Group #36324299 — subgroups

Subgroups — #36324299

Order 21376.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×#19128765	C₂	1	2	yes	maximal
2	C₂×#19128764	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₁₆₇	C₂	1	1	yes	maximal, characteristic
4	#24923134	C₂	1	2	yes	maximal
5	C₂×#106×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₆₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₆₇	C₂²	1	1	yes	characteristic
9	#19128765	—	2	4	no	—
10	#19128764	—	2	2	no	—
11	C₈×Dic₁₆₇	C₄	1	4	yes	complemented
12	C₁₆₇⋊#95	—	2	4	no	—
13	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	2	yes	—
15	#106×C₁₆₇	—	2	2	no	—
16	C₂²×Dic₁₆₇	—	2	2	no	—
17	C₂²×Dic₁₆₇	D₄	1	2	yes	—
18	C₂×C₁₆₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₁₆₇⋊C₈	D₄	1	2	yes	—
20	C₂×C₁₆₇⋊C₈	Q₈	1	2	yes	—
21	C₄⋊Dic₁₆₇	—	2	2	no	—
22	C₄⋊Dic₁₆₇	—	2	2	no	—
23	C₄×Dic₁₆₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₆₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₆₇	—	2	2	no	—
26	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
29	C₂×C₈×C₁₆₇	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₆₇	—	2	2	no	—
31	C₄⋊C₄×C₁₆₇	—	2	2	no	—
32	C₂×Dic₁₆₇	—	2	4	no	—
33	C₂×Dic₁₆₇	—	4	2	no	—
34	C₂×Dic₁₆₇	—	4	4	no	—
35	C₂×Dic₁₆₇	C₄⋊C₄	1	4	yes	—
36	C₁₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₂³×C₁₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₆₇	—	4	2	no	—
39	C₂×C₄×C₁₆₇	—	4	4	no	—
40	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₁₆₇	C₄⋊C₄	1	4	yes	—
47	C₂²×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₆₇	#127	1	1	yes	characteristic
49	C₂²×C₁₆₇	C₄⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₆₇	#135	1	4	yes	—
51	C₄×C₁₆₇	—	8	2	no	—
52	C₄×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₆₇	C₄⋊Q₈	1	1	yes	characteristic
55	C₄×C₁₆₇	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₆₇	—	2	4	no	—
57	Dic₁₆₇	—	8	2	no	—
58	C₂×C₁₆₇	#384	1	1	yes	characteristic
59	C₂×C₁₆₇	#427	1	2	yes	—
60	C₂×C₁₆₇	#495	1	2	yes	—
61	C₂×C₁₆₇	#499	1	2	yes	—
62	C₁₆₇	#1846	1	1	yes	characteristic, complemented, Sylow
63	#1846	—	167	1	no	maximal, Sylow
64	C₂×#107	—	167	2	no	—
65	C₂×#106	—	167	1	no	—
66	C₂×#106	D₁₆₇	1	1	yes	characteristic
67	C₂×C₄×C₈	—	167	1	no	—
68	#382	—	167	2	no	—
69	C₂²×C₈	—	167	1	no	—
70	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
71	C₂×C₄⋊C₄	—	167	2	no	—
72	C₂×C₄⋊C₄	C₂×D₁₆₇	1	2	yes	—
73	C₂×C₄²	—	167	1	no	—
74	#107	—	334	4	no	—
75	#106	—	2	2	no	—
76	#106	—	334	2	no	—
77	C₄×C₈	—	167	4	no	—
78	#95	—	334	4	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	334	2	no	—
81	C₂²×C₄	—	167	2	no	—
82	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
83	C₂×C₈	—	167	2	no	—
84	C₂×C₈	—	167	2	no	—
85	C₂×C₈	—	167	2	no	—
86	C₂×C₈	C₄×D₁₆₇	1	2	yes	—
87	C₂×C₈	C₂²⋊D₁₆₇	1	2	yes	—
88	C₂×C₈	Dic₃₃₄	1	2	yes	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	334	2	no	—
92	C₄⋊C₄	—	334	2	no	—
93	C₄²	—	167	2	no	—
94	C₄²	—	167	2	no	—
95	C₂³	D₄×D₁₆₇	1	1	yes	characteristic
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	334	4	no	—
99	C₂×C₄	—	668	4	no	—
100	C₂×C₄	—	668	2	no	—
101	C₂×C₄	—	167	4	no	—
102	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
103	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₁₆₇	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₆₇	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₃₃₄	1	1	yes	characteristic
108	C₈	—	167	4	no	—
109	C₈	Dic₁₆₇⋊C₄	1	4	yes	—
110	C₂²	C₄⋊C₄×D₁₆₇	1	1	yes	characteristic
111	C₂²	#19128828	1	1	yes	characteristic
112	C₂²	#19128831	1	1	yes	characteristic
113	C₂²	#19128853	1	4	yes	—
114	C₄	—	8	2	no	—
115	C₄	—	1336	2	no	—
116	C₄	—	334	4	no	—
117	C₄	C₄⋊C₄×D₁₆₇	1	1	yes	characteristic
118	C₄	C₂×Dic₁₆₇⋊C₄	1	1	yes	characteristic
119	C₄	#19128830	1	1	yes	characteristic
120	C₄	#19128831	1	1	yes	characteristic
121	C₂	#24923138	1	1	yes	characteristic
122	C₂	#24923304	1	2	yes	—
123	C₂	#24923650	1	2	yes	—
124	C₂	#24923669	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128765

