Group #83056 — subgroups

Subgroups — #83056

Order 384.

472 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 181 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×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×#870	C₂	1	1	yes	maximal, characteristic
3	#4977	C₂	1	2	yes	maximal
4	#4976	C₂	1	8	yes	maximal
5	#4975	C₂	1	2	yes	maximal
6	C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
7	#2859	—	3	1	no	maximal, Sylow
8	C₃×#125	—	2	4	no	—
9	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
14	C₂×Dic₃⋊C₄	C₂²	1	8	yes	—
15	C₂×C₄×Dic₃	C₂²	1	4	yes	—
16	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
18	#870	—	2	4	no	—
19	C₃⋊#95	C₂²	1	8	yes	—
20	C₂×#125	—	3	1	no	—
21	C₂×#125	D₃	1	1	yes	characteristic
22	#389	—	3	8	no	—
23	#384	—	3	2	no	—
24	#382	—	3	2	no	—
25	C₄³	—	3	1	no	—
26	C₂²×C₃×C₄	C₂³	1	4	yes	—
27	C₂²×C₃×C₄	C₂³	1	2	yes	—
28	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂²×Dic₃	C₂³	1	4	yes	—
30	C₂²×Dic₃	C₂³	1	2	yes	—
31	C₂²×Dic₃	C₂³	1	2	yes	—
32	C₃×C₄⋊C₄	—	2	8	no	—
33	C₃×C₄⋊C₄	—	2	4	no	—
34	C₃×C₄²	D₄	1	4	yes	—
35	C₄⋊Dic₃	—	2	4	no	—
36	Dic₃⋊C₄	—	2	8	no	—
37	Dic₃⋊C₄	—	2	16	no	—
38	C₄×Dic₃	—	2	8	no	—
39	C₄×Dic₃	Q₈	1	4	yes	—
40	C₄×Dic₃	Q₈	1	4	yes	—
41	#125	—	2	4	no	—
42	#125	—	6	4	no	—
43	C₂×C₄⋊C₄	—	3	2	no	—
44	C₂×C₄⋊C₄	—	3	4	no	—
45	C₂×C₄⋊C₄	—	3	8	no	—
46	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
47	C₂×C₄⋊C₄	C₂×D₃	1	4	yes	—
48	C₂×C₄²	—	3	1	no	—
49	C₂×C₄²	—	3	1	no	—
50	C₂×C₄²	—	3	4	no	—
51	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
52	#95	—	3	8	no	—
53	C₂³×C₃	C₂⁴	1	1	yes	characteristic
54	C₂×C₃×C₄	—	2	4	no	—
55	C₂×C₃×C₄	—	2	4	no	—
56	C₂×C₃×C₄	—	2	8	no	—
57	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
58	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
59	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
60	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
61	C₂×Dic₃	—	2	4	no	—
62	C₂×Dic₃	—	2	4	no	—
63	C₂×Dic₃	—	2	8	no	—
64	C₂×Dic₃	C₂×Q₈	1	4	yes	—
65	C₂×Dic₃	C₂×Q₈	1	4	yes	—
66	C₂×Dic₃	#41	1	8	yes	—
67	C₂²×C₄	—	3	2	no	—
68	C₂²×C₄	—	3	2	no	—
69	C₂²×C₄	—	3	4	no	—
70	C₂²×C₄	C₂²×D₃	1	4	yes	—
71	C₂²×C₄	C₂²×D₃	1	2	yes	—
72	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
73	C₄⋊C₄	—	2	8	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	6	16	no	—
76	C₄⋊C₄	—	6	8	no	—
77	C₄⋊C₄	—	6	4	no	—
78	C₄²	—	3	4	no	—
79	C₄²	—	3	4	no	—
80	C₄²	—	6	8	no	—
81	C₄²	#67	1	4	yes	—
82	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
83	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
84	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₃	C₂×#41	1	4	yes	—
86	C₃×C₄	—	2	4	no	—
87	C₃×C₄	—	4	4	no	—
88	C₃×C₄	C₄⋊Q₈	1	4	yes	—
89	Dic₃	—	2	4	no	—
90	Dic₃	—	4	4	no	—
91	Dic₃	#125	1	8	yes	complemented
92	C₂³	C₂³×D₃	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	3	4	no	—
97	C₂×C₄	—	3	8	no	—
98	C₂×C₄	—	3	4	no	—
99	C₂×C₄	—	6	4	no	—
100	C₂×C₄	—	6	8	no	—
101	C₂×C₄	—	6	4	no	—
102	C₂×C₄	D₃×Q₈	1	2	yes	—
103	C₂×C₄	D₃×Q₈	1	2	yes	—
104	C₂×C₄	C₂×#67	1	4	yes	—
105	C₂×C₄	C₂×#67	1	2	yes	—
106	C₂×C₃	C₂×#125	1	2	yes	—
107	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₂×C₃	#532	1	2	yes	—
109	C₂×C₃	#533	1	2	yes	—
110	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
111	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
112	C₂²	D₃×#41	1	4	yes	—
113	C₂²	C₂²×#67	1	1	yes	characteristic
114	C₄	—	2	4	no	—
115	C₄	—	3	8	no	—
116	C₄	—	4	4	no	—
117	C₄	—	6	4	no	—
118	C₄	—	12	4	no	—
119	C₄	#944	1	4	yes	—
120	C₃	#2859	1	1	yes	characteristic, complemented, Sylow
121	C₂	D₃×#125	1	2	yes	—
122	C₂	#5688	1	2	yes	—
123	C₂	C₂×#944	1	1	yes	characteristic
124	C₂	#5832	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#125

