Group #10711551 — subgroups

Subgroups — #10711551

Order 768.

302 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 155 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×#1726	C₂	1	1	yes	maximal, characteristic
2	#69743	C₂	1	2	yes	maximal
3	#67658	C₂	1	4	yes	maximal
4	#12884	—	3	1	no	maximal, Sylow
5	C₂×C₃×#97	C₄	1	2	yes	—
6	C₂×C₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₃×#376	C₄	1	1	yes	characteristic
10	C₂×#812	C₂²	1	4	yes	—
11	C₄⋊(C₃⋊C₁₆)	—	2	4	no	—
12	#2150	—	3	2	no	—
13	#1726	D₃	1	1	yes	characteristic
14	#1376	—	3	4	no	—
15	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
19	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₃×#97	—	2	4	no	—
21	C₃×#97	C₈	1	4	yes	—
22	C₃×C₄×C₈	—	2	4	no	—
23	C₃×C₄×C₈	D₄	1	2	yes	—
24	C₃×C₄×C₈	Q₈	1	2	yes	—
25	C₃×#95	—	2	2	no	—
26	#812	—	2	8	no	—
27	C₂×C₃⋊C₁₆	—	2	4	no	—
28	C₂×M₃₂	—	3	4	no	—
29	C₂×#97	Dic₃	1	2	yes	—
30	C₂×#97	Dic₃	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
32	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
33	#376	Dic₃	1	1	yes	characteristic
34	C₄⋊C₁₆	—	6	4	no	—
35	C₂²×C₃×C₄	—	2	2	no	—
36	C₂²×C₃×C₄	M₁₆	1	2	yes	—
37	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₃×C₈	—	2	2	no	—
39	C₂×C₃×C₈	—	2	2	no	—
40	C₂×C₃×C₈	—	2	2	no	—
41	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
42	C₂×C₃×C₈	C₂×C₈	1	4	yes	—
43	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₃×C₈	#41	1	4	yes	—
46	C₃×C₄²	—	2	2	no	—
47	C₃×C₄²	C₄⋊C₄	1	2	yes	—
48	C₃×C₄²	C₂×C₈	1	2	yes	—
49	C₃×C₄²	M₁₆	1	4	yes	—
50	C₃⋊C₁₆	—	4	4	no	—
51	C₂²×C₈	C₂×Dic₃	1	2	yes	—
52	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
53	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
54	C₂×C₄²	C₂×Dic₃	1	2	yes	—
55	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
56	M₃₂	—	6	8	no	—
57	C₂×C₁₆	—	6	4	no	—
58	#97	—	2	4	no	—
59	#97	C₃⋊C₈	1	4	yes	—
60	C₄×C₈	—	2	4	no	—
61	C₄×C₈	Dic₆	1	2	yes	—
62	C₄×C₈	D₁₂	1	2	yes	—
63	#95	—	2	2	no	—
64	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
65	C₂×C₃×C₄	—	2	4	no	—
66	C₂×C₃×C₄	—	4	2	no	—
67	C₂×C₃×C₄	—	4	4	no	—
68	C₂×C₃×C₄	C₄⋊C₈	1	4	yes	—
69	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₄	#117	1	2	yes	—
71	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
73	C₃×C₈	—	2	2	no	—
74	C₃×C₈	—	2	4	no	—
75	C₃×C₈	#124	1	2	yes	—
76	C₃×C₈	#125	1	2	yes	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	#208	1	2	yes	—
79	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	C₂×C₃⋊C₈	1	4	yes	—
84	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
85	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
86	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
87	C₂×C₈	#235	1	4	yes	—
88	C₄²	—	2	2	no	—
89	C₄²	C₂×C₃⋊C₈	1	2	yes	—
90	C₄²	#208	1	4	yes	—
91	C₄²	C₄⋊Dic₃	1	2	yes	—
92	C₁₆	—	12	4	no	—
93	C₂²×C₃	—	2	2	no	—
94	C₂²×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₂²×C₃	#432	1	2	yes	—
96	C₃×C₄	—	2	4	no	—
97	C₃×C₄	—	4	4	no	—
98	C₃×C₄	#382	1	1	yes	characteristic
99	C₃×C₄	#423	1	1	yes	characteristic
100	C₃×C₄	#431	1	2	yes	—
101	C₂³	C₂×#208	1	1	yes	characteristic
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
106	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
107	C₂×C₄	C₂×#208	1	2	yes	—
108	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
109	C₂×C₄	#926	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	2	4	no	—
112	C₈	#870	1	2	yes	—
113	C₈	#875	1	2	yes	—
114	C₂×C₃	—	2	2	no	—
115	C₂×C₃	#1831	1	1	yes	characteristic
116	C₂×C₃	#2153	1	2	yes	—
117	C₂²	—	2	2	no	—
118	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
119	C₂²	#4926	1	2	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	4	4	no	—
122	C₄	#4924	1	1	yes	characteristic
123	C₄	#4925	1	2	yes	—
124	C₄	#4936	1	1	yes	characteristic
125	C₃	#12884	1	1	yes	characteristic, complemented, Sylow
126	C₂	—	2	2	no	—
127	C₂	#69392	1	1	yes	characteristic
128	C₂	#69885	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#1726

