Group #70740 — subgroups

Subgroups — #70740

Order 384.

212 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#4454	C₂	1	2	yes	maximal, complemented
5	C₈×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
6	#1450	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊C₈	C₄	1	2	yes	—
9	C₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₃×C₂²⋊C₈	C₄	1	2	yes	—
11	C₃×C₄×C₈	C₂²	1	2	yes	—
12	#830	C₄	1	2	yes	—
13	C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
14	C₄×C₃⋊C₈	C₄	1	2	yes	—
15	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₃⋊C₈	C₂²	1	2	yes	—
18	#431	—	3	1	no	—
19	#431	D₃	1	1	yes	characteristic
20	C₂×C₄×C₈	—	3	1	no	—
21	#322	—	3	2	no	—
22	C₈²	—	3	2	no	—
23	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
27	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄²	C₂³	1	1	yes	characteristic
31	C₂×C₃⋊C₈	C₈	1	8	yes	complemented
32	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
33	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₈	—	3	2	no	—
36	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₄⋊C₈	—	3	2	no	—
38	C₄⋊C₈	Dic₃	1	2	yes	—
39	C₂²⋊C₈	—	3	2	no	—
40	C₂²⋊C₈	Dic₃	1	2	yes	—
41	C₄×C₈	—	3	2	no	—
42	C₄×C₈	—	3	1	no	—
43	C₄×C₈	—	3	2	no	—
44	C₄×C₈	—	3	1	no	—
45	C₄×C₈	C₂×D₃	1	2	yes	—
46	C₂³×C₃	C₄²	1	1	yes	characteristic
47	C₂×C₃×C₄	—	2	2	no	—
48	C₂×C₃×C₄	—	2	2	no	—
49	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
50	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
51	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
52	C₂×C₃×C₄	C₂×C₈	1	4	yes	—
53	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₈	—	2	4	no	—
57	C₃⋊C₈	—	2	4	no	—
58	C₃⋊C₈	C₂×C₈	1	4	yes	—
59	C₃⋊C₈	C₂×C₈	1	4	yes	—
60	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
61	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
62	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
63	C₂×C₈	—	3	8	no	—
64	C₂×C₈	—	3	2	no	—
65	C₂×C₈	—	3	2	no	—
66	C₂×C₈	—	3	4	no	—
67	C₂×C₈	C₂×Dic₃	1	4	yes	—
68	C₄²	C₄×D₃	1	1	yes	characteristic
69	C₄²	C₄×D₃	1	1	yes	characteristic
70	C₄²	C₂×Dic₃	1	1	yes	characteristic
71	C₄²	C₂²×D₃	1	1	yes	characteristic
72	C₂²×C₃	—	2	1	no	—
73	C₂²×C₃	—	2	1	no	—
74	C₂²×C₃	C₄×C₈	1	2	yes	—
75	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
76	C₃×C₄	—	2	1	no	—
77	C₃×C₄	—	2	1	no	—
78	C₃×C₄	C₄×C₈	1	2	yes	—
79	C₃×C₄	C₂²×C₈	1	2	yes	—
80	C₃×C₄	#131	1	2	yes	—
81	C₃×C₄	#131	1	2	yes	—
82	C₂³	C₄×Dic₃	1	1	yes	characteristic
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	D₃×C₈	1	4	yes	—
86	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
87	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
88	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
91	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
92	C₈	—	2	4	no	—
93	C₈	—	3	4	no	—
94	C₈	—	3	4	no	—
95	C₈	—	6	4	no	—
96	C₂×C₃	—	2	2	no	—
97	C₂×C₃	C₂×C₄×C₈	1	1	yes	characteristic
98	C₂×C₃	#405	1	1	yes	characteristic
99	C₂×C₃	#405	1	1	yes	characteristic
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	C₈×Dic₃	1	2	yes	—
103	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	C₈×Dic₃	1	2	yes	—
107	C₄	C₂×D₃×C₈	1	2	yes	—
108	C₄	#907	1	2	yes	—
109	C₄	#948	1	2	yes	—
110	C₃	#1450	1	1	yes	characteristic, complemented, Sylow
111	C₂	—	2	2	no	—
112	C₂	#4997	1	1	yes	characteristic
113	C₂	C₂×C₈×Dic₃	1	1	yes	characteristic
114	C₂	#5122	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#431

