Group #10709916 — subgroups

Subgroups — #10709916

Order 768.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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×#1709	C₂	1	1	yes	maximal, characteristic, complemented
2	#71313	C₂	1	2	yes	maximal, complemented
3	#71290	C₂	1	2	yes	maximal, complemented
4	#68062	C₂	1	1	yes	maximal, characteristic, complemented
5	#67437	C₂	1	1	yes	maximal, characteristic, complemented
6	#12519	—	3	1	no	maximal, Sylow
7	C₃×#494	C₂²	1	1	yes	characteristic
8	C₃×#493	C₂²	1	1	yes	characteristic
9	C₃×#486	—	2	2	no	—
10	C₂×#828	—	2	4	no	—
11	C₂×#826	—	2	4	no	—
12	#4695	—	2	2	no	—
13	#4695	C₂²	1	2	yes	—
14	C₃×C₈²	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
16	#2248	—	3	2	no	—
17	#2246	—	3	2	no	—
18	#1712	—	3	1	no	—
19	#1709	D₃	1	1	yes	characteristic, complemented
20	#1311	—	3	1	no	—
21	C₂×C₃×Q₁₆	—	2	2	no	—
22	C₂×C₃×D₈	—	2	2	no	—
23	C₃×C₄⋊Q₈	—	2	1	no	—
24	C₃×#127	—	2	1	no	—
25	C₂×D₂₄	—	2	2	no	—
26	C₂×D₂₄	—	2	2	no	—
27	C₂×D₂₄	—	4	4	no	—
28	#874	—	2	2	no	—
29	C₃×#102	—	4	4	no	—
30	C₃×C₄×C₈	D₄	1	2	yes	—
31	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
32	#828	—	4	8	no	—
33	#826	—	4	8	no	—
34	C₂×C₃⋊C₁₆	D₄	1	4	yes	—
35	C₂×SD₃₂	—	6	4	no	—
36	C₂×D₁₆	—	6	4	no	—
37	#494	C₂×D₃	1	1	yes	characteristic
38	#493	—	3	2	no	—
39	#493	—	6	2	no	—
40	#493	C₂×D₃	1	1	yes	characteristic
41	#486	—	2	2	no	—
42	C₄×C₁₆	—	3	2	no	—
43	C₈²	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×Q₈	—	4	1	no	—
45	C₂×C₃×D₄	—	4	1	no	—
46	C₂×C₃×D₄	—	4	2	no	—
47	C₂×D₁₂	—	4	2	no	—
48	C₂×D₁₂	—	4	4	no	—
49	C₃×Q₁₆	—	4	4	no	—
50	C₃×D₈	—	4	4	no	—
51	C₂×C₃×C₈	D₈	1	4	yes	—
52	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
53	C₃×C₄⋊C₄	—	4	2	no	—
54	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
55	D₂₄	—	4	4	no	—
56	D₂₄	—	4	4	no	—
57	D₂₄	—	8	8	no	—
58	C₃⋊C₁₆	D₈	1	8	yes	complemented
59	C₂×Q₁₆	—	2	2	no	—
60	C₂×D₈	—	2	2	no	—
61	C₂×D₈	—	6	2	no	—
62	C₂×D₈	—	6	2	no	—
63	C₂×D₈	—	12	4	no	—
64	C₄⋊Q₈	—	2	1	no	—
65	#127	—	2	1	no	—
66	#127	—	6	2	no	—
67	SD₃₂	—	12	8	no	—
68	D₁₆	—	12	8	no	—
69	C₂×C₁₆	—	3	4	no	—
70	#102	—	4	4	no	—
71	C₄×C₈	#67	1	2	yes	—
72	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
73	C₂³×C₃	—	8	1	no	—
74	C₂²×D₃	—	8	2	no	—
75	C₃×Q₈	—	8	2	no	—
76	C₃×D₄	—	8	2	no	—
77	C₃×D₄	—	8	4	no	—
78	C₂×C₃×C₄	—	8	1	no	—
79	C₂×C₃×C₄	#127	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
81	D₁₂	—	8	2	no	—
82	D₁₂	—	8	2	no	—
83	D₁₂	—	8	8	no	—
84	C₃×C₈	—	2	4	no	—
85	C₃×C₈	C₂×D₈	1	4	yes	—
86	C₂×Q₈	—	4	1	no	—
87	C₂×D₄	—	4	2	no	—
88	C₂×D₄	—	4	1	no	—
89	C₂×D₄	—	12	4	no	—
90	C₂×D₄	—	12	2	no	—
91	Q₁₆	—	4	4	no	—
92	D₈	—	4	4	no	—
93	D₈	—	12	4	no	—
94	D₈	—	12	4	no	—
95	D₈	—	24	8	no	—
96	C₂×C₈	#213	1	4	yes	—
97	C₂×C₈	D₃×D₄	1	2	yes	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄²	C₂×#67	1	1	yes	characteristic
100	C₁₆	—	3	8	no	—
101	C₂²×C₃	—	16	1	no	—
102	C₂²×C₃	—	16	2	no	—
103	C₂²×C₃	#493	1	1	yes	characteristic
104	C₂×D₃	—	16	2	no	—
105	C₂×D₃	—	16	4	no	—
106	C₃×C₄	—	16	1	no	—
107	C₃×C₄	#493	1	2	yes	—
108	C₃×C₄	#509	1	4	yes	—
109	C₂³	—	8	1	no	—
110	C₂³	—	24	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	8	2	no	—
113	D₄	—	8	4	no	—
114	D₄	—	24	2	no	—
115	D₄	—	24	2	no	—
116	D₄	—	24	8	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	C₂×#213	1	2	yes	—
119	C₂×C₄	#940	1	1	yes	characteristic
120	C₈	—	2	4	no	—
121	C₈	D₃×D₈	1	4	yes	—
122	C₂×C₃	—	32	1	no	—
123	C₂×C₃	#1712	1	1	yes	characteristic
124	C₂×C₃	#2250	1	2	yes	—
125	D₃	—	32	2	no	—
126	C₂²	—	16	1	no	—
127	C₂²	—	16	2	no	—
128	C₂²	—	48	2	no	—
129	C₂²	—	48	4	no	—
130	C₂²	#5073	1	1	yes	characteristic
131	C₄	—	16	1	no	—
132	C₄	#5151	1	2	yes	—
133	C₄	#5193	1	4	yes	—
134	C₃	#12519	1	1	yes	characteristic, complemented, Sylow
135	C₂	—	32	1	no	—
136	C₂	—	96	2	no	—
137	C₂	#71276	1	1	yes	characteristic
138	C₂	#71321	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#1709

