Group #71269 — subgroups

Subgroups — #71269

Order 384.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#5722	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#492	C₂	1	1	yes	maximal, characteristic
3	#5075	C₂	1	2	yes	maximal
4	#4546	C₂	1	1	yes	maximal, characteristic, complemented
5	#4456	C₂	1	2	yes	maximal, complemented
6	#1623	—	3	1	no	maximal, Sylow
7	C₂×C₃×SD₁₆	—	2	2	no	—
8	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃×#127	C₂²	1	1	yes	characteristic
10	#944	—	2	2	no	—
11	C₂×#214	—	2	2	no	—
12	#888	—	2	2	no	—
13	#887	—	2	2	no	—
14	C₄×Dic₆	C₂²	1	2	yes	—
15	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	#832	—	2	2	no	—
17	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	Q₈²	—	3	1	no	—
19	#492	D₃	1	1	yes	characteristic
20	#460	—	3	2	no	—
21	#333	—	3	1	no	—
22	Q₈⋊C₈	—	3	2	no	—
23	C₂×C₃×Q₈	D₄	1	2	yes	—
24	C₂×C₃×D₄	—	2	2	no	—
25	C₂×C₃×D₄	—	2	1	no	—
26	C₂×Dic₆	—	2	2	no	—
27	C₂×Dic₆	D₄	1	2	yes	—
28	C₃×SD₁₆	—	4	4	no	—
29	C₂×C₃×C₈	—	2	1	no	—
30	C₃×C₄⋊C₄	—	2	2	no	—
31	C₃×C₄²	C₂³	1	1	yes	characteristic
32	#214	—	4	4	no	—
33	C₄⋊Dic₃	D₄	1	2	yes	—
34	Dic₃⋊C₄	—	2	4	no	—
35	Dic₃⋊C₄	—	2	2	no	—
36	C₄×Dic₃	—	2	2	no	—
37	C₄×Dic₃	—	2	2	no	—
38	C₂×C₃⋊C₈	—	2	2	no	—
39	C₂×SD₁₆	—	2	2	no	—
40	C₂×SD₁₆	—	6	2	no	—
41	C₄⋊Q₈	—	6	2	no	—
42	C₄⋊Q₈	—	6	2	no	—
43	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
44	#127	C₂×D₃	1	1	yes	characteristic
45	C₄×Q₈	—	3	2	no	—
46	C₄×Q₈	—	6	2	no	—
47	C₄⋊C₈	—	3	2	no	—
48	#102	—	6	2	no	—
49	C₄×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂³×C₃	—	4	1	no	—
51	C₃×Q₈	—	4	2	no	—
52	C₃×D₄	—	4	2	no	—
53	C₃×D₄	—	4	2	no	—
54	C₃×D₄	—	4	2	no	—
55	C₂×C₃×C₄	—	2	2	no	—
56	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
57	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₃	—	2	2	no	—
59	C₂×Dic₃	—	2	2	no	—
60	Dic₆	—	2	2	no	—
61	Dic₆	—	8	2	no	—
62	Dic₆	SD₁₆	1	4	yes	complemented
63	C₃×C₈	—	2	2	no	—
64	C₃⋊C₈	—	4	2	no	—
65	C₂×Q₈	—	3	2	no	—
66	C₂×Q₈	—	6	2	no	—
67	C₂×Q₈	#67	1	2	yes	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	1	no	—
70	SD₁₆	—	4	4	no	—
71	SD₁₆	—	12	4	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	6	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	3	2	no	—
76	C₄⋊C₄	—	6	4	no	—
77	C₄⋊C₄	—	6	2	no	—
78	C₄²	—	6	2	no	—
79	C₄²	—	6	2	no	—
80	C₄²	C₂²×D₃	1	1	yes	characteristic
81	C₂²×C₃	—	8	2	no	—
82	C₂²×C₃	—	8	1	no	—
83	C₂²×C₃	#120	1	1	yes	characteristic
84	C₃×C₄	—	2	1	no	—
85	C₃×C₄	—	4	2	no	—
86	C₃×C₄	C₂×SD₁₆	1	2	yes	—
87	C₃×C₄	#137	1	2	yes	—
88	Dic₃	—	2	4	no	—
89	Dic₃	—	4	2	no	—
90	C₂³	—	4	1	no	—
91	Q₈	—	3	4	no	—
92	Q₈	—	4	2	no	—
93	Q₈	—	6	2	no	—
94	Q₈	—	24	2	no	—
95	D₄	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	6	2	no	—
100	C₂×C₄	—	6	2	no	—
101	C₂×C₄	D₃×D₄	1	2	yes	—
102	C₂×C₄	C₂×#67	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	—	12	2	no	—
105	C₂×C₃	—	16	1	no	—
106	C₂×C₃	#448	1	2	yes	—
107	C₂×C₃	#453	1	1	yes	characteristic
108	C₂²	—	8	2	no	—
109	C₂²	—	8	1	no	—
110	C₂²	#937	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	4	2	no	—
113	C₄	—	6	4	no	—
114	C₄	—	12	2	no	—
115	C₄	D₃×SD₁₆	1	2	yes	—
116	C₄	#915	1	2	yes	—
117	C₃	#1623	1	1	yes	characteristic, complemented, Sylow
118	C₂	—	16	1	no	—
119	C₂	#5077	1	1	yes	characteristic
120	C₂	#5170	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#5722

