Group #23773928 — subgroups

Subgroups — #23773928

Order 10112.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#19099194	C₂	1	2	yes	maximal
2	#19098970	C₂	1	4	yes	maximal, complemented
3	#19098963	C₂	1	2	yes	maximal, complemented
4	#19098960	C₂	1	2	yes	maximal, complemented
5	#19098603	C₂	1	2	yes	maximal
6	#19098601	C₂	1	1	yes	maximal, characteristic
7	#19098518	C₂	1	1	yes	maximal, characteristic
8	C₂×#106×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#11929608	C₂²	1	1	yes	characteristic
10	#14579160	—	2	2	no	—
11	#14579160	C₂²	1	4	yes	—
12	#14579152	—	2	2	no	—
13	#14579152	C₂²	1	4	yes	—
14	#14579140	—	2	4	no	—
15	#14579120	C₂²	1	2	yes	—
16	#14579120	C₂²	1	1	yes	characteristic
17	#14579095	C₂²	1	2	yes	—
18	#14579094	C₂²	1	4	yes	—
19	#14579093	C₂²	1	1	yes	characteristic
20	#14579093	C₂²	1	1	yes	characteristic
21	#14579092	C₂²	1	4	yes	—
22	#14579071	C₂²	1	2	yes	—
23	C₈×Dic₇₉	C₂²	1	2	yes	—
24	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
26	#106×C₇₉	C₂²	1	2	yes	—
27	#106×C₇₉	C₂²	1	2	yes	—
28	C₂×Dic₁₅₈	—	2	4	no	—
29	#11929608	Q₈	1	4	yes	complemented
30	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
31	C₄⋊Dic₇₉	—	2	4	no	—
32	C₄⋊Dic₇₉	D₄	1	2	yes	—
33	C₄⋊Dic₇₉	C₂³	1	2	yes	—
34	C₄⋊Dic₇₉	C₂³	1	2	yes	—
35	Dic₇₉⋊C₄	—	2	4	no	—
36	Dic₇₉⋊C₄	—	2	4	no	—
37	C₂²⋊Dic₇₉	—	2	2	no	—
38	C₂²⋊Dic₇₉	D₄	1	2	yes	—
39	C₄×Dic₇₉	—	2	2	no	—
40	C₄×Dic₇₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₇₉	—	2	2	no	—
42	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₉	Q₈	1	4	yes	complemented
44	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₇₉	—	2	2	no	—
47	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
48	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	—	2	4	no	—
50	C₂×C₄×C₇₉	—	8	2	no	—
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₇₉	—	2	2	no	—
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	C₂×D₄	1	4	yes	—
58	Dic₁₅₈	—	8	4	no	—
59	C₈×C₇₉	C₂×Q₈	1	2	yes	—
60	C₈×C₇₉	C₂×Q₈	1	2	yes	—
61	C₇₉⋊C₈	C₂×Q₈	1	4	yes	—
62	C₂²×C₇₉	—	2	2	no	—
63	C₂²×C₇₉	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₇₉	—	4	4	no	—
66	C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
67	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
69	Dic₇₉	—	2	4	no	—
70	Dic₇₉	—	4	2	no	—
71	Dic₇₉	—	4	2	no	—
72	C₂×C₇₉	—	2	2	no	—
73	C₂×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₇₉	#577	1	2	yes	—
75	#3163	—	79	1	no	maximal, Sylow
76	C₇₉	#3163	1	1	yes	characteristic, complemented, Sylow
77	#544	—	79	2	no	—
78	#501	—	79	4	no	—
79	#499	—	79	2	no	—
80	#498	—	79	2	no	—
81	#428	—	79	2	no	—
82	#427	—	79	1	no	—
83	C₂×#106	D₇₉	1	1	yes	characteristic
84	#405	—	79	1	no	—
85	C₂×M₁₆	—	79	1	no	—
86	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
87	C₄⋊Q₈	—	79	4	no	—
88	C₄⋊Q₈	—	158	2	no	—
89	#125	—	79	4	no	—
90	#125	—	158	2	no	—
91	C₂²⋊Q₈	—	158	4	no	—
92	#117	—	79	1	no	—
93	#117	—	79	2	no	—
94	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
95	#107	—	79	2	no	—
96	#107	—	79	4	no	—
97	#106	—	79	1	no	—
98	#106	—	79	1	no	—
99	#106	—	79	4	no	—
100	#106	C₂×D₇₉	1	2	yes	—
101	#106	C₂×D₇₉	1	2	yes	—
102	#97	—	79	2	no	—
103	C₄×C₈	—	79	2	no	—
104	C₂×Q₈	—	158	4	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
107	M₁₆	—	79	4	no	—
108	C₂×C₈	—	79	2	no	—
109	C₂×C₈	Dic₁₅₈	1	4	yes	complemented
110	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	79	2	no	—
114	C₄⋊C₄	—	79	2	no	—
115	C₄⋊C₄	—	79	2	no	—
116	C₄⋊C₄	—	158	4	no	—
117	C₄⋊C₄	—	158	4	no	—
118	C₄⋊C₄	—	158	4	no	—
119	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
120	C₂²⋊C₄	—	79	2	no	—
121	C₂²⋊C₄	—	158	2	no	—
122	C₄²	—	79	2	no	—
123	C₄²	—	158	2	no	—
124	C₂³	D₄×D₇₉	1	1	yes	characteristic
125	Q₈	—	632	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	8	2	no	—
128	C₂×C₄	—	79	4	no	—
129	C₂×C₄	—	158	2	no	—
130	C₂×C₄	—	158	2	no	—
131	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
132	C₂×C₄	Q₈×D₇₉	1	2	yes	—
133	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
134	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
135	C₈	—	79	4	no	—
136	C₈	C₂×Dic₁₅₈	1	2	yes	—
137	C₈	C₂×Dic₁₅₈	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#14579160	1	2	yes	—
140	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
141	C₄	—	4	4	no	—
142	C₄	—	316	2	no	—
143	C₄	—	316	2	no	—
144	C₄	—	158	4	no	—
145	C₄	#14579160	1	2	yes	—
146	C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
147	C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#14579160	1	1	yes	characteristic
150	C₂	#19099369	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19099194

