Group #23764152 — subgroups

Subgroups — #23764152

Order 10112.

200 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×#14579096	C₂	1	1	yes	maximal, characteristic
2	#19098599	C₂	1	1	yes	maximal, characteristic
3	#19098591	C₂	1	1	yes	maximal, characteristic
4	#19098590	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579072	C₂	1	1	yes	maximal, characteristic
6	#19098323	C₂	1	1	yes	maximal, characteristic
7	#339×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11929608	C₂²	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	#14579120	C₂²	1	1	yes	characteristic
12	#14579096	—	2	2	no	—
13	#14579094	C₄	1	2	yes	complemented
14	#14579092	C₄	1	2	yes	complemented
15	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
16	Dic₇₉⋊C₈	—	2	2	no	—
17	#14579072	—	2	2	no	—
18	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
19	#117×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
21	#11929608	—	2	1	no	—
22	#11929608	—	2	1	no	—
23	C₂×C₇₉⋊C₈	—	2	1	no	—
24	C₂×C₇₉⋊C₈	—	2	1	no	—
25	C₂×C₇₉⋊C₈	—	2	1	no	—
26	C₂×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
29	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
30	C₂×C₇₉⋊C₈	Q₈	1	2	yes	—
31	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
32	C₂²⋊Dic₇₉	—	2	1	no	—
33	C₄×Dic₇₉	—	2	1	no	—
34	C₂²×C₄×C₇₉	—	2	1	no	—
35	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	M₁₆×C₇₉	—	2	1	no	—
37	M₁₆×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
40	C₂²⋊C₄×C₇₉	—	2	1	no	—
41	C₄²×C₇₉	—	2	1	no	—
42	C₄²×C₇₉	—	2	1	no	—
43	C₄²×C₇₉	Q₈	1	2	yes	—
44	C₂³×C₇₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₇₉	—	2	2	no	—
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	4	2	no	—
48	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
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×Dic₇₉	—	2	2	no	—
55	C₈×C₇₉	—	4	1	no	—
56	C₇₉⋊C₈	—	2	2	no	—
57	C₇₉⋊C₈	—	4	1	no	—
58	C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₇₉	—	2	2	no	—
61	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₇₉	#125	1	1	yes	characteristic
64	C₄×C₇₉	—	4	2	no	—
65	C₄×C₇₉	—	4	2	no	—
66	C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
68	C₄×C₇₉	#121	1	1	yes	characteristic
69	C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
70	Dic₇₉	—	4	2	no	—
71	C₂×C₇₉	—	2	1	no	—
72	C₂×C₇₉	—	2	1	no	—
73	C₂×C₇₉	#389	1	1	yes	characteristic
74	C₂×C₇₉	#444	1	2	yes	—
75	#1946	—	79	1	no	maximal, Sylow
76	C₇₉	#1946	1	1	yes	characteristic, complemented, Sylow
77	C₂×#108	—	79	1	no	—
78	#427	—	79	1	no	—
79	#424	—	79	1	no	—
80	#424	—	79	1	no	—
81	C₂×#97	—	79	1	no	—
82	#339	—	79	1	no	—
83	#339	D₇₉	1	1	yes	characteristic
84	C₂×M₁₆	—	79	1	no	—
85	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
86	C₂²×C₈	—	79	1	no	—
87	C₂²×C₈	—	79	1	no	—
88	#117	—	79	1	no	—
89	#117	C₂×D₇₉	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
91	#108	—	158	2	no	—
92	#107	—	79	2	no	—
93	#106	—	79	2	no	—
94	C₄⋊C₈	—	158	2	no	—
95	C₄⋊C₈	—	158	2	no	—
96	#97	—	158	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	158	1	no	—
102	M₁₆	—	158	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	79	1	no	—
105	C₂×C₈	—	79	1	no	—
106	C₂×C₈	—	79	2	no	—
107	C₂×C₈	—	79	2	no	—
108	C₂×C₈	—	79	2	no	—
109	C₂×C₈	—	158	1	no	—
110	C₂×C₈	—	158	1	no	—
111	C₂×C₈	—	158	1	no	—
112	C₄⋊C₄	—	79	2	no	—
113	C₄⋊C₄	C₄×D₇₉	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	158	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	158	1	no	—
119	C₄²	Dic₁₅₈	1	2	yes	—
120	C₂³	#11929623	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	158	2	no	—
125	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
126	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
127	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
130	C₂×C₄	#11929624	1	1	yes	characteristic
131	C₈	—	4	1	no	—
132	C₈	—	316	1	no	—
133	C₈	—	79	2	no	—
134	C₈	—	79	2	no	—
135	C₈	—	158	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
138	C₂²	#14579138	1	1	yes	characteristic
139	C₂²	#14579151	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	316	2	no	—
143	C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
144	C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
145	C₄	#14579134	1	1	yes	characteristic
146	C₄	#14579160	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#19098460	1	1	yes	characteristic
150	C₂	#19098682	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579096

