Group #10694578 — subgroups

Subgroups — #10694578

Order 768.

370 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₃×#1835	C₂	1	1	yes	maximal, characteristic
3	#69726	C₂	1	1	yes	maximal, characteristic
4	#69546	C₂	1	4	yes	maximal
5	#9759	—	3	1	no	maximal, Sylow
6	#5470	C₂²	1	2	yes	—
7	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₃×#382	C₂²	1	2	yes	—
9	#5106	—	2	2	no	—
10	#4925	C₂²	1	2	yes	—
11	#4683	—	2	2	no	—
12	#4682	C₄	1	2	yes	complemented
13	#4680	C₄	1	2	yes	complemented
14	C₃×#340	—	2	2	no	—
15	#4454	—	2	2	no	—
16	#4452	—	2	8	no	—
17	#3160	—	3	1	no	—
18	#1835	D₃	1	1	yes	characteristic
19	#1518	—	3	4	no	—
20	#1452	—	3	1	no	—
21	C₂²×C₃×C₈	D₄	1	2	yes	—
22	C₂×C₃×C₄⋊C₄	—	2	2	no	—
23	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
24	#926	—	2	2	no	—
25	#924	—	2	4	no	—
26	#870	—	2	2	no	—
27	#869	C₂×C₄	1	4	yes	—
28	C₄×Dic₆	—	2	4	no	—
29	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄×C₈	D₄	1	2	yes	—
32	C₃×#95	—	2	4	no	—
33	#830	D₄	1	4	yes	—
34	#818	—	2	4	no	—
35	#817	—	2	4	no	—
36	C₄⋊(C₃⋊C₈)	—	2	2	no	—
37	C₄×C₃⋊C₈	—	2	2	no	—
38	#533	—	3	2	no	—
39	#500	—	3	2	no	—
40	#499	—	6	2	no	—
41	#498	—	3	2	no	—
42	#431	—	3	2	no	—
43	#427	—	6	2	no	—
44	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
45	#382	C₂×D₃	1	2	yes	—
46	#340	—	2	2	no	—
47	#328	—	6	8	no	—
48	#322	—	6	2	no	—
49	C₂²×C₃×C₄	—	4	2	no	—
50	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
51	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₆	—	2	4	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	—	4	4	no	—
57	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
59	C₃×C₄²	#41	1	2	yes	—
60	#217	—	4	2	no	—
61	C₄⋊Dic₃	—	2	4	no	—
62	C₄⋊Dic₃	—	2	4	no	—
63	Dic₃⋊C₄	—	4	4	no	—
64	C₄×Dic₃	—	4	2	no	—
65	C₂×C₃⋊C₈	—	2	4	no	—
66	C₂²×C₈	#67	1	2	yes	—
67	C₄⋊Q₈	—	3	4	no	—
68	#125	—	6	2	no	—
69	C₂²⋊Q₈	—	6	4	no	—
70	C₄×Q₈	—	6	4	no	—
71	#117	—	6	2	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
74	#107	—	6	4	no	—
75	#106	—	6	4	no	—
76	C₄⋊C₈	—	6	2	no	—
77	C₂²⋊C₈	—	3	4	no	—
78	C₄×C₈	—	6	2	no	—
79	C₄×C₈	C₄×D₃	1	1	yes	characteristic
80	C₄×C₈	C₄×D₃	1	1	yes	characteristic
81	C₄×C₈	D₁₂	1	2	yes	—
82	#95	—	2	4	no	—
83	C₂³×C₃	#127	1	1	yes	characteristic
84	C₂×C₃×C₄	—	4	2	no	—
85	C₂×C₃×C₄	—	8	4	no	—
86	C₂×C₃×C₄	—	8	2	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	—	4	4	no	—
93	Dic₆	—	8	4	no	—
94	C₃×C₈	—	2	2	no	—
95	C₃×C₈	—	2	2	no	—
96	C₃⋊C₈	—	4	4	no	—
97	C₂×Q₈	—	6	4	no	—
98	C₂²×C₄	—	4	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	(C₂×D₃)⋊C₄	1	2	yes	—
104	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
105	C₄⋊C₄	—	4	4	no	—
106	C₄⋊C₄	—	6	4	no	—
107	C₄⋊C₄	—	6	4	no	—
108	C₄⋊C₄	—	12	4	no	—
109	C₂²⋊C₄	—	12	2	no	—
110	C₄²	—	12	2	no	—
111	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
112	C₄²	C₂×D₁₂	1	1	yes	characteristic
113	C₄²	#239	1	2	yes	—
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	—	16	2	no	—
120	C₃×C₄	#419	1	2	yes	—
121	C₃×C₄	#491	1	2	yes	—
122	Dic₃	—	8	4	no	—
123	C₂³	#940	1	1	yes	characteristic
124	Q₈	—	24	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	8	4	no	—
127	C₂×C₄	—	8	2	no	—
128	C₂×C₄	—	12	4	no	—
129	C₂×C₄	#893	1	2	yes	—
130	C₂×C₄	#895	1	2	yes	—
131	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
132	C₂×C₄	#947	1	1	yes	characteristic
133	C₂×C₄	#951	1	4	yes	—
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	#4988	1	1	yes	characteristic
142	C₂²	#5060	1	2	yes	—
143	C₄	—	2	2	no	—
144	C₄	—	2	2	no	—
145	C₄	—	16	2	no	—
146	C₄	—	24	4	no	—
147	C₄	#5004	1	2	yes	—
148	C₄	#5069	1	2	yes	—
149	C₃	#9759	1	1	yes	characteristic, complemented, Sylow
150	C₂	—	2	2	no	—
151	C₂	#69552	1	1	yes	characteristic
152	C₂	#70658	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#78442

