Group #10709920 — subgroups

Subgroups — #10709920

Order 768.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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	#69066	C₂	1	2	yes	maximal, complemented
2	#69061	C₂	1	2	yes	maximal
3	#68065	C₂	1	1	yes	maximal, characteristic, complemented
4	#68060	C₂	1	1	yes	maximal, characteristic
5	C₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	#12520	—	3	1	no	maximal, Sylow
7	C₂×Dic₂₄	—	2	4	no	—
8	C₂×#800	—	2	4	no	—
9	#4699	C₂²	1	2	yes	—
10	#4699	C₂²	1	1	yes	characteristic
11	#4695	C₂²	1	1	yes	characteristic
12	#4694	—	2	2	no	—
13	#4682	—	2	2	no	—
14	C₃×C₄×C₁₆	C₂²	1	2	yes	—
15	C₃×C₈²	C₂²	1	1	yes	characteristic
16	#2248	—	3	2	no	—
17	#2247	—	3	2	no	—
18	#1713	—	3	1	no	—
19	#1709	—	3	1	no	—
20	#1311	D₃	1	1	yes	characteristic, complemented
21	C₂×Dic₁₂	—	2	2	no	—
22	C₂×Dic₁₂	—	2	2	no	—
23	C₂×Dic₁₂	—	2	2	no	—
24	C₂×D₂₄	—	2	2	no	—
25	#874	—	2	1	no	—
26	#869	—	2	2	no	—
27	#869	—	2	1	no	—
28	C₂×C₃×C₁₆	D₄	1	4	yes	—
29	C₃×C₄×C₈	D₄	1	2	yes	—
30	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
31	#821	—	4	4	no	—
32	#818	—	4	4	no	—
33	Dic₂₄	—	4	8	no	—
34	#800	—	4	8	no	—
35	C₂×Q₃₂	—	6	4	no	—
36	C₂×SD₃₂	—	6	4	no	—
37	#500	—	6	2	no	—
38	#494	—	3	1	no	—
39	#494	—	3	2	no	—
40	#493	—	3	1	no	—
41	#486	—	6	2	no	—
42	C₄×C₁₆	C₂×D₃	1	2	yes	—
43	C₈²	C₂×D₃	1	1	yes	characteristic
44	C₂×D₁₂	—	4	1	no	—
45	C₂×D₁₂	—	4	2	no	—
46	C₂×Dic₆	—	4	2	no	—
47	C₂×Dic₆	—	4	1	no	—
48	C₂×C₃×C₈	D₈	1	4	yes	—
49	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
50	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
51	C₄⋊Dic₃	—	4	4	no	—
52	C₄⋊Dic₃	—	4	2	no	—
53	Dic₁₂	—	4	4	no	—
54	Dic₁₂	—	4	4	no	—
55	Dic₁₂	—	4	4	no	—
56	D₂₄	—	4	4	no	—
57	C₃×C₁₆	D₈	1	8	yes	—
58	C₂×Q₁₆	—	6	2	no	—
59	C₂×Q₁₆	—	6	2	no	—
60	C₂×Q₁₆	—	6	2	no	—
61	C₂×D₈	—	6	2	no	—
62	C₄⋊Q₈	—	6	1	no	—
63	C₄⋊Q₈	—	6	2	no	—
64	#127	—	6	1	no	—
65	Q₃₂	—	12	8	no	—
66	SD₃₂	—	12	8	no	—
67	C₂×C₁₆	D₁₂	1	4	yes	—
68	#107	—	12	4	no	—
69	#102	—	12	4	no	—
70	C₄×C₈	D₁₂	1	2	yes	—
71	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
72	C₂²×D₃	—	8	1	no	—
73	C₂×C₃×C₄	#127	1	1	yes	characteristic
74	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
75	C₂×Dic₃	—	8	2	no	—
76	C₂×Dic₃	—	8	1	no	—
77	D₁₂	—	8	2	no	—
78	D₁₂	—	8	4	no	—
79	Dic₆	—	8	2	no	—
80	Dic₆	—	8	2	no	—
81	Dic₆	—	8	2	no	—
82	C₃×C₈	—	2	4	no	—
83	C₃×C₈	C₂×D₈	1	4	yes	—
84	C₂×Q₈	—	12	1	no	—
85	C₂×Q₈	—	12	2	no	—
86	C₂×D₄	—	12	1	no	—
87	C₂×D₄	—	12	2	no	—
88	Q₁₆	—	12	4	no	—
89	Q₁₆	—	12	4	no	—
90	Q₁₆	—	12	4	no	—
91	D₈	—	12	4	no	—
92	C₂×C₈	D₂₄	1	4	yes	—
93	C₂×C₈	C₂×D₁₂	1	2	yes	—
94	C₄⋊C₄	—	12	2	no	—
95	C₄⋊C₄	—	12	4	no	—
96	C₄²	C₂×D₁₂	1	1	yes	characteristic
97	C₁₆	D₂₄	1	8	yes	—
98	C₂²×C₃	#493	1	1	yes	characteristic
99	C₂×D₃	—	16	1	no	—
100	C₂×D₃	—	16	2	no	—
101	C₃×C₄	#493	1	2	yes	—
102	C₃×C₄	#510	1	4	yes	—
103	Dic₃	—	16	2	no	—
104	Dic₃	—	16	1	no	—
105	C₂³	—	24	1	no	—
106	Q₈	—	24	2	no	—
107	Q₈	—	24	2	no	—
108	Q₈	—	24	2	no	—
109	D₄	—	24	4	no	—
110	D₄	—	24	2	no	—
111	C₂×C₄	—	24	1	no	—
112	C₂×C₄	—	24	2	no	—
113	C₂×C₄	#874	1	1	yes	characteristic
114	C₂×C₄	C₂×D₂₄	1	2	yes	—
115	C₈	—	2	4	no	—
116	C₈	C₂×D₂₄	1	4	yes	—
117	C₂×C₃	#1712	1	1	yes	characteristic
118	C₂×C₃	#2251	1	2	yes	—
119	D₃	—	32	1	no	—
120	C₂²	—	48	2	no	—
121	C₂²	—	48	1	no	—
122	C₂²	#4695	1	1	yes	characteristic
123	C₄	—	48	1	no	—
124	C₄	—	48	2	no	—
125	C₄	#4695	1	2	yes	—
126	C₄	#4909	1	4	yes	—
127	C₃	#12520	1	1	yes	characteristic, complemented, Sylow
128	C₂	—	96	1	no	—
129	C₂	#68062	1	1	yes	characteristic
130	C₂	#69084	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#69066

