Group #36320326 — subgroups

Subgroups — #1966×C₁₆₇ #36320326

Order 21376.

292 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 120 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×C₄⋊Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
2	#423×C₁₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#101×C₁₆₇	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₁₆₇	C₂²	1	2	yes	—
6	C₂×M₁₆×C₁₆₇	C₂²	1	2	yes	—
7	C₂×M₁₆×C₁₆₇	C₂²	1	2	yes	—
8	C₄⋊Q₈×C₁₆₇	—	2	1	no	—
9	C₄⋊Q₈×C₁₆₇	—	2	1	no	—
10	C₄⋊Q₈×C₁₆₇	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₆₇	C₄	1	4	yes	—
12	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₆₇	—	2	2	no	—
14	#101×C₁₆₇	—	2	8	no	—
15	#97×C₁₆₇	—	2	1	no	—
16	C₄×C₈×C₁₆₇	—	2	1	no	—
17	C₂×Q₈×C₁₆₇	—	2	4	no	—
18	C₂×Q₈×C₁₆₇	—	2	2	no	—
19	C₂×Q₈×C₁₆₇	—	2	2	no	—
20	C₂×Q₈×C₁₆₇	—	2	4	no	—
21	C₂²×C₄×C₁₆₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₆₇	—	2	2	no	—
25	M₁₆×C₁₆₇	—	2	4	no	—
26	M₁₆×C₁₆₇	D₄	1	4	yes	—
27	C₂×C₈×C₁₆₇	—	2	2	no	—
28	C₂×C₈×C₁₆₇	—	2	2	no	—
29	C₄⋊C₄×C₁₆₇	—	2	4	no	—
30	C₄⋊C₄×C₁₆₇	—	2	4	no	—
31	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
32	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
33	C₄²×C₁₆₇	D₄	1	2	yes	—
34	C₂³×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₆₇	—	4	4	no	—
36	Q₈×C₁₆₇	—	4	2	no	—
37	Q₈×C₁₆₇	—	4	2	no	—
38	C₂×C₄×C₁₆₇	—	2	4	no	—
39	C₂×C₄×C₁₆₇	—	2	4	no	—
40	C₂×C₄×C₁₆₇	—	2	8	no	—
41	C₂×C₄×C₁₆₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₆₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₆₇	#41	1	2	yes	—
47	C₈×C₁₆₇	—	2	4	no	—
48	C₈×C₁₆₇	—	4	2	no	—
49	C₂²×C₁₆₇	—	2	2	no	—
50	C₂²×C₁₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₆₇	C₄×D₄	1	2	yes	—
52	C₄×C₁₆₇	—	2	2	no	—
53	C₄×C₁₆₇	—	4	4	no	—
54	C₄×C₁₆₇	#101	1	4	yes	complemented
55	C₄×C₁₆₇	#121	1	2	yes	—
56	C₄×C₁₆₇	#124	1	1	yes	characteristic
57	C₄×C₁₆₇	#127	1	1	yes	characteristic
58	C₂×C₁₆₇	—	2	2	no	—
59	C₂×C₁₆₇	#390	1	1	yes	characteristic
60	C₂×C₁₆₇	C₂×#101	1	2	yes	—
61	C₁₆₇	#1966	1	1	yes	characteristic, direct product factor, Sylow
62	#1966	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₄⋊Q₈	C₂×C₁₆₇	1	1	yes	characteristic
64	#423	C₂×C₁₆₇	1	1	yes	characteristic
65	C₂×#101	C₂×C₁₆₇	1	4	yes	—
66	C₄×M₁₆	C₂×C₁₆₇	1	1	yes	characteristic
67	C₂²×Q₈	C₂²×C₁₆₇	1	2	yes	—
68	C₂×M₁₆	C₂²×C₁₆₇	1	2	yes	—
69	C₂×M₁₆	C₂²×C₁₆₇	1	2	yes	—
70	C₄⋊Q₈	—	2	1	no	—
71	C₄⋊Q₈	—	2	1	no	—
72	C₄⋊Q₈	—	2	2	no	—
73	C₂×C₄⋊C₄	C₄×C₁₆₇	1	4	yes	—
74	C₂×C₄²	C₂²×C₁₆₇	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	#101	—	2	8	no	—
77	#97	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂²×C₄	C₂×C₄×C₁₆₇	1	4	yes	—
84	C₂²×C₄	C₂×C₄×C₁₆₇	1	2	yes	—
85	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	4	no	—
88	M₁₆	D₄×C₁₆₇	1	4	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄²	D₄×C₁₆₇	1	1	yes	characteristic
94	C₄²	D₄×C₁₆₇	1	1	yes	characteristic
95	C₄²	D₄×C₁₆₇	1	2	yes	—
96	C₂³	C₂²×C₄×C₁₆₇	1	1	yes	characteristic
97	Q₈	—	4	4	no	—
98	Q₈	—	4	2	no	—
99	Q₈	—	4	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	C₂²⋊C₄×C₁₆₇	1	2	yes	—
104	C₂×C₄	C₂²⋊C₄×C₁₆₇	1	2	yes	—
105	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
106	C₂×C₄	C₂×D₄×C₁₆₇	1	2	yes	—
107	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
108	C₂×C₄	#41×C₁₆₇	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	4	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×C₂²⋊C₄×C₁₆₇	1	1	yes	characteristic
113	C₂²	C₄×D₄×C₁₆₇	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	#101×C₁₆₇	1	4	yes	complemented
117	C₄	#121×C₁₆₇	1	2	yes	—
118	C₄	#124×C₁₆₇	1	1	yes	characteristic
119	C₄	#127×C₁₆₇	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#390×C₁₆₇	1	1	yes	characteristic
122	C₂	C₂×#101×C₁₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₁₆₇

