Group #36325890 — subgroups

Subgroups — C₂×Dic₁₆₇⋊C₁₆ #36325890

Order 21376.

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

normal subgroups only

This group has 184 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	C₂²×C₁₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₆₇	C₂	1	1	yes	maximal, characteristic
3	Dic₁₆₇⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×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	C₂×C₁₆₇⋊C₁₆	—	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	C₂×C₁₆×C₁₆₇	—	2	2	no	—
13	C₂×C₁₆×C₁₆₇	C₂²	1	2	yes	—
14	C₂²×Dic₁₆₇	C₈	1	1	yes	characteristic
15	C₂²×Dic₁₆₇	C₈	1	1	yes	characteristic
16	C₂×C₁₆₇⋊C₈	—	2	2	no	—
17	C₂×C₁₆₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₆₇	C₈	1	2	yes	—
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	2	no	—
28	C₂×Dic₁₆₇	C₁₆	1	4	yes	complemented
29	C₂×Dic₁₆₇	C₂×C₈	1	2	yes	—
30	C₂×Dic₁₆₇	C₂×C₈	1	2	yes	—
31	C₁₆₇⋊C₈	—	2	2	no	—
32	C₂³×C₁₆₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₆₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₁₆₇	M₁₆	1	2	yes	—
36	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₁₆₇	C₄⋊C₄	1	2	yes	—
38	C₈×C₁₆₇	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₁₆₇	C₄⋊C₈	1	2	yes	—
41	C₂²×C₁₆₇	C₂×C₁₆	1	2	yes	—
42	C₂²×C₁₆₇	M₃₂	1	2	yes	—
43	C₂²×C₁₆₇	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₆₇	C₄⋊C₈	1	2	yes	—
45	C₄×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₁₆₇	C₂×M₁₆	1	1	yes	characteristic
47	Dic₁₆₇	—	2	2	no	—
48	Dic₁₆₇	C₂×C₁₆	1	4	yes	complemented
49	C₂×C₁₆₇	C₄⋊C₁₆	1	4	yes	complemented
50	C₂×C₁₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₁₆₇	C₂²×C₁₆	1	1	yes	characteristic
52	C₂×C₁₆₇	C₂×M₃₂	1	1	yes	characteristic
53	C₁₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₄⋊C₁₆	—	167	1	no	maximal, Sylow
55	C₂²×C₁₆	—	167	1	no	—
56	C₂²×C₁₆	D₁₆₇	1	1	yes	characteristic
57	C₂×C₄×C₈	—	167	1	no	—
58	C₄⋊C₁₆	—	167	4	no	—
59	C₂²×C₈	—	167	1	no	—
60	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
61	C₂×C₄²	—	167	1	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	334	2	no	—
64	C₂×C₁₆	—	167	2	no	—
65	C₂×C₁₆	C₂×D₁₆₇	1	2	yes	—
66	C₄×C₈	—	167	2	no	—
67	C₄×C₈	—	167	2	no	—
68	C₂²×C₄	—	167	1	no	—
69	C₂²×C₄	—	167	1	no	—
70	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
71	C₂×C₈	—	334	2	no	—
72	C₂×C₈	—	167	2	no	—
73	C₂×C₈	C₄×D₁₆₇	1	1	yes	characteristic
74	C₂×C₈	C₂²⋊D₁₆₇	1	2	yes	—
75	C₂×C₈	Dic₃₃₄	1	2	yes	—
76	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
77	C₄²	—	167	2	no	—
78	C₄²	—	167	2	no	—
79	C₁₆	—	2	2	no	—
80	C₁₆	—	334	2	no	—
81	C₂³	C₈×D₁₆₇	1	1	yes	characteristic
82	C₂×C₄	—	334	2	no	—
83	C₂×C₄	—	167	2	no	—
84	C₂×C₄	—	167	2	no	—
85	C₂×C₄	—	167	4	no	—
86	C₂×C₄	Dic₁₆₇⋊C₄	1	2	yes	—
87	C₂×C₄	C₈×D₁₆₇	1	1	yes	characteristic
88	C₂×C₄	#14581933	1	2	yes	—
89	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
90	C₈	—	334	2	no	—
91	C₈	Dic₁₆₇⋊C₄	1	2	yes	—
92	C₈	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
93	C₈	C₂×Dic₃₃₄	1	1	yes	characteristic
94	C₂²	Dic₁₆₇⋊C₈	1	2	yes	—
95	C₂²	C₁₆×D₁₆₇	1	2	yes	—
96	C₂²	#19128771	1	2	yes	—
97	C₂²	C₂×C₈×D₁₆₇	1	1	yes	characteristic
98	C₄	—	334	2	no	—
99	C₄	—	167	4	no	—
100	C₄	Dic₁₆₇⋊C₈	1	2	yes	—
101	C₄	C₂×Dic₁₆₇⋊C₄	1	1	yes	characteristic
102	C₄	C₂×#14581933	1	1	yes	characteristic
103	C₂	Dic₁₆₇⋊C₁₆	1	4	yes	direct product factor
104	C₂	C₂×Dic₁₆₇⋊C₈	1	1	yes	characteristic
105	C₂	C₂×C₁₆×D₁₆₇	1	1	yes	characteristic
106	C₂	C₂×#19128771	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆₇⋊C₁₆

