Group #36324257 — subgroups

Subgroups — #36324257

Order 21376.

216 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 101 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⋊(C₁₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#24923115	C₂	1	1	yes	maximal, characteristic
3	#24923023	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₆₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₆₇	C₂²	1	2	yes	—
7	C₄⋊(C₁₆₇⋊C₈)	—	2	2	no	—
8	C₁₆₇⋊#95	C₄	1	2	yes	—
9	C₁₆₇⋊#95	C₄	1	2	yes	—
10	C₂²×C₈×C₁₆₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₆₇	—	2	2	no	—
13	C₂²×Dic₁₆₇	C₂×C₄	1	4	yes	—
14	C₂×C₁₆₇⋊C₈	—	2	2	no	—
15	C₂×C₁₆₇⋊C₈	—	2	4	no	—
16	C₄×Dic₁₆₇	—	2	2	no	—
17	C₄×Dic₁₆₇	—	2	2	no	—
18	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₁₆₇	—	2	2	no	—
22	C₂×C₈×C₁₆₇	—	2	4	no	—
23	C₄²×C₁₆₇	D₄	1	2	yes	—
24	C₄²×C₁₆₇	Q₈	1	2	yes	—
25	C₂×Dic₁₆₇	—	2	4	no	—
26	C₂×Dic₁₆₇	—	2	4	no	—
27	C₂×Dic₁₆₇	—	2	4	no	—
28	C₁₆₇⋊C₈	—	4	2	no	—
29	C₂³×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₆₇	—	2	2	no	—
31	C₂×C₄×C₁₆₇	—	2	2	no	—
32	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₆₇	#41	1	2	yes	—
37	C₂×C₄×C₁₆₇	#41	1	2	yes	—
38	C₈×C₁₆₇	—	4	2	no	—
39	C₂²×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₆₇	#117	1	2	yes	—
41	C₂²×C₁₆₇	#131	1	2	yes	—
42	C₂²×C₁₆₇	#131	1	2	yes	—
43	C₄×C₁₆₇	—	2	2	no	—
44	C₄×C₁₆₇	—	2	2	no	—
45	C₄×C₁₆₇	#124	1	1	yes	characteristic
46	C₄×C₁₆₇	#124	1	1	yes	characteristic
47	C₄×C₁₆₇	#125	1	2	yes	—
48	Dic₁₆₇	—	4	4	no	—
49	C₂×C₁₆₇	#382	1	1	yes	characteristic
50	C₂×C₁₆₇	#424	1	1	yes	characteristic
51	C₂×C₁₆₇	#424	1	1	yes	characteristic
52	C₂×C₁₆₇	#433	1	4	yes	—
53	C₁₆₇	#1832	1	1	yes	characteristic, complemented, Sylow
54	#1832	—	167	1	no	maximal, Sylow
55	C₂×C₄⋊C₈	—	167	1	no	—
56	C₂×C₄⋊C₈	D₁₆₇	1	1	yes	characteristic
57	#376	—	167	1	no	—
58	#336	—	167	4	no	—
59	C₂²×C₈	—	167	2	no	—
60	C₂²×C₈	C₂×D₁₆₇	1	2	yes	—
61	C₂×C₄²	—	167	2	no	—
62	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	334	2	no	—
65	#95	—	167	2	no	—
66	#95	—	167	2	no	—
67	C₂²×C₄	—	167	4	no	—
68	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
69	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	334	2	no	—
74	C₂×C₈	—	334	4	no	—
75	C₄²	—	334	2	no	—
76	C₄²	—	334	2	no	—
77	C₄²	C₂²⋊D₁₆₇	1	2	yes	—
78	C₄²	Dic₃₃₄	1	2	yes	—
79	C₂³	C₂×C₄×D₁₆₇	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	334	4	no	—
83	C₂×C₄	—	334	4	no	—
84	C₂×C₄	—	334	4	no	—
85	C₂×C₄	Dic₁₆₇⋊C₄	1	2	yes	—
86	C₂×C₄	Dic₁₆₇⋊C₄	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₃₃₄	1	1	yes	characteristic
89	C₂×C₄	#14581949	1	2	yes	—
90	C₂×C₄	#14581951	1	2	yes	—
91	C₈	—	4	2	no	—
92	C₈	—	668	2	no	—
93	C₂²	C₂×Dic₁₆₇⋊C₄	1	1	yes	characteristic
94	C₂²	#19128790	1	2	yes	—
95	C₂²	#19128838	1	2	yes	—
96	C₂²	#19128840	1	2	yes	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	668	4	no	—
100	C₄	#19128819	1	1	yes	characteristic
101	C₄	#19128821	1	1	yes	characteristic
102	C₄	#19128823	1	2	yes	—
103	C₂	#24923134	1	1	yes	characteristic
104	C₂	#24923295	1	1	yes	characteristic
105	C₂	#24923297	1	1	yes	characteristic
106	C₂	#24923329	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₆₇⋊C₈)