C₂

yes

maximal

C₂×#19128764

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₆₇

C₂

yes

maximal, characteristic

#24923134

C₂

yes

maximal

C₂×#106×C₁₆₇

C₂

yes

maximal, characteristic

C₂²×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₆₇

C₂²

yes

—

C₂×C₄×Dic₁₆₇

C₂²

yes

characteristic

#19128765

—

#19128764

—

C₈×Dic₁₆₇

C₄

yes

complemented

C₁₆₇⋊#95

—

C₂²×C₈×C₁₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₆₇

C₂²

yes

—

#106×C₁₆₇

—

C₂²×Dic₁₆₇

—

C₂²×Dic₁₆₇

D₄

yes

—

C₂×C₁₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₆₇⋊C₈

D₄

yes

—

C₂×C₁₆₇⋊C₈

Q₈

yes

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₄×Dic₁₆₇

C₂×C₄

yes

—

C₄×Dic₁₆₇

Q₈

yes

—

C₂²×C₄×C₁₆₇

—

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₈×C₁₆₇

C₂×C₄

yes

—

C₂×C₈×C₁₆₇

D₄

yes

—

C₂×C₈×C₁₆₇

Q₈

yes

—

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

C₄⋊C₄

yes

—

C₁₆₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₇

C₂×Q₈

yes

characteristic

C₈×C₁₆₇

C₄⋊C₄

yes

—

C₂²×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₆₇

#127

yes

characteristic

C₂²×C₁₆₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₆₇

#135

yes

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₆₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₆₇

C₄⋊Q₈

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

#384

yes

characteristic

C₂×C₁₆₇

#427

yes

—

C₂×C₁₆₇

#495

yes

—

C₂×C₁₆₇

#499

yes

—

C₁₆₇

#1846

yes

characteristic, complemented, Sylow

#1846

—

167

maximal, Sylow

C₂×#107

—

167

—

C₂×#106

—

167

—

C₂×#106

D₁₆₇

yes

characteristic

C₂×C₄×C₈

—

167

—

#382

—

167

—

C₂²×C₈

—

167

—

C₂²×C₈

C₂×D₁₆₇

yes

characteristic

C₂×C₄⋊C₄

—

167

—

C₂×C₄⋊C₄

C₂×D₁₆₇

yes

—

C₂×C₄²

—

167

—

#107

—

334

—

#106

—

#106

—

334

—

C₄×C₈

—

167

—

#95

—

334

—

C₂²×C₄

—

C₂²×C₄

—

334

—

C₂²×C₄

—

167

—

C₂²×C₄

C₂²×D₁₆₇

yes

characteristic

C₂×C₈

—

167

—

C₂×C₈

—

167

—

C₂×C₈

—

167

—

C₂×C₈

C₄×D₁₆₇

yes

—

C₂×C₈

C₂²⋊D₁₆₇

yes

—

C₂×C₈

Dic₃₃₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

334

—

C₄⋊C₄

—

334

—

C₄²

—

167

—

C₄²

—

167

—

C₂³

D₄×D₁₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

—

668

—

100

C₂×C₄

—

668

—

101

C₂×C₄

—

167

—

102

C₂×C₄

C₂×C₄×D₁₆₇

yes

characteristic

103

C₂×C₄

D₄×D₁₆₇

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₁₆₇

yes

characteristic

105

C₂×C₄

Q₈×D₁₆₇

yes

characteristic

106

C₂×C₄

Q₈×D₁₆₇

yes

characteristic

107

C₂×C₄

C₂×Dic₃₃₄

yes

characteristic

108

C₈

—

167

—

109

C₈

Dic₁₆₇⋊C₄

yes

—

110

C₂²

C₄⋊C₄×D₁₆₇

yes

characteristic

111

C₂²

#19128828

yes

characteristic

112

C₂²

#19128831

yes

characteristic

113

C₂²

#19128853

yes

—

114

C₄

—

115

C₄

—

1336

—

116

C₄

—

334

—

117

C₄

C₄⋊C₄×D₁₆₇

yes

characteristic

118

C₄

C₂×Dic₁₆₇⋊C₄

yes

characteristic

119

C₄

#19128830

yes

characteristic

120

C₄

#19128831

yes

characteristic

121

C₂

#24923138

yes

characteristic

122

C₂

#24923304

yes

—

123

C₂

#24923650

yes

—

124

C₂

#24923669

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₆₇
commutator	C₂×C₄×C₁₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₇
Fitting	C₂×#106×C₁₆₇	C₂
socle, prime radical	C₂³×C₁₆₇	C₂×D₄
cosocle	C₂²×C₄×C₁₆₇	C₂³
hypercenter	C₂×#106	D₁₆₇
hypocenter	C₁₆₇	#1846
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₆₇

commutator

C₂×C₄×C₁₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₇

Fitting

C₂×#106×C₁₆₇

C₂

socle, prime radical

C₂³×C₁₆₇

C₂×D₄

cosocle

C₂²×C₄×C₁₆₇

C₂³

D₁₆₇

C₁₆₇

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
11. C₈×Dic₁₆₇	C₄	semidirect
62. C₁₆₇	#1846	semidirect

H ≅ G/N

type

11. C₈×Dic₁₆₇

C₄

semidirect

62. C₁₆₇

#1846

semidirect

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

subgroup	index
C₂×#19128765	2
C₂×#19128764	2
C₂×C₈×Dic₁₆₇	2
#24923134	2
C₂×#106×C₁₆₇	2
#1846	167

subgroup

index

C₂×C₈×Dic₁₆₇

C₂×#106×C₁₆₇

167

Subgroups — #36324299

Conjugacy classes of subgroups

Complements

Maximal subgroups