Group #10695155 — subgroups

Subgroups — #10695155

Order 768.

406 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 155 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	#78442	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic
3	#69792	C₂	1	1	yes	maximal, characteristic
4	#69738	C₂	1	4	yes	maximal
5	#9861	—	3	1	no	maximal, Sylow
6	#5470	C₂²	1	2	yes	—
7	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#819	—	2	2	no	—
10	#5106	—	2	2	no	—
11	#4924	C₂²	1	2	yes	—
12	#4683	—	2	2	no	—
13	#4682	C₄	1	2	yes	—
14	#4680	C₄	1	2	yes	—
15	C₃×#322	—	2	2	no	—
16	#4456	—	2	8	no	—
17	#3160	—	3	1	no	—
18	#1848	—	3	1	no	—
19	#1493	—	3	4	no	—
20	C₂×#322	D₃	1	1	yes	characteristic
21	C₂²×C₃×C₈	—	2	2	no	—
22	C₂²×C₃×C₈	D₄	1	2	yes	—
23	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
24	#926	—	2	2	no	—
25	#924	—	2	4	no	—
26	C₂×#208	—	2	2	no	—
27	#870	—	2	2	no	—
28	#869	C₂×C₄	1	4	yes	—
29	C₄×Dic₆	—	2	4	no	—
30	C₃×C₄×C₈	—	2	2	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	4	yes	—
34	C₃×C₄×C₈	D₄	1	2	yes	—
35	#819	—	4	4	no	—
36	#818	—	2	4	no	—
37	#817	—	2	4	no	—
38	C₄⋊(C₃⋊C₈)	—	2	4	no	—
39	#533	—	3	2	no	—
40	#500	—	3	2	no	—
41	#499	—	6	2	no	—
42	#498	—	3	2	no	—
43	C₂×#108	—	6	2	no	—
44	#427	—	6	2	no	—
45	#423	—	3	2	no	—
46	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
47	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
48	Q₈⋊C₈	—	6	8	no	—
49	#322	—	2	2	no	—
50	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₃×C₄	#41	1	2	yes	—
52	C₂×Dic₆	—	2	4	no	—
53	C₂×C₃×C₈	—	2	4	no	—
54	C₂×C₃×C₈	—	4	4	no	—
55	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
57	C₂×C₃×C₈	SD₁₆	1	4	yes	—
58	C₂×C₃×C₈	Q₁₆	1	4	yes	—
59	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
60	C₃×C₄²	C₂×D₄	1	2	yes	—
61	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
62	#217	—	4	2	no	—
63	C₄⋊Dic₃	—	2	4	no	—
64	C₄⋊Dic₃	—	2	4	no	—
65	Dic₃⋊C₄	—	4	4	no	—
66	C₄×Dic₃	—	4	2	no	—
67	#208	—	4	4	no	—
68	C₂×C₃⋊C₈	—	4	2	no	—
69	C₂×M₁₆	—	6	2	no	—
70	C₂²×C₈	—	2	2	no	—
71	C₂²×C₈	D₁₂	1	2	yes	—
72	C₄⋊Q₈	—	3	4	no	—
73	#125	—	6	2	no	—
74	C₂²⋊Q₈	—	6	4	no	—
75	C₄×Q₈	—	6	4	no	—
76	#117	—	6	2	no	—
77	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
78	#108	—	12	4	no	—
79	#107	—	6	4	no	—
80	#106	—	6	4	no	—
81	C₄⋊C₈	—	6	4	no	—
82	C₄×C₈	—	2	2	no	—
83	C₄×C₈	C₄×D₃	1	1	yes	characteristic
84	C₄×C₈	C₄×D₃	1	1	yes	characteristic
85	C₄×C₈	D₁₂	1	4	yes	—
86	C₄×C₈	#67	1	2	yes	—
87	C₂³×C₃	#124	1	1	yes	characteristic
88	C₂×C₃×C₄	—	2	4	no	—
89	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
91	C₂×C₃×C₄	#121	1	2	yes	—
92	C₂×C₃×C₄	#127	1	1	yes	characteristic
93	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
94	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
95	C₂×Dic₃	—	4	4	no	—
96	Dic₆	—	4	8	no	—
97	C₃×C₈	—	4	4	no	—
98	C₃×C₈	#103	1	4	yes	—
99	C₃×C₈	#103	1	4	yes	—
100	C₃⋊C₈	—	8	2	no	—
101	C₂×Q₈	—	6	4	no	—
102	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
103	C₂²×C₄	#235	1	2	yes	—
104	M₁₆	—	12	4	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	4	4	no	—
107	C₂×C₈	—	12	2	no	—
108	C₂×C₈	#204	1	4	yes	—
109	C₂×C₈	Dic₁₂	1	4	yes	—
110	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
111	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
112	C₄⋊C₄	—	6	4	no	—
113	C₄⋊C₄	—	6	4	no	—
114	C₄⋊C₄	—	12	4	no	—
115	C₂²⋊C₄	—	12	2	no	—
116	C₄²	—	12	2	no	—
117	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
118	C₄²	C₂×D₁₂	1	2	yes	—
119	C₄²	C₂×#67	1	1	yes	characteristic
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	#390	1	1	yes	characteristic
122	C₂²×C₃	#487	1	2	yes	—
123	C₃×C₄	—	2	2	no	—
124	C₃×C₄	C₂×#103	1	2	yes	—
125	C₃×C₄	#444	1	4	yes	—
126	C₃×C₄	#492	1	1	yes	characteristic
127	C₃×C₄	#494	1	1	yes	characteristic
128	Dic₃	—	8	4	no	—
129	C₂³	#875	1	1	yes	characteristic
130	Q₈	—	12	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	12	4	no	—
133	C₂×C₄	C₄×D₁₂	1	2	yes	—
134	C₂×C₄	#874	1	1	yes	characteristic
135	C₂×C₄	C₂×#204	1	2	yes	—
136	C₂×C₄	C₂×Dic₁₂	1	2	yes	—
137	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
138	C₂×C₄	#930	1	2	yes	—
139	C₈	—	4	4	no	—
140	C₈	—	24	2	no	—
141	C₈	#816	1	4	yes	—
142	C₈	#816	1	4	yes	—
143	C₂×C₃	—	2	2	no	—
144	C₂×C₃	#1956	1	1	yes	characteristic
145	C₂×C₃	#1975	1	2	yes	—
146	C₂²	—	2	2	no	—
147	C₂²	#4681	1	2	yes	—
148	C₂²	#4939	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	24	4	no	—
151	C₄	#4693	1	1	yes	characteristic
152	C₄	#4699	1	1	yes	characteristic
153	C₄	#4717	1	4	yes	—
154	C₄	C₂×#816	1	2	yes	—
155	C₃	#9861	1	1	yes	characteristic, complemented, Sylow
156	C₂	—	2	2	no	—
157	C₂	#69771	1	1	yes	characteristic
158	C₂	#69845	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#78442