C₂

yes

maximal, characteristic

C₃×#1835

C₂

yes

maximal, characteristic

#69726

C₂

yes

maximal, characteristic

#69546

C₂

yes

—

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₃×#382

C₂²

yes

—

#5106

—

#4925

C₂²

yes

—

#4683

—

#4682

C₄

yes

complemented

#4680

C₄

yes

—

—

—

—

D₃

yes

characteristic

#1518

—

#1452

—

C₂²×C₃×C₈

D₄

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

C₂³

yes

—

—

—

C₂×C₄

yes

—

C₄×Dic₆

—

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

—

C₃×#95

—

#830

D₄

yes

—

#818

—

#817

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

#382

C₂×D₃

yes

—

#340

—

#328

—

#322

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

#41

yes

—

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₈

#67

yes

—

C₄⋊Q₈

—

#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

C₄×D₃

yes

characteristic

C₄×C₈

C₄×D₃

yes

characteristic

C₄×C₈

D₁₂

yes

—

#95

—

C₂³×C₃

#127

yes

characteristic

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

#124

yes

characteristic

C₂×C₃×C₄

#137

yes

—

C₂×Dic₃

—

Dic₆

—

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

(C₂×D₃)⋊C₄

yes

—

104

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₂×C₄×D₃

yes

characteristic

112

C₄²

C₂×D₁₂

yes

characteristic

113

C₄²

#239

yes

—

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

#893

yes

—

130

C₂×C₄

#895

yes

—

131

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

132

C₂×C₄

#947

yes

characteristic

133

C₂×C₄

#951

yes

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₂×C₃

—

138

C₂×C₃

#1960

yes

characteristic

139

C₂×C₃

#1976

yes

—

140

C₂²

—

141

C₂²

#4988

yes

characteristic

142

C₂²

#5060

yes

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#5004

yes

—

148

C₄

#5069

yes

—

149

C₃

#9759

yes

characteristic, complemented, Sylow

150

C₂

—

151

C₂

#69552

yes

characteristic

152

C₂

#70658

yes

—

	subgroup	G/H
center	C₂²	#4988
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1835	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#127
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1835	D₃
hypocenter	C₃	#9759
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

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 ◁ #1835

N	H ≅ G/N	type
12. #4682	C₄	semidirect
13. #4680	C₄	semidirect
149. C₃	#9759	semidirect

H ≅ G/N

type

12. #4682

C₄

semidirect

13. #4680

C₄

semidirect

149. C₃

#9759

semidirect

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

subgroup	index
#78442	2
C₃×#1835	2
#69726	2
#69546	2
#9759	3

subgroup

index

Subgroups — #10694578

Conjugacy classes of subgroups

Complements

Maximal subgroups