Group #23764500 — subgroups

Subgroups — #23764500

Order 10112.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#11929614	C₂	1	2	yes	maximal
2	C₂×#14579085	C₂	1	2	yes	maximal
3	#19098469	C₂	1	1	yes	maximal, characteristic
4	#19098318	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
7	C₂×#11929614	—	2	4	no	—
8	C₂×#11929614	—	2	4	no	—
9	C₂×#11929614	—	2	4	no	—
10	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
12	#14579085	—	2	4	no	—
13	C₇₉⋊#95	—	2	2	no	—
14	C₂²×Q₈×C₇₉	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₇₉	—	2	4	no	—
16	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
17	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₅₈	—	2	2	no	—
19	C₂×Dic₁₅₈	—	2	2	no	—
20	C₂×Dic₁₅₈	—	2	1	no	—
21	C₂×Dic₁₅₈	D₄	1	2	yes	—
22	C₂²×Dic₇₉	—	2	1	no	—
23	C₂²×Dic₇₉	—	2	1	no	—
24	#11929614	—	2	16	no	—
25	C₂×C₇₉⋊C₈	—	2	2	no	—
26	C₂×C₇₉⋊C₈	—	2	2	no	—
27	C₂×C₇₉⋊C₈	D₄	1	4	yes	—
28	C₄⋊Dic₇₉	—	2	2	no	—
29	C₂×Q₈×C₇₉	—	2	2	no	—
30	C₂×Q₈×C₇₉	—	2	2	no	—
31	C₂×Q₈×C₇₉	—	2	2	no	—
32	C₂×Q₈×C₇₉	—	2	2	no	—
33	C₂×Q₈×C₇₉	—	2	2	no	—
34	C₂×Q₈×C₇₉	D₄	1	4	yes	—
35	C₂²×C₄×C₇₉	—	2	2	no	—
36	C₂²×C₄×C₇₉	D₄	1	2	yes	—
37	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₉	—	2	8	no	—
39	C₄²×C₇₉	—	2	2	no	—
40	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₇₉	—	2	4	no	—
42	Q₈×C₇₉	—	2	4	no	—
43	Q₈×C₇₉	—	4	2	no	—
44	Q₈×C₇₉	—	4	2	no	—
45	C₂×C₄×C₇₉	—	2	2	no	—
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	2	4	no	—
48	C₂×C₄×C₇₉	—	4	2	no	—
49	C₂×C₄×C₇₉	—	4	2	no	—
50	C₂×C₄×C₇₉	Q₁₆	1	4	yes	complemented
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₂×Dic₇₉	—	2	2	no	—
56	C₂×Dic₇₉	—	4	1	no	—
57	C₂×Dic₇₉	—	4	1	no	—
58	C₂×Dic₇₉	—	4	1	no	—
59	C₂×Dic₇₉	—	4	2	no	—
60	Dic₁₅₈	—	2	4	no	—
61	Dic₁₅₈	—	4	2	no	—
62	C₇₉⋊C₈	—	2	4	no	—
63	C₂²×C₇₉	#120	1	1	yes	characteristic
64	C₂²×C₇₉	#121	1	1	yes	characteristic
65	C₂²×C₇₉	#127	1	1	yes	characteristic
66	C₂²×C₇₉	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₇₉	#137	1	2	yes	—
68	C₄×C₇₉	—	2	4	no	—
69	C₄×C₇₉	—	4	4	no	—
70	C₄×C₇₉	#120	1	2	yes	—
71	C₄×C₇₉	#121	1	2	yes	—
72	Dic₇₉	—	4	2	no	—
73	Dic₇₉	—	8	1	no	—
74	C₂×C₇₉	#392	1	1	yes	characteristic
75	C₂×C₇₉	#451	1	2	yes	—
76	C₂×C₇₉	#462	1	2	yes	—
77	C₂×C₇₉	#494	1	1	yes	characteristic
78	C₂×C₇₉	#497	1	1	yes	characteristic
79	#2016	—	79	1	no	maximal, Sylow
80	C₇₉	#2016	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₁₆	—	79	2	no	—
82	C₂×C₄⋊Q₈	D₇₉	1	1	yes	characteristic
83	C₂×#103	—	79	2	no	—
84	#391	—	79	1	no	—
85	#336	—	79	1	no	—
86	C₂²×Q₈	—	79	1	no	—
87	C₂²×Q₈	C₂×D₇₉	1	2	yes	—
88	C₂×Q₁₆	—	158	4	no	—
89	C₂×Q₁₆	—	158	4	no	—
90	C₂×Q₁₆	—	158	4	no	—
91	C₂²×C₈	—	79	2	no	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	79	1	no	—
95	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
96	#103	—	158	4	no	—
97	#95	—	158	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	79	2	no	—
104	C₂×Q₈	—	158	2	no	—
105	C₂×Q₈	—	158	2	no	—
106	C₂×Q₈	—	158	1	no	—
107	C₂×Q₈	C₂²⋊D₇₉	1	4	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	158	1	no	—
110	C₂²×C₄	—	158	1	no	—
111	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
113	Q₁₆	—	158	16	no	—
114	C₂×C₈	—	79	4	no	—
115	C₂×C₈	—	158	2	no	—
116	C₂×C₈	—	158	2	no	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	158	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	316	2	no	—
126	Q₈	—	158	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	316	1	no	—
133	C₂×C₄	—	316	1	no	—
134	C₂×C₄	—	316	1	no	—
135	C₂×C₄	—	316	2	no	—
136	C₂×C₄	—	158	2	no	—
137	C₂×C₄	#11929614	1	4	yes	complemented
138	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
139	C₂×C₄	D₄×D₇₉	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
141	C₂×C₄	#11929623	1	1	yes	characteristic
142	C₈	—	158	4	no	—
143	C₂²	#14579130	1	1	yes	characteristic
144	C₂²	#14579132	1	1	yes	characteristic
145	C₂²	#14579157	1	1	yes	characteristic
146	C₂²	C₂×#11929614	1	2	yes	—
147	C₂²	#14579192	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	4	no	—
150	C₄	—	316	2	no	—
151	C₄	—	632	1	no	—
152	C₄	#14579129	1	2	yes	—
153	C₄	#14579133	1	2	yes	—
154	C₂	#19098471	1	1	yes	characteristic
155	C₂	#19098719	1	2	yes	—
156	C₂	#19098770	1	2	yes	—
157	C₂	#19098942	1	1	yes	characteristic
158	C₂	#19098955	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11929614

