Group #23768612 — subgroups

Subgroups — #23768612

Order 10112.

468 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14579161	C₂	1	1	yes	maximal, characteristic
2	#19098502	C₂	1	4	yes	maximal
3	#19098470	C₂	1	2	yes	maximal
4	#19098469	C₂	1	4	yes	maximal
5	#19098428	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
9	#14579161	—	2	4	no	—
10	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
13	C₂×C₄×Dic₇₉	C₂²	1	2	yes	—
14	C₇₉⋊#95	C₂²	1	2	yes	—
15	C₇₉⋊#95	C₂²	1	8	yes	—
16	C₇₉⋊#95	C₂²	1	8	yes	—
17	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₇₉	—	2	4	no	—
19	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₇₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₅₈	—	2	4	no	—
24	C₂×Dic₁₅₈	—	2	8	no	—
25	C₂²×Dic₇₉	C₂³	1	4	yes	—
26	C₂²×Dic₇₉	C₂³	1	4	yes	—
27	C₄⋊Dic₇₉	—	2	4	no	—
28	C₄⋊Dic₇₉	—	2	4	no	—
29	C₄⋊Dic₇₉	—	2	4	no	—
30	C₄×Dic₇₉	—	2	4	no	—
31	C₂×Q₈×C₇₉	—	2	2	no	—
32	C₂×Q₈×C₇₉	—	2	2	no	—
33	C₂×Q₈×C₇₉	Q₈	1	4	yes	complemented
34	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₇₉	—	2	4	no	—
40	C₄⋊C₄×C₇₉	—	2	2	no	—
41	C₄²×C₇₉	—	2	4	no	—
42	C₄²×C₇₉	D₄	1	4	yes	—
43	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
44	Q₈×C₇₉	—	2	8	no	—
45	C₂×C₄×C₇₉	—	2	2	no	—
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	2	2	no	—
48	C₂×C₄×C₇₉	—	2	2	no	—
49	C₂×C₄×C₇₉	—	2	4	no	—
50	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
54	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₇₉	#41	1	2	yes	—
56	C₂×C₄×C₇₉	#41	1	4	yes	—
57	C₂×Dic₇₉	—	2	4	no	—
58	C₂×Dic₇₉	—	2	4	no	—
59	C₂×Dic₇₉	—	2	4	no	—
60	C₂×Dic₇₉	—	2	8	no	—
61	C₂×Dic₇₉	—	2	4	no	—
62	Dic₁₅₈	—	4	8	no	—
63	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
64	C₂²×C₇₉	C₂²×Q₈	1	1	yes	characteristic
65	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
66	C₂²×C₇₉	C₂×#41	1	2	yes	—
67	C₂²×C₇₉	2₋¹⁺⁴	1	2	yes	—
68	C₄×C₇₉	—	2	4	no	—
69	C₄×C₇₉	—	2	2	no	—
70	C₄×C₇₉	—	4	2	no	—
71	C₄×C₇₉	C₂²⋊Q₈	1	4	yes	—
72	C₄×C₇₉	#124	1	4	yes	—
73	Dic₇₉	—	4	4	no	—
74	Dic₇₉	—	4	4	no	—
75	C₂×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₇₉	C₂×#124	1	1	yes	characteristic
77	C₂×C₇₉	#536	1	1	yes	characteristic
78	C₂×C₇₉	#556	1	2	yes	—
79	C₂×C₇₉	#557	1	2	yes	—
80	#2568	—	79	1	no	maximal, Sylow
81	C₇₉	#2568	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊Q₈	—	79	1	no	—
83	C₂×C₄×Q₈	D₇₉	1	1	yes	characteristic
84	#400	—	79	4	no	—
85	#391	—	79	2	no	—
86	#391	—	79	4	no	—
87	#382	—	79	2	no	—
88	C₄×C₄⋊C₄	—	79	1	no	—
89	C₂²×Q₈	—	79	2	no	—
90	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
91	C₄⋊Q₈	—	158	4	no	—
92	C₄×Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	—	79	2	no	—
94	C₂×C₄⋊C₄	—	79	2	no	—
95	C₂×C₄⋊C₄	—	79	2	no	—
96	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
98	C₂×C₄²	—	79	2	no	—
99	C₂×C₄²	C₂×D₇₉	1	2	yes	—
100	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
101	#95	—	79	2	no	—
102	#95	—	79	8	no	—
103	#95	—	79	8	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	158	4	no	—
107	C₂×Q₈	—	158	8	no	—
108	C₂×Q₈	Dic₁₅₈	1	4	yes	complemented
109	C₂²×C₄	—	79	4	no	—
110	C₂²×C₄	—	79	4	no	—
111	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
114	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	158	4	no	—
119	C₄⋊C₄	—	158	4	no	—
120	C₄⋊C₄	—	158	4	no	—
121	C₄²	—	2	4	no	—
122	C₄²	—	158	4	no	—
123	C₄²	C₂²⋊D₇₉	1	4	yes	—
124	C₂³	C₂³×D₇₉	1	1	yes	characteristic
125	Q₈	—	2	8	no	—
126	Q₈	—	316	8	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	158	4	no	—
133	C₂×C₄	—	158	4	no	—
134	C₂×C₄	—	158	8	no	—
135	C₂×C₄	—	158	4	no	—
136	C₂×C₄	—	158	4	no	—
137	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
140	C₂×C₄	C₂×Dic₁₅₈	1	4	yes	—
141	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
142	C₂×C₄	#11929624	1	2	yes	—
143	C₂×C₄	#11929625	1	4	yes	—
144	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
145	C₂²	C₂²×Dic₁₅₈	1	1	yes	characteristic
146	C₂²	C₂×#11929624	1	1	yes	characteristic
147	C₂²	C₂×#11929625	1	2	yes	—
148	C₂²	#14579208	1	2	yes	—
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	316	4	no	—
153	C₄	—	316	4	no	—
154	C₄	#14579140	1	4	yes	—
155	C₄	#14579150	1	4	yes	—
156	C₂	C₂×#14579140	1	1	yes	characteristic
157	C₂	C₂×#14579150	1	1	yes	characteristic
158	C₂	#19099134	1	1	yes	characteristic
159	C₂	#19099286	1	2	yes	—
160	C₂	#19099292	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic

