Group #10694686 — subgroups

Subgroups — #10694686

Order 768.

454 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 119 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×#4699	C₂	1	1	yes	maximal, characteristic
2	C₃×#1837	C₂	1	1	yes	maximal, characteristic
3	#70506	C₂	1	4	yes	maximal
4	#69726	C₂	1	1	yes	maximal, characteristic
5	#9779	—	3	1	no	maximal, Sylow
6	C₂²×Dic₁₂	—	2	2	no	—
7	C₂×#869	C₂²	1	2	yes	—
8	C₃×#423	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	#4925	C₂²	1	2	yes	—
11	#4699	—	2	2	no	—
12	#4699	C₄	1	4	yes	—
13	C₃×#341	—	2	2	no	—
14	#4482	—	2	8	no	—
15	#4454	—	2	2	no	—
16	C₂×#494	—	3	1	no	—
17	#1837	D₃	1	1	yes	characteristic
18	#1546	—	3	4	no	—
19	#1452	—	3	1	no	—
20	C₂²×Dic₆	—	2	2	no	—
21	C₂×C₃×M₁₆	—	2	2	no	—
22	C₂²×C₃×C₈	D₄	1	2	yes	—
23	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₃	—	2	4	no	—
25	C₂×Dic₁₂	—	2	8	no	—
26	C₂×Dic₁₂	—	4	8	no	—
27	#869	—	2	4	no	—
28	#869	—	2	2	no	—
29	#869	C₂×C₄	1	4	yes	—
30	C₃×C₄⋊C₈	—	2	4	no	—
31	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
32	C₃×C₄×C₈	D₄	1	2	yes	—
33	#830	—	2	2	no	—
34	C₄⋊(C₃⋊C₈)	D₄	1	4	yes	—
35	C₄×C₃⋊C₈	—	2	2	no	—
36	C₂²×Q₁₆	—	6	2	no	—
37	C₂×C₄⋊Q₈	—	3	2	no	—
38	#494	—	3	4	no	—
39	#494	—	6	2	no	—
40	#431	—	3	2	no	—
41	#423	C₂×D₃	1	2	yes	—
42	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
43	#341	—	2	2	no	—
44	#332	—	6	8	no	—
45	#322	—	6	2	no	—
46	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₃×C₄	#41	1	2	yes	—
48	C₂²×Dic₃	—	4	2	no	—
49	C₂×Dic₆	—	2	4	no	—
50	C₂×Dic₆	—	4	8	no	—
51	C₂×Dic₆	—	4	2	no	—
52	C₃×M₁₆	—	4	4	no	—
53	C₂×C₃×C₈	—	2	4	no	—
54	C₂×C₃×C₈	—	4	2	no	—
55	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
56	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
58	C₃×C₄²	C₂×D₄	1	2	yes	—
59	C₄⋊Dic₃	—	2	8	no	—
60	C₄⋊Dic₃	—	4	4	no	—
61	C₂×C₃⋊C₈	—	2	4	no	—
62	Dic₁₂	—	4	16	no	—
63	C₂²×Q₈	—	6	2	no	—
64	C₂×Q₁₆	—	6	8	no	—
65	C₂×Q₁₆	—	12	8	no	—
66	C₂×M₁₆	—	2	2	no	—
67	C₂²×C₈	#67	1	2	yes	—
68	C₄⋊Q₈	—	3	4	no	—
69	C₄⋊Q₈	—	6	2	no	—
70	C₄⋊Q₈	—	6	4	no	—
71	C₂×C₄⋊C₄	—	6	4	no	—
72	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
73	C₄⋊C₈	—	2	4	no	—
74	C₄⋊C₈	—	3	4	no	—
75	C₂²⋊C₈	—	6	2	no	—
76	C₄×C₈	—	6	2	no	—
77	C₄×C₈	C₄×D₃	1	2	yes	—
78	C₄×C₈	D₁₂	1	2	yes	—
79	C₂³×C₃	#124	1	1	yes	characteristic
80	C₂×C₃×C₄	—	2	4	no	—
81	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
83	C₂×C₃×C₄	#121	1	2	yes	—
84	C₂×C₃×C₄	#127	1	1	yes	characteristic
85	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
86	C₂×Dic₃	—	4	4	no	—
87	C₂×Dic₃	—	8	4	no	—
88	Dic₆	—	4	8	no	—
89	Dic₆	—	8	4	no	—
90	C₃×C₈	—	2	4	no	—
91	C₃×C₈	—	8	2	no	—
92	C₃⋊C₈	—	4	4	no	—
93	C₂×Q₈	—	6	4	no	—
94	C₂×Q₈	—	12	2	no	—
95	C₂×Q₈	—	12	8	no	—
96	C₂²×C₄	—	12	2	no	—
97	C₂²×C₄	#239	1	2	yes	—
98	C₂²×C₄	C₂×#67	1	1	yes	characteristic
99	Q₁₆	—	12	16	no	—
100	M₁₆	—	4	4	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	6	4	no	—
104	C₂×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
105	C₄⋊C₄	—	6	8	no	—
106	C₄⋊C₄	—	12	4	no	—
107	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
108	C₄²	C₂×D₁₂	1	1	yes	characteristic
109	C₄²	D₃×D₄	1	2	yes	—
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	#418	1	2	yes	—
115	C₃×C₄	#473	1	4	yes	—
116	C₃×C₄	#496	1	2	yes	—
117	Dic₃	—	8	4	no	—
118	C₂³	#947	1	1	yes	characteristic
119	Q₈	—	12	8	no	—
120	Q₈	—	24	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	12	4	no	—
123	C₂×C₄	—	24	4	no	—
124	C₂×C₄	#893	1	2	yes	—
125	C₂×C₄	#895	1	2	yes	—
126	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
127	C₂×C₄	#940	1	1	yes	characteristic
128	C₂×C₄	#949	1	4	yes	—
129	C₈	—	2	4	no	—
130	C₈	—	8	2	no	—
131	C₈	—	12	4	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	#4988	1	1	yes	characteristic
137	C₂²	#5067	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	24	4	no	—
140	C₄	#4896	1	4	yes	—
141	C₄	#5001	1	2	yes	—
142	C₄	#5058	1	2	yes	—
143	C₃	#9779	1	1	yes	characteristic, complemented, Sylow
144	C₂	—	2	2	no	—
145	C₂	#69553	1	1	yes	characteristic
146	C₂	#70710	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4699

