Group #10694688 — subgroups

Subgroups — #10694688

Order 768.

454 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×C₃×#494	C₂	1	1	yes	maximal, characteristic
2	#71773	C₂	1	4	yes	maximal
3	#69791	C₂	1	1	yes	maximal, characteristic
4	#69726	C₂	1	1	yes	maximal, characteristic
5	#9779	—	3	1	no	maximal, Sylow
6	C₂²×C₃×Q₁₆	—	2	2	no	—
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₃×#494	—	2	2	no	—
9	C₃×#494	C₄	1	4	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	#4925	C₂²	1	2	yes	—
12	#4924	C₂²	1	2	yes	—
13	#4552	—	2	2	no	—
14	#4547	—	2	8	no	—
15	#4454	—	2	2	no	—
16	C₂×#494	D₃	1	1	yes	characteristic
17	#1837	—	3	1	no	—
18	#1546	—	3	4	no	—
19	#1452	—	3	1	no	—
20	C₂²×C₃×Q₈	—	2	2	no	—
21	C₂×C₃×Q₁₆	—	2	8	no	—
22	C₂×C₃×Q₁₆	—	4	8	no	—
23	C₂²×C₃×C₈	D₄	1	2	yes	—
24	C₃×C₄⋊Q₈	—	2	4	no	—
25	C₃×C₄⋊Q₈	—	2	2	no	—
26	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₃×C₄⋊C₄	—	2	4	no	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×#208	—	2	2	no	—
30	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
31	C₃×C₄×C₈	D₄	1	2	yes	—
32	#830	—	2	2	no	—
33	C₄⋊(C₃⋊C₈)	—	2	4	no	—
34	C₄⋊(C₃⋊C₈)	D₄	1	4	yes	—
35	C₄×C₃⋊C₈	—	2	2	no	—
36	C₂²×Q₁₆	—	2	2	no	—
37	C₂×C₄⋊Q₈	C₂×D₃	1	2	yes	—
38	#494	—	2	2	no	—
39	#494	Dic₃	1	4	yes	—
40	#431	—	3	2	no	—
41	#423	—	3	2	no	—
42	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
43	#341	—	6	2	no	—
44	#332	—	6	8	no	—
45	#322	—	6	2	no	—
46	C₂×C₃×Q₈	—	2	4	no	—
47	C₂×C₃×Q₈	—	4	2	no	—
48	C₂×C₃×Q₈	—	4	8	no	—
49	C₂²×C₃×C₄	—	4	2	no	—
50	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₃×C₄	#41	1	2	yes	—
52	C₃×Q₁₆	—	4	16	no	—
53	C₂×C₃×C₈	—	2	4	no	—
54	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
55	C₃×C₄⋊C₄	—	2	8	no	—
56	C₃×C₄⋊C₄	—	4	4	no	—
57	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₄²	C₂×D₄	1	2	yes	—
59	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
60	#208	—	4	4	no	—
61	C₂×C₃⋊C₈	—	2	4	no	—
62	C₂×C₃⋊C₈	—	4	2	no	—
63	C₂²×Q₈	—	2	2	no	—
64	C₂×Q₁₆	—	2	8	no	—
65	C₂×Q₁₆	—	4	8	no	—
66	C₂×M₁₆	—	6	2	no	—
67	C₂²×C₈	#67	1	2	yes	—
68	C₄⋊Q₈	—	2	4	no	—
69	C₄⋊Q₈	—	2	2	no	—
70	C₄⋊Q₈	C₂×Dic₃	1	4	yes	—
71	C₂×C₄⋊C₄	—	2	4	no	—
72	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
73	C₄⋊C₈	—	3	4	no	—
74	C₄⋊C₈	—	6	4	no	—
75	C₂²⋊C₈	—	6	2	no	—
76	C₄×C₈	—	6	2	no	—
77	C₄×C₈	C₂×Dic₃	1	2	yes	—
78	C₄×C₈	#67	1	2	yes	—
79	C₂³×C₃	#124	1	1	yes	characteristic
80	C₃×Q₈	—	4	8	no	—
81	C₃×Q₈	—	8	4	no	—
82	C₂×C₃×C₄	—	2	4	no	—
83	C₂×C₃×C₄	—	4	4	no	—
84	C₂×C₃×C₄	—	8	4	no	—
85	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
87	C₂×C₃×C₄	#121	1	2	yes	—
88	C₂×C₃×C₄	#127	1	1	yes	characteristic
89	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
90	C₃×C₈	—	2	4	no	—
91	C₃⋊C₈	—	4	4	no	—
92	C₃⋊C₈	—	8	2	no	—
93	C₂×Q₈	—	2	4	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	4	8	no	—
96	C₂²×C₄	—	4	2	no	—
97	C₂²×C₄	#237	1	2	yes	—
98	C₂²×C₄	C₂×#67	1	1	yes	characteristic
99	Q₁₆	—	4	16	no	—
100	M₁₆	—	12	4	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	6	4	no	—
103	C₂×C₈	—	12	2	no	—
104	C₂×C₈	#217	1	4	yes	—
105	C₄⋊C₄	—	2	8	no	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄²	D₃×D₄	1	2	yes	—
108	C₄²	C₂²×Dic₃	1	1	yes	characteristic
109	C₄²	C₂×#67	1	1	yes	characteristic
110	C₂²×C₃	—	2	2	no	—
111	C₂²×C₃	#390	1	1	yes	characteristic
112	C₂²×C₃	#490	1	2	yes	—
113	C₃×C₄	—	2	2	no	—
114	C₃×C₄	—	8	4	no	—
115	C₃×C₄	#418	1	2	yes	—
116	C₃×C₄	#473	1	4	yes	—
117	C₃×C₄	#496	1	2	yes	—
118	C₂³	#936	1	1	yes	characteristic
119	Q₈	—	4	8	no	—
120	Q₈	—	8	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	4	4	no	—
123	C₂×C₄	—	8	4	no	—
124	C₂×C₄	#932	1	4	yes	—
125	C₂×C₄	D₄×Dic₃	1	2	yes	—
126	C₂×C₄	#938	1	2	yes	—
127	C₂×C₄	#940	1	1	yes	characteristic
128	C₂×C₄	C₂×#217	1	1	yes	characteristic
129	C₈	—	2	4	no	—
130	C₈	—	12	4	no	—
131	C₈	—	24	2	no	—
132	C₂×C₃	—	2	2	no	—
133	C₂×C₃	#1961	1	1	yes	characteristic
134	C₂×C₃	#1979	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	#5071	1	2	yes	—
137	C₂²	#5221	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	8	4	no	—
140	C₄	#5074	1	2	yes	—
141	C₄	#5192	1	4	yes	—
142	C₄	#5215	1	2	yes	—
143	C₃	#9779	1	1	yes	characteristic, complemented, Sylow
144	C₂	—	2	2	no	—
145	C₂	#71765	1	1	yes	characteristic
146	C₂	#72069	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#494

