Group #10694580 — subgroups

Subgroups — #10694580

Order 768.

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

normal subgroups only

This group has 370 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	C₃×#3160	C₂	1	1	yes	maximal, characteristic
2	#71838	C₂	1	4	yes	maximal
3	#69763	C₂	1	1	yes	maximal, characteristic
4	#69726	C₂	1	1	yes	maximal, characteristic
5	#9759	—	3	1	no	maximal, Sylow
6	C₃×#533	C₂²	1	2	yes	—
7	C₃×#500	C₄	1	2	yes	complemented
8	C₃×#499	—	2	2	no	—
9	C₃×#498	C₄	1	2	yes	complemented
10	C₃×#427	—	2	2	no	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#4936	C₂²	1	2	yes	—
13	#4925	C₂²	1	2	yes	—
14	#4551	—	2	2	no	—
15	#4538	—	2	8	no	—
16	#4454	—	2	2	no	—
17	#3160	D₃	1	1	yes	characteristic
18	#1835	—	3	1	no	—
19	#1518	—	3	4	no	—
20	#1452	—	3	1	no	—
21	C₂²×C₃×C₈	D₄	1	2	yes	—
22	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
23	C₃×#125	—	2	2	no	—
24	C₃×C₂²⋊Q₈	—	2	4	no	—
25	C₃×C₄×Q₈	—	2	4	no	—
26	C₃×#117	—	2	2	no	—
27	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₃	—	2	2	no	—
29	C₃×#107	—	2	4	no	—
30	C₃×#106	—	2	4	no	—
31	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄×C₈	D₄	1	2	yes	—
34	C₃⋊#95	—	2	4	no	—
35	#830	D₄	1	4	yes	—
36	C₄⋊(C₃⋊C₈)	—	2	2	no	—
37	C₄×C₃⋊C₈	—	2	2	no	—
38	#533	C₂×D₃	1	2	yes	—
39	#500	Dic₃	1	2	yes	complemented
40	#499	—	2	2	no	—
41	#498	Dic₃	1	2	yes	complemented
42	#431	—	3	2	no	—
43	#427	—	2	2	no	—
44	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
45	#382	—	3	2	no	—
46	#340	—	6	2	no	—
47	#328	—	6	8	no	—
48	#322	—	6	2	no	—
49	C₂×C₃×Q₈	—	2	4	no	—
50	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
51	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₃	—	4	2	no	—
53	C₂×C₃×C₈	—	2	4	no	—
54	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
55	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
56	C₃×C₄⋊C₄	—	2	4	no	—
57	C₃×C₄⋊C₄	—	2	4	no	—
58	C₃×C₄⋊C₄	—	4	4	no	—
59	C₃×C₂²⋊C₄	—	4	2	no	—
60	C₃×C₄²	—	4	2	no	—
61	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
63	C₃×C₄²	#41	1	2	yes	—
64	C₄⋊Dic₃	—	4	4	no	—
65	C₂×C₃⋊C₈	—	2	4	no	—
66	C₂²×C₈	#67	1	2	yes	—
67	C₄⋊Q₈	C₂×Dic₃	1	4	yes	—
68	#125	—	2	2	no	—
69	C₂²⋊Q₈	—	2	4	no	—
70	C₄×Q₈	—	2	4	no	—
71	#117	—	2	2	no	—
72	C₂×C₄⋊C₄	—	6	2	no	—
73	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
74	#107	—	2	4	no	—
75	#106	—	2	4	no	—
76	C₄⋊C₈	—	6	2	no	—
77	C₂²⋊C₈	—	3	4	no	—
78	C₄×C₈	—	6	2	no	—
79	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
80	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
81	C₄×C₈	#67	1	2	yes	—
82	#95	—	6	4	no	—
83	C₂³×C₃	#127	1	1	yes	characteristic
84	C₃×Q₈	—	8	4	no	—
85	C₂×C₃×C₄	—	4	2	no	—
86	C₂×C₃×C₄	—	4	4	no	—
87	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
89	C₂×C₃×C₄	#121	1	2	yes	—
90	C₂×C₃×C₄	#124	1	1	yes	characteristic
91	C₂×C₃×C₄	#137	1	4	yes	—
92	C₂×Dic₃	—	8	4	no	—
93	C₂×Dic₃	—	8	2	no	—
94	C₃×C₈	—	2	2	no	—
95	C₃×C₈	—	2	2	no	—
96	C₃⋊C₈	—	4	4	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂²×C₄	—	12	2	no	—
99	C₂²×C₄	D₃×D₄	1	2	yes	—
100	C₂²×C₄	C₂×#67	1	1	yes	characteristic
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	6	4	no	—
103	C₂×C₈	#217	1	2	yes	—
104	C₂×C₈	#217	1	2	yes	—
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	4	4	no	—
108	C₄⋊C₄	—	12	4	no	—
109	C₂²⋊C₄	—	4	2	no	—
110	C₄²	—	4	2	no	—
111	C₄²	#237	1	2	yes	—
112	C₄²	C₂²×Dic₃	1	1	yes	characteristic
113	C₄²	C₂×#67	1	1	yes	characteristic
114	C₂²×C₃	—	2	2	no	—
115	C₂²×C₃	#390	1	1	yes	characteristic
116	C₂²×C₃	#497	1	2	yes	—
117	C₃×C₄	—	2	2	no	—
118	C₃×C₄	—	2	2	no	—
119	C₃×C₄	—	8	4	no	—
120	C₃×C₄	#419	1	2	yes	—
121	C₃×C₄	#491	1	2	yes	—
122	Dic₃	—	16	2	no	—
123	C₂³	#940	1	1	yes	characteristic
124	Q₈	—	8	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	24	4	no	—
128	C₂×C₄	—	24	2	no	—
129	C₂×C₄	D₄×Dic₃	1	2	yes	—
130	C₂×C₄	#936	1	1	yes	characteristic
131	C₂×C₄	#938	1	2	yes	—
132	C₂×C₄	#942	1	4	yes	—
133	C₂×C₄	C₂×#217	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	2	2	no	—
136	C₈	—	12	4	no	—
137	C₂×C₃	—	2	2	no	—
138	C₂×C₃	#1960	1	1	yes	characteristic
139	C₂×C₃	#1976	1	2	yes	—
140	C₂²	—	2	2	no	—
141	C₂²	#5086	1	2	yes	—
142	C₂²	#5221	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	2	2	no	—
145	C₄	—	8	4	no	—
146	C₄	—	48	2	no	—
147	C₄	#5082	1	2	yes	—
148	C₄	#5225	1	2	yes	—
149	C₃	#9759	1	1	yes	characteristic, complemented, Sylow
150	C₂	—	2	2	no	—
151	C₂	#71786	1	1	yes	characteristic
152	C₂	#72086	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#3160

