Group #23771998 — subgroups

Subgroups — #23771998

Order 10112.

476 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 253 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×#14579168	C₂	1	1	yes	maximal, characteristic
2	#19099046	C₂	1	1	yes	maximal, characteristic
3	#19098606	C₂	1	3	yes	maximal, complemented
4	#19098603	C₂	1	3	yes	maximal, complemented
5	#19098599	C₂	1	3	yes	maximal, complemented
6	C₂×#14579094	C₂	1	3	yes	maximal, complemented
7	#520×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#14579168	C₄	1	8	yes	complemented
9	C₂×#11929608	C₂²	1	3	yes	—
10	C₂²×C₇₉⋊C₈	C₂²	1	3	yes	—
11	Q₈×Dic₇₉	—	2	1	no	—
12	D₄×Dic₇₉	—	2	3	no	—
13	#14579120	C₂²	1	3	yes	—
14	C₂×C₄⋊Dic₇₉	C₂²	1	3	yes	—
15	#14579094	C₂²	1	3	yes	—
16	#14579094	C₂²	1	6	yes	—
17	#14579094	C₂²	1	1	yes	characteristic
18	#14579092	C₂²	1	3	yes	—
19	#14579092	C₂²	1	3	yes	—
20	C₂×#41×C₇₉	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₇₉	—	2	1	no	—
22	C₄×D₄×C₇₉	—	2	3	no	—
23	#117×C₇₉	C₂²	1	3	yes	—
24	C₂×C₄⋊C₄×C₇₉	C₂²	1	3	yes	—
25	C₂²×Dic₇₉	—	2	3	no	—
26	#11929608	C₂×C₄	1	12	yes	complemented
27	C₂×C₇₉⋊C₈	C₂×C₄	1	12	yes	complemented
28	C₂×C₇₉⋊C₈	C₂³	1	3	yes	—
29	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
30	C₄⋊Dic₇₉	—	2	3	no	—
31	C₄⋊Dic₇₉	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₇₉	C₂³	1	3	yes	—
33	C₂²⋊Dic₇₉	—	2	3	no	—
34	C₄×Dic₇₉	—	2	3	no	—
35	#41×C₇₉	C₂×C₄	1	4	yes	—
36	#41×C₇₉	Q₈	1	4	yes	—
37	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
38	C₂×D₄×C₇₉	D₄	1	3	yes	—
39	C₂²×C₄×C₇₉	—	2	3	no	—
40	C₂²×C₄×C₇₉	C₂³	1	3	yes	—
41	C₄⋊C₄×C₇₉	—	2	3	no	—
42	C₄⋊C₄×C₇₉	C₂³	1	3	yes	—
43	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
44	C₂²⋊C₄×C₇₉	—	2	3	no	—
45	C₄²×C₇₉	—	2	3	no	—
46	C₂³×C₇₉	C₂×D₄	1	3	yes	—
47	Q₈×C₇₉	C₄⋊C₄	1	4	yes	—
48	D₄×C₇₉	C₄⋊C₄	1	12	yes	—
49	C₂×C₄×C₇₉	—	2	1	no	—
50	C₂×C₄×C₇₉	—	2	3	no	—
51	C₂×C₄×C₇₉	—	8	3	no	—
52	C₂×C₄×C₇₉	C₂²×C₄	1	6	yes	—
53	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₇₉	C₂×Q₈	1	6	yes	—
55	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₇₉	—	2	1	no	—
57	C₂×Dic₇₉	—	2	3	no	—
58	C₂×Dic₇₉	—	8	3	no	—
59	C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
60	C₇₉⋊C₈	C₂²×C₄	1	6	yes	—
61	C₂²×C₇₉	—	2	3	no	—
62	C₂²×C₇₉	—	2	3	no	—
63	C₂²×C₇₉	C₂×C₄⋊C₄	1	6	yes	—
64	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₇₉	—	4	3	no	—
66	C₄×C₇₉	—	4	1	no	—
67	C₄×C₇₉	C₂×C₄⋊C₄	1	6	yes	—
68	C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
70	Dic₇₉	—	4	3	no	—
71	Dic₇₉	—	4	1	no	—
72	C₂×C₇₉	—	2	6	no	—
73	C₂×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₇₉	#576	1	1	yes	characteristic
75	C₂×C₇₉	#578	1	1	yes	characteristic
76	#2913	—	79	1	no	maximal, Sylow
77	C₇₉	#2913	1	1	yes	characteristic, complemented, Sylow
78	C₂×#131	—	79	1	no	—
79	#520	—	79	1	no	—