C₂

yes

maximal, characteristic

#19098599

C₂

yes

maximal, characteristic

#19098591

C₂

yes

maximal, characteristic

#19098590

C₂

yes

maximal, characteristic

C₂×#14579072

C₂

yes

maximal, characteristic

#19098323

C₂

yes

maximal, characteristic

#339×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

#14579120

C₂²

yes

characteristic

#14579096

—

#14579094

C₄

yes

complemented

#14579092

C₄

yes

complemented

C₄⋊(C₇₉⋊C₈)

—

Dic₇₉⋊C₈

—

#14579072

—

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×C₇₉⋊C₈

Q₈

yes

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

Q₈

yes

—

C₂³×C₇₉

#41

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

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

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

C₄⋊C₄

yes

—

C₇₉⋊C₈

C₄⋊C₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×D₄

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

#125

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₉

C₄×Q₈

yes

characteristic

C₄×C₇₉

#121

yes

characteristic

C₄×C₇₉

C₄⋊Q₈

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#389

yes

characteristic

C₂×C₇₉

#444

yes

—

#1946

—

maximal, Sylow

C₇₉

#1946

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×#97

—

#339

—

#339

D₇₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₇₉

yes

characteristic

C₂×C₄²

C₂×D₇₉

yes

characteristic

#108

—

158

—

#107

—

#106

—

C₄⋊C₈

—

158

—

C₄⋊C₈

—

158

—

#97

—

158

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

158

—

102

M₁₆

—

158

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

158

—

110

C₂×C₈

—

158

—

111

C₂×C₈

—

158

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₄×D₇₉

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

158

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

158

—

119

C₄²

Dic₁₅₈

yes

—

120

C₂³

#11929623

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

158

—

125

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

126

C₂×C₄

D₄×D₇₉

yes

characteristic

127

C₂×C₄

D₄×D₇₉

yes

characteristic

128

C₂×C₄

Q₈×D₇₉

yes

characteristic

129

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

130

C₂×C₄

#11929624

yes

characteristic

131

C₈

—

132

C₈

—

316

—

133

C₈

—

134

C₈

—

135

C₈

—

158

—

136

C₂²

—

137

C₂²

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

138

C₂²

#14579138

yes

characteristic

139

C₂²

#14579151

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

316

—

143

C₄

C₄⋊C₄×D₇₉

yes

characteristic

144

C₄

C₄×Dic₁₅₈

yes

characteristic

145

C₄

#14579134

yes

characteristic

146

C₄

#14579160

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#19098460

yes

characteristic

150

C₂

#19098682

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#339×C₇₉	C₂
socle	C₂²×C₇₉	C₄×D₄
prime radical	C₂³×C₇₉	#41
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#339	D₇₉
hypocenter	C₇₉	#1946
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

subgroup

G/H

center

C₂×C₄

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#339×C₇₉

C₂

socle

C₂²×C₇₉

C₄×D₄

prime radical

C₂³×C₇₉

#41

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

N	H ≅ G/N	type
13. #14579094	C₄	semidirect
14. #14579092	C₄	semidirect
76. C₇₉	#1946	semidirect

H ≅ G/N

type

13. #14579094

C₄

semidirect

14. #14579092

C₄

semidirect

76. C₇₉

#1946

semidirect

Subgroups — #23764152

Conjugacy classes of subgroups

Complements