C₂

yes

maximal, characteristic

#24923115

C₂

yes

maximal, characteristic

#24923023

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₆₇

C₂

yes

maximal, characteristic

C₂²×C₁₆₇⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₆₇

C₂²

yes

—

C₄⋊(C₁₆₇⋊C₈)

—

C₁₆₇⋊#95

C₄

yes

—

C₁₆₇⋊#95

C₄

yes

—

C₂²×C₈×C₁₆₇

C₂²

yes

—

C₂×C₄²×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₇

—

C₂²×Dic₁₆₇

C₂×C₄

yes

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₄×Dic₁₆₇

—

C₄×Dic₁₆₇

—

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₄²×C₁₆₇

D₄

yes

—

C₄²×C₁₆₇

Q₈

yes

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₇

#41

yes

—

C₂×C₄×C₁₆₇

#41

yes

—

C₈×C₁₆₇

—

C₂²×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₆₇

#117

yes

—

C₂²×C₁₆₇

#131

yes

—

C₂²×C₁₆₇

#131

yes

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

yes

C₄×C₁₆₇

yes

C₄×C₁₆₇

yes

—

Dic₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

—

C₁₆₇

#1832

yes

characteristic, complemented, Sylow

#1832

—

167

maximal, Sylow

C₂×C₄⋊C₈

—

167

—

C₂×C₄⋊C₈

D₁₆₇

yes

characteristic

#376

—

167

—

#336

—

167

—

C₂²×C₈

—

167

—

C₂²×C₈

C₂×D₁₆₇

yes

—

C₂×C₄²

—

167

—

C₂×C₄²

C₂×D₁₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

334

—

#95

—

167

—

#95

—

167

—

C₂²×C₄

—

167

—

C₂²×C₄

C₄×D₁₆₇

yes

characteristic

C₂²×C₄

C₄×D₁₆₇

yes

characteristic

C₂²×C₄

C₂²×D₁₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

334

—

C₂×C₈

—

334

—

C₄²

—

334

—

C₄²

—

334

—

C₄²

C₂²⋊D₁₆₇

yes

—

C₄²

Dic₃₃₄

yes

—

C₂³

C₂×C₄×D₁₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

—

334

—

C₂×C₄

—

334

—

C₂×C₄

Dic₁₆₇⋊C₄

yes

—

C₂×C₄

Dic₁₆₇⋊C₄

yes

—

C₂×C₄

C₂×C₂²⋊D₁₆₇

yes

characteristic

C₂×C₄

C₂×Dic₃₃₄

yes

characteristic

C₂×C₄

#14581949

yes

—

C₂×C₄

#14581951

yes

—

C₈

—

C₈

—

668

—

C₂²

C₂×Dic₁₆₇⋊C₄

yes

characteristic

C₂²

#19128790

yes

—

C₂²

#19128838

yes

—

C₂²

#19128840

yes

—

C₄

—

C₄

—

C₄

—

668

—

100

C₄

#19128819

yes

characteristic

101

C₄

#19128821

yes

characteristic

102

C₄

#19128823

yes

—

103

C₂

#24923134

yes

characteristic

104

C₂

#24923295

yes

characteristic

105

C₂

#24923297

yes

characteristic

106

C₂

#24923329

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₆₇
commutator, socle, prime radical	C₂³×C₁₆₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₆₇	C₂
cosocle	C₂²×C₄×C₁₆₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₆₇
hypocenter	C₁₆₇	#1832
LCS	G ▷ C₂³×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₆₇

commutator, socle, prime radical

C₂³×C₁₆₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₆₇

C₂

cosocle

C₂²×C₄×C₁₆₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₆₇

C₁₆₇

G ▷ C₂³×C₁₆₇ ▷ C₁₆₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
53. C₁₆₇	#1832	semidirect

H ≅ G/N

type

53. C₁₆₇

#1832

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₆₇⋊C₈)	2
#24923115	2
#24923023	2
C₂×C₄⋊C₈×C₁₆₇	2
#1832	167

subgroup

index

C₂×C₄⋊(C₁₆₇⋊C₈)

#24923115

#24923023

C₂×C₄⋊C₈×C₁₆₇

#1832

167

Subgroups — #36324257

Conjugacy classes of subgroups

Complements

Maximal subgroups