Group #71963 — subgroups

Subgroups — #71963

Order 384.

260 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#812	C₂	1	2	yes	maximal, complemented
3	#4549	C₂	1	4	yes	maximal, complemented
4	#2112	—	3	1	no	maximal, Sylow
5	C₂³×C₃×C₄	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₈	—	2	4	no	—
7	C₂²×C₃×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₈	C₄	1	2	yes	—
9	C₂²×C₃×C₈	C₂²	1	2	yes	—
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	#812	—	2	4	no	—
12	C₂×C₃⋊C₁₆	C₂²	1	4	yes	complemented
13	C₂³×C₈	D₃	1	1	yes	characteristic
14	C₂×M₃₂	—	3	2	no	—
15	C₂²⋊C₁₆	—	3	4	no	—
16	C₂⁴×C₃	C₈	1	1	yes	characteristic
17	C₂²×C₃×C₄	—	2	2	no	—
18	C₂²×C₃×C₄	—	2	2	no	—
19	C₂²×C₃×C₄	C₈	1	1	yes	characteristic
20	C₂²×C₃×C₄	C₈	1	2	yes	—
21	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₈	—	2	2	no	—
24	C₂×C₃×C₈	—	2	4	no	—
25	C₂×C₃×C₈	—	2	4	no	—
26	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₈	D₄	1	4	yes	—
29	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
30	C₃⋊C₁₆	—	2	4	no	—
31	C₂³×C₄	Dic₃	1	1	yes	characteristic
32	C₂²×C₈	—	2	4	no	—
33	C₂²×C₈	Dic₃	1	1	yes	characteristic
34	C₂²×C₈	Dic₃	1	2	yes	—
35	C₂²×C₈	C₂×D₃	1	2	yes	—
36	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
37	M₃₂	—	6	4	no	—
38	C₂×C₁₆	—	3	4	no	—
39	C₂³×C₃	—	2	2	no	—
40	C₂³×C₃	—	2	4	no	—
41	C₂³×C₃	C₂×C₈	1	1	yes	characteristic
42	C₂³×C₃	C₂×C₈	1	2	yes	—
43	C₂×C₃×C₄	—	2	2	no	—
44	C₂×C₃×C₄	—	2	4	no	—
45	C₂×C₃×C₄	—	2	2	no	—
46	C₂×C₃×C₄	—	2	2	no	—
47	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
48	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
49	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
50	C₂×C₃×C₄	M₁₆	1	2	yes	—
51	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₃×C₈	—	2	2	no	—
53	C₃×C₈	C₂²⋊C₄	1	2	yes	—
54	C₃×C₈	C₂×D₄	1	2	yes	—
55	C₂⁴	C₃⋊C₈	1	1	yes	characteristic
56	C₂²×C₄	—	2	2	no	—
57	C₂²×C₄	—	2	2	no	—
58	C₂²×C₄	C₃⋊C₈	1	1	yes	characteristic
59	C₂²×C₄	C₃⋊C₈	1	2	yes	—
60	C₂²×C₄	C₂×Dic₃	1	2	yes	—
61	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	4	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	C₂×Dic₃	1	2	yes	—
66	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
67	C₂×C₈	#67	1	4	yes	—
68	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
69	C₁₆	—	6	4	no	—
70	C₂²×C₃	—	2	8	no	—
71	C₂²×C₃	—	2	2	no	—
72	C₂²×C₃	—	2	2	no	—
73	C₂²×C₃	—	2	2	no	—
74	C₂²×C₃	C₂²⋊C₈	1	2	yes	—
75	C₂²×C₃	M₃₂	1	4	yes	complemented
76	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
77	C₃×C₄	—	2	2	no	—
78	C₃×C₄	C₂²⋊C₈	1	2	yes	—
79	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
81	C₂³	—	2	4	no	—
82	C₂³	—	2	2	no	—
83	C₂³	C₂×C₃⋊C₈	1	2	yes	—
84	C₂³	C₂×C₃⋊C₈	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	C₂×C₃⋊C₈	1	2	yes	—
90	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
91	C₂×C₄	#208	1	2	yes	—
92	C₂×C₄	#217	1	2	yes	—
93	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
94	C₈	—	2	2	no	—
95	C₈	#217	1	2	yes	—
96	C₈	C₂×#67	1	2	yes	—
97	C₂×C₃	—	2	2	no	—
98	C₂×C₃	—	2	4	no	—
99	C₂×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₃	C₂×M₃₂	1	2	yes	—
101	C₂²	—	2	8	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	#812	1	4	yes	complemented
106	C₂²	#830	1	2	yes	—
107	C₂²	C₂²×C₃⋊C₈	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	#830	1	2	yes	—
110	C₄	C₂×#208	1	1	yes	characteristic
111	C₄	C₂×#217	1	1	yes	characteristic
112	C₃	#2112	1	1	yes	characteristic, complemented, Sylow
113	C₂	—	2	4	no	—
114	C₂	—	2	2	no	—
115	C₂	C₂×#812	1	2	yes	—
116	C₂	C₂×#830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₃×C₈