C₂

yes

maximal, characteristic

C₃×#1837

C₂

yes

maximal, characteristic

#70506

C₂

yes

maximal

#69726

C₂

yes

maximal, characteristic

#9779

—

maximal, Sylow

C₂²×Dic₁₂

—

C₂×#869

C₂²

yes

—

C₃×#423

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

#4925

C₂²

yes

—

#4699

—

#4699

C₄

yes

—

—

—

—

—

D₃

yes

characteristic

#1546

—

#1452

—

C₂²×Dic₆

—

C₂×C₃×M₁₆

—

C₂²×C₃×C₈

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

#869

—

#869

—

#869

C₂×C₄

yes

—

C₃×C₄⋊C₈

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

#830

—

C₄⋊(C₃⋊C₈)

D₄

yes

—

C₄×C₃⋊C₈

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

—

—

—

C₂×D₃

yes

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

#41

yes

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

#67

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₃

yes

—

C₄×C₈

D₁₂

yes

—

C₂³×C₃

#124

yes

characteristic

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

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#239

yes

—

C₂²×C₄

C₂×#67

yes

characteristic

Q₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

C₂×C₄×D₃

yes

characteristic

108

C₄²

C₂×D₁₂

yes

characteristic

109

C₄²

D₃×D₄

yes

—

110

C₂²×C₃

—

111

C₂²×C₃

#390

yes

characteristic

112

C₂²×C₃

#490

yes

—

113

C₃×C₄

—

114

C₃×C₄

#418

yes

—

115

C₃×C₄

#473

yes

—

116

C₃×C₄

#496

yes

—

117

Dic₃

—

118

C₂³

#947

yes

characteristic

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#893

yes

—

125

C₂×C₄

#895

yes

—

126

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

127

C₂×C₄

#940

yes

characteristic

128

C₂×C₄

#949

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂×C₃

—

133

C₂×C₃

#1961

yes

characteristic

134

C₂×C₃

#1979

yes

—

135

C₂²

—

136

C₂²

#4988

yes

characteristic

137

C₂²

#5067

yes

—

138

C₄

—

139

C₄

—

140

C₄

#4896

yes

—

141

C₄

#5001

yes

—

142

C₄

#5058

yes

—

143

C₃

#9779

yes

characteristic, complemented, Sylow

144

C₂

—

145

C₂

#69553

yes

characteristic

146

C₂

#70710

yes

—

	subgroup	G/H
center	C₂²	#4988
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1837	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#124
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1837	D₃
hypocenter	C₃	#9779
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#4988

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

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×#4699	2
C₃×#1837	2
#70506	2
#69726	2
#9779	3

subgroup

index

Subgroups — #10694686

Conjugacy classes of subgroups

Complements

Maximal subgroups