C₂

yes

maximal

C₂×#14579085

C₂

yes

maximal

#19098469

C₂

yes

maximal, characteristic

#19098318

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

—

—

C₂²×C₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

#14579085

—

C₇₉⋊#95

—

C₂²×Q₈×C₇₉

C₂²

yes

—

C₄⋊Q₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

#11929614

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

Q₁₆

yes

complemented

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

C₂×Q₁₆

yes

—

C₂²×C₇₉

#137

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#120

yes

—

C₄×C₇₉

#121

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#392

yes

characteristic

C₂×C₇₉

#451

yes

—

C₂×C₇₉

#462

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

D₇₉

yes

—

—

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇₉

yes

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#103

—

158

—

#95

—

158

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

158

—

105

C₂×Q₈

—

158

—

106

C₂×Q₈

—

158

—

107

C₂×Q₈

C₂²⋊D₇₉

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

158

—

110

C₂²×C₄

—

158

—

111

C₂²×C₄

C₂²⋊D₇₉

yes

—

112

C₂²×C₄

C₂²×D₇₉

yes

characteristic

113

Q₁₆

—

158

—

114

C₂×C₈

—

115

C₂×C₈

—

158

—

116

C₂×C₈

—

158

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

158

—

119

C₄²

—

120

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

316

—

126

Q₈

—

158

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

316

—

133

C₂×C₄

—

316

—

134

C₂×C₄

—

316

—

135

C₂×C₄

—

316

—

136

C₂×C₄

—

158

—

137

C₂×C₄

#11929614

yes

complemented

138

C₂×C₄

D₄×D₇₉

yes

characteristic

139

C₂×C₄

D₄×D₇₉

yes

—

140

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

141

C₂×C₄

#11929623

yes

characteristic

142

C₈

—

158

—

143

C₂²

#14579130

yes

characteristic

144

C₂²

#14579132

yes

characteristic

145

C₂²

#14579157

yes

characteristic

146

C₂²

C₂×#11929614

yes

—

147

C₂²

#14579192

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

316

—

151

C₄

—

632

—

152

C₄

#14579129

yes

—

153

C₄

#14579133

yes

—

154

C₂

#19098471

yes

characteristic

155

C₂

#19098719

yes

—

156

C₂

#19098770

yes

—

157

C₂

#19098942

yes

characteristic

158

C₂

#19098955

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×C₄⋊Q₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₇₉
hypocenter	C₇₉	#2016
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×C₄⋊Q₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₇₉

hypocenter

C₇₉

#2016

LCS

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
50. C₂×C₄×C₇₉	Q₁₆	semidirect
80. C₇₉	#2016	semidirect
137. C₂×C₄	#11929614	semidirect

H ≅ G/N

type

50. C₂×C₄×C₇₉

Q₁₆

semidirect

80. C₇₉

#2016

semidirect

137. C₂×C₄

#11929614

semidirect

Subgroups — #23764500

Conjugacy classes of subgroups

Complements