Group #71810 — subgroups

Subgroups — #71810

Order 384.

254 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 85 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×#532	C₂	1	1	yes	maximal, characteristic
2	#4924	C₂	1	2	yes	maximal, complemented
3	#4546	C₂	1	2	yes	maximal, complemented
4	#4544	C₂	1	2	yes	maximal, complemented
5	#1560	—	3	1	no	maximal, Sylow
6	C₂×C₃×#41	C₄	1	2	yes	—
7	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₃×#127	C₂²	1	1	yes	characteristic
9	C₃×#124	C₄	1	2	yes	—
10	C₃×#121	—	2	2	no	—
11	C₃×C₄×D₄	—	2	2	no	—
12	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₂×#208	—	2	2	no	—
14	C₄⋊(C₃⋊C₈)	—	2	2	no	—
15	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
16	#532	D₃	1	1	yes	characteristic
17	#423	—	3	2	no	—
18	#333	—	3	2	no	—
19	#331	—	3	2	no	—
20	C₃×#41	—	4	4	no	—
21	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
22	C₂×C₃×D₄	—	2	2	no	—
23	C₂×C₃×D₄	—	2	2	no	—
24	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄	—	2	2	no	—
26	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄	D₄	1	2	yes	—
28	C₃×C₄⋊C₄	—	2	2	no	—
29	C₃×C₂²⋊C₄	—	2	4	no	—
30	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄²	D₄	1	2	yes	—
32	C₃×C₄²	C₂³	1	1	yes	characteristic
33	#208	—	4	4	no	—
34	C₂×C₃⋊C₈	—	2	4	no	—
35	C₂×#41	Dic₃	1	2	yes	—
36	C₂×M₁₆	—	6	2	no	—
37	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
38	#127	C₂×D₃	1	1	yes	characteristic
39	#124	Dic₃	1	2	yes	—
40	#121	—	2	2	no	—
41	C₄×D₄	—	2	2	no	—
42	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₄⋊C₈	—	3	4	no	—
44	C₄⋊C₈	—	6	2	no	—
45	C₂³×C₃	—	2	2	no	—
46	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₃×Q₈	—	4	2	no	—
48	C₃×D₄	—	4	2	no	—
49	C₃×D₄	—	4	4	no	—
50	C₃×D₄	—	4	2	no	—
51	C₃×D₄	—	4	2	no	—
52	C₂×C₃×C₄	—	2	2	no	—
53	C₂×C₃×C₄	—	2	2	no	—
54	C₂×C₃×C₄	—	2	2	no	—
55	C₂×C₃×C₄	—	2	2	no	—
56	C₂×C₃×C₄	—	4	4	no	—
57	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
61	C₃⋊C₈	—	4	4	no	—
62	#41	—	4	4	no	—
63	C₂×Q₈	C₂×Dic₃	1	2	yes	—
64	C₂×D₄	—	2	2	no	—
65	C₂×D₄	—	2	2	no	—
66	C₂×D₄	C₂×Dic₃	1	2	yes	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
69	C₂²×C₄	#67	1	2	yes	—
70	M₁₆	—	12	4	no	—
71	C₂×C₈	—	6	4	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₂²⋊C₄	—	2	4	no	—
74	C₄²	C₂×Dic₃	1	1	yes	characteristic
75	C₄²	#67	1	2	yes	—
76	C₄²	C₂²×D₃	1	1	yes	characteristic
77	C₂²×C₃	—	2	1	no	—
78	C₂²×C₃	—	2	2	no	—
79	C₂²×C₃	—	4	2	no	—
80	C₂²×C₃	—	4	4	no	—
81	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₃×C₄	—	2	1	no	—
83	C₃×C₄	—	2	2	no	—
84	C₃×C₄	—	4	2	no	—
85	C₃×C₄	#100	1	2	yes	—
86	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
87	C₃×C₄	#137	1	2	yes	—
88	C₂³	—	2	2	no	—
89	C₂³	#217	1	1	yes	characteristic
90	Q₈	—	4	2	no	—
91	D₄	—	4	2	no	—
92	D₄	—	4	4	no	—
93	D₄	—	4	2	no	—
94	D₄	—	4	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	#217	1	2	yes	—
101	C₂×C₄	#217	1	1	yes	characteristic
102	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
103	C₂×C₄	C₂×#67	1	2	yes	—
104	C₈	—	12	4	no	—
105	C₂×C₃	—	4	1	no	—
106	C₂×C₃	—	8	2	no	—
107	C₂×C₃	C₂×#100	1	1	yes	characteristic
108	C₂×C₃	#417	1	2	yes	—
109	C₂²	—	2	1	no	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	2	no	—
112	C₂²	—	4	4	no	—
113	C₂²	C₂×#217	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	4	2	no	—
117	C₄	#833	1	2	yes	—
118	C₄	#949	1	2	yes	—
119	C₄	#951	1	2	yes	—
120	C₃	#1560	1	1	yes	characteristic, complemented, Sylow
121	C₂	—	4	1	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×#833	1	1	yes	characteristic
124	C₂	#5232	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#532

