Group #74265 — subgroups

Subgroups — #74265

Order 384.

424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×#383	C₂	1	1	yes	maximal, characteristic
3	#4937	C₂	1	1	yes	maximal, characteristic
4	#4654	C₂	1	4	yes	maximal
5	#4653	C₂	1	4	yes	maximal
6	#4646	C₂	1	4	yes	maximal
7	#2381	—	3	1	no	maximal, Sylow
8	C₂²×Dic₆	C₂²	1	2	yes	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄×Dic₃	C₂²	1	4	yes	—
14	#869	C₄	1	8	yes	complemented
15	C₃×#95	C₂²	1	2	yes	—
16	C₃×#95	C₂²	1	2	yes	—
17	C₃×#95	C₂²	1	2	yes	—
18	C₃⋊#95	C₂²	1	2	yes	—
19	C₃⋊#95	C₂²	1	8	yes	—
20	C₃⋊#95	C₂²	1	2	yes	—
21	C₃⋊#95	C₂²	1	2	yes	—
22	C₂×C₄⋊Q₈	—	3	1	no	—
23	#391	—	3	4	no	—
24	#387	—	3	4	no	—
25	#387	—	3	4	no	—
26	#383	—	3	1	no	—
27	#383	D₃	1	1	yes	characteristic
28	C₂²×C₃×C₄	C₂³	1	4	yes	—
29	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂²×Dic₃	C₂³	1	4	yes	—
33	C₂²×Dic₃	C₂³	1	4	yes	—
34	C₂×Dic₆	—	4	2	no	—
35	C₂×Dic₆	—	4	2	no	—
36	C₂×Dic₆	C₂×C₄	1	8	yes	—
37	C₃×C₄²	C₂×C₄	1	4	yes	—
38	C₄⋊Dic₃	C₂×C₄	1	8	yes	—
39	C₄⋊Dic₃	C₂×C₄	1	8	yes	—
40	Dic₃⋊C₄	—	4	4	no	—
41	C₄×Dic₃	—	4	4	no	—
42	C₂²×Q₈	—	3	2	no	—
43	C₄⋊Q₈	—	3	8	no	—
44	C₂×C₄⋊C₄	—	3	2	no	—
45	C₂×C₄⋊C₄	—	3	2	no	—
46	C₂×C₄⋊C₄	—	3	4	no	—
47	C₂×C₄²	—	3	4	no	—
48	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
49	#95	—	3	2	no	—
50	#95	—	3	2	no	—
51	#95	—	3	2	no	—
52	#95	—	3	8	no	—
53	#95	C₂×D₃	1	2	yes	—
54	#95	C₂×D₃	1	2	yes	—
55	#95	C₂×D₃	1	2	yes	—
56	C₂³×C₃	C₂⁴	1	1	yes	characteristic
57	C₂×C₃×C₄	—	2	2	no	—
58	C₂×C₃×C₄	—	2	2	no	—
59	C₂×C₃×C₄	—	2	2	no	—
60	C₂×C₃×C₄	—	2	4	no	—
61	C₂×C₃×C₄	—	2	4	no	—
62	C₂×C₃×C₄	—	2	4	no	—
63	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
64	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
65	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
66	C₂×Dic₃	—	2	4	no	—
67	C₂×Dic₃	—	2	4	no	—
68	C₂×Dic₃	—	2	4	no	—
69	C₂×Dic₃	—	2	4	no	—
70	C₂×Dic₃	—	2	4	no	—
71	C₂×Dic₃	C₂²×C₄	1	8	yes	—
72	Dic₆	—	4	8	no	—
73	C₂×Q₈	—	3	8	no	—
74	C₂×Q₈	—	12	2	no	—
75	C₂×Q₈	—	12	2	no	—
76	C₂²×C₄	—	3	4	no	—
77	C₂²×C₄	—	3	4	no	—
78	C₂²×C₄	C₂²×D₃	1	4	yes	—
79	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
80	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
82	C₄⋊C₄	—	3	8	no	—
83	C₄⋊C₄	—	3	8	no	—
84	C₄⋊C₄	—	12	4	no	—
85	C₄²	—	12	4	no	—
86	C₄²	C₄×D₃	1	4	yes	—
87	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
88	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
89	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
90	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
91	C₃×C₄	—	2	2	no	—
92	C₃×C₄	—	2	2	no	—
93	C₃×C₄	—	2	2	no	—
94	C₃×C₄	—	4	4	no	—
95	Dic₃	—	2	8	no	—
96	Dic₃	—	4	4	no	—
97	C₂³	C₂³×D₃	1	1	yes	characteristic
98	Q₈	—	12	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	3	8	no	—
106	C₂×C₄	—	6	4	no	—
107	C₂×C₄	—	6	4	no	—
108	C₂×C₄	—	6	4	no	—
109	C₂×C₄	—	6	4	no	—
110	C₂×C₄	—	6	4	no	—
111	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
113	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
114	C₂×C₃	#518	1	1	yes	characteristic
115	C₂×C₃	#519	1	1	yes	characteristic
116	C₂×C₃	#519	1	1	yes	characteristic
117	C₂×C₃	#563	1	4	yes	—
118	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
119	C₂²	#1009	1	2	yes	—
120	C₂²	#1010	1	2	yes	—
121	C₂²	#1010	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	6	8	no	—
127	C₄	—	12	4	no	—
128	C₃	#2381	1	1	yes	characteristic, complemented, Sylow
129	C₂	#5516	1	1	yes	characteristic
130	C₂	#5516	1	1	yes	characteristic
131	C₂	#5521	1	1	yes	characteristic
132	C₂	#5712	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#869

