Group #23764908 — subgroups

Subgroups — #23764908

Order 10112.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#19098606	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579094	C₂	1	1	yes	maximal, characteristic
3	#19098590	C₂	1	1	yes	maximal, characteristic
4	#19098332	C₂	1	1	yes	maximal, characteristic
5	#19098331	C₂	1	1	yes	maximal, characteristic
6	#19098327	C₂	1	1	yes	maximal, characteristic
7	#428×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11929608	C₂²	1	1	yes	characteristic
9	C₂×#11929608	C₂²	1	1	yes	characteristic
10	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
12	#14579094	—	2	1	no	—
13	#14579094	—	2	2	no	—
14	#14579092	—	2	1	no	—
15	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
16	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	#117×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
19	#107×C₇₉	—	2	1	no	—
20	#106×C₇₉	—	2	1	no	—
21	C₂²×Dic₇₉	—	2	1	no	—
22	#11929608	—	2	1	no	—
23	#11929608	—	2	1	no	—
24	#11929608	—	2	1	no	—
25	#11929608	Q₈	1	2	yes	—
26	C₂×C₇₉⋊C₈	—	2	1	no	—
27	C₂×C₇₉⋊C₈	—	2	1	no	—
28	C₂×C₇₉⋊C₈	—	2	1	no	—
29	C₂×C₇₉⋊C₈	—	2	1	no	—
30	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
31	C₄⋊Dic₇₉	—	2	1	no	—
32	C₄⋊Dic₇₉	—	2	1	no	—
33	C₂²×C₄×C₇₉	—	2	1	no	—
34	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
35	M₁₆×C₇₉	—	2	1	no	—
36	M₁₆×C₇₉	Q₈	1	2	yes	—
37	C₂×C₈×C₇₉	—	2	1	no	—
38	C₄⋊C₄×C₇₉	—	2	1	no	—
39	C₄⋊C₄×C₇₉	—	2	1	no	—
40	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
42	C₂²⋊C₄×C₇₉	—	2	1	no	—
43	C₄²×C₇₉	—	2	1	no	—
44	C₂³×C₇₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₇₉	—	2	1	no	—
46	C₂×C₄×C₇₉	—	2	1	no	—
47	C₂×C₄×C₇₉	—	4	1	no	—
48	C₂×C₄×C₇₉	—	8	1	no	—
49	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₂×Dic₇₉	—	4	1	no	—
56	C₂×Dic₇₉	—	8	1	no	—
57	C₈×C₇₉	—	2	2	no	—
58	C₇₉⋊C₈	—	2	2	no	—
59	C₇₉⋊C₈	—	2	2	no	—
60	C₇₉⋊C₈	—	4	1	no	—
61	C₂²×C₇₉	—	2	1	no	—
62	C₂²×C₇₉	—	2	1	no	—
63	C₂²×C₇₉	#121	1	1	yes	characteristic
64	C₂²×C₇₉	#125	1	1	yes	characteristic
65	C₂²×C₇₉	#125	1	1	yes	characteristic
66	C₄×C₇₉	—	4	1	no	—
67	C₄×C₇₉	—	4	1	no	—
68	C₄×C₇₉	—	8	1	no	—
69	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
70	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₄×C₇₉	#123	1	1	yes	characteristic
72	C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
73	Dic₇₉	—	8	1	no	—
74	C₂×C₇₉	—	2	1	no	—
75	C₂×C₇₉	—	2	1	no	—
76	C₂×C₇₉	#398	1	1	yes	characteristic
77	C₂×C₇₉	#472	1	1	yes	characteristic
78	C₂×C₇₉	#473	1	1	yes	characteristic
79	#2089	—	79	1	no	maximal, Sylow
80	C₇₉	#2089	1	1	yes	characteristic, complemented, Sylow
81	#428	—	79	1	no	—
82	#428	D₇₉	1	1	yes	characteristic
83	C₂×#107	—	79	1	no	—
84	#424	—	79	1	no	—
85	#343	—	79	1	no	—
86	#343	—	79	1	no	—
87	#341	—	79	1	no	—
88	C₂×M₁₆	—	79	1	no	—
89	C₂×M₁₆	—	79	1	no	—
90	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
91	C₂²×C₈	—	79	1	no	—
92	#117	C₂×D₇₉	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	79	1	no	—
94	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
95	#107	—	2	1	no	—
96	#107	—	158	2	no	—
97	#107	—	158	1	no	—
98	#106	—	2	1	no	—
99	#106	—	158	1	no	—
100	C₄⋊C₈	—	158	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	158	1	no	—
103	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
104	M₁₆	—	2	1	no	—
105	M₁₆	—	79	2	no	—
106	M₁₆	—	158	1	no	—
107	M₁₆	—	158	1	no	—
108	M₁₆	—	158	1	no	—
109	M₁₆	Dic₁₅₈	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	79	2	no	—
112	C₂×C₈	—	158	1	no	—
113	C₂×C₈	—	158	1	no	—
114	C₂×C₈	—	158	1	no	—
115	C₂×C₈	—	158	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	158	1	no	—
119	C₄⋊C₄	—	158	1	no	—
120	C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
121	C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₂³	#11929623	1	1	yes	characteristic
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	316	1	no	—
130	C₂×C₄	—	632	1	no	—
131	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
133	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
134	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
135	C₂×C₄	#11929624	1	1	yes	characteristic
136	C₂×C₄	#11929625	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	316	1	no	—
139	C₈	—	158	2	no	—
140	C₈	—	158	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#14579133	1	1	yes	characteristic
144	C₂²	#14579151	1	1	yes	characteristic
145	C₂²	#14579152	1	1	yes	characteristic
146	C₄	—	4	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	8	1	no	—
149	C₄	—	632	1	no	—
150	C₄	#14579137	1	1	yes	characteristic
151	C₄	#14579140	1	1	yes	characteristic
152	C₄	#14579146	1	1	yes	characteristic
153	C₄	#14579160	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#19098496	1	1	yes	characteristic
157	C₂	#19098831	1	1	yes	characteristic
158	C₂	#19098836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098606

