Group #67741 — subgroups

Subgroups — #67741

Order 384.

152 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₃×#431	C₂	1	1	yes	maximal, characteristic
2	#4925	C₂	1	1	yes	maximal, characteristic
3	C₂×#803	C₂	1	1	yes	maximal, characteristic
4	#1281	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
6	C₂²×C₃⋊C₈	C₄	1	2	yes	—
7	C₃×C₄⋊C₈	—	2	1	no	—
8	C₃×C₂²⋊C₈	C₄	1	2	yes	—
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	#803	—	2	1	no	—
14	C₄×C₃⋊C₈	—	2	1	no	—
15	#431	—	3	1	no	—
16	#431	D₃	1	1	yes	characteristic
17	C₂×#97	—	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	C₃×C₄²	D₄	1	1	yes	characteristic
23	C₃×C₄²	D₄	1	1	yes	characteristic
24	C₃×C₄²	D₄	1	1	yes	characteristic
25	C₃×C₄²	Q₈	1	1	yes	characteristic
26	C₂×C₃⋊C₈	—	2	2	no	—
27	C₂×C₃⋊C₈	—	2	2	no	—
28	C₂×C₃⋊C₈	—	2	2	no	—
29	C₂×C₃⋊C₈	C₈	1	4	yes	complemented
30	C₂²×C₈	—	3	2	no	—
31	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
32	C₄⋊C₈	—	2	1	no	—
33	C₄⋊C₈	—	6	1	no	—
34	C₂²⋊C₈	—	3	2	no	—
35	C₂²⋊C₈	Dic₃	1	2	yes	—
36	#97	—	6	1	no	—
37	#97	—	6	1	no	—
38	C₄×C₈	—	2	1	no	—
39	C₄×C₈	—	6	1	no	—
40	C₂³×C₃	C₄²	1	1	yes	characteristic
41	C₂×C₃×C₄	—	4	1	no	—
42	C₂×C₃×C₄	—	4	1	no	—
43	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
50	C₂×C₃×C₄	M₁₆	1	2	yes	—
51	C₃×C₈	—	4	2	no	—
52	C₃⋊C₈	—	2	4	no	—
53	C₃⋊C₈	—	4	2	no	—
54	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
55	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	3	4	no	—
59	C₂×C₈	—	6	2	no	—
60	C₂×C₈	—	6	2	no	—
61	C₂×C₈	—	6	2	no	—
62	C₄²	Dic₆	1	1	yes	characteristic
63	C₄²	D₁₂	1	1	yes	characteristic
64	C₄²	#67	1	1	yes	characteristic
65	C₄²	#67	1	1	yes	characteristic
66	C₂²×C₃	—	2	1	no	—
67	C₂²×C₃	—	2	1	no	—
68	C₂²×C₃	#95	1	1	yes	characteristic
69	C₂²×C₃	C₄×C₈	1	1	yes	characteristic
70	C₂²×C₃	#97	1	1	yes	characteristic
71	C₃×C₄	—	2	1	no	—
72	C₃×C₄	—	2	1	no	—
73	C₃×C₄	—	2	1	no	—
74	C₃×C₄	—	2	1	no	—
75	C₃×C₄	C₂²⋊C₈	1	2	yes	—
76	C₃×C₄	C₄⋊C₈	1	2	yes	—
77	C₂³	C₄×Dic₃	1	1	yes	characteristic
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	D₃×C₈	1	2	yes	—
81	C₂×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
82	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
83	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
84	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
85	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
86	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	#217	1	1	yes	characteristic
88	C₈	—	4	2	no	—
89	C₈	—	6	4	no	—
90	C₈	—	12	2	no	—
91	C₂×C₃	—	2	1	no	—
92	C₂×C₃	—	2	1	no	—
93	C₂×C₃	#336	1	1	yes	characteristic
94	C₂×C₃	#344	1	1	yes	characteristic
95	C₂×C₃	#344	1	1	yes	characteristic
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	C₈×Dic₃	1	1	yes	characteristic
99	C₂²	#815	1	1	yes	characteristic
100	C₂²	C₃⋊#95	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	Dic₃⋊C₈	1	2	yes	—
106	C₄	(C₂×D₃)⋊C₈	1	2	yes	—
107	C₃	#1281	1	1	yes	characteristic, complemented, Sylow
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	#4533	1	1	yes	characteristic
111	C₂	#4550	1	1	yes	characteristic
112	C₂	#4559	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#431

C₂

yes

maximal, characteristic

#4925

C₂

yes

maximal, characteristic

C₂×#803

C₂

yes

maximal, characteristic

#1281

—

maximal, Sylow

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₄

yes

—

C₃×C₄⋊C₈

—

C₃×C₂²⋊C₈

C₄

yes

—

C₃×C₄×C₈

—

#830

C₄

yes

—

C₄⋊(C₃⋊C₈)

—

#803

—

#803

—

C₄×C₃⋊C₈

—

#431

—

#431

D₃

yes

characteristic

C₂×#97

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂×C₃×C₈

—

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

Q₈

yes

characteristic

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

C₈

yes

complemented

C₂²×C₈

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₃

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

C₄²

yes

characteristic

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

C₂×C₈

—

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

—

C₃×C₄

C₂²⋊C₈

yes

—

C₃×C₄

C₄⋊C₈

yes

—

C₂³

C₄×Dic₃

yes

characteristic

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

C₂×C₃

yes

C₂²

—

C₂²

—

C₂²

C₈×Dic₃

yes

characteristic

C₂²

#815

yes

characteristic

100

C₂²

C₃⋊#95

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

Dic₃⋊C₈

yes

—

106

C₄

(C₂×D₃)⋊C₈

yes

—

107

C₃

#1281

yes

characteristic, complemented, Sylow

108

C₂

—

109

C₂

—

110

C₂

#4533

yes

characteristic

111

C₂

#4550

yes

characteristic

112

C₂

#4559

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₃×#431	C₂
socle	C₂²×C₃	#95
prime radical	C₂³×C₃	C₄²
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	#431	D₃
hypocenter	C₃	#1281
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₃)⋊C₄

commutator

C₂²×C₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₃

C₂

C₂²×C₃

C₂³×C₃

C₄²

cosocle

C₂×C₃×C₄²

C₂²

D₃

C₃

G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
29. C₂×C₃⋊C₈	C₈	semidirect
107. C₃	#1281	semidirect

H ≅ G/N

type

29. C₂×C₃⋊C₈

C₈

semidirect

107. C₃

#1281

semidirect

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

subgroup	index
C₃×#431	2
#4925	2
C₂×#803	2
#1281	3

subgroup

index

Subgroups — #67741

Conjugacy classes of subgroups

Complements

Maximal subgroups