Group #23773912 — subgroups

Subgroups — #23773912

Order 10112.

356 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 153 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	#19099116	C₂	1	2	yes	maximal
2	#19098967	C₂	1	2	yes	maximal, complemented
3	#19098964	C₂	1	2	yes	maximal, complemented
4	#19098963	C₂	1	2	yes	maximal, complemented
5	#19098960	C₂	1	2	yes	maximal, complemented
6	#19098601	C₂	1	1	yes	maximal, characteristic
7	#19098599	C₂	1	2	yes	maximal
8	C₂×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
9	#427×C₇₉	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	#14579160	C₂²	1	4	yes	—
12	#14579152	C₂²	1	4	yes	—
13	#14579138	—	2	4	no	—
14	C₄×Dic₁₅₈	—	2	4	no	—
15	#14579120	C₂²	1	2	yes	—
16	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
17	#14579095	C₂²	1	2	yes	—
18	#14579094	C₂²	1	4	yes	—
19	#14579093	C₂²	1	2	yes	—
20	#14579092	C₂²	1	4	yes	—
21	C₈×Dic₇₉	C₂²	1	1	yes	characteristic
22	C₈×Dic₇₉	C₂²	1	1	yes	characteristic
23	C₈×Dic₇₉	C₂²	1	1	yes	characteristic
24	C₈×Dic₇₉	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
26	#117×C₇₉	C₂²	1	2	yes	—
27	#107×C₇₉	C₂²	1	2	yes	—
28	#106×C₇₉	C₂²	1	2	yes	—
29	C₂×Dic₁₅₈	—	2	4	no	—
30	C₂²×Dic₇₉	D₄	1	2	yes	—
31	C₂×C₇₉⋊C₈	Q₈	1	4	yes	complemented
32	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
33	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
34	C₄⋊Dic₇₉	—	2	4	no	—
35	C₄⋊Dic₇₉	C₂³	1	4	yes	—
36	Dic₇₉⋊C₄	—	2	4	no	—
37	Dic₇₉⋊C₄	—	2	4	no	—
38	C₂²⋊Dic₇₉	—	2	2	no	—
39	C₄×Dic₇₉	—	2	2	no	—
40	C₄×Dic₇₉	D₄	1	2	yes	—
41	C₄×Dic₇₉	C₂³	1	1	yes	characteristic
42	C₄×Dic₇₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₇₉	Q₈	1	4	yes	complemented
45	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
48	C₂²⋊C₄×C₇₉	—	2	2	no	—
49	C₄²×C₇₉	—	2	2	no	—
50	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	—	2	4	no	—
52	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₇₉	—	2	4	no	—
57	C₂×Dic₇₉	—	2	4	no	—
58	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
59	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
60	Dic₁₅₈	—	4	8	no	—
61	C₈×C₇₉	C₂×Q₈	1	2	yes	—
62	C₈×C₇₉	C₂×Q₈	1	2	yes	—
63	C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
64	C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
65	C₂²×C₇₉	—	2	2	no	—
66	C₂²×C₇₉	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₇₉	—	4	4	no	—
69	C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
70	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
71	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
72	Dic₇₉	—	2	2	no	—
73	Dic₇₉	—	4	4	no	—
74	Dic₇₉	#135	1	4	yes	—
75	C₂×C₇₉	—	2	2	no	—
76	C₂×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₇₉	C₂×#135	1	2	yes	—
78	#3160	—	79	1	no	maximal, Sylow
79	C₇₉	#3160	1	1	yes	characteristic, complemented, Sylow
80	#533	—	79	2	no	—
81	#500	—	79	2	no	—
82	#499	—	79	2	no	—
83	#499	—	79	2	no	—
84	#498	—	79	2	no	—
85	#427	—	79	1	no	—
86	#427	—	79	2	no	—
87	#427	D₇₉	1	1	yes	characteristic
88	C₂×C₄×C₈	—	79	1	no	—
89	C₂²×C₈	—	79	1	no	—
90	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
91	C₄⋊Q₈	—	79	4	no	—
92	#125	—	79	4	no	—
93	C₂²⋊Q₈	—	158	4	no	—
94	C₄×Q₈	—	158	4	no	—
95	#117	—	79	2	no	—
96	#117	C₂×D₇₉	1	2	yes	—
97	C₂×C₄²	—	79	1	no	—
98	#107	—	79	2	no	—
99	#107	—	79	4	no	—
100	#107	C₂×D₇₉	1	2	yes	—
101	#106	—	79	2	no	—
102	#106	—	79	4	no	—
103	#106	C₂×D₇₉	1	2	yes	—
104	C₄×C₈	—	79	1	no	—
105	C₄×C₈	—	79	1	no	—
106	C₄×C₈	—	79	1	no	—
107	C₄×C₈	—	79	1	no	—
108	C₂×Q₈	—	158	4	no	—
109	C₂²×C₄	—	79	2	no	—
110	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
111	C₂×C₈	—	79	4	no	—
112	C₂×C₈	—	79	1	no	—
113	C₂×C₈	—	79	1	no	—
114	C₂×C₈	Dic₁₅₈	1	4	yes	complemented
115	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
117	C₄⋊C₄	—	79	4	no	—
118	C₄⋊C₄	—	158	4	no	—
119	C₄⋊C₄	—	158	4	no	—
120	C₄⋊C₄	—	158	4	no	—
121	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	158	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	79	2	no	—
126	C₄²	—	79	1	no	—
127	C₄²	—	79	1	no	—
128	C₄²	—	158	2	no	—
129	C₂³	D₄×D₇₉	1	1	yes	characteristic
130	Q₈	—	316	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	79	2	no	—
133	C₂×C₄	—	79	2	no	—
134	C₂×C₄	—	158	4	no	—
135	C₂×C₄	—	158	4	no	—
136	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
137	C₂×C₄	Q₈×D₇₉	1	2	yes	—
138	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
139	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
140	C₈	—	79	2	no	—
141	C₈	—	79	2	no	—
142	C₈	C₂×Dic₁₅₈	1	2	yes	—
143	C₈	C₂×Dic₁₅₈	1	2	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	#14579160	1	2	yes	—
146	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
147	C₄	—	4	4	no	—
148	C₄	—	316	4	no	—
149	C₄	—	79	4	no	—
150	C₄	—	158	2	no	—
151	C₄	#14579160	1	2	yes	—
152	C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
153	C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
154	C₂	—	2	2	no	—
155	C₂	C₂×#14579160	1	1	yes	characteristic
156	C₂	#135×D₇₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19099116