C₂

yes

maximal, characteristic

C₂×#812

C₂

yes

maximal, complemented

#4549

C₂

yes

maximal, complemented

#2112

—

maximal, Sylow

C₂³×C₃×C₄

C₄

yes

characteristic

C₂²×C₃×C₈

—

C₂²×C₃×C₈

C₄

yes

characteristic

C₂²×C₃×C₈

C₄

yes

—

C₂²×C₃×C₈

C₂²

yes

—

C₂²×C₃×C₈

C₂²

yes

characteristic

#812

—

C₂×C₃⋊C₁₆

C₂²

yes

complemented

C₂³×C₈

D₃

yes

characteristic

C₂×M₃₂

—

C₂²⋊C₁₆

—

C₂⁴×C₃

C₈

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₈

yes

characteristic

C₂²×C₃×C₄

C₈

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₄

yes

—

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

D₄

yes

—

C₂×C₃×C₈

C₂³

yes

characteristic

C₃⋊C₁₆

—

C₂³×C₄

Dic₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₃

yes

characteristic

C₂²×C₈

Dic₃

yes

—

C₂²×C₈

C₂×D₃

yes

—

C₂²×C₈

C₂×D₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂×C₈

yes

characteristic

C₂³×C₃

C₂×C₈

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂×C₈

yes

—

C₂×C₃×C₄

C₂×C₈

yes

characteristic

C₂×C₃×C₄

M₁₆

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₃×C₈

—

C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₈

C₂×D₄

yes

—

C₂⁴

C₃⋊C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₃⋊C₈

yes

characteristic

C₂²×C₄

C₃⋊C₈

yes

—

C₂²×C₄

C₂×Dic₃

yes

—

C₂²×C₄

C₂×Dic₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃

yes

—

C₂×C₈

C₂×Dic₃

yes

characteristic

C₂×C₈

#67

yes

—

C₂×C₈

C₂²×D₃

yes

characteristic

C₁₆

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²⋊C₈

yes

—

C₂²×C₃

M₃₂

yes

complemented

C₂²×C₃

C₂²×C₈

yes

characteristic

C₃×C₄

—

C₃×C₄

C₂²⋊C₈

yes

—

C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₃×C₄

C₂×M₁₆

yes

characteristic

C₂³

—

C₂³

—

C₂³

C₂×C₃⋊C₈

yes

—

C₂³

C₂×C₃⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₃⋊C₈

yes

—

C₂×C₄

C₂×C₃⋊C₈

yes

characteristic

C₂×C₄

#208

yes

—

C₂×C₄

#217

yes

—

C₂×C₄

C₂²×Dic₃

yes

characteristic

C₈

—

C₈

#217

yes

—

C₈

C₂×#67

yes

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₂×C₂²⋊C₈

yes

characteristic

100

C₂×C₃

C₂×M₃₂

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

#812

yes

complemented

106

C₂²

#830

yes

—

107

C₂²

C₂²×C₃⋊C₈

yes

characteristic

108

C₄

—

109

C₄

#830

yes

—

110

C₄

C₂×#208

yes

characteristic

111

C₄

C₂×#217

yes

characteristic

112

C₃

#2112

yes

characteristic, complemented, Sylow

113

C₂

—

114

C₂

—

115

C₂

C₂×#812

yes

—

116

C₂

C₂×#830

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₈
Fitting	C₂³×C₃×C₈	C₂
prime radical	C₂⁴×C₃	C₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂³×C₈	D₃
hypocenter	C₃	#2112
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₃

commutator, socle

C₂²×C₃

C₂²×C₈

Fitting

C₂³×C₃×C₈

C₂

prime radical

C₂⁴×C₃

C₈

cosocle

C₂×C₃×C₈

C₂³

hypercenter

C₂³×C₈

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂×C₈ ◁ C₂³×C₈

N	H ≅ G/N	type
2. C₂×#812	C₂	semidirect
3. #4549	C₂	semidirect
12. C₂×C₃⋊C₁₆	C₂²	semidirect
75. C₂²×C₃	M₃₂	semidirect
105. C₂²	#812	semidirect
112. C₃	#2112	semidirect

H ≅ G/N

type

2. C₂×#812

C₂

semidirect

3. #4549

C₂

semidirect

12. C₂×C₃⋊C₁₆

C₂²

semidirect

75. C₂²×C₃

M₃₂

semidirect

105. C₂²

#812

semidirect

112. C₃

#2112

semidirect

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

subgroup	index
C₂³×C₃×C₈	2
C₂×#812	2
#4549	2
#2112	3

subgroup

index

C₂³×C₃×C₈

C₂×#812

#4549

#2112

Subgroups — #71963

Conjugacy classes of subgroups

Complements

Maximal subgroups