—

C₂×C₈

—

167

—

C₂×C₈

—

167

—

C₂×C₈

C₂×Dic₁₆₇

yes

—

C₄²

C₄×D₁₆₇

yes

characteristic

C₄²

C₄×D₁₆₇

yes

characteristic

C₄²

C₂×Dic₁₆₇

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

#14581934

yes

—

C₂×C₄

C₂×C₄×D₁₆₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₆₇

yes

characteristic

C₂×C₄

C₂²×Dic₁₆₇

yes

characteristic

C₈

—

100

C₈

—

334

—

101

C₈

—

334

—

102

C₈

—

167

—

103

C₂²

—

104

C₂²

—

105

C₂²

#19128743

yes

—

106

C₂²

C₂×C₄×Dic₁₆₇

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#19128743

yes

—

110

C₄

M₁₆×D₁₆₇

yes

—

111

C₄

C₂×#14581934

yes

—

112

C₄

#19128840

yes

—

113

C₂

—

114

C₂

C₂×#19128743

yes

characteristic

115

C₂

M₁₆×Dic₁₆₇

yes

characteristic

116

C₂

#24923227

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₆₇
commutator, socle	C₂²×C₁₆₇	C₂×C₄²
Fitting	C₄×M₁₆×C₁₆₇	C₂
prime radical	C₂³×C₁₆₇	C₄²
cosocle	C₄²×C₁₆₇	C₂³
hypercenter	C₄×M₁₆	D₁₆₇
hypocenter	C₁₆₇	#1451
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₄²

C₂²×D₁₆₇

commutator, socle

C₂²×C₁₆₇

C₂×C₄²

Fitting

C₄×M₁₆×C₁₆₇

C₂

prime radical

C₂³×C₁₆₇

C₄²

cosocle

C₄²×C₁₆₇

C₂³

hypercenter

C₄×M₁₆

D₁₆₇

C₁₆₇

G ▷ C₂²×C₁₆₇ ▷ C₁₆₇

C₁ ◁ C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #24922991	C₂	semidirect
4. #24922991	C₂	semidirect
32. C₁₆₇⋊C₈	M₁₆	semidirect
58. C₁₆₇	#1451	semidirect

H ≅ G/N

type

3. #24922991

C₂

semidirect

4. #24922991

C₂

semidirect

32. C₁₆₇⋊C₈

M₁₆

semidirect

58. C₁₆₇

#1451

semidirect

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

subgroup	index
#24923322	2
C₂×C₄×C₁₆₇⋊C₈	2
#24922991	2
C₄×M₁₆×C₁₆₇	2
#1451	167

subgroup

index

#24923322

C₂×C₄×C₁₆₇⋊C₈

#24922991

C₄×M₁₆×C₁₆₇

#1451

167

Subgroups — #36322425

Conjugacy classes of subgroups

Complements

Maximal subgroups