Group #36319949 — subgroups

Subgroups — #1589×C₁₆₇ #36319949

Order 21376.

180 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 88 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	C₄×D₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
2	#436×C₁₆₇	C₂	1	2	yes	maximal, complemented
3	#335×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊C₈×C₁₆₇	C₂	1	2	yes	maximal, complemented
5	C₈²×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈×C₁₆₇	C₄	1	1	yes	characteristic
7	C₂×M₁₆×C₁₆₇	—	2	2	no	—
8	C₄×D₄×C₁₆₇	C₂²	1	2	yes	—
9	#107×C₁₆₇	C₄	1	1	yes	characteristic
10	C₄⋊C₈×C₁₆₇	C₂²	1	2	yes	—
11	#102×C₁₆₇	C₄	1	2	yes	—
12	C₂²⋊C₈×C₁₆₇	—	2	2	no	—
13	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₆₇	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₁₆₇	—	2	2	no	—
18	D₈×C₁₆₇	—	2	2	no	—
19	M₁₆×C₁₆₇	—	4	4	no	—
20	C₂×C₈×C₁₆₇	—	2	1	no	—
21	C₂×C₈×C₁₆₇	—	2	2	no	—
22	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₁₆₇	C₂×C₄	1	2	yes	—
27	C₂²⋊C₄×C₁₆₇	—	2	2	no	—
28	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
29	C₂³×C₁₆₇	—	2	2	no	—
30	D₄×C₁₆₇	—	2	2	no	—
31	D₄×C₁₆₇	—	2	2	no	—
32	C₂×C₄×C₁₆₇	—	2	2	no	—
33	C₂×C₄×C₁₆₇	—	4	2	no	—
34	C₂×C₄×C₁₆₇	—	4	2	no	—
35	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
38	C₈×C₁₆₇	—	2	2	no	—
39	C₈×C₁₆₇	—	2	1	no	—
40	C₈×C₁₆₇	—	2	1	no	—
41	C₈×C₁₆₇	—	4	2	no	—
42	C₈×C₁₆₇	M₁₆	1	2	yes	complemented
43	C₈×C₁₆₇	D₈	1	2	yes	complemented
44	C₂²×C₁₆₇	—	4	2	no	—
45	C₂²×C₁₆₇	—	4	2	no	—
46	C₂²×C₁₆₇	—	4	2	no	—
47	C₂²×C₁₆₇	C₄×D₄	1	1	yes	characteristic
48	C₄×C₁₆₇	—	2	1	no	—
49	C₄×C₁₆₇	—	4	2	no	—
50	C₄×C₁₆₇	C₂×M₁₆	1	1	yes	characteristic
51	C₄×C₁₆₇	#131	1	1	yes	characteristic
52	C₄×C₁₆₇	C₂×D₈	1	1	yes	characteristic
53	C₄×C₁₆₇	#135	1	1	yes	characteristic
54	C₂×C₁₆₇	—	8	2	no	—
55	C₂×C₁₆₇	#436	1	1	yes	characteristic
56	C₂×C₁₆₇	C₄×D₈	1	1	yes	characteristic
57	C₂×C₁₆₇	#443	1	1	yes	characteristic
58	C₁₆₇	#1589	1	1	yes	characteristic, direct product factor, Sylow
59	#1589	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₄×D₈	C₂×C₁₆₇	1	1	yes	characteristic
61	#436	C₂×C₁₆₇	1	2	yes	complemented
62	#335	C₂×C₁₆₇	1	1	yes	characteristic, complemented
63	D₄⋊C₈	C₂×C₁₆₇	1	2	yes	complemented
64	C₈²	C₂×C₁₆₇	1	1	yes	characteristic, complemented
65	C₂×D₈	C₄×C₁₆₇	1	1	yes	characteristic
66	C₂×M₁₆	—	2	2	no	—
67	C₄×D₄	C₂²×C₁₆₇	1	2	yes	—
68	#107	C₄×C₁₆₇	1	1	yes	characteristic
69	C₄⋊C₈	C₂²×C₁₆₇	1	2	yes	—
70	#102	C₄×C₁₆₇	1	2	yes	—
71	C₂²⋊C₈	—	2	2	no	—
72	C₄×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
73	C₄×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
74	C₄×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
75	C₂×D₄	C₂×C₄×C₁₆₇	1	2	yes	—
76	C₂²×C₄	—	2	2	no	—
77	D₈	—	2	2	no	—
78	M₁₆	—	4	4	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	C₂×C₄×C₁₆₇	1	1	yes	characteristic
82	C₂×C₈	C₂×C₄×C₁₆₇	1	1	yes	characteristic
83	C₂×C₈	D₄×C₁₆₇	1	1	yes	characteristic
84	C₂×C₈	D₄×C₁₆₇	1	1	yes	characteristic
85	C₄⋊C₄	C₂×C₄×C₁₆₇	1	2	yes	—
86	C₂²⋊C₄	—	2	2	no	—
87	C₄²	C₂³×C₁₆₇	1	1	yes	characteristic
88	C₂³	—	2	2	no	—
89	D₄	—	2	2	no	—
90	D₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	C₂²×C₄×C₁₆₇	1	1	yes	characteristic
95	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
96	C₂×C₄	#41×C₁₆₇	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₈	—	4	2	no	—
101	C₈	M₁₆×C₁₆₇	1	2	yes	complemented
102	C₈	D₈×C₁₆₇	1	2	yes	complemented
103	C₂²	—	4	2	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	C₄×D₄×C₁₆₇	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	4	2	no	—
109	C₄	C₂×M₁₆×C₁₆₇	1	1	yes	characteristic
110	C₄	#131×C₁₆₇	1	1	yes	characteristic
111	C₄	C₂×D₈×C₁₆₇	1	1	yes	characteristic
112	C₄	#135×C₁₆₇	1	1	yes	characteristic
113	C₂	—	8	2	no	—
114	C₂	#436×C₁₆₇	1	1	yes	characteristic
115	C₂	C₄×D₈×C₁₆₇	1	1	yes	characteristic
116	C₂	#443×C₁₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₈×C₁₆₇

