Group #67706 — subgroups

Subgroups — #67706

Order 384.

160 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 81 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×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#4925	C₂	1	1	yes	maximal, characteristic
3	C₄×#208	C₂	1	1	yes	maximal, characteristic
4	#1283	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	C₄	1	2	yes	—
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂×#208	C₄	1	2	yes	—
8	C₃×C₄⋊C₈	—	2	1	no	—
9	C₃×C₄⋊C₈	—	2	1	no	—
10	#830	C₄	1	2	yes	—
11	C₄⋊(C₃⋊C₈)	—	2	1	no	—
12	#803	—	2	1	no	—
13	C₄×C₃⋊C₈	—	2	1	no	—
14	C₄×C₃⋊C₈	—	2	1	no	—
15	#431	—	3	1	no	—
16	C₂×C₄⋊C₈	D₃	1	1	yes	characteristic
17	C₄×M₁₆	—	3	1	no	—
18	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₈	—	2	2	no	—
22	C₂×C₃×C₈	—	4	2	no	—
23	C₃×C₄²	D₄	1	1	yes	characteristic
24	C₃×C₄²	D₄	1	1	yes	characteristic
25	C₃×C₄²	D₄	1	1	yes	characteristic
26	C₃×C₄²	Q₈	1	1	yes	characteristic
27	#208	—	2	2	no	—
28	#208	C₈	1	4	yes	complemented
29	C₂×C₃⋊C₈	—	2	2	no	—
30	C₂×C₃⋊C₈	—	2	2	no	—
31	C₂×M₁₆	—	3	2	no	—
32	C₂²×C₈	Dic₃	1	2	yes	—
33	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	2	1	no	—
36	C₄⋊C₈	—	6	1	no	—
37	C₂²⋊C₈	—	3	2	no	—
38	#97	—	6	1	no	—
39	C₄×C₈	—	6	1	no	—
40	C₄×C₈	—	6	1	no	—
41	C₂³×C₃	C₄²	1	1	yes	characteristic
42	C₂×C₃×C₄	—	2	1	no	—
43	C₂×C₃×C₄	—	2	1	no	—
44	C₂×C₃×C₄	—	4	1	no	—
45	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
52	C₂×C₃×C₄	M₁₆	1	2	yes	—
53	C₃×C₈	—	4	2	no	—
54	C₃⋊C₈	—	2	4	no	—
55	C₃⋊C₈	—	4	2	no	—
56	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
57	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
58	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
59	M₁₆	—	3	4	no	—
60	M₁₆	—	6	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	—	6	2	no	—
64	C₂×C₈	—	6	2	no	—
65	C₄²	Dic₆	1	1	yes	characteristic
66	C₄²	D₁₂	1	1	yes	characteristic
67	C₄²	#67	1	1	yes	characteristic
68	C₄²	#67	1	1	yes	characteristic
69	C₂²×C₃	—	2	1	no	—
70	C₂²×C₃	—	2	1	no	—
71	C₂²×C₃	#95	1	1	yes	characteristic
72	C₂²×C₃	C₄×C₈	1	1	yes	characteristic
73	C₂²×C₃	#97	1	1	yes	characteristic
74	C₃×C₄	—	2	1	no	—
75	C₃×C₄	—	2	1	no	—
76	C₃×C₄	—	2	1	no	—
77	C₃×C₄	C₂²⋊C₈	1	2	yes	—
78	C₃×C₄	#100	1	1	yes	characteristic
79	C₃×C₄	#101	1	1	yes	characteristic
80	C₃×C₄	C₄⋊C₈	1	2	yes	—
81	C₂³	C₄×Dic₃	1	1	yes	characteristic
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	D₃×C₈	1	2	yes	—
86	C₂×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
87	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
88	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
90	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
91	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
92	C₂×C₄	#217	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	6	4	no	—
95	C₈	—	12	2	no	—
96	C₂×C₃	—	2	1	no	—
97	C₂×C₃	—	2	1	no	—
98	C₂×C₃	#336	1	1	yes	characteristic
99	C₂×C₃	#343	1	1	yes	characteristic
100	C₂×C₃	#344	1	1	yes	characteristic
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	C₈×Dic₃	1	1	yes	characteristic
104	C₂²	#815	1	1	yes	characteristic
105	C₂²	C₃⋊#95	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	Dic₃⋊C₈	1	2	yes	—
110	C₄	(C₂×D₃)⋊C₈	1	2	yes	—
111	C₄	#833	1	1	yes	characteristic
112	C₄	#836	1	1	yes	characteristic
113	C₃	#1283	1	1	yes	characteristic, complemented, Sylow
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#4530	1	1	yes	characteristic
117	C₂	#4550	1	1	yes	characteristic
118	C₂	#4556	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊C₈