C₂

yes

maximal, characteristic

#69743

C₂

yes

maximal

#67658

C₂

yes

maximal

#12884

—

maximal, Sylow

C₂×C₃×#97

C₄

yes

—

C₂×C₃×#97

C₄

yes

characteristic

C₂×C₃×C₄×C₈

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₃×#376

C₄

yes

characteristic

C₂×#812

C₂²

yes

—

C₄⋊(C₃⋊C₁₆)

—

#2150

—

#1726

D₃

yes

characteristic

#1376

—

C₂²×C₃×C₈

C₂×C₄

yes

characteristic

C₂²×C₃×C₈

C₂×C₄

yes

—

C₂²×C₃×C₈

C₂³

yes

characteristic

C₂×C₃×C₄²

C₂×C₄

yes

—

C₂×C₃×C₄²

C₂×C₄

yes

characteristic

C₃×#97

—

C₃×#97

C₈

yes

—

C₃×C₄×C₈

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

Q₈

yes

—

C₃×#95

—

#812

—

C₂×C₃⋊C₁₆

—

C₂×M₃₂

—

C₂×#97

Dic₃

yes

—

C₂×#97

Dic₃

yes

characteristic

C₂×C₄×C₈

C₂×D₃

yes

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

#376

Dic₃

yes

characteristic

C₄⋊C₁₆

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

M₁₆

yes

—

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄⋊C₄

yes

—

C₂×C₃×C₈

C₂×C₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

C₂×Q₈

yes

characteristic

C₂×C₃×C₈

#41

yes

—

C₃×C₄²

—

C₃×C₄²

C₄⋊C₄

yes

—

C₃×C₄²

C₂×C₈

yes

—

C₃×C₄²

M₁₆

yes

—

C₃⋊C₁₆

—

C₂²×C₈

C₂×Dic₃

yes

—

C₂²×C₈

C₂×Dic₃

yes

characteristic

C₂²×C₈

C₂²×D₃

yes

characteristic

C₂×C₄²

C₂×Dic₃

yes

—

C₂×C₄²

C₂×Dic₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#97

—

#97

C₃⋊C₈

yes

—

C₄×C₈

—

C₄×C₈

Dic₆

yes

—

C₄×C₈

D₁₂

yes

—

#95

—

C₂³×C₃

C₂×M₁₆

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₈

yes

—

C₂×C₃×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

#117

yes

—

C₂×C₃×C₄

C₂²×C₈

yes

characteristic

C₂×C₃×C₄

C₂×M₁₆

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

#124

yes

—

C₃×C₈

#125

yes

—

C₂²×C₄

—

C₂²×C₄

#208

yes

—

C₂²×C₄

C₂²×Dic₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₃⋊C₈

yes

—

C₂×C₈

C₄⋊Dic₃

yes

—

C₂×C₈

C₂×Dic₆

yes

characteristic

C₂×C₈

C₂×D₁₂

yes

characteristic

C₂×C₈

#235

yes

—

C₄²

—

C₄²

C₂×C₃⋊C₈

yes

—

C₄²

#208

yes

—

C₄²

C₄⋊Dic₃

yes

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₃

#432

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

100

C₃×C₄

#431

yes

—

101

C₂³

C₂×#208

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₄⋊(C₃⋊C₈)

yes

—

106

C₂×C₄

C₂²×C₃⋊C₈

yes

characteristic

107

C₂×C₄

C₂×#208

yes

—

108

C₂×C₄

C₂×C₄⋊Dic₃

yes

characteristic

109

C₂×C₄

#926

yes

—

110

C₈

—

111

C₈

—

112

C₈

#870

yes

—

113

C₈

#875

yes

—

114

C₂×C₃

—

115

C₂×C₃

#1831

yes

characteristic

116

C₂×C₃

#2153

yes

—

117

C₂²

—

118

C₂²

C₂×C₄⋊(C₃⋊C₈)

yes

characteristic

119

C₂²

#4926

yes

—

120

C₄

—

121

C₄

—

122

C₄

#4924

yes

characteristic

123

C₄

#4925

yes

—

124

C₄

#4936

yes

characteristic

125

C₃

#12884

yes

characteristic, complemented, Sylow

126

C₂

—

127

C₂

#69392

yes

characteristic

128

C₂

#69885

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1726	C₂
socle	C₂²×C₃	C₂×C₄⋊C₈
prime radical	C₂³×C₃	C₂×M₁₆
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1726	D₃
hypocenter	C₃	#12884
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₂×D₁₂

commutator

C₂×C₃×C₄

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₃

Fitting

C₃×#1726

C₂

socle

C₂²×C₃

C₂×C₄⋊C₈

prime radical

C₂³×C₃

C₂×M₁₆

cosocle

C₂²×C₃×C₈

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

N	H ≅ G/N	type
125. C₃	#12884	semidirect

H ≅ G/N

type

125. C₃

#12884

semidirect

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

subgroup	index
C₃×#1726	2
#69743	2
#67658	2
#12884	3

subgroup

index

Subgroups — #10711551

Conjugacy classes of subgroups

Complements

Maximal subgroups