C₂

yes

maximal, characteristic

#71773

C₂

yes

maximal

#69791

C₂

yes

maximal, characteristic

#69726

C₂

yes

maximal, characteristic

#9779

—

maximal, Sylow

C₂²×C₃×Q₁₆

—

C₂×C₃×C₄⋊Q₈

C₂²

yes

—

C₃×#494

—

C₃×#494

C₄

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

#4925

C₂²

yes

—

#4924

C₂²

yes

—

—

—

—

D₃

yes

—

—

—

C₂²×C₃×Q₈

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₂²×C₃×C₈

D₄

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×#208

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

#830

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

D₄

yes

—

C₄×C₃⋊C₈

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₃

yes

—

#494

—

#494

Dic₃

yes

—

#431

—

#423

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

#41

yes

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

#67

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×Dic₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×Dic₃

yes

—

C₄×C₈

#67

yes

—

C₂³×C₃

#124

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄×D₄

yes

—

C₂×C₃×C₄

#121

yes

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#237

yes

—

C₂²×C₄

C₂×#67

yes

characteristic

Q₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

#217

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

D₃×D₄

yes

—

108

C₄²

C₂²×Dic₃

yes

characteristic

109

C₄²

C₂×#67

yes

characteristic

110

C₂²×C₃

—

111

C₂²×C₃

#390

yes

characteristic

112

C₂²×C₃

#490

yes

—

113

C₃×C₄

—

114

C₃×C₄

—

115

C₃×C₄

#418

yes

—

116

C₃×C₄

#473

yes

—

117

C₃×C₄

#496

yes

—

118

C₂³

#936

yes

characteristic

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#932

yes

—

125

C₂×C₄

D₄×Dic₃

yes

—

126

C₂×C₄

#938

yes

—

127

C₂×C₄

#940

yes

characteristic

128

C₂×C₄

C₂×#217

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂×C₃

—

133

C₂×C₃

#1961

yes

characteristic

134

C₂×C₃

#1979

yes

—

135

C₂²

—

136

C₂²

#5071

yes

—

137

C₂²

#5221

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

#5074

yes

—

141

C₄

#5192

yes

—

142

C₄

#5215

yes

—

143

C₃

#9779

yes

characteristic, complemented, Sylow

144

C₂

—

145

C₂

#71765

yes

characteristic

146

C₂

#72069

yes

—

	subgroup	G/H
center	C₂²	#5221
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#494	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#124
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#494	D₃
hypocenter	C₃	#9779
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#5221

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₂×C₃×#494

C₂

C₂²×C₃

C₂³×C₃

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
143. C₃	#9779	semidirect

H ≅ G/N

type

143. C₃

#9779

semidirect

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

subgroup	index
C₂×C₃×#494	2
#71773	2
#69791	2
#69726	2
#9779	3

subgroup

index

C₂×C₃×#494

Subgroups — #10694688

Conjugacy classes of subgroups

Complements

Maximal subgroups