#19098502

C₂

yes

maximal

#19098470

C₂

yes

maximal

#19098469

C₂

yes

maximal

#19098428

C₂

yes

maximal

C₄×C₄⋊Dic₇₉

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

#14579161

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₄×Q₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

Q₈

yes

complemented

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

D₄

yes

—

C₂³×C₇₉

C₂⁴

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂²×C₇₉

C₂²×Q₈

yes

characteristic

C₂²×C₇₉

C₂×#41

yes

characteristic

C₂²×C₇₉

C₂×#41

yes

—

C₂²×C₇₉

2₋¹⁺⁴

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

C₄×C₇₉

#124

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

C₂×C₂²⋊Q₈

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#557

yes

—

#2568

—

maximal, Sylow

C₇₉

#2568

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₇₉

yes

—

—

—

—

C₄×C₄⋊C₄

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇₉

yes

characteristic

C₄⋊Q₈

—

158

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₉

yes

—

100

C₂×C₄²

C₂×D₇₉

yes

characteristic

101

#95

—

102

#95

—

103

#95

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

158

—

107

C₂×Q₈

—

158

—

108

C₂×Q₈

Dic₁₅₈

yes

complemented

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₇₉

yes

—

112

C₂²×C₄

C₂²×D₇₉

yes

characteristic

113

C₂²×C₄

C₂²×D₇₉

yes

—

114

C₂²×C₄

C₂²×D₇₉

yes

characteristic

115

C₂²×C₄

C₂²×D₇₉

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

158

—

119

C₄⋊C₄

—

158

—

120

C₄⋊C₄

—

158

—

121

C₄²

—

122

C₄²

—

158

—

123

C₄²

C₂²⋊D₇₉

yes

—

124

C₂³

C₂³×D₇₉

yes

characteristic

125

Q₈

—

126

Q₈

—

316

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

158

—

133

C₂×C₄

—

158

—

134

C₂×C₄

—

158

—

135

C₂×C₄

—

158

—

136

C₂×C₄

—

158

—

137

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

138

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

139

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

140

C₂×C₄

C₂×Dic₁₅₈

yes

—

141

C₂×C₄

C₂×Dic₁₅₈

yes

—

142

C₂×C₄

#11929624

yes

—

143

C₂×C₄

#11929625

yes

—

144

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

145

C₂²

C₂²×Dic₁₅₈

yes

characteristic

146

C₂²

C₂×#11929624

yes

characteristic

147

C₂²

C₂×#11929625

yes

—

148

C₂²

#14579208

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

316

—

153

C₄

—

316

—

154

C₄

#14579140

yes

—

155

C₄

#14579150

yes

—

156

C₂

C₂×#14579140

yes

characteristic

157

C₂

C₂×#14579150

yes

characteristic

158

C₂

#19099134

yes

characteristic

159

C₂

#19099286

yes

—

160

C₂

#19099292

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, prime radical, cosocle	C₂³×C₇₉	C₂⁴
Fitting	C₂×C₄×Q₈×C₇₉	C₂
hypercenter	C₂×C₄×Q₈	D₇₉
hypocenter	C₇₉	#2568
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator, socle, prime radical, cosocle

C₂³×C₇₉

C₂⁴

Fitting

C₂×C₄×Q₈×C₇₉

C₂

hypercenter

C₂×C₄×Q₈

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
33. C₂×Q₈×C₇₉	Q₈	semidirect
81. C₇₉	#2568	semidirect
108. C₂×Q₈	Dic₁₅₈	semidirect

H ≅ G/N

type

33. C₂×Q₈×C₇₉

Q₈

semidirect

81. C₇₉

#2568

semidirect

108. C₂×Q₈

Dic₁₅₈

semidirect

Subgroups — #23768612

Conjugacy classes of subgroups

Complements