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₆₇

C₂

yes

maximal, characteristic

Dic₁₆₇⋊C₁₆

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₁₆₇

C₂

yes

maximal, characteristic

C₂²×C₁₆₇⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₁₆₇

C₄

yes

characteristic

C₂×C₁₆₇⋊C₁₆

—

C₂×C₁₆₇⋊C₁₆

C₂²

yes

—

C₈×Dic₁₆₇

C₄

yes

—

C₈×Dic₁₆₇

C₂²

yes

—

C₂²×C₈×C₁₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₆₇

—

C₂×C₁₆×C₁₆₇

C₂²

yes

—

C₂²×Dic₁₆₇

C₈

yes

characteristic

C₂²×Dic₁₆₇

C₈

yes

characteristic

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₆₇

C₈

yes

—

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

C₁₆

yes

complemented

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

C₄⋊C₈

yes

—

C₂²×C₁₆₇

C₂×C₁₆

yes

—

C₂²×C₁₆₇

M₃₂

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

C₂×C₁₆

yes

complemented

C₂×C₁₆₇

C₄⋊C₁₆

yes

complemented

C₂×C₁₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₆₇

C₂²×C₁₆

yes

characteristic

C₂×C₁₆₇

C₂×M₃₂

yes

characteristic

C₁₆₇

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₁₆

—

167

maximal, Sylow

C₂²×C₁₆

—

167

—

C₂²×C₁₆

D₁₆₇

yes

characteristic

C₂×C₄×C₈

—

167

—

C₄⋊C₁₆

—

167

—

C₂²×C₈

—

167

—

C₂²×C₈

C₂×D₁₆₇

yes

characteristic

C₂×C₄²

—

167

—

C₂×C₁₆

—

C₂×C₁₆

—

334

—

C₂×C₁₆

—

167

—

C₂×C₁₆

C₂×D₁₆₇

yes

—

C₄×C₈

—

167

—

C₄×C₈

—

167

—

C₂²×C₄

—

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

—

C₁₆

—

C₁₆

—

334

—

C₂³

C₈×D₁₆₇

yes

characteristic

C₂×C₄

—

334

—

C₂×C₄

—

167

—

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

Dic₁₆₇⋊C₈

yes

—

C₂²

C₁₆×D₁₆₇

yes

—

C₂²

#19128771

yes

—

C₂²

C₂×C₈×D₁₆₇

yes

characteristic

C₄

—

334

—

C₄

—

167

—

100

C₄

Dic₁₆₇⋊C₈

yes

—

101

C₄

C₂×Dic₁₆₇⋊C₄

yes

characteristic

102

C₄

C₂×#14581933

yes

characteristic

103

C₂

Dic₁₆₇⋊C₁₆

yes

direct product factor

104

C₂

C₂×Dic₁₆₇⋊C₈

yes

characteristic

105

C₂

C₂×C₁₆×D₁₆₇

yes

characteristic

106

C₂

C₂×#19128771

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₆₇
commutator	C₂×C₁₆₇	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting	C₂²×C₁₆×C₁₆₇	C₂
socle, prime radical	C₂³×C₁₆₇	C₂×C₈
cosocle	C₂×C₈×C₁₆₇	C₂³
hypercenter	C₂²×C₁₆	D₁₆₇
hypocenter	C₁₆₇	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²×C₈

C₂×D₁₆₇

commutator

C₂×C₁₆₇

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₁₆₇

Fitting

C₂²×C₁₆×C₁₆₇

C₂

socle, prime radical

C₂³×C₁₆₇

C₂×C₈

cosocle

C₂×C₈×C₁₆₇

C₂³

hypercenter

C₂²×C₁₆

D₁₆₇

hypocenter

C₁₆₇

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. Dic₁₆₇⋊C₁₆	C₂	direct (N × H)
28. C₂×Dic₁₆₇	C₁₆	semidirect
48. Dic₁₆₇	C₂×C₁₆	semidirect
49. C₂×C₁₆₇	C₄⋊C₁₆	semidirect
53. C₁₆₇	C₂×C₄⋊C₁₆	semidirect
103. C₂	Dic₁₆₇⋊C₁₆	direct (N × H)

H ≅ G/N

type

3. Dic₁₆₇⋊C₁₆

C₂

direct (N × H)

28. C₂×Dic₁₆₇

C₁₆

semidirect

48. Dic₁₆₇

C₂×C₁₆

semidirect

49. C₂×C₁₆₇

C₄⋊C₁₆

semidirect

53. C₁₆₇

C₂×C₄⋊C₁₆

semidirect

103. C₂

Dic₁₆₇⋊C₁₆

direct (N × H)

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

subgroup	index
C₂²×C₁₆₇⋊C₁₆	2
C₂×C₈×Dic₁₆₇	2
Dic₁₆₇⋊C₁₆	2
C₂²×C₁₆×C₁₆₇	2
C₂×C₄⋊C₁₆	167

subgroup

index

C₂²×C₁₆₇⋊C₁₆

C₂×C₈×Dic₁₆₇

Dic₁₆₇⋊C₁₆

C₂²×C₁₆×C₁₆₇

C₂×C₄⋊C₁₆

167

Subgroups — C₂×Dic₁₆₇⋊C₁₆ #36325890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×Dic167⋊C16 #36325890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₁₆₇⋊C₁₆ #36325890