C₂

yes

maximal, characteristic

C₂×#14579094

C₂

yes

maximal, characteristic

#19098590

C₂

yes

maximal, characteristic

#19098332

C₂

yes

maximal, characteristic

#19098331

C₂

yes

maximal, characteristic

#19098327

C₂

yes

maximal, characteristic

#428×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

—

—

—

C₄⋊(C₇₉⋊C₈)

—

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

#107×C₇₉

—

#106×C₇₉

—

C₂²×Dic₇₉

—

—

—

—

Q₈

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

Q₈

yes

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

#41

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

#123

yes

characteristic

C₄×C₇₉

C₄⋊Q₈

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#428

—

#428

D₇₉

yes

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₉

yes

characteristic

C₂²×C₈

—

#117

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

#107

—

#107

—

158

—

#107

—

158

—

#106

—

#106

—

158

—

100

C₄⋊C₈

—

158

—

101

C₂²×C₄

—

102

C₂²×C₄

—

158

—

103

C₂²×C₄

C₂²×D₇₉

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

158

—

107

M₁₆

—

158

—

108

M₁₆

—

158

—

109

M₁₆

Dic₁₅₈

yes

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

158

—

113

C₂×C₈

—

158

—

114

C₂×C₈

—

158

—

115

C₂×C₈

—

158

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

158

—

119

C₄⋊C₄

—

158

—

120

C₄⋊C₄

C₂²⋊D₇₉

yes

characteristic

121

C₄⋊C₄

C₂²⋊D₇₉

yes

characteristic

122

C₂²⋊C₄

—

123

C₄²

—

124

C₂³

#11929623

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

316

—

130

C₂×C₄

—

632

—

131

C₂×C₄

D₄×D₇₉

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

133

C₂×C₄

Q₈×D₇₉

yes

characteristic

134

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

135

C₂×C₄

#11929624

yes

characteristic

136

C₂×C₄

#11929625

yes

characteristic

137

C₈

—

138

C₈

—

316

—

139

C₈

—

158

—

140

C₈

—

158

—

141

C₂²

—

142

C₂²

—

143

C₂²

#14579133

yes

characteristic

144

C₂²

#14579151

yes

characteristic

145

C₂²

#14579152

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

632

—

150

C₄

#14579137

yes

characteristic

151

C₄

#14579140

yes

characteristic

152

C₄

#14579146

yes

characteristic

153

C₄

#14579160

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#19098496

yes

characteristic

157

C₂

#19098831

yes

characteristic

158

C₂

#19098836

yes

characteristic

	subgroup	G/H
center	C₂²	#14579133
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#428×C₇₉	C₂
socle	C₂²×C₇₉	#121
prime radical	C₂³×C₇₉	#41
hypercenter	#428	D₇₉
hypocenter	C₇₉	#2089
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

subgroup

G/H

center

C₂²

#14579133

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#428×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

N	H ≅ G/N	type
80. C₇₉	#2089	semidirect

H ≅ G/N

type

80. C₇₉

#2089

semidirect

Subgroups — #23764908

Conjugacy classes of subgroups

Complements