Group #36323813 — subgroups

Subgroups — #36323813

Order 21376.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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	#24923672	C₂	1	1	yes	maximal, characteristic
2	#24923672	C₂	1	1	yes	maximal, characteristic
3	#24923665	C₂	1	1	yes	maximal, characteristic
4	#24923621	C₂	1	1	yes	maximal, characteristic
5	#24923620	C₂	1	1	yes	maximal, characteristic
6	#24922991	C₂	1	1	yes	maximal, characteristic
7	#494×C₁₆₇	C₂	1	1	yes	maximal, characteristic
8	#19128832	C₂²	1	1	yes	characteristic
9	#19128832	C₂²	1	1	yes	characteristic
10	#19128765	—	2	2	no	—
11	#19128765	—	2	2	no	—
12	#19128763	—	2	2	no	—
13	#19128757	—	2	2	no	—
14	#19128755	—	2	2	no	—
15	C₄×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₆₇	—	2	2	no	—
18	C₄⋊Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₃₄	—	2	1	no	—
22	C₂×Dic₃₃₄	—	2	1	no	—
23	C₂×C₁₆₇⋊C₈	D₄	1	2	yes	—
24	C₂×C₁₆₇⋊C₈	D₄	1	2	yes	—
25	C₄⋊Dic₁₆₇	—	2	1	no	—
26	C₄⋊Dic₁₆₇	—	2	1	no	—
27	C₄⋊Dic₁₆₇	—	2	1	no	—
28	C₄⋊Dic₁₆₇	—	2	1	no	—
29	C₂×Q₈×C₁₆₇	—	2	1	no	—
30	C₂×Q₈×C₁₆₇	—	2	1	no	—
31	Q₁₆×C₁₆₇	—	4	4	no	—
32	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₆₇	—	2	1	no	—
34	C₄⋊C₄×C₁₆₇	—	2	1	no	—
35	C₄⋊C₄×C₁₆₇	—	2	1	no	—
36	C₄⋊C₄×C₁₆₇	—	2	1	no	—
37	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
38	Dic₃₃₄	—	4	2	no	—
39	Dic₃₃₄	—	4	2	no	—
40	C₂×Dic₁₆₇	—	4	1	no	—
41	C₂×Dic₁₆₇	—	4	1	no	—
42	C₁₆₇⋊C₈	—	2	2	no	—
43	C₁₆₇⋊C₈	Q₁₆	1	4	yes	complemented
44	Q₈×C₁₆₇	—	4	2	no	—
45	Q₈×C₁₆₇	—	4	2	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	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₆₇	—	2	2	no	—
52	C₂²×C₁₆₇	#127	1	1	yes	characteristic
53	C₄×C₁₆₇	—	8	1	no	—
54	C₄×C₁₆₇	—	8	1	no	—
55	C₄×C₁₆₇	C₂×Q₁₆	1	2	yes	—
56	C₄×C₁₆₇	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₁₆₇	#137	1	2	yes	—
58	Dic₁₆₇	—	8	1	no	—
59	Dic₁₆₇	—	8	1	no	—
60	C₂×C₁₆₇	#494	1	1	yes	characteristic
61	C₂×C₁₆₇	#496	1	1	yes	characteristic
62	C₂×C₁₆₇	#497	1	1	yes	characteristic
63	C₁₆₇	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	167	1	no	maximal, Sylow
65	#500	—	167	1	no	—
66	#500	—	167	1	no	—
67	#498	—	167	1	no	—
68	#494	D₁₆₇	1	1	yes	characteristic
69	#487	—	167	1	no	—
70	#487	—	167	1	no	—
71	#322	—	167	1	no	—
72	C₂×Q₁₆	—	2	2	no	—
73	C₄⋊Q₈	—	167	1	no	—
74	C₄⋊Q₈	—	167	1	no	—
75	C₄⋊Q₈	C₂×D₁₆₇	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₆₇	1	1	yes	characteristic
77	#107	—	334	2	no	—
78	#107	—	334	2	no	—
79	#106	—	334	2	no	—
80	#103	—	334	2	no	—
81	#103	—	334	2	no	—
82	C₄×C₈	—	167	1	no	—
83	C₄×C₈	—	167	1	no	—
84	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	334	1	no	—
88	C₂×Q₈	—	334	1	no	—
89	Q₁₆	—	4	4	no	—
90	C₂×C₈	—	167	2	no	—
91	C₂×C₈	—	167	2	no	—
92	C₂×C₈	C₂²⋊D₁₆₇	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	334	1	no	—
98	C₄⋊C₄	—	334	1	no	—
99	C₄⋊C₄	—	334	1	no	—
100	C₄⋊C₄	—	334	1	no	—
101	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	668	2	no	—
105	Q₈	—	668	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	668	1	no	—
109	C₂×C₄	—	668	1	no	—
110	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	334	2	no	—
115	C₈	—	167	4	no	—
116	C₂²	#19128828	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	1336	1	no	—
120	C₄	—	1336	1	no	—
121	C₄	Q₁₆×D₁₆₇	1	2	yes	—
122	C₄	#19128858	1	2	yes	—
123	C₄	#19128864	1	2	yes	—
124	C₂	#24923647	1	1	yes	characteristic
125	C₂	#24923656	1	1	yes	characteristic
126	C₂	#24923662	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24923672