C₂

yes

maximal

#19098967

C₂

yes

maximal, complemented

#19098964

C₂

yes

maximal, complemented

#19098963

C₂

yes

maximal, complemented

#19098960

C₂

yes

maximal, complemented

#19098601

C₂

yes

maximal, characteristic

#19098599

C₂

yes

maximal

C₂×C₈×Dic₇₉

C₂

yes

maximal, characteristic

#427×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

#14579160

C₂²

yes

—

#14579152

C₂²

yes

—

#14579138

—

C₄×Dic₁₅₈

—

#14579120

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579095

C₂²

yes

—

#14579094

C₂²

yes

—

#14579093

C₂²

yes

—

#14579092

C₂²

yes

—

C₈×Dic₇₉

C₂²

yes

characteristic

C₈×Dic₇₉

C₂²

yes

characteristic

C₈×Dic₇₉

C₂²

yes

characteristic

C₈×Dic₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

—

#106×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

D₄

yes

—

C₂×C₇₉⋊C₈

Q₈

yes

complemented

C₂×C₇₉⋊C₈

C₂³

yes

characteristic

C₂×C₇₉⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

D₄

yes

—

C₄×Dic₇₉

C₂³

yes

characteristic

C₄×Dic₇₉

C₂³

yes

characteristic

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

C₂³

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

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

—

C₂×Dic₇₉

C₂×D₄

yes

—

Dic₁₅₈

—

C₈×C₇₉

C₂×Q₈

yes

—

C₈×C₇₉

C₂×Q₈

yes

—

C₇₉⋊C₈

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

#135

yes

—

C₂×C₇₉

—

C₂×C₇₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₉

C₂×#135

yes

—

#3160

—

maximal, Sylow

C₇₉

#3160

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

D₇₉

yes

characteristic

C₂×C₄×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

—

158

—

#117

—

#117

C₂×D₇₉

yes

—

C₂×C₄²

—

#107

—

#107

—

100

#107

C₂×D₇₉

yes

—

101

#106

—

102

#106

—

103

#106

C₂×D₇₉

yes

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₂×Q₈

—

158

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²×D₇₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

Dic₁₅₈

yes

complemented

115

C₂×C₈

C₂²×D₇₉

yes

characteristic

116

C₂×C₈

C₂²×D₇₉

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

158

—

119

C₄⋊C₄

—

158

—

120

C₄⋊C₄

—

158

—

121

C₄⋊C₄

C₂²×D₇₉

yes

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

158

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

158

—

129

C₂³

D₄×D₇₉

yes

characteristic

130

Q₈

—

316

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

158

—

135

C₂×C₄

—

158

—

136

C₂×C₄

D₄×D₇₉

yes

characteristic

137

C₂×C₄

Q₈×D₇₉

yes

—

138

C₂×C₄

C₂×Dic₁₅₈

yes

—

139

C₂×C₄

C₂³×D₇₉

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

C₂×Dic₁₅₈

yes

—

143

C₈

C₂×Dic₁₅₈

yes

—

144

C₂²

—

145

C₂²

#14579160

yes

—

146

C₂²

C₂×D₄×D₇₉

yes

characteristic

147

C₄

—

148

C₄

—

316

—

149

C₄

—

150

C₄

—

158

—

151

C₄

#14579160

yes

—

152

C₄

C₂×Q₈×D₇₉

yes

characteristic

153

C₄

C₂²×Dic₁₅₈

yes

characteristic

154

C₂

—

155

C₂

C₂×#14579160

yes

characteristic

156

C₂

#135×D₇₉

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#427×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	#427	D₇₉
hypocenter	C₇₉	#3160
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#427×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂³×C₇₉

C₂×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #19098967	C₂	semidirect
3. #19098964	C₂	semidirect
4. #19098963	C₂	semidirect
5. #19098960	C₂	semidirect
31. C₂×C₇₉⋊C₈	Q₈	semidirect
44. C₂×C₈×C₇₉	Q₈	semidirect
79. C₇₉	#3160	semidirect
114. C₂×C₈	Dic₁₅₈	semidirect

H ≅ G/N

type

2. #19098967

C₂

semidirect

3. #19098964

C₂

semidirect

4. #19098963

C₂

semidirect

5. #19098960

C₂

semidirect

31. C₂×C₇₉⋊C₈

Q₈

semidirect

44. C₂×C₈×C₇₉

Q₈

semidirect

79. C₇₉

#3160

semidirect

114. C₂×C₈

Dic₁₅₈

semidirect

Subgroups — #23773912

Conjugacy classes of subgroups

Complements