Group #10709957 — subgroups

Subgroups — #10709957

Order 768.

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

normal subgroups only

This group has 274 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×#1713	C₂	1	1	yes	maximal, characteristic
2	#71312	C₂	1	2	yes	maximal
3	#71310	C₂	1	2	yes	maximal
4	#68063	C₂	1	1	yes	maximal, characteristic
5	#67438	C₂	1	1	yes	maximal, characteristic
6	#12528	—	3	1	no	maximal, Sylow
7	C₃×#500	—	2	1	no	—
8	C₃×#500	—	2	1	no	—
9	C₃×#494	C₂²	1	2	yes	—
10	C₂×#829	—	2	4	no	—
11	#4699	C₂²	1	1	yes	characteristic
12	#4682	C₂²	1	1	yes	characteristic
13	#4681	—	2	1	no	—
14	#4680	—	2	1	no	—
15	C₃×C₈²	C₂²	1	1	yes	characteristic
16	#4564	—	2	4	no	—
17	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
18	#2247	—	3	2	no	—
19	#2241	—	3	2	no	—
20	#1713	D₃	1	1	yes	characteristic
21	#1707	—	3	1	no	—
22	#1312	—	3	1	no	—
23	C₂×C₃×Q₁₆	—	2	2	no	—
24	C₂×C₃×Q₁₆	—	2	2	no	—
25	C₃×C₄⋊Q₈	—	2	2	no	—
26	C₂×Dic₁₂	—	2	2	no	—
27	#869	—	2	1	no	—
28	#869	—	2	1	no	—
29	C₃×#107	—	4	2	no	—
30	C₃×#107	—	4	2	no	—
31	C₃×C₄×C₈	D₄	1	1	yes	characteristic
32	C₃×C₄×C₈	D₄	1	1	yes	characteristic
33	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
34	#829	—	4	8	no	—
35	#818	—	2	2	no	—
36	#817	—	4	2	no	—
37	#816	—	4	2	no	—
38	C₂×C₃⋊C₁₆	D₄	1	4	yes	—
39	C₂×Q₃₂	—	6	4	no	—
40	#500	—	2	1	no	—
41	#500	—	2	1	no	—
42	#500	—	3	1	no	—
43	#498	—	6	1	no	—
44	#494	—	3	1	no	—
45	#494	C₂×D₃	1	2	yes	—
46	#487	—	6	1	no	—
47	#358	—	6	4	no	—
48	C₄×C₁₆	—	3	2	no	—
49	C₈²	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×Q₈	—	4	2	no	—
51	C₂×Dic₆	—	4	1	no	—
52	C₂×Dic₆	—	4	1	no	—
53	C₃×Q₁₆	—	4	4	no	—
54	C₃×Q₁₆	—	4	4	no	—
55	C₂×C₃×C₈	D₈	1	2	yes	—
56	C₂×C₃×C₈	D₈	1	2	yes	—
57	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
58	C₃×C₄⋊C₄	—	4	2	no	—
59	C₃×C₄⋊C₄	—	4	2	no	—
60	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Dic₃	—	4	1	no	—
62	C₄⋊Dic₃	—	4	1	no	—
63	C₄⋊Dic₃	—	4	1	no	—
64	C₄⋊Dic₃	—	4	1	no	—
65	Dic₁₂	—	4	4	no	—
66	C₃⋊C₁₆	—	2	4	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₂×Q₁₆	—	2	2	no	—
69	C₂×Q₁₆	—	6	2	no	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	—	6	1	no	—
72	C₄⋊Q₈	—	6	1	no	—
73	Q₃₂	—	12	8	no	—
74	C₂×C₁₆	—	3	4	no	—
75	#107	—	4	2	no	—
76	#107	—	4	2	no	—
77	#107	—	6	2	no	—
78	#106	—	12	2	no	—
79	#103	—	12	2	no	—
80	C₄×C₈	#67	1	1	yes	characteristic
81	C₄×C₈	#67	1	1	yes	characteristic
82	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
83	C₃×Q₈	—	8	2	no	—
84	C₃×Q₈	—	8	2	no	—
85	C₂×C₃×C₄	—	8	2	no	—
86	C₂×C₃×C₄	#127	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
89	C₂×Dic₃	—	8	1	no	—
90	C₂×Dic₃	—	8	1	no	—
91	Dic₆	—	8	2	no	—
92	Dic₆	—	8	2	no	—
93	C₃×C₈	—	2	2	no	—
94	C₃×C₈	Q₃₂	1	4	yes	complemented
95	C₃×C₈	AGL(1,ℤ/8)	1	4	yes	—
96	C₂×Q₈	—	4	2	no	—
97	C₂×Q₈	—	12	1	no	—
98	C₂×Q₈	—	12	1	no	—
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	4	4	no	—
101	Q₁₆	—	12	4	no	—
102	C₂×C₈	#213	1	2	yes	—
103	C₂×C₈	#213	1	2	yes	—
104	C₂×C₈	D₃×D₄	1	2	yes	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	12	1	no	—
108	C₄⋊C₄	—	12	1	no	—
109	C₄⋊C₄	—	12	1	no	—
110	C₄⋊C₄	—	12	1	no	—
111	C₄²	C₂×#67	1	1	yes	characteristic
112	C₁₆	—	6	4	no	—
113	C₂²×C₃	#493	1	1	yes	characteristic
114	C₃×C₄	—	16	2	no	—
115	C₃×C₄	#496	1	2	yes	—
116	C₃×C₄	C₂×Q₃₂	1	2	yes	—
117	C₃×C₄	#510	1	2	yes	—
118	Dic₃	—	16	1	no	—
119	Dic₃	—	16	1	no	—
120	Q₈	—	8	2	no	—
121	Q₈	—	8	2	no	—
122	Q₈	—	24	2	no	—
123	Q₈	—	24	2	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	—	24	1	no	—
126	C₂×C₄	—	24	1	no	—
127	C₂×C₄	C₂×#213	1	1	yes	characteristic
128	C₂×C₄	C₂×#213	1	1	yes	characteristic
129	C₂×C₄	#940	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	#829	1	4	yes	complemented
132	C₈	#911	1	4	yes	—
133	C₂×C₃	#1721	1	1	yes	characteristic
134	C₂×C₃	#2247	1	1	yes	characteristic
135	C₂×C₃	#2251	1	1	yes	characteristic
136	C₂²	#5073	1	1	yes	characteristic
137	C₄	—	16	2	no	—
138	C₄	—	48	1	no	—
139	C₄	—	48	1	no	—
140	C₄	#5154	1	2	yes	—
141	C₄	#5179	1	2	yes	—
142	C₄	C₂×#829	1	2	yes	—
143	C₃	#12528	1	1	yes	characteristic, complemented, Sylow
144	C₂	#71280	1	1	yes	characteristic
145	C₂	#71312	1	1	yes	characteristic
146	C₂	#71314	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#1713

