Group #36326382 — subgroups

Subgroups — #36326382

Order 21376.

132 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 47 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	#24923670	C₂	1	1	yes	maximal, characteristic
2	#24923647	C₂	1	1	yes	maximal, characteristic
3	#24923074	C₂	1	1	yes	maximal, characteristic
4	#24923073	C₂	1	1	yes	maximal, characteristic
5	#24923072	C₂	1	1	yes	maximal, characteristic
6	#24923044	C₂	1	1	yes	maximal, characteristic
7	#358×C₁₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₆₈	C₂²	1	1	yes	characteristic
9	C₂×#14581940	—	2	1	no	—
10	#19128830	—	2	1	no	—
11	#19128823	—	2	1	no	—
12	C₂×C₁₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#19128766	C₂²	1	1	yes	characteristic
14	#19128763	—	2	1	no	—
15	C₈×Dic₁₆₇	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
18	#107×C₁₆₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₃₄	—	2	1	no	—
20	Dic₆₆₈	—	2	1	no	—
21	Dic₆₆₈	—	2	1	no	—
22	#14581940	—	4	2	no	—
23	C₂×C₁₆₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₆₇	—	2	1	no	—
25	C₄⋊Dic₁₆₇	—	4	1	no	—
26	Dic₁₆₇⋊C₄	—	4	1	no	—
27	Dic₁₆₇⋊C₄	—	4	1	no	—
28	Dic₁₆₇⋊C₄	—	4	1	no	—
29	C₄×Dic₁₆₇	D₄	1	1	yes	characteristic
30	C₁₆₇⋊C₁₆	—	2	1	no	—
31	C₂×Q₈×C₁₆₇	—	2	1	no	—
32	Q₁₆×C₁₆₇	—	2	1	no	—
33	Q₁₆×C₁₆₇	—	2	1	no	—
34	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₆₇	—	2	1	no	—
36	C₁₆×C₁₆₇	—	2	1	no	—
37	Dic₃₃₄	—	4	1	no	—
38	Dic₃₃₄	—	4	1	no	—
39	C₂×Dic₁₆₇	—	4	1	no	—
40	C₂×Dic₁₆₇	—	4	1	no	—
41	C₂×Dic₁₆₇	D₈	1	1	yes	characteristic
42	C₂×Dic₁₆₇	D₈	1	1	yes	characteristic
43	C₁₆₇⋊C₈	SD₁₆	1	2	yes	—
44	Q₈×C₁₆₇	—	4	1	no	—
45	Q₈×C₁₆₇	—	4	1	no	—
46	C₂×C₄×C₁₆₇	—	4	1	no	—
47	C₂×C₄×C₁₆₇	—	4	1	no	—
48	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₆₇	#41	1	1	yes	characteristic
50	C₈×C₁₆₇	#41	1	1	yes	characteristic
51	C₂²×C₁₆₇	C₂×D₈	1	1	yes	characteristic
52	C₄×C₁₆₇	—	8	1	no	—
53	C₄×C₁₆₇	—	8	1	no	—
54	C₄×C₁₆₇	#124	1	1	yes	characteristic
55	C₄×C₁₆₇	C₂×SD₁₆	1	1	yes	characteristic
56	Dic₁₆₇	—	2	1	no	—
57	Dic₁₆₇	—	2	1	no	—
58	Dic₁₆₇	—	8	1	no	—
59	Dic₁₆₇	—	8	1	no	—
60	C₂×C₁₆₇	#486	1	1	yes	characteristic
61	C₂×C₁₆₇	#510	1	1	yes	characteristic
62	C₂×C₁₆₇	#510	1	1	yes	characteristic
63	C₁₆₇	#2245	1	1	yes	characteristic, complemented, Sylow
64	#2245	—	167	1	no	maximal, Sylow
65	#499	—	167	1	no	—
66	#494	—	167	1	no	—
67	#358	—	167	1	no	—
68	#358	—	167	1	no	—
69	#358	—	167	1	no	—
70	#358	D₁₆₇	1	1	yes	characteristic
71	#346	—	167	1	no	—
72	C₂×Q₁₆	—	334	1	no	—
73	C₂×Q₁₆	—	167	1	no	—
74	C₂×Q₁₆	C₂×D₁₆₇	1	1	yes	characteristic
75	C₄⋊Q₈	—	334	1	no	—
76	#125	—	334	1	no	—
77	C₂×C₁₆	—	167	1	no	—
78	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
79	#107	—	167	1	no	—
80	#107	C₂×D₁₆₇	1	1	yes	characteristic
81	#106	—	334	1	no	—
82	C₄×C₈	—	167	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	334	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	334	1	no	—
88	Q₁₆	—	334	1	no	—
89	Q₁₆	—	668	2	no	—
90	C₂×C₈	—	167	1	no	—
91	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	334	1	no	—
94	C₄⋊C₄	—	668	1	no	—
95	C₄⋊C₄	—	668	1	no	—
96	C₄⋊C₄	—	668	1	no	—
97	C₄⋊C₄	—	668	1	no	—
98	C₄²	—	167	1	no	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	334	1	no	—
101	Q₈	—	4	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	668	1	no	—
104	Q₈	—	668	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	668	1	no	—
108	C₂×C₄	—	668	1	no	—
109	C₂×C₄	—	167	1	no	—
110	C₂×C₄	—	167	1	no	—
111	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
112	C₈	—	167	2	no	—
113	C₈	#14581949	1	1	yes	characteristic
114	C₈	#14581950	1	1	yes	characteristic
115	C₂²	D₈×D₁₆₇	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	—	1336	1	no	—
119	C₄	—	1336	1	no	—
120	C₄	—	334	1	no	—
121	C₄	—	334	1	no	—
122	C₄	#19128818	1	1	yes	characteristic
123	C₄	SD₁₆×D₁₆₇	1	1	yes	characteristic
124	C₂	#24923614	1	1	yes	characteristic
125	C₂	#24923708	1	1	yes	characteristic
126	C₂	#24923711	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24923670

