Group #70443 — subgroups

Subgroups — #70443

Order 384.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#4550	C₂	1	4	yes	maximal
5	#1831	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₃	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₃	C₄	1	2	yes	—
10	C₂×C₄×Dic₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃×C₄⋊C₈	—	2	2	no	—
14	C₄⋊(C₃⋊C₈)	—	2	2	no	—
15	C₂×C₄⋊C₈	—	3	1	no	—
16	C₂×C₄⋊C₈	D₃	1	1	yes	characteristic
17	C₄³	—	3	1	no	—
18	#336	—	3	4	no	—
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃	C₂×C₄	1	2	yes	—
23	C₂²×Dic₃	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₈	—	2	2	no	—
25	C₂×C₃×C₈	—	2	4	no	—
26	C₃×C₄²	D₄	1	2	yes	—
27	C₃×C₄²	Q₈	1	2	yes	—
28	C₄×Dic₃	—	2	2	no	—
29	C₄×Dic₃	—	2	2	no	—
30	C₄×Dic₃	—	2	2	no	—
31	C₄×Dic₃	—	2	4	no	—
32	C₄×Dic₃	C₈	1	4	yes	complemented
33	C₂×C₃⋊C₈	—	2	2	no	—
34	C₂×C₃⋊C₈	—	2	4	no	—
35	C₂²×C₈	—	3	2	no	—
36	C₂²×C₈	C₂×D₃	1	2	yes	—
37	C₂×C₄²	—	3	1	no	—
38	C₂×C₄²	—	3	2	no	—
39	C₂×C₄²	—	3	1	no	—
40	C₂×C₄²	—	3	2	no	—
41	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	—	6	2	no	—
44	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₃×C₄	—	2	2	no	—
46	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
47	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
48	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
49	C₂×C₃×C₄	M₁₆	1	2	yes	—
50	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₃×C₄	#41	1	2	yes	—
53	C₂×C₃×C₄	#41	1	2	yes	—
54	C₂×Dic₃	—	2	2	no	—
55	C₂×Dic₃	—	2	2	no	—
56	C₂×Dic₃	—	2	2	no	—
57	C₂×Dic₃	—	2	2	no	—
58	C₂×Dic₃	C₂×C₈	1	4	yes	—
59	C₂×Dic₃	M₁₆	1	4	yes	—
60	C₃×C₈	—	4	2	no	—
61	C₃⋊C₈	—	4	2	no	—
62	C₂²×C₄	—	3	2	no	—
63	C₂²×C₄	—	3	2	no	—
64	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
65	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
66	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	6	4	no	—
70	C₂×C₈	—	6	2	no	—
71	C₄²	—	3	4	no	—
72	C₄²	—	6	4	no	—
73	C₄²	—	6	2	no	—
74	C₄²	—	6	2	no	—
75	C₄²	—	6	2	no	—
76	C₄²	Dic₆	1	2	yes	—
77	C₄²	#67	1	2	yes	—
78	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₃	#117	1	2	yes	—
80	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
81	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂²×C₃	C₂×M₁₆	1	2	yes	—
83	C₃×C₄	—	2	2	no	—
84	C₃×C₄	C₄⋊C₈	1	4	yes	—
85	C₃×C₄	#124	1	1	yes	characteristic
86	C₃×C₄	#124	1	1	yes	characteristic
87	C₃×C₄	#125	1	2	yes	—
88	Dic₃	—	2	4	no	—
89	Dic₃	—	2	4	no	—
90	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	3	4	no	—
93	C₂×C₄	—	3	4	no	—
94	C₂×C₄	—	6	2	no	—
95	C₂×C₄	—	6	2	no	—
96	C₂×C₄	—	6	2	no	—
97	C₂×C₄	—	6	2	no	—
98	C₂×C₄	D₃×C₈	1	2	yes	—
99	C₂×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
100	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
101	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
102	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
103	C₂×C₄	#237	1	2	yes	—
104	C₂×C₄	#239	1	2	yes	—
105	C₂×C₄	C₂×#67	1	1	yes	characteristic
106	C₈	—	4	2	no	—
107	C₈	—	12	2	no	—
108	C₂×C₃	#382	1	1	yes	characteristic
109	C₂×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₃	#423	1	1	yes	characteristic
111	C₂×C₃	#431	1	2	yes	—
112	C₂×C₃	#432	1	2	yes	—
113	C₂²	#892	1	2	yes	—
114	C₂²	C₂×D₃×C₈	1	1	yes	characteristic
115	C₂²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
116	C₂²	D₃×M₁₆	1	2	yes	—
117	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	6	4	no	—
120	C₄	—	6	4	no	—
121	C₄	Dic₃⋊C₈	1	4	yes	—
122	C₄	#890	1	2	yes	—
123	C₄	#936	1	1	yes	characteristic
124	C₄	#947	1	1	yes	characteristic
125	C₃	#1831	1	1	yes	characteristic, complemented, Sylow
126	C₂	#4833	1	2	yes	—
127	C₂	#4835	1	2	yes	—
128	C₂	#4977	1	1	yes	characteristic
129	C₂	C₂×Dic₃⋊C₈	1	1	yes	characteristic
130	C₂	#5118	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₄²×Dic₃