C₂

yes

maximal, characteristic

#4925

C₂

yes

maximal, characteristic

C₄×#208

C₂

yes

maximal, characteristic

#1283

—

maximal, Sylow

C₂²×C₃×C₈

C₄

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

C₄

yes

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

#830

C₄

yes

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

#431

—

C₂×C₄⋊C₈

D₃

yes

characteristic

C₄×M₁₆

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

Q₈

yes

characteristic

#208

—

#208

C₈

yes

complemented

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

Dic₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

C₄²

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂×C₈

yes

—

C₂×C₃×C₄

M₁₆

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂²×C₄

C₄×D₃

yes

characteristic

C₂²×C₄

C₄×D₃

yes

characteristic

C₂²×C₄

C₂×Dic₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

Dic₆

yes

characteristic

C₄²

D₁₂

yes

C₄²

yes

C₄²

yes

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#95

yes

characteristic

C₂²×C₃

C₄×C₈

yes

characteristic

C₂²×C₃

#97

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂²⋊C₈

yes

—

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

C₄⋊C₈

yes

—

C₂³

C₄×Dic₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₃×C₈

yes

—

C₂×C₄

ΓL(1,𝔽₂₅)

yes

—

C₂×C₄

Dic₃⋊C₄

yes

characteristic

C₂×C₄

Dic₃⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₃

yes

characteristic

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

C₂×C₄

#217

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

yes

C₂×C₃

yes

100

C₂×C₃

#344

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

C₈×Dic₃

yes

characteristic

104

C₂²

#815

yes

characteristic

105

C₂²

C₃⋊#95

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

Dic₃⋊C₈

yes

—

110

C₄

(C₂×D₃)⋊C₈

yes

—

111

C₄

#833

yes

characteristic

112

C₄

#836

yes

characteristic

113

C₃

#1283

yes

characteristic, complemented, Sylow

114

C₂

—

115

C₂

—

116

C₂

#4530

yes

characteristic

117

C₂

#4550

yes

characteristic

118

C₂

#4556

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#217
commutator	C₂²×C₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₂×C₃×C₄⋊C₈	C₂
socle	C₂²×C₃	#95
prime radical	C₂³×C₃	C₄²
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#1283
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

#217

commutator

C₂²×C₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₃

Fitting

C₂×C₃×C₄⋊C₈

C₂

socle

C₂²×C₃

#95

prime radical

C₂³×C₃

C₄²

cosocle

C₂×C₃×C₄²

C₂²

hypercenter

C₂×C₄⋊C₈

D₃

hypocenter

C₃

#1283

LCS

G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
28. #208	C₈	semidirect
113. C₃	#1283	semidirect

H ≅ G/N

type

28. #208

C₈

semidirect

113. C₃

#1283

semidirect

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

subgroup	index
C₂×C₃×C₄⋊C₈	2
#4925	2
C₄×#208	2
#1283	3

subgroup

index

C₂×C₃×C₄⋊C₈

#4925

C₄×#208

#1283

Subgroups — #67706

Conjugacy classes of subgroups

Complements

Maximal subgroups