C₂

yes

maximal, complemented

#69061

C₂

yes

maximal

#68065

C₂

yes

maximal, characteristic, complemented

#68060

C₂

yes

maximal, characteristic

C₃×#1311

C₂

yes

maximal, characteristic, complemented

#12520

—

maximal, Sylow

C₂×Dic₂₄

—

C₂×#800

—

#4699

C₂²

yes

—

#4699

C₂²

yes

characteristic

#4695

C₂²

yes

characteristic

#4694

—

#4682

—

C₃×C₄×C₁₆

C₂²

yes

—

C₃×C₈²

C₂²

yes

—

—

—

—

D₃

yes

characteristic, complemented

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×D₂₄

—

#874

—

#869

—

#869

—

C₂×C₃×C₁₆

D₄

yes

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

C₂³

yes

characteristic

#821

—

#818

—

Dic₂₄

—

#800

—

C₂×Q₃₂

—

C₂×SD₃₂

—

—

—

—

—

—

C₄×C₁₆

C₂×D₃

yes

—

C₈²

C₂×D₃

yes

characteristic

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

D₂₄

—

C₃×C₁₆

D₈

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

C₂×C₁₆

D₁₂

yes

—

#107

—

#102

—

C₄×C₈

D₁₂

yes

—

C₄×C₈

C₂²×D₃

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

C₂×D₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

D₈

—

C₂×C₈

D₂₄

yes

—

C₂×C₈

C₂×D₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₁₂

yes

characteristic

C₁₆

D₂₄

yes

—

C₂²×C₃

#493

yes

characteristic

C₂×D₃

—

100

C₂×D₃

—

101

C₃×C₄

#493

yes

—

102

C₃×C₄

#510

yes

—

103

Dic₃

—

104

Dic₃

—

105

C₂³

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#874

yes

characteristic

114

C₂×C₄

C₂×D₂₄

yes

—

115

C₈

—

116

C₈

C₂×D₂₄

yes

—

117

C₂×C₃

#1712

yes

characteristic

118

C₂×C₃

#2251

yes

—

119

D₃

—

120

C₂²

—

121

C₂²

—

122

C₂²

#4695

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

#4695

yes

—

126

C₄

#4909

yes

—

127

C₃

#12520

yes

characteristic, complemented, Sylow

128

C₂

—

129

C₂

#68062

yes

characteristic

130

C₂

#69084

yes

—

	subgroup	G/H
center	C₂²	#4695
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1311	C₂
socle	C₂²×C₃	#493
prime radical	#4695	C₂²
hypercenter	#1311	D₃
hypocenter	C₃	#12520
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#4695

commutator, cosocle

C₃×C₄×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₃

C₂

C₂²×C₃

C₂²

D₃

C₃

G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #69066	C₂	semidirect
3. #68065	C₂	semidirect
5. C₃×#1311	C₂	semidirect
20. #1311	D₃	semidirect
127. C₃	#12520	semidirect

H ≅ G/N

type

1. #69066

C₂

semidirect

3. #68065

C₂

semidirect

5. C₃×#1311

C₂

semidirect

20. #1311

D₃

semidirect

127. C₃

#12520

semidirect

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

subgroup	index
#69066	2
#69061	2
#68065	2
#68060	2
C₃×#1311	2
#12520	3

subgroup

index

Subgroups — #10709920

Conjugacy classes of subgroups

Complements

Maximal subgroups