C₂

yes

maximal, characteristic

#436×C₁₆₇

C₂

yes

maximal, complemented

#335×C₁₆₇

C₂

yes

maximal, characteristic, complemented

D₄⋊C₈×C₁₆₇

C₂

yes

maximal, complemented

C₈²×C₁₆₇

C₂

yes

maximal, characteristic, complemented

C₂×D₈×C₁₆₇

C₄

yes

characteristic

C₂×M₁₆×C₁₆₇

—

C₄×D₄×C₁₆₇

C₂²

yes

—

#107×C₁₆₇

C₄

yes

characteristic

C₄⋊C₈×C₁₆₇

C₂²

yes

—

#102×C₁₆₇

C₄

yes

—

C₂²⋊C₈×C₁₆₇

—

C₄×C₈×C₁₆₇

C₂²

yes

characteristic

C₄×C₈×C₁₆₇

C₂²

yes

characteristic

C₄×C₈×C₁₆₇

C₂²

yes

characteristic

C₂×D₄×C₁₆₇

C₂×C₄

yes

—

C₂²×C₄×C₁₆₇

—

D₈×C₁₆₇

—

M₁₆×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₇

D₄

yes

characteristic

C₂×C₈×C₁₆₇

D₄

yes

characteristic

C₄⋊C₄×C₁₆₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₆₇

—

C₄²×C₁₆₇

C₂³

yes

characteristic

C₂³×C₁₆₇

—

D₄×C₁₆₇

—

D₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

#41

yes

characteristic

C₈×C₁₆₇

—

C₈×C₁₆₇

—

C₈×C₁₆₇

—

C₈×C₁₆₇

—

C₈×C₁₆₇

M₁₆

yes

complemented

C₈×C₁₆₇

D₈

yes

complemented

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₄×D₄

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×M₁₆

yes

characteristic

C₄×C₁₆₇

#131

yes

characteristic

C₄×C₁₆₇

C₂×D₈

yes

characteristic

C₄×C₁₆₇

#135

yes

characteristic

C₂×C₁₆₇

—

C₂×C₁₆₇

#436

yes

characteristic

C₂×C₁₆₇

C₄×D₈

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, direct product factor, Sylow