C₂

yes

maximal, characteristic, complemented

#71313

C₂

yes

maximal, complemented

#71290

C₂

yes

maximal, complemented

#68062

C₂

yes

maximal, characteristic, complemented

#67437

C₂

yes

maximal, characteristic, complemented

#12519

—

maximal, Sylow

C₃×#494

C₂²

yes

characteristic

C₃×#493

C₂²

yes

—

—

—

—

C₂²

yes

—

C₃×C₈²

C₂²

yes

characteristic

C₄×C₃⋊C₁₆

C₂²

yes

—

—

—

—

D₃

yes

characteristic, complemented

#1311

—

C₂×C₃×Q₁₆

—

C₂×C₃×D₈

—

C₃×C₄⋊Q₈

—

C₃×#127

—

C₂×D₂₄

—

C₂×D₂₄

—

C₂×D₂₄

—

#874

—

C₃×#102

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

C₂³

yes

characteristic

#828

—

#826

—

C₂×C₃⋊C₁₆

D₄

yes

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

C₂×D₃

yes

—

—

C₂×D₃

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×D₁₂

—

C₂×D₁₂

—

C₃×Q₁₆

—

C₃×D₈

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₄

yes

characteristic

D₂₄

—

D₂₄

—

D₂₄

—

C₃⋊C₁₆

D₈

yes

complemented

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

#102

—

C₄×C₈

#67

yes

—

C₄×C₈

C₂²×D₃

yes

characteristic

C₂³×C₃

—

C₂²×D₃

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

—

D₁₂

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₃×C₈

C₂×D₈

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

D₈

—

D₈

—

D₈

—

C₂×C₈

#213

yes

—

C₂×C₈

D₃×D₄

yes

—

C₄⋊C₄

—

C₄²

C₂×#67

yes

characteristic

100

C₁₆

—

101

C₂²×C₃

—

102

C₂²×C₃

—

103

C₂²×C₃

#493

yes

characteristic

104

C₂×D₃

—

105

C₂×D₃

—

106

C₃×C₄

—

107

C₃×C₄

#493

yes

—

108

C₃×C₄

#509

yes

—

109

C₂³

—

110

C₂³

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂×#213

yes

—

119

C₂×C₄

#940

yes

characteristic

120

C₈

—

121

C₈

D₃×D₈

yes

—

122

C₂×C₃

—

123

C₂×C₃

#1712

yes

characteristic

124

C₂×C₃

#2250

yes

—

125

D₃

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

#5073

yes

characteristic

131

C₄

—

132

C₄

#5151

yes

—

133

C₄

#5193

yes

—

134

C₃

#12519

yes

characteristic, complemented, Sylow

135

C₂

—

136

C₂

—

137

C₂

#71276

yes

characteristic

138

C₂

#71321

yes

—

	subgroup	G/H
center	C₂²	#5073
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1709	C₂
socle	C₂²×C₃	#493
hypercenter	#1709	D₃
hypocenter	C₃	#12519
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#5073

commutator, cosocle

C₃×C₄×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₃

C₂

C₂²×C₃

D₃

C₃

G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. C₃×#1709	C₂	semidirect
2. #71313	C₂	semidirect
3. #71290	C₂	semidirect
4. #68062	C₂	semidirect
5. #67437	C₂	semidirect
19. #1709	D₃	semidirect
58. C₃⋊C₁₆	D₈	semidirect
134. C₃	#12519	semidirect

H ≅ G/N

type

1. C₃×#1709

C₂

semidirect

2. #71313

C₂

semidirect

3. #71290

C₂

semidirect

4. #68062

C₂

semidirect

5. #67437

C₂

semidirect

19. #1709

D₃

semidirect

58. C₃⋊C₁₆

D₈

semidirect

134. C₃

#12519

semidirect

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

subgroup	index
C₃×#1709	2
#71313	2
#71290	2
#68062	2
#67437	2
#12519	3

subgroup

index

Subgroups — #10709916

Conjugacy classes of subgroups

Complements

Maximal subgroups