C₂

yes

maximal, characteristic

#24923672

C₂

yes

maximal, characteristic

#24923665

C₂

yes

maximal, characteristic

#24923621

C₂

yes

maximal, characteristic

#24923620

C₂

yes

maximal, characteristic

#24922991

C₂

yes

maximal, characteristic

#494×C₁₆₇

C₂

yes

maximal, characteristic

#19128832

C₂²

yes

characteristic

#19128832

C₂²

yes

—

—

—

—

—

C₄×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₁₆₇

—

C₄⋊Q₈×C₁₆₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₆₇

C₂²

yes

characteristic

C₄×C₈×C₁₆₇

C₂²

yes

characteristic

C₂×Dic₃₃₄

—

C₂×Dic₃₃₄

—

C₂×C₁₆₇⋊C₈

D₄

yes

—

C₂×C₁₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

Q₁₆×C₁₆₇

—

C₂×C₈×C₁₆₇

D₄

yes

—

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₄²×C₁₆₇

C₂³

yes

characteristic

Dic₃₃₄

—

Dic₃₃₄

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

Q₁₆

yes

complemented

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₈×C₁₆₇

—

C₂²×C₁₆₇

#127

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×Q₁₆

yes

—

C₄×C₁₆₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₆₇

#137

yes

—

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#1723

—

167

maximal, Sylow

#500

—

167

—

#500

—

167

—

#498

—

167

—

#494

D₁₆₇

yes

characteristic

#487

—

167

—

#487

—

167

—

#322

—

167

—

C₂×Q₁₆

—

C₄⋊Q₈

—

167

—

C₄⋊Q₈

—

167

—

C₄⋊Q₈

C₂×D₁₆₇

yes

characteristic

C₄⋊Q₈

C₂×D₁₆₇

yes

characteristic

#107

—

334

—

#107

—

334

—

#106

—

334

—

#103

—

334

—

#103

—

334

—

C₄×C₈

—

167

—

C₄×C₈

—

167

—

C₄×C₈

C₂×D₁₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

334

—

C₂×Q₈

—

334

—

Q₁₆

—

C₂×C₈

—

167

—

C₂×C₈

—

167

—

C₂×C₈

C₂²⋊D₁₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

334

—

C₄⋊C₄

—

334

—

C₄⋊C₄

—

334

—

100

C₄⋊C₄

—

334

—

101

C₄²

C₂²×D₁₆₇

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

668

—

105

Q₈

—

668

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

668

—

109

C₂×C₄

—

668

—

110

C₂×C₄

D₄×D₁₆₇

yes

characteristic

111

C₂×C₄

D₄×D₁₆₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₆₇

yes

characteristic

113

C₈

—

114

C₈

—

334

—

115

C₈

—

167

—

116

C₂²

#19128828

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

1336

—

120

C₄

—

1336

—

121

C₄

Q₁₆×D₁₆₇

yes

—

122

C₄

#19128858

yes

—

123

C₄

#19128864

yes

—

124

C₂

#24923647

yes

characteristic

125

C₂

#24923656

yes

characteristic

126

C₂

#24923662

yes

characteristic

	subgroup	G/H
center	C₂²	#19128828
commutator, cosocle	C₄²×C₁₆₇	C₂³
Frattini	C₄²	C₂²×D₁₆₇
Fitting	#494×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	#127
hypercenter	#494	D₁₆₇
hypocenter	C₁₆₇	#1723
LCS	G ▷ C₄²×C₁₆₇ ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#19128828

commutator, cosocle

C₄²×C₁₆₇

C₂³

Frattini

C₄²

C₂²×D₁₆₇

Fitting

#494×C₁₆₇

C₂

C₂²×C₁₆₇

D₁₆₇

C₁₆₇

G ▷ C₄²×C₁₆₇ ▷ C₂²×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
43. C₁₆₇⋊C₈	Q₁₆	semidirect
63. C₁₆₇	#1723	semidirect

H ≅ G/N

type

43. C₁₆₇⋊C₈

Q₁₆

semidirect

63. C₁₆₇

#1723

semidirect

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

subgroup	index
#24923672	2
#24923665	2
#24923621	2
#24923620	2
#24922991	2
#494×C₁₆₇	2
#1723	167

subgroup

index

#494×C₁₆₇

167

Subgroups — #36323813

Conjugacy classes of subgroups

Complements

Maximal subgroups