C₂

yes

maximal, characteristic

#4925

C₂

yes

maximal, characteristic

C₂×C₄×C₃⋊C₈

C₂

yes

maximal, characteristic

#4454

C₂

yes

maximal, complemented

C₈×C₃⋊C₈

C₂

yes

maximal, complemented

#1450

—

maximal, Sylow

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₄

yes

—

C₃×C₄⋊C₈

C₄

yes

—

C₃×C₂²⋊C₈

C₄

yes

—

C₃×C₄×C₈

C₂²

yes

—

#830

C₄

yes

—

C₄⋊(C₃⋊C₈)

C₄

yes

—

C₄×C₃⋊C₈

C₄

yes

—

C₄×C₃⋊C₈

C₂²

yes

characteristic

C₄×C₃⋊C₈

C₂²

yes

characteristic

C₄×C₃⋊C₈

C₂²

yes

—

#431

—

#431

D₃

yes

characteristic

C₂×C₄×C₈

—

#322

—

C₈²

—

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×C₄

yes

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

C₂³

yes

characteristic

C₂×C₃⋊C₈

C₈

yes

complemented

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃

yes

—

C₂³×C₃

C₄²

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄²

yes

characteristic

C₂×C₃×C₄

C₄²

yes

characteristic

C₂×C₃×C₄

C₄²

yes

characteristic

C₂×C₃×C₄

C₂×C₈

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

C₂×C₈

yes

—

C₃⋊C₈

C₂×C₈

yes

—

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

C₂×Dic₃

yes

—

C₄²

C₄×D₃

yes

characteristic

C₄²

C₄×D₃

yes

characteristic

C₄²

C₂×Dic₃

yes

characteristic

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₄×C₈

yes

—

C₂²×C₃

C₂×C₄²

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄×C₈

yes

—

C₃×C₄

C₂²×C₈

yes

—

C₃×C₄

#131

yes

—

C₃×C₄

#131

yes

—

C₂³

C₄×Dic₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₃×C₈

yes

—

C₂×C₄

C₄×Dic₃

yes

characteristic

C₂×C₄

C₄×Dic₃

yes

characteristic

C₂×C₄

C₄×Dic₃

yes

characteristic

C₂×C₄

C₂×C₄×D₃

yes

characteristic

C₂×C₄

C₂×C₄×D₃

yes

characteristic

C₂×C₄

C₂²×Dic₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

C₂×C₄×C₈

yes

C₂×C₃

yes

C₂×C₃

yes

100

C₂²

—

101

C₂²

—

102

C₂²

C₈×Dic₃

yes

—

103

C₂²

C₂×C₄×Dic₃

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

C₈×Dic₃

yes

—

107

C₄

C₂×D₃×C₈

yes

—

108

C₄

#907

yes

—

109

C₄

#948

yes

—

110

C₃

#1450

yes

characteristic, complemented, Sylow

111

C₂

—

112

C₂

#4997

yes

characteristic

113

C₂

C₂×C₈×Dic₃

yes

characteristic

114

C₂

#5122

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₃
commutator	C₂×C₃	C₂×C₄×C₈
Fitting	C₃×#431	C₂
socle	C₂²×C₃	C₂×C₄²
prime radical	C₂³×C₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	#431	D₃
hypocenter	C₃	#1450
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₃

commutator

C₂×C₃

C₂×C₄×C₈

Fitting

C₃×#431

C₂

socle

C₂²×C₃

C₂×C₄²

prime radical

C₂³×C₃

C₄²

cosocle

C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂×C₃ ▷ C₃

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
4. #4454	C₂	semidirect
5. C₈×C₃⋊C₈	C₂	semidirect
31. C₂×C₃⋊C₈	C₈	semidirect
110. C₃	#1450	semidirect

H ≅ G/N

type

4. #4454

C₂

semidirect

5. C₈×C₃⋊C₈

C₂

semidirect

31. C₂×C₃⋊C₈

C₈

semidirect

110. C₃

#1450

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×C₄×C₃⋊C₈	2
#4454	2
C₈×C₃⋊C₈	2
#1450	3

subgroup

index

C₂×C₄×C₃⋊C₈

C₈×C₃⋊C₈

Subgroups — #70740

Conjugacy classes of subgroups

Complements

Maximal subgroups