C₂

yes

maximal, characteristic

#4924

C₂

yes

maximal, complemented

#4546

C₂

yes

maximal, complemented

#4544

C₂

yes

maximal, complemented

#1560

—

maximal, Sylow

C₂×C₃×#41

C₄

yes

—

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₃×#127

C₂²

yes

characteristic

C₃×#124

C₄

yes

—

C₃×#121

—

C₃×C₄×D₄

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

C₂²

yes

—

#532

D₃

yes

—

—

—

C₃×#41

—

C₂×C₃×Q₈

C₂×C₄

yes

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₂×C₄

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

D₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂×#41

Dic₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₃

yes

characteristic

#127

C₂×D₃

yes

characteristic

#124

Dic₃

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂³×C₃

—

C₂³×C₃

C₂²⋊C₄

yes

characteristic

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

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

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₃⋊C₈

—

#41

—

C₂×Q₈

C₂×Dic₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃

yes

characteristic

C₂²×C₄

#67

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₃

yes

characteristic

C₄²

#67

yes

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#100

yes

—

C₃×C₄

AGL(1,ℤ/8)

yes

—

C₃×C₄

#137

yes

—

C₂³

—

C₂³

#217

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#217

yes

—

101

C₂×C₄

#217

yes

characteristic

102

C₂×C₄

C₂²×Dic₃

yes

characteristic

103

C₂×C₄

C₂×#67

yes

—

104

C₈

—

105

C₂×C₃

—

106

C₂×C₃

—

107

C₂×C₃

C₂×#100

yes

characteristic

108

C₂×C₃

#417

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×#217

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#833

yes

—

118

C₄

#949

yes

—

119

C₄

#951

yes

—

120

C₃

#1560

yes

characteristic, complemented, Sylow

121

C₂

—

122

C₂

—

123

C₂

C₂×#833

yes

characteristic

124

C₂

#5232

yes

—

	subgroup	G/H
center	C₂²	C₂×#217
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting, prime radical	C₃×#532	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
cosocle	C₃×C₄²	C₂³
hypercenter	#532	D₃
hypocenter	C₃	#1560
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×#217

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₄²

C₂²×D₃

Fitting, prime radical

C₃×#532

C₂

socle

C₂²×C₃

C₂×C₂²⋊C₄

cosocle

C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
2. #4924	C₂	semidirect
3. #4546	C₂	semidirect
4. #4544	C₂	semidirect
120. C₃	#1560	semidirect

H ≅ G/N

type

2. #4924

C₂

semidirect

3. #4546

C₂

semidirect

4. #4544

C₂

semidirect

120. C₃

#1560

semidirect

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

subgroup	index
C₃×#532	2
#4924	2
#4546	2
#4544	2
#1560	3

subgroup

index

Subgroups — #71810

Conjugacy classes of subgroups

Complements

Maximal subgroups