C₂

yes

maximal, characteristic

C₃×#383

C₂

yes

maximal, characteristic

#4937

C₂

yes

maximal, characteristic

#4654

C₂

yes

maximal

#4653

C₂

yes

maximal

#4646

C₂

yes

maximal

#2381

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

#869

C₄

yes

complemented

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₃×#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₂×C₄⋊Q₈

—

—

—

—

—

D₃

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

C₂×C₄

yes

—

C₃×C₄²

C₂×C₄

yes

—

C₄⋊Dic₃

C₂×C₄

yes

—

C₄⋊Dic₃

C₂×C₄

yes

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

—

—

—

—

C₂×D₃

yes

—

#95

C₂×D₃

yes

—

#95

C₂×D₃

yes

—

C₂³×C₃

C₂⁴

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂²×C₄

yes

—

Dic₆

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₄×D₃

yes

—

C₂²×C₃

C₂³×C₄

yes

characteristic

C₂²×C₃

H(5,𝔽₂)

yes

—

C₂²×C₃

2₋¹⁺⁴

yes

—

C₂²×C₃

2₋¹⁺⁴

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₂³

C₂³×D₃

yes

characteristic

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₄×D₃

yes

—

112

C₂×C₄

C₂×C₄×D₃

yes

—

113

C₂×C₄

C₂×C₄×D₃

yes

—

114

C₂×C₃

#518

yes

characteristic

115

C₂×C₃

#519

yes

characteristic

116

C₂×C₃

#519

yes

characteristic

117

C₂×C₃

#563

yes

—

118

C₂²

C₂²×C₄×D₃

yes

characteristic

119

C₂²

#1009

yes

—

120

C₂²

#1010

yes

—

121

C₂²

#1010

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₃

#2381

yes

characteristic, complemented, Sylow

129

C₂

#5516

yes

characteristic

130

C₂

#5516

yes

characteristic

131

C₂

#5521

yes

characteristic

132

C₂

#5712

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting	C₃×#383	C₂
socle, prime radical, cosocle	C₂³×C₃	C₂⁴
hypercenter	#383	D₃
hypocenter	C₃	#2381
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator

C₂²×C₃

C₂³×C₄

Fitting

C₃×#383

C₂

socle, prime radical, cosocle

C₂³×C₃

C₂⁴

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
14. #869	C₄	semidirect
128. C₃	#2381	semidirect

H ≅ G/N

type

14. #869

C₄

semidirect

128. C₃

#2381

semidirect

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

subgroup	index
C₂×#869	2
C₃×#383	2
#4937	2
#4654	2
#4653	2
#4646	2
#2381	3

subgroup

index

Subgroups — #74265

Conjugacy classes of subgroups

Complements

Maximal subgroups