Group #36325893 — subgroups

Subgroups — #36325893

Order 21376.

172 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 97 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×#19128772	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₆₇	C₂	1	1	yes	maximal, characteristic
3	Dic₁₆₇⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₆₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₆₇	C₄	1	1	yes	characteristic
7	#19128772	—	2	2	no	—
8	C₂×C₁₆₇⋊C₁₆	C₂²	1	2	yes	—
9	C₈×Dic₁₆₇	C₄	1	2	yes	—
10	C₈×Dic₁₆₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
12	M₃₂×C₁₆₇	—	2	2	no	—
13	C₂×C₁₆×C₁₆₇	C₂²	1	2	yes	—
14	C₂²×Dic₁₆₇	C₈	1	2	yes	—
15	C₂×C₁₆₇⋊C₈	—	2	2	no	—
16	C₂×C₁₆₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₆₇	C₈	1	1	yes	characteristic
18	C₄×Dic₁₆₇	C₈	1	1	yes	characteristic
19	C₄×Dic₁₆₇	C₂×C₄	1	2	yes	—
20	C₁₆₇⋊C₁₆	—	2	2	no	—
21	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
24	C₂×C₈×C₁₆₇	Q₈	1	2	yes	—
25	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
26	C₁₆×C₁₆₇	—	2	2	no	—
27	C₂×Dic₁₆₇	—	2	4	no	—
28	C₂×Dic₁₆₇	C₂×C₈	1	2	yes	—
29	C₂×Dic₁₆₇	C₂×C₈	1	2	yes	—
30	C₁₆₇⋊C₈	—	2	2	no	—
31	C₂³×C₁₆₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₁₆₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₆₇	M₁₆	1	2	yes	—
35	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₆₇	C₄⋊C₄	1	2	yes	—
37	C₈×C₁₆₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₁₆₇	—	2	2	no	—
40	C₂²×C₁₆₇	C₄⋊C₈	1	2	yes	—
41	C₂²×C₁₆₇	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₆₇	C₄⋊C₈	1	2	yes	—
43	C₄×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₁₆₇	C₂×M₁₆	1	1	yes	characteristic
45	Dic₁₆₇	—	2	2	no	—
46	Dic₁₆₇	M₃₂	1	4	yes	complemented
47	C₂×C₁₆₇	—	2	2	no	—
48	C₂×C₁₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₁₆₇	C₂×M₃₂	1	2	yes	—
50	C₁₆₇	#2150	1	1	yes	characteristic, complemented, Sylow
51	#2150	—	167	1	no	maximal, Sylow
52	C₂×M₃₂	—	167	1	no	—
53	C₂×M₃₂	D₁₆₇	1	1	yes	characteristic
54	C₂×C₄×C₈	—	167	1	no	—
55	C₄⋊C₁₆	—	167	4	no	—
56	C₂²×C₈	—	167	1	no	—
57	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
58	C₂×C₄²	—	167	1	no	—
59	M₃₂	—	2	2	no	—
60	M₃₂	—	334	2	no	—
61	C₂×C₁₆	—	167	2	no	—
62	C₂×C₁₆	C₂×D₁₆₇	1	2	yes	—
63	C₄×C₈	—	167	2	no	—
64	C₄×C₈	—	167	2	no	—
65	C₂²×C₄	—	167	2	no	—
66	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
67	C₂×C₈	—	334	2	no	—
68	C₂×C₈	—	167	2	no	—
69	C₂×C₈	C₄×D₁₆₇	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊D₁₆₇	1	2	yes	—
71	C₂×C₈	Dic₃₃₄	1	2	yes	—
72	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
73	C₄²	—	167	1	no	—
74	C₄²	—	167	1	no	—
75	C₄²	—	167	2	no	—
76	C₁₆	—	2	2	no	—
77	C₁₆	—	334	2	no	—
78	C₂³	C₈×D₁₆₇	1	1	yes	characteristic
79	C₂×C₄	—	334	4	no	—
80	C₂×C₄	—	167	2	no	—
81	C₂×C₄	—	167	2	no	—
82	C₂×C₄	Dic₁₆₇⋊C₄	1	2	yes	—
83	C₂×C₄	C₈×D₁₆₇	1	1	yes	characteristic
84	C₂×C₄	#14581933	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
86	C₈	—	334	2	no	—
87	C₈	Dic₁₆₇⋊C₄	1	2	yes	—
88	C₈	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
89	C₈	C₂×Dic₃₃₄	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	Dic₁₆₇⋊C₈	1	2	yes	—
92	C₂²	C₂×C₈×D₁₆₇	1	1	yes	characteristic
93	C₄	—	334	2	no	—
94	C₄	—	167	4	no	—
95	C₄	Dic₁₆₇⋊C₈	1	2	yes	—
96	C₄	C₂×Dic₁₆₇⋊C₄	1	1	yes	characteristic
97	C₄	C₂×#14581933	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	C₂×Dic₁₆₇⋊C₈	1	1	yes	characteristic
100	C₂	M₃₂×D₁₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128772