C₂

yes

maximal, characteristic

C₂×C₃×#322

C₂

yes

maximal, characteristic

#69792

C₂

yes

maximal, characteristic

#69738

C₂

yes

—

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

—

—

C₂²

yes

—

#4683

—

#4682

C₄

yes

—

#4680

C₄

yes

—

—

—

—

—

—

D₃

yes

characteristic

C₂²×C₃×C₈

—

C₂²×C₃×C₈

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

—

—

—

—

C₂×C₄

yes

—

C₄×Dic₆

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

D₄

yes

—

#819

—

#818

—

#817

—

C₄⋊(C₃⋊C₈)

—

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₃

yes

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

Q₈⋊C₈

—

#322

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

#41

yes

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

SD₁₆

yes

—

C₂×C₃×C₈

Q₁₆

yes

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

#208

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₁₂

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²×D₃

yes

—

—

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₃

yes

characteristic

C₄×C₈

C₄×D₃

yes

characteristic

C₄×C₈

D₁₂

yes

—

C₄×C₈

#67

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

C₂×SD₁₆

yes

—

C₂×C₃×C₄

C₂×Q₁₆

yes

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

#103

yes

—

C₃×C₈

#103

yes

—

100

C₃⋊C₈

—

101

C₂×Q₈

—

102

C₂²×C₄

C₂×D₁₂

yes

characteristic

103

C₂²×C₄

#235

yes

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

#204

yes

—

109

C₂×C₈

Dic₁₂

yes

—

110

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

111

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

C₂×C₄×D₃

yes

characteristic

118

C₄²

C₂×D₁₂

yes

—

119

C₄²

C₂×#67

yes

characteristic

120

C₂²×C₃

—

121

C₂²×C₃

#390

yes

characteristic

122

C₂²×C₃

#487

yes

—

123

C₃×C₄

—

124

C₃×C₄

C₂×#103

yes

—

125

C₃×C₄

#444

yes

—

126

C₃×C₄

#492

yes

characteristic

127

C₃×C₄

#494

yes

characteristic

128

Dic₃

—

129

C₂³

#875

yes

characteristic

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄×D₁₂

yes

—

134

C₂×C₄

#874

yes

characteristic

135

C₂×C₄

C₂×#204

yes

—

136

C₂×C₄

C₂×Dic₁₂

yes

—

137

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

138

C₂×C₄

#930

yes

—

139

C₈

—

140

C₈

—

141

C₈

#816

yes

—

142

C₈

#816

yes

—

143

C₂×C₃

—

144

C₂×C₃

#1956

yes

characteristic

145

C₂×C₃

#1975

yes

—

146

C₂²

—

147

C₂²

#4681

yes

—

148

C₂²

#4939

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

#4693

yes

characteristic

152

C₄

#4699

yes

characteristic

153

C₄

#4717

yes

—

154

C₄

C₂×#816

yes

—

155

C₃

#9861

yes

characteristic, complemented, Sylow

156

C₂

—

157

C₂

#69771

yes

characteristic

158

C₂

#69845

yes

—

	subgroup	G/H
center	C₂×C₄	#874
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#322	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#124
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#322	D₃
hypocenter	C₃	#9861
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#874

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₂×C₃×#322

C₂

C₂²×C₃

C₂³×C₃

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
155. C₃	#9861	semidirect

H ≅ G/N

type

155. C₃

#9861

semidirect

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

subgroup	index
#78442	2
C₂×C₃×#322	2
#69792	2
#69738	2
#9861	3

subgroup

index

C₂×C₃×#322

Subgroups — #10695155

Conjugacy classes of subgroups

Complements

Maximal subgroups