Group #77240 — subgroups

Subgroups — #77240

Order 384.

432 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 149 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×#395	C₂	1	1	yes	maximal, characteristic
2	#5230	C₂	1	2	yes	maximal
3	#4979	C₂	1	3	yes	maximal
4	#4973	C₂	1	3	yes	maximal
5	#4646	C₂	1	6	yes	maximal
6	#2811	—	3	1	no	maximal, Sylow
7	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×Dic₆	C₂²	1	1	yes	characteristic
9	C₂²×Dic₆	C₂²	1	3	yes	—
10	C₂×C₃×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×Dic₃⋊C₄	C₂²	1	6	yes	—
12	C₂×C₄×Dic₃	C₂²	1	3	yes	—
13	C₃×#95	C₂²	1	3	yes	—
14	C₃⋊#95	C₂²	1	6	yes	—
15	C₃⋊#95	C₂²	1	6	yes	—
16	C₃⋊#95	C₂²	1	3	yes	—
17	#395	—	3	2	no	—
18	#395	—	3	3	no	—
19	#395	D₃	1	1	yes	characteristic
20	#391	—	3	3	no	—
21	#391	—	3	6	no	—
22	C₂×C₃×Q₈	—	4	3	no	—
23	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂²×C₃×C₄	C₂³	1	3	yes	—
25	C₂²×C₃×C₄	C₂³	1	3	yes	—
26	C₂²×Dic₃	C₂³	1	2	yes	—
27	C₂²×Dic₃	C₂³	1	6	yes	—
28	C₂×Dic₆	—	4	3	no	—
29	C₂×Dic₆	—	4	3	no	—
30	C₂×Dic₆	—	4	3	no	—
31	C₂×Dic₆	Q₈	1	12	yes	complemented
32	C₃×C₄⋊C₄	—	4	3	no	—
33	Dic₃⋊C₄	—	4	6	no	—
34	C₄×Dic₃	—	4	3	no	—
35	C₂²×Q₈	—	3	3	no	—
36	C₂²×Q₈	—	3	1	no	—
37	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
38	C₂×C₄⋊C₄	—	3	6	no	—
39	C₂×C₄⋊C₄	C₂×D₃	1	3	yes	—
40	C₂×C₄²	—	3	3	no	—
41	#95	—	3	3	no	—
42	#95	—	3	6	no	—
43	#95	—	3	6	no	—
44	#95	C₂×D₃	1	3	yes	—
45	C₂³×C₃	C₂⁴	1	1	yes	characteristic
46	C₃×Q₈	—	4	4	no	—
47	C₂×C₃×C₄	—	2	3	no	—
48	C₂×C₃×C₄	—	2	3	no	—
49	C₂×C₃×C₄	—	2	6	no	—
50	C₂×C₃×C₄	—	2	3	no	—
51	C₂×C₃×C₄	—	2	3	no	—
52	C₂×C₃×C₄	C₂×Q₈	1	6	yes	—
53	C₂×Dic₃	—	2	6	no	—
54	C₂×Dic₃	—	2	6	no	—
55	C₂×Dic₃	—	2	6	no	—
56	C₂×Dic₃	C₂×Q₈	1	12	yes	—
57	Dic₆	—	4	12	no	—
58	Dic₆	—	4	4	no	—
59	C₂×Q₈	—	3	12	no	—
60	C₂×Q₈	—	4	3	no	—
61	C₂×Q₈	—	12	3	no	—
62	C₂×Q₈	—	12	3	no	—
63	C₂×Q₈	—	12	3	no	—
64	C₂²×C₄	—	3	6	no	—
65	C₂²×C₄	—	3	2	no	—
66	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₂²×C₄	C₂²×D₃	1	3	yes	—
68	C₂²×C₄	C₂²×D₃	1	3	yes	—
69	C₄⋊C₄	—	4	3	no	—
70	C₄⋊C₄	—	12	6	no	—
71	C₄²	—	12	3	no	—
72	C₂²×C₃	C₂²×Q₈	1	3	yes	—
73	C₂²×C₃	H(5,𝔽₂)	1	3	yes	—
74	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
75	C₃×C₄	—	2	6	no	—
76	C₃×C₄	—	4	1	no	—
77	C₃×C₄	—	4	3	no	—
78	Dic₃	—	2	12	no	—
79	Dic₃	—	4	2	no	—
80	C₂³	C₂³×D₃	1	1	yes	characteristic
81	Q₈	—	4	4	no	—
82	Q₈	—	12	12	no	—
83	Q₈	—	12	4	no	—
84	C₂×C₄	—	2	3	no	—
85	C₂×C₄	—	2	3	no	—
86	C₂×C₄	—	2	3	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	6	no	—
89	C₂×C₄	—	3	12	no	—
90	C₂×C₄	—	6	6	no	—
91	C₂×C₄	—	6	6	no	—
92	C₂×C₄	—	6	6	no	—
93	C₂×C₄	D₃×Q₈	1	6	yes	—
94	C₂×C₃	#543	1	3	yes	—
95	C₂×C₃	Q₈²	1	3	yes	—
96	C₂×C₃	H(3,𝔽₄)	1	1	yes	characteristic
97	C₂²	#1004	1	3	yes	—
98	C₂²	C₂×D₃×Q₈	1	3	yes	—
99	C₂²	#1009	1	1	yes	characteristic
100	C₄	—	2	6	no	—
101	C₄	—	4	3	no	—
102	C₄	—	4	1	no	—
103	C₄	—	6	12	no	—
104	C₄	—	12	2	no	—
105	C₃	#2811	1	1	yes	characteristic, complemented, Sylow
106	C₂	#5594	1	1	yes	characteristic
107	C₂	#5634	1	3	yes	—
108	C₂	#5722	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#395

C₂

yes

maximal, characteristic

#5230

C₂

yes

maximal

#4979

C₂

yes

maximal

#4973

C₂

yes

maximal

#4646

C₂

yes

maximal

#2811

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

characteristic

C₂²×Dic₆

C₂²

yes

characteristic

C₂²×Dic₆

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

#395

—

#395

—

#395

D₃

yes

characteristic

#391

—

#391

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

Q₈

yes

complemented

C₃×C₄⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

—

—

—

C₂×D₃

yes

—

C₂³×C₃

C₂⁴

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂×Q₈

yes

—

Dic₆

—

Dic₆

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂²×C₃

C₂²×Q₈

yes

—

C₂²×C₃

H(5,𝔽₂)

yes

—

C₂²×C₃

H(5,𝔽₂)

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₂³

C₂³×D₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₃×Q₈

yes

—

C₂×C₃

#543

yes

—

C₂×C₃

Q₈²

yes

—

C₂×C₃

H(3,𝔽₄)

yes

characteristic

C₂²

#1004

yes

—

C₂²

C₂×D₃×Q₈

yes

—

C₂²

#1009

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₃

#2811

yes

characteristic, complemented, Sylow

106

C₂

#5594

yes

characteristic

107

C₂

#5634

yes

—

108

C₂

#5722

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#395	C₂
hypercenter	#395	D₃
hypocenter	C₃	#2811
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator, socle, prime radical, cosocle

C₂³×C₃

C₂⁴

C₂

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
31. C₂×Dic₆	Q₈	semidirect
105. C₃	#2811	semidirect

H ≅ G/N

type

31. C₂×Dic₆

Q₈

semidirect

105. C₃

#2811

semidirect

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

subgroup	index
C₃×#395	2
#5230	2
#4979	2
#4973	2
#4646	2
#2811	3

subgroup

index

Subgroups — #77240

Conjugacy classes of subgroups

Complements

Maximal subgroups