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₆₇

C₂

yes

maximal, characteristic

Dic₁₆₇⋊C₁₆

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₆₇

C₂

yes

maximal, characteristic

C₂²×C₁₆₇⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₁₆₇

C₄

yes

characteristic

#19128772

—

C₂×C₁₆₇⋊C₁₆

C₂²

yes

—

C₈×Dic₁₆₇

C₄

yes

—

C₈×Dic₁₆₇

C₂²

yes

—

C₂²×C₈×C₁₆₇

C₂²

yes

characteristic

M₃₂×C₁₆₇

—

C₂×C₁₆×C₁₆₇

C₂²

yes

—

C₂²×Dic₁₆₇

C₈

yes

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₆₇

C₈

yes

characteristic

C₄×Dic₁₆₇

C₈

yes

characteristic

C₄×Dic₁₆₇

C₂×C₄

yes

—

C₁₆₇⋊C₁₆

—

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₇

D₄

yes

—

C₂×C₈×C₁₆₇

Q₈

yes

—

C₂×C₈×C₁₆₇

C₂³

yes

characteristic

C₁₆×C₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

C₂×C₈

yes

—

C₂×Dic₁₆₇

C₂×C₈

yes

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₆₇

M₁₆

yes

—

C₂×C₄×C₁₆₇

C₂²×C₄

yes

characteristic

C₈×C₁₆₇

C₄⋊C₄

yes

—

C₈×C₁₆₇

C₂×D₄

yes

characteristic

C₈×C₁₆₇

C₂×Q₈

yes

characteristic

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₄⋊C₈

yes

—

C₂²×C₁₆₇

C₂²×C₈

yes

characteristic

C₄×C₁₆₇

C₄⋊C₈

yes

—

C₄×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₆₇

C₂×M₁₆

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

M₃₂

yes

complemented

C₂×C₁₆₇

—

C₂×C₁₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₆₇

C₂×M₃₂

yes

—

C₁₆₇

#2150

yes

characteristic, complemented, Sylow

#2150

—

167

maximal, Sylow

C₂×M₃₂

—

167

—

C₂×M₃₂

D₁₆₇

yes

characteristic

C₂×C₄×C₈

—

167

—

C₄⋊C₁₆

—

167

—

C₂²×C₈

—

167

—

C₂²×C₈

C₂×D₁₆₇

yes

characteristic

C₂×C₄²

—

167

—

M₃₂

—

M₃₂

—

334

—

C₂×C₁₆

—

167

—

C₂×C₁₆

C₂×D₁₆₇

yes

—

C₄×C₈

—

167

—

C₄×C₈

—

167

—

C₂²×C₄

—

167

—

C₂²×C₄

C₄×D₁₆₇

yes

characteristic

C₂×C₈

—

334

—

C₂×C₈

—

167

—

C₂×C₈

C₄×D₁₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₆₇

yes

—

C₂×C₈

Dic₃₃₄

yes

—

C₂×C₈

C₂²×D₁₆₇

yes

characteristic

C₄²

—

167

—

C₄²

—

167

—

C₄²

—

167

—

C₁₆

—

C₁₆

—

334

—

C₂³

C₈×D₁₆₇

yes

characteristic

C₂×C₄

—

334

—

C₂×C₄

—

167

—

C₂×C₄

—

167

—

C₂×C₄

Dic₁₆₇⋊C₄

yes

—

C₂×C₄

C₈×D₁₆₇

yes

characteristic

C₂×C₄

#14581933

yes

—

C₂×C₄

C₂×C₄×D₁₆₇

yes

characteristic

C₈

—

334

—

C₈

Dic₁₆₇⋊C₄

yes

—

C₈

C₂×C₂²⋊D₁₆₇

yes

characteristic

C₈

C₂×Dic₃₃₄

yes

characteristic

C₂²

—

C₂²

Dic₁₆₇⋊C₈

yes

—

C₂²

C₂×C₈×D₁₆₇

yes

characteristic

C₄

—

334

—

C₄

—

167

—

C₄

Dic₁₆₇⋊C₈

yes

—

C₄

C₂×Dic₁₆₇⋊C₄

yes

characteristic

C₄

C₂×#14581933

yes

characteristic

C₂

—

C₂

C₂×Dic₁₆₇⋊C₈

yes

characteristic

100

C₂

M₃₂×D₁₆₇

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₆₇
commutator, socle	C₂²×C₁₆₇	C₂²×C₈
Fitting	C₂×M₃₂×C₁₆₇	C₂
prime radical	C₂³×C₁₆₇	C₂×C₈
cosocle	C₂×C₈×C₁₆₇	C₂³
hypercenter	C₂×M₃₂	D₁₆₇
hypocenter	C₁₆₇	#2150
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₆₇

commutator, socle

C₂²×C₁₆₇

C₂²×C₈

Fitting

C₂×M₃₂×C₁₆₇

C₂

prime radical

C₂³×C₁₆₇

C₂×C₈

cosocle

C₂×C₈×C₁₆₇

C₂³

hypercenter

C₂×M₃₂

D₁₆₇

C₁₆₇

G ▷ C₂²×C₁₆₇ ▷ C₁₆₇

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. Dic₁₆₇⋊C₁₆	C₂	semidirect
46. Dic₁₆₇	M₃₂	semidirect
50. C₁₆₇	#2150	semidirect

H ≅ G/N

type

3. Dic₁₆₇⋊C₁₆

C₂

semidirect

46. Dic₁₆₇

M₃₂

semidirect

50. C₁₆₇

#2150

semidirect

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

subgroup	index
C₂×#19128772	2
C₂×C₈×Dic₁₆₇	2
Dic₁₆₇⋊C₁₆	2
C₂×M₃₂×C₁₆₇	2
#2150	167

subgroup

index

C₂×#19128772

C₂×C₈×Dic₁₆₇

Dic₁₆₇⋊C₁₆

C₂×M₃₂×C₁₆₇

#2150

167

Subgroups — #36325893

Conjugacy classes of subgroups

Complements

Maximal subgroups