C₂

yes

maximal, characteristic

#423×C₁₆₇

C₂

yes

maximal, characteristic

C₂×#101×C₁₆₇

C₂

yes

maximal

C₄×M₁₆×C₁₆₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₆₇

C₂²

yes

—

C₂×M₁₆×C₁₆₇

C₂²

yes

—

C₂×M₁₆×C₁₆₇

C₂²

yes

—

C₄⋊Q₈×C₁₆₇

—

C₄⋊Q₈×C₁₆₇

—

C₄⋊Q₈×C₁₆₇

—

C₂×C₄⋊C₄×C₁₆₇

C₄

yes

—

C₂×C₄²×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₇

—

#101×C₁₆₇

—

#97×C₁₆₇

—

C₄×C₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂²×C₄×C₁₆₇

C₂×C₄

yes

—

C₂²×C₄×C₁₆₇

C₂×C₄

yes

—

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

M₁₆×C₁₆₇

—

M₁₆×C₁₆₇

—

M₁₆×C₁₆₇

D₄

yes

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₄²×C₁₆₇

D₄

yes

characteristic

C₄²×C₁₆₇

D₄

yes

characteristic

C₄²×C₁₆₇

D₄

yes

—

C₂³×C₁₆₇

C₂²×C₄

yes

characteristic

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

#41

yes

—

C₈×C₁₆₇

—

C₈×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₆₇

C₄×D₄

yes

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

#101

yes

complemented

C₄×C₁₆₇

#121

yes

—

C₄×C₁₆₇

yes

C₄×C₁₆₇

yes

C₂×C₁₆₇

—

C₂×C₁₆₇

#390

yes

characteristic

C₂×C₁₆₇

C₂×#101

yes

—

C₁₆₇

#1966

yes

characteristic, direct product factor, Sylow

#1966

C₁₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₁₆₇

yes

characteristic

#423

C₂×C₁₆₇

yes

characteristic

C₂×#101

C₂×C₁₆₇

yes

—

C₄×M₁₆

C₂×C₁₆₇

yes

characteristic

C₂²×Q₈

C₂²×C₁₆₇

yes

—

C₂×M₁₆

C₂²×C₁₆₇

yes

—

C₂×M₁₆

C₂²×C₁₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

C₄×C₁₆₇

yes

—

C₂×C₄²

C₂²×C₁₆₇

yes

characteristic

C₄⋊C₈

—

#101

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×C₄×C₁₆₇

yes

—

C₂²×C₄

C₂×C₄×C₁₆₇

yes

—

C₂²×C₄

C₂³×C₁₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

D₄×C₁₆₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₄×C₁₆₇

yes

characteristic

C₄²

D₄×C₁₆₇

yes

characteristic

C₄²

D₄×C₁₆₇

yes

—

C₂³

C₂²×C₄×C₁₆₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²⋊C₄×C₁₆₇

yes

—

104

C₂×C₄

C₂²⋊C₄×C₁₆₇

yes

—

105

C₂×C₄

C₂×D₄×C₁₆₇

yes

characteristic

106

C₂×C₄

C₂×D₄×C₁₆₇

yes

—

107

C₂×C₄

C₂×D₄×C₁₆₇

yes

characteristic

108

C₂×C₄

#41×C₁₆₇

yes

—

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

C₂×C₂²⋊C₄×C₁₆₇

yes

characteristic

113

C₂²

C₄×D₄×C₁₆₇

yes

—

114

C₄

—

115

C₄

—

116

C₄

#101×C₁₆₇

yes

complemented

117

C₄

#121×C₁₆₇

yes

—

118

C₄

#124×C₁₆₇

yes

characteristic

119

C₄

#127×C₁₆₇

yes

characteristic

120

C₂

—

121

C₂

#390×C₁₆₇

yes

characteristic

122

C₂

C₂×#101×C₁₆₇

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₆₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₁₆₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₆₇
prime radical	C₂³×C₁₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆₇ ◁ C₂×C₄²×C₁₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₆₇

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₁₆₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₆₇

prime radical

C₂³×C₁₆₇

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₆₇ ◁ C₂×C₄²×C₁₆₇ ◁ G

N	H ≅ G/N	type
54. C₄×C₁₆₇	#101	semidirect
61. C₁₆₇	#1966	direct (N × H)
62. #1966	C₁₆₇	direct (N × H)
116. C₄	#101×C₁₆₇	semidirect

H ≅ G/N

type

54. C₄×C₁₆₇

#101

semidirect

61. C₁₆₇

#1966

direct (N × H)

62. #1966

C₁₆₇

direct (N × H)

116. C₄

#101×C₁₆₇

semidirect

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

subgroup	index
C₂×C₄⋊Q₈×C₁₆₇	2
#423×C₁₆₇	2
C₂×#101×C₁₆₇	2
C₄×M₁₆×C₁₆₇	2
#1966	167

subgroup

index

C₂×C₄⋊Q₈×C₁₆₇

#423×C₁₆₇

C₂×#101×C₁₆₇

C₄×M₁₆×C₁₆₇

#1966

167

Subgroups — #1966×C₁₆₇ #36320326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1966×C167 #36320326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1966×C₁₆₇ #36320326