C₂

yes

maximal, characteristic

#71838

C₂

yes

maximal

#69763

C₂

yes

maximal, characteristic

#69726

C₂

yes

maximal, characteristic

#9759

—

maximal, Sylow

C₃×#533

C₂²

yes

—

C₃×#500

C₄

yes

complemented

C₃×#499

—

C₃×#498

C₄

yes

complemented

C₃×#427

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

#4936

C₂²

yes

—

#4925

C₂²

yes

—

—

—

—

D₃

yes

—

—

—

C₂²×C₃×C₈

D₄

yes

—

C₃×C₄⋊Q₈

C₂×C₄

yes

—

C₃×#125

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×#117

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₃×#107

—

C₃×#106

—

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

—

C₃⋊#95

—

#830

D₄

yes

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

#533

C₂×D₃

yes

—

#500

Dic₃

yes

complemented

#499

—

#498

Dic₃

yes

complemented

#431

—

#427

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂×D₄

yes

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

#41

yes

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₈

#67

yes

—

C₄⋊Q₈

C₂×Dic₃

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×Dic₃

yes

characteristic

C₄×C₈

C₂×Dic₃

yes

characteristic

C₄×C₈

#67

yes

—

#95

—

C₂³×C₃

#127

yes

characteristic

C₃×Q₈

—

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

#124

yes

characteristic

C₂×C₃×C₄

#137

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₃×D₄

yes

—

100

C₂²×C₄

C₂×#67

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

#217

yes

—

104

C₂×C₈

#217

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

#237

yes

—

112

C₄²

C₂²×Dic₃

yes

characteristic

113

C₄²

C₂×#67

yes

characteristic

114

C₂²×C₃

—

115

C₂²×C₃

#390

yes

characteristic

116

C₂²×C₃

#497

yes

—

117

C₃×C₄

—

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃×C₄

#419

yes

—

121

C₃×C₄

#491

yes

—

122

Dic₃

—

123

C₂³

#940

yes

characteristic

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₄×Dic₃

yes

—

130

C₂×C₄

#936

yes

characteristic

131

C₂×C₄

#938

yes

—

132

C₂×C₄

#942

yes

—

133

C₂×C₄

C₂×#217

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₂×C₃

—

138

C₂×C₃

#1960

yes

characteristic

139

C₂×C₃

#1976

yes

—

140

C₂²

—

141

C₂²

#5086

yes

—

142

C₂²

#5221

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#5082

yes

—

148

C₄

#5225

yes

—

149

C₃

#9759

yes

characteristic, complemented, Sylow

150

C₂

—

151

C₂

#71786

yes

characteristic

152

C₂

#72086

yes

—

	subgroup	G/H
center	C₂²	#5221
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#3160	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#127
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#3160	D₃
hypocenter	C₃	#9759
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂²

#5221

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 ◁ #3160

N	H ≅ G/N	type
7. C₃×#500	C₄	semidirect
9. C₃×#498	C₄	semidirect
39. #500	Dic₃	semidirect
41. #498	Dic₃	semidirect
149. C₃	#9759	semidirect

H ≅ G/N

type

7. C₃×#500

C₄

semidirect

9. C₃×#498

C₄

semidirect

39. #500

Dic₃

semidirect

41. #498

Dic₃

semidirect

149. C₃

#9759

semidirect

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

subgroup	index
C₃×#3160	2
#71838	2
#69763	2
#69726	2
#9759	3

subgroup

index

Subgroups — #10694580

Conjugacy classes of subgroups

Complements

Maximal subgroups