C₂

yes

maximal, characteristic

C₂×#870

C₂

yes

maximal, characteristic

#4977

C₂

yes

maximal

#4976

C₂

yes

maximal

#4975

C₂

yes

maximal

C₄²×Dic₃

C₂

yes

maximal, characteristic

#2859

—

maximal, Sylow

C₃×#125

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

characteristic

#870

—

C₃⋊#95

C₂²

yes

—

C₂×#125

—

C₂×#125

D₃

yes

—

—

—

C₄³

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

Q₈

yes

—

C₄×Dic₃

Q₈

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

#95

—

C₂³×C₃

C₂⁴

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂×Q₈

yes

—

C₂×Dic₃

C₂×Q₈

yes

—

C₂×Dic₃

#41

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#67

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

C₂×#41

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄⋊Q₈

yes

—

Dic₃

—

Dic₃

—

Dic₃

#125

yes

complemented

C₂³

C₂³×D₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

D₃×Q₈

yes

—

103

C₂×C₄

D₃×Q₈

yes

—

104

C₂×C₄

C₂×#67

yes

—

105

C₂×C₄

C₂×#67

yes

—

106

C₂×C₃

C₂×#125

yes

—

107

C₂×C₃

C₂×C₄⋊Q₈

yes

characteristic

108

C₂×C₃

#532

yes

—

109

C₂×C₃

#533

yes

—

110

C₂²

C₂×D₃×Q₈

yes

characteristic

111

C₂²

C₂×D₃×Q₈

yes

characteristic

112

C₂²

D₃×#41

yes

—

113

C₂²

C₂²×#67

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#944

yes

—

120

C₃

#2859

yes

characteristic, complemented, Sylow

121

C₂

D₃×#125

yes

—

122

C₂

#5688

yes

—

123

C₂

C₂×#944

yes

characteristic

124

C₂

#5832

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₂×C₃×#125	C₂
hypercenter	C₂×#125	D₃
hypocenter	C₃	#2859
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator, socle, prime radical, cosocle

C₂³×C₃

C₂⁴

Fitting

C₂×C₃×#125

C₂

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
91. Dic₃	#125	semidirect
120. C₃	#2859	semidirect

H ≅ G/N

type

91. Dic₃

#125

semidirect

120. C₃

#2859

semidirect

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

subgroup	index
C₂×C₃×#125	2
C₂×#870	2
#4977	2
#4976	2
#4975	2
C₄²×Dic₃	2
#2859	3

subgroup

index

C₂×C₃×#125

C₄²×Dic₃

Subgroups — #83056

Conjugacy classes of subgroups

Complements

Maximal subgroups