C₂

yes

maximal, characteristic, complemented

C₃×#492

C₂

yes

maximal, characteristic

#5075

C₂

yes

maximal

#4546

C₂

yes

maximal, characteristic, complemented

#4456

C₂

yes

maximal, complemented

#1623

—

maximal, Sylow

C₂×C₃×SD₁₆

—

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₃×#127

C₂²

yes

—

—

—

—

C₄×Dic₆

C₂²

yes

—

C₃×C₄×C₈

C₂²

yes

characteristic

#832

—

C₄⋊(C₃⋊C₈)

C₂²

yes

—

Q₈²

—

#492

D₃

yes

characteristic

#460

—

#333

—

Q₈⋊C₈

—

C₂×C₃×Q₈

D₄

yes

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×Dic₆

—

C₂×Dic₆

D₄

yes

—

C₃×SD₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂³

yes

characteristic

#214

—

C₄⋊Dic₃

D₄

yes

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

characteristic

#127

C₂×D₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄⋊C₈

—

#102

—

C₄×C₈

C₂×D₃

yes

characteristic

C₂³×C₃

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

SD₁₆

yes

complemented

C₃×C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#67

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#120

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×SD₁₆

yes

—

C₃×C₄

#137

yes

—

Dic₃

—

Dic₃

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

D₃×D₄

yes

—

102

C₂×C₄

C₂×#67

yes

characteristic

103

C₈

—

104

C₈

—

105

C₂×C₃

—

106

C₂×C₃

#448

yes

—

107

C₂×C₃

#453

yes

characteristic

108

C₂²

—

109

C₂²

—

110

C₂²

#937

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

D₃×SD₁₆

yes

—

116

C₄

#915

yes

—

117

C₃

#1623

yes

characteristic, complemented, Sylow

118

C₂

—

119

C₂

#5077

yes

characteristic

120

C₂

#5170

yes

—

	subgroup	G/H
center	C₂²	#937
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#492	C₂
socle	C₂²×C₃	#120
prime radical	C₃×#127	C₂²
hypercenter	#492	D₃
hypocenter	C₃	#1623
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#937

commutator, cosocle

C₃×C₄²

C₂³

Frattini

C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂²

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #5722	C₂	semidirect
4. #4546	C₂	semidirect
5. #4456	C₂	semidirect
62. Dic₆	SD₁₆	semidirect
117. C₃	#1623	semidirect

H ≅ G/N

type

1. #5722

C₂

semidirect

4. #4546

C₂

semidirect

5. #4456

C₂

semidirect

62. Dic₆

SD₁₆

semidirect

117. C₃

#1623

semidirect

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

subgroup	index
#5722	2
C₃×#492	2
#5075	2
#4546	2
#4456	2
#1623	3

subgroup

index

Subgroups — #71269

Conjugacy classes of subgroups

Complements

Maximal subgroups