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#4550

C₂

yes

maximal

#1831

—

maximal, Sylow

C₂²×C₃×C₈

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₄

yes

characteristic

C₂×C₄×Dic₃

C₄

yes

—

C₂×C₄×Dic₃

C₄

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

—

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₃

yes

characteristic

C₄³

—

#336

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂×C₄

yes

—

C₂²×Dic₃

C₂×C₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

D₄

yes

—

C₃×C₄²

Q₈

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

C₈

yes

complemented

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂×C₈

yes

—

C₂×C₃×C₄

M₁₆

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂×C₈

yes

—

C₂×Dic₃

M₁₆

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃

yes

characteristic

C₂²×C₄

C₄×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₆

yes

—

C₄²

#67

yes

—

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃

#117

yes

—

C₂²×C₃

C₂²×C₈

yes

characteristic

C₂²×C₃

C₂×M₁₆

yes

characteristic

C₂²×C₃

C₂×M₁₆

yes

—

C₃×C₄

—

C₃×C₄

C₄⋊C₈

yes

—

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

—

Dic₃

—

Dic₃

—

C₂³

C₂×C₄×D₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₃×C₈

yes

—

C₂×C₄

ΓL(1,𝔽₂₅)

yes

—

100

C₂×C₄

Dic₃⋊C₄

yes

—

101

C₂×C₄

Dic₃⋊C₄

yes

—

102

C₂×C₄

C₂×Dic₆

yes

characteristic

103

C₂×C₄

#237

yes

—

104

C₂×C₄

#239

yes

—

105

C₂×C₄

C₂×#67

yes

characteristic

106

C₈

—

107

C₈

—

108

C₂×C₃

#382

yes

characteristic

109

C₂×C₃

C₂×C₄⋊C₈

yes

characteristic

110

C₂×C₃

#423

yes

characteristic

111

C₂×C₃

#431

yes

—

112

C₂×C₃

#432

yes

—

113

C₂²

#892

yes

—

114

C₂²

C₂×D₃×C₈

yes

characteristic

115

C₂²

C₂×ΓL(1,𝔽₂₅)

yes

characteristic

116

C₂²

D₃×M₁₆

yes

—

117

C₂²

C₂×Dic₃⋊C₄

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

Dic₃⋊C₈

yes

—

122

C₄

#890

yes

—

123

C₄

#936

yes

characteristic

124

C₄

#947

yes

characteristic

125

C₃

#1831

yes

characteristic, complemented, Sylow

126

C₂

#4833

yes

—

127

C₂

#4835

yes

—

128

C₂

#4977

yes

characteristic

129

C₂

C₂×Dic₃⋊C₈

yes

characteristic

130

C₂

#5118

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₈
Fitting	C₂×C₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#1831
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃

commutator

C₂²×C₃

C₂²×C₈

Fitting

C₂×C₃×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₃

C₂²×C₄

cosocle

C₂²×C₃×C₄

C₂³

hypercenter

C₂×C₄⋊C₈

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
32. C₄×Dic₃	C₈	semidirect
125. C₃	#1831	semidirect

H ≅ G/N

type

32. C₄×Dic₃

C₈

semidirect

125. C₃

#1831

semidirect

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

subgroup	index
C₂×C₃×C₄⋊C₈	2
C₄²×Dic₃	2
C₂×C₄⋊(C₃⋊C₈)	2
#4550	2
#1831	3

subgroup

index

C₂×C₃×C₄⋊C₈

C₄²×Dic₃

C₂×C₄⋊(C₃⋊C₈)

#4550

#1831

Subgroups — #70443

Conjugacy classes of subgroups

Complements

Maximal subgroups