C₂

yes

maximal, characteristic

#24923647

C₂

yes

maximal, characteristic

#24923074

C₂

yes

maximal, characteristic

#24923073

C₂

yes

maximal, characteristic

#24923072

C₂

yes

maximal, characteristic

#24923044

C₂

yes

maximal, characteristic

#358×C₁₆₇

C₂

yes

maximal, characteristic

C₂×Dic₆₆₈

C₂²

yes

—

—

—

C₂×C₁₆₇⋊C₁₆

C₂²

yes

characteristic

#19128766

C₂²

yes

characteristic

#19128763

—

C₈×Dic₁₆₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₆₇

C₂²

yes

characteristic

#107×C₁₆₇

C₂²

yes

characteristic

C₂×Dic₃₃₄

—

Dic₆₆₈

—

Dic₆₆₈

—

#14581940

—

C₂×C₁₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

Dic₁₆₇⋊C₄

—

Dic₁₆₇⋊C₄

—

Dic₁₆₇⋊C₄

—

C₄×Dic₁₆₇

D₄

yes

characteristic

C₁₆₇⋊C₁₆

—

C₂×Q₈×C₁₆₇

—

Q₁₆×C₁₆₇

—

Q₁₆×C₁₆₇

—

C₂×C₈×C₁₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₇

—

C₁₆×C₁₆₇

—

Dic₃₃₄

—

Dic₃₃₄

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

D₈

yes

characteristic

C₂×Dic₁₆₇

D₈

yes

characteristic

C₁₆₇⋊C₈

SD₁₆

yes

—

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

C₈×C₁₆₇

yes

C₈×C₁₆₇

yes

C₂²×C₁₆₇

C₂×D₈

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

#124

yes

characteristic

C₄×C₁₆₇

C₂×SD₁₆

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#2245

—

167

maximal, Sylow

#499

—

167

—

#494

—

167

—

#358

—

167

—

#358

—

167

—

#358

—

167

—

#358

D₁₆₇

yes

characteristic

#346

—

167

—

C₂×Q₁₆

—

334

—

C₂×Q₁₆

—

167

—

C₂×Q₁₆

C₂×D₁₆₇

yes

characteristic

C₄⋊Q₈

—

334

—

#125

—

334

—

C₂×C₁₆

—

167

—

C₂×C₁₆

C₂×D₁₆₇

yes

characteristic

#107

—

167

—

#107

C₂×D₁₆₇

yes

characteristic

#106

—

334

—

C₄×C₈

—

167

—

C₂×Q₈

—

C₂×Q₈

—

334

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

334

—

Q₁₆

—

334

—

Q₁₆

—

668

—

C₂×C₈

—

167

—

C₂×C₈

C₂²×D₁₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

334

—

C₄⋊C₄

—

668

—

C₄⋊C₄

—

668

—

C₄⋊C₄

—

668

—

C₄⋊C₄

—

668

—

C₄²

—

167

—

C₁₆

—

100

C₁₆

—

334

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

668

—

104

Q₈

—

668

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

668

—

108

C₂×C₄

—

668

—

109

C₂×C₄

—

167

—

110

C₂×C₄

—

167

—

111

C₂×C₄

D₄×D₁₆₇

yes

characteristic

112

C₈

—

167

—

113

C₈

#14581949

yes

characteristic

114

C₈

#14581950

yes

characteristic

115

C₂²

D₈×D₁₆₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

1336

—

119

C₄

—

1336

—

120

C₄

—

334

—

121

C₄

—

334

—

122

C₄

#19128818

yes

characteristic

123

C₄

SD₁₆×D₁₆₇

yes

characteristic

124

C₂

#24923614

yes

characteristic

125

C₂

#24923708

yes

characteristic

126

C₂

#24923711

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₆₇
commutator, cosocle	C₂×C₈×C₁₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting	#358×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₂×D₈
hypercenter	#358	D₁₆₇
hypocenter	C₁₆₇	#2245
LCS	G ▷ C₂×C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₆₇

commutator, cosocle

C₂×C₈×C₁₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₆₇

Fitting

#358×C₁₆₇

C₂

socle, prime radical

C₂²×C₁₆₇

C₂×D₈

D₁₆₇

C₁₆₇

G ▷ C₂×C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
63. C₁₆₇	#2245	semidirect

H ≅ G/N

type

63. C₁₆₇

#2245

semidirect

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

subgroup	index
#24923670	2
#24923647	2
#24923074	2
#24923073	2
#24923072	2
#24923044	2
#358×C₁₆₇	2
#2245	167

subgroup

index

#358×C₁₆₇

167

Subgroups — #36326382

Conjugacy classes of subgroups

Complements

Maximal subgroups