C₂

yes

maximal

#19098970

C₂

yes

maximal, complemented

#19098963

C₂

yes

maximal, complemented

#19098960

C₂

yes

maximal, complemented

#19098603

C₂

yes

maximal

#19098601

C₂

yes

maximal, characteristic

#19098518

C₂

yes

maximal, characteristic

C₂×#106×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₂²

yes

characteristic

#14579160

—

#14579160

C₂²

yes

—

#14579152

—

#14579152

C₂²

yes

—

#14579140

—

#14579120

C₂²

yes

—

#14579120

C₂²

yes

characteristic

#14579095

C₂²

yes

—

#14579094

C₂²

yes

—

#14579093

C₂²

yes

characteristic

#14579093

C₂²

yes

characteristic

#14579092

C₂²

yes

—

#14579071

C₂²

yes

—

C₈×Dic₇₉

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

#106×C₇₉

C₂²

yes

—

#106×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

#11929608

Q₈

yes

complemented

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

D₄

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

Q₈

yes

complemented

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

Dic₁₅₈

—

C₈×C₇₉

C₂×Q₈

yes

—

C₈×C₇₉

C₂×Q₈

yes

—

C₇₉⋊C₈

C₂×Q₈

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄⋊Q₈

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₄⋊Q₈

yes

—

C₄×C₇₉

C₂²×Q₈

yes

characteristic

C₄×C₇₉

C₂²×Q₈

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₉

#577

yes

—

#3163

—

maximal, Sylow

C₇₉

#3163

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₉

yes

characteristic

#405

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

#125

—

#125

—

158

—

C₂²⋊Q₈

—

158

—

#117

—

#117

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

—

—

—

—

—

100

#106

C₂×D₇₉

yes

—

101

#106

C₂×D₇₉

yes

—

102

#97

—

103

C₄×C₈

—

104

C₂×Q₈

—

158

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₇₉

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

Dic₁₅₈

yes

complemented

110

C₂×C₈

C₂²×D₇₉

yes

characteristic

111

C₂×C₈

C₂²×D₇₉

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

158

—

117

C₄⋊C₄

—

158

—

118

C₄⋊C₄

—

158

—

119

C₄⋊C₄

C₂²×D₇₉

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

158

—

122

C₄²

—

123

C₄²

—

158

—

124

C₂³

D₄×D₇₉

yes

characteristic

125

Q₈

—

632

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

158

—

130

C₂×C₄

—

158

—

131

C₂×C₄

D₄×D₇₉

yes

characteristic

132

C₂×C₄

Q₈×D₇₉

yes

—

133

C₂×C₄

C₂×Dic₁₅₈

yes

—

134

C₂×C₄

C₂³×D₇₉

yes

characteristic

135

C₈

—

136

C₈

C₂×Dic₁₅₈

yes

—

137

C₈

C₂×Dic₁₅₈

yes

—

138

C₂²

—

139

C₂²

#14579160

yes

—

140

C₂²

C₂×D₄×D₇₉

yes

characteristic

141

C₄

—

142

C₄

—

316

—

143

C₄

—

316

—

144

C₄

—

158

—

145

C₄

#14579160

yes

—

146

C₄

C₂×Q₈×D₇₉

yes

characteristic

147

C₄

C₂²×Dic₁₅₈

yes

characteristic

148

C₂

—

149

C₂

C₂×#14579160

yes

characteristic

150

C₂

#19099369

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂×#106×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	C₂×#106	D₇₉
hypocenter	C₇₉	#3163
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂×#106×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂³×C₇₉

C₂×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #19098970	C₂	semidirect
3. #19098963	C₂	semidirect
4. #19098960	C₂	semidirect
29. #11929608	Q₈	semidirect
43. C₂×C₈×C₇₉	Q₈	semidirect
76. C₇₉	#3163	semidirect
109. C₂×C₈	Dic₁₅₈	semidirect

H ≅ G/N

type

2. #19098970

C₂

semidirect

3. #19098963

C₂

semidirect

4. #19098960

C₂

semidirect

29. #11929608

Q₈

semidirect

43. C₂×C₈×C₇₉

Q₈

semidirect

76. C₇₉

#3163

semidirect

109. C₂×C₈

Dic₁₅₈

semidirect

Subgroups — #23773928

Conjugacy classes of subgroups

Complements