#1589

C₁₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₄×D₈

C₂×C₁₆₇

yes

characteristic

#436

C₂×C₁₆₇

yes

complemented

#335

C₂×C₁₆₇

yes

characteristic, complemented

D₄⋊C₈

C₂×C₁₆₇

yes

complemented

C₈²

C₂×C₁₆₇

yes

characteristic, complemented

C₂×D₈

C₄×C₁₆₇

yes

characteristic

C₂×M₁₆

—

C₄×D₄

C₂²×C₁₆₇

yes

—

#107

C₄×C₁₆₇

yes

characteristic

C₄⋊C₈

C₂²×C₁₆₇

yes

—

#102

C₄×C₁₆₇

yes

—

C₂²⋊C₈

—

C₄×C₈

C₂²×C₁₆₇

yes

characteristic

C₄×C₈

C₂²×C₁₆₇

yes

characteristic

C₄×C₈

C₂²×C₁₆₇

yes

characteristic

C₂×D₄

C₂×C₄×C₁₆₇

yes

—

C₂²×C₄

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₆₇

yes

characteristic

C₂×C₈

C₂×C₄×C₁₆₇

yes

characteristic

C₂×C₈

D₄×C₁₆₇

yes

characteristic

C₂×C₈

D₄×C₁₆₇

yes

characteristic

C₄⋊C₄

C₂×C₄×C₁₆₇

yes

—

C₂²⋊C₄

—

C₄²

C₂³×C₁₆₇

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₁₆₇

yes

characteristic

C₂×C₄

C₂×D₄×C₁₆₇

yes

characteristic

C₂×C₄

#41×C₁₆₇

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

M₁₆×C₁₆₇

yes

complemented

102

C₈

D₈×C₁₆₇

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₄×D₄×C₁₆₇

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

C₂×M₁₆×C₁₆₇

yes

characteristic

110

C₄

#131×C₁₆₇

yes

characteristic

111

C₄

C₂×D₈×C₁₆₇

yes

characteristic

112

C₄

#135×C₁₆₇

yes

characteristic

113

C₂

—

114

C₂

#436×C₁₆₇

yes

characteristic

115

C₂

C₄×D₈×C₁₆₇

yes

characteristic

116

C₂

#443×C₁₆₇

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₆₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₆₇
Frattini, cosocle	C₄²	C₂³×C₁₆₇
socle	C₂²×C₁₆₇	C₄×D₄
prime radical	C₂×D₈×C₁₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₆₇ ◁ C₄×C₈×C₁₆₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₆₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₆₇

Frattini, cosocle

C₄²

C₂³×C₁₆₇

socle

C₂²×C₁₆₇

C₄×D₄

prime radical

C₂×D₈×C₁₆₇

C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₆₇ ◁ C₄×C₈×C₁₆₇ ◁ G

N	H ≅ G/N	type
2. #436×C₁₆₇	C₂	semidirect
3. #335×C₁₆₇	C₂	semidirect
4. D₄⋊C₈×C₁₆₇	C₂	semidirect
5. C₈²×C₁₆₇	C₂	semidirect
42. C₈×C₁₆₇	M₁₆	semidirect
43. C₈×C₁₆₇	D₈	semidirect
58. C₁₆₇	#1589	direct (N × H)
59. #1589	C₁₆₇	direct (N × H)
61. #436	C₂×C₁₆₇	semidirect
62. #335	C₂×C₁₆₇	semidirect
63. D₄⋊C₈	C₂×C₁₆₇	semidirect
64. C₈²	C₂×C₁₆₇	semidirect
101. C₈	M₁₆×C₁₆₇	semidirect
102. C₈	D₈×C₁₆₇	semidirect

H ≅ G/N

type

2. #436×C₁₆₇

C₂

semidirect

3. #335×C₁₆₇

C₂

semidirect

4. D₄⋊C₈×C₁₆₇

C₂

semidirect

5. C₈²×C₁₆₇

C₂

semidirect

42. C₈×C₁₆₇

M₁₆

semidirect

43. C₈×C₁₆₇

D₈

semidirect

58. C₁₆₇

#1589

direct (N × H)

59. #1589

C₁₆₇

direct (N × H)

61. #436

C₂×C₁₆₇

semidirect

62. #335

C₂×C₁₆₇

semidirect

63. D₄⋊C₈

C₂×C₁₆₇

semidirect

64. C₈²

C₂×C₁₆₇

semidirect

101. C₈

M₁₆×C₁₆₇

semidirect

102. C₈

D₈×C₁₆₇

semidirect

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

subgroup	index
C₄×D₈×C₁₆₇	2
#436×C₁₆₇	2
#335×C₁₆₇	2
D₄⋊C₈×C₁₆₇	2
C₈²×C₁₆₇	2
#1589	167

subgroup

index

C₄×D₈×C₁₆₇

#436×C₁₆₇

#335×C₁₆₇

D₄⋊C₈×C₁₆₇

C₈²×C₁₆₇

#1589

167

Subgroups — #1589×C₁₆₇ #36319949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1589×C167 #36319949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1589×C₁₆₇ #36319949