C₂

yes

maximal, characteristic

#71312

C₂

yes

maximal

#71310

C₂

yes

maximal

#68063

C₂

yes

maximal, characteristic

#67438

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂×#829

—

#4699

C₂²

yes

characteristic

#4682

C₂²

yes

characteristic

#4681

—

#4680

—

C₃×C₈²

C₂²

yes

characteristic

#4564

—

C₄×C₃⋊C₁₆

C₂²

yes

—

#2247

—

#2241

—

#1713

D₃

yes

characteristic

#1707

—

#1312

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₃×C₄⋊Q₈

—

C₂×Dic₁₂

—

—

—

—

—

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

C₂³

yes

—

—

—

—

C₂×C₃⋊C₁₆

D₄

yes

—

C₂×Q₃₂

—

—

—

—

—

—

C₂×D₃

yes

—

#487

—

#358

—

C₄×C₁₆

—

C₈²

C₂×D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

C₃⋊C₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

—

—

—

—

—

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

C₂²×D₃

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

Q₃₂

yes

complemented

C₃×C₈

AGL(1,ℤ/8)

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

#213

yes

—

103

C₂×C₈

#213

yes

—

104

C₂×C₈

D₃×D₄

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

C₂×#67

yes

characteristic

112

C₁₆

—

113

C₂²×C₃

#493

yes

characteristic

114

C₃×C₄

—

115

C₃×C₄

#496

yes

—

116

C₃×C₄

C₂×Q₃₂

yes

—

117

C₃×C₄

#510

yes

—

118

Dic₃

—

119

Dic₃

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂×#213

yes

characteristic

128

C₂×C₄

C₂×#213

yes

characteristic

129

C₂×C₄

#940

yes

characteristic

130

C₈

—

131

C₈

#829

yes

complemented

132

C₈

#911

yes

—

133

C₂×C₃

#1721

yes

characteristic

134

C₂×C₃

#2247

yes

characteristic

135

C₂×C₃

#2251

yes

characteristic

136

C₂²

#5073

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#5154

yes

—

141

C₄

#5179

yes

—

142

C₄

C₂×#829

yes

—

143

C₃

#12528

yes

characteristic, complemented, Sylow

144

C₂

#71280

yes

characteristic

145

C₂

#71312

yes

characteristic

146

C₂

#71314

yes

characteristic

	subgroup	G/H
center	C₂²	#5073
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1713	C₂
socle, prime radical	C₂²×C₃	#493
hypercenter	#1713	D₃
hypocenter	C₃	#12528
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

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 ◁ #1713

N	H ≅ G/N	type
94. C₃×C₈	Q₃₂	semidirect
131. C₈	#829	semidirect
143. C₃	#12528	semidirect

H ≅ G/N

type

94. C₃×C₈

Q₃₂

semidirect

131. C₈

#829

semidirect

143. C₃

#12528

semidirect

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

subgroup	index
C₃×#1713	2
#71312	2
#71310	2
#68063	2
#67438	2
#12528	3

subgroup

index

Subgroups — #10709957

Conjugacy classes of subgroups

Complements

Maximal subgroups