80	#520	D₇₉	1	1	yes	characteristic
81	#428	—	79	3	no	—
82	#428	—	79	3	no	—
83	#427	—	79	3	no	—
84	C₂×#107	—	79	3	no	—
85	C₂×#41	C₂×D₇₉	1	1	yes	characteristic
86	#131	—	79	8	no	—
87	C₂×M₁₆	—	79	3	no	—
88	C₂²×C₈	—	79	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	—	158	1	no	—
91	C₄×D₄	—	2	3	no	—
92	C₄×D₄	—	158	3	no	—
93	#117	—	79	3	no	—
94	#117	C₂×D₇₉	1	3	yes	—
95	C₂×C₄⋊C₄	—	79	3	no	—
96	C₂×C₄⋊C₄	C₂×D₇₉	1	3	yes	—
97	#107	—	79	3	no	—
98	#107	—	79	6	no	—
99	#107	—	79	1	no	—
100	#106	—	79	3	no	—
101	#106	—	79	3	no	—
102	#41	Dic₁₅₈	1	4	yes	—
103	#41	C₄×D₇₉	1	4	yes	—
104	C₂×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
105	C₂×D₄	C₂²⋊D₇₉	1	3	yes	—
106	C₂²×C₄	—	2	3	no	—
107	C₂²×C₄	—	158	3	no	—
108	C₂²×C₄	C₂²×D₇₉	1	3	yes	—
109	M₁₆	—	79	12	no	—
110	C₂×C₈	—	79	12	no	—
111	C₂×C₈	—	79	3	no	—
112	C₂×C₈	—	79	1	no	—
113	C₄⋊C₄	—	2	3	no	—
114	C₄⋊C₄	—	79	1	no	—
115	C₄⋊C₄	—	79	3	no	—
116	C₄⋊C₄	—	158	3	no	—
117	C₄⋊C₄	C₂²×D₇₉	1	1	yes	characteristic
118	C₄⋊C₄	C₂²×D₇₉	1	3	yes	—
119	C₂²⋊C₄	—	2	3	no	—
120	C₂²⋊C₄	—	158	3	no	—
121	C₄²	—	2	3	no	—
122	C₄²	—	158	3	no	—
123	C₂³	C₂×C₂²⋊D₇₉	1	3	yes	—
124	Q₈	Dic₇₉⋊C₄	1	4	yes	—
125	D₄	Dic₇₉⋊C₄	1	12	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	8	3	no	—
129	C₂×C₄	—	632	3	no	—
130	C₂×C₄	—	158	1	no	—
131	C₂×C₄	—	158	3	no	—
132	C₂×C₄	C₂×C₄×D₇₉	1	6	yes	—
133	C₂×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
134	C₂×C₄	C₂×Dic₁₅₈	1	6	yes	—
135	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
136	C₈	—	79	2	no	—
137	C₈	—	79	6	no	—
138	C₂²	—	2	3	no	—
139	C₂²	—	2	3	no	—
140	C₂²	C₂×Dic₇₉⋊C₄	1	6	yes	—
141	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
142	C₄	—	4	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	316	3	no	—
145	C₄	—	316	1	no	—
146	C₄	C₂×Dic₇₉⋊C₄	1	6	yes	—
147	C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
148	C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
149	C₂	—	2	6	no	—
150	C₂	C₂²×Dic₇₉⋊C₄	1	1	yes	characteristic
151	C₂	#19099366	1	1	yes	characteristic
152	C₂	#19099378	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579168

C₂

yes

maximal, characteristic

#19099046

C₂

yes

maximal, characteristic

#19098606

C₂

yes

maximal, complemented

#19098603

C₂

yes

maximal, complemented

#19098599

C₂

yes

maximal, complemented

C₂×#14579094

C₂

yes

maximal, complemented

#520×C₇₉

C₂

yes

maximal, characteristic

#14579168

C₄

yes

complemented

C₂×#11929608

C₂²

yes

—

C₂²×C₇₉⋊C₈

C₂²

yes

—

Q₈×Dic₇₉

—

D₄×Dic₇₉

—

#14579120

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

#14579094

C₂²

yes

—

#14579094

C₂²

yes

—

#14579094

C₂²

yes

characteristic

#14579092

C₂²

yes

—

#14579092

C₂²

yes

—

C₂×#41×C₇₉

C₂²

yes

characteristic

C₄×Q₈×C₇₉

—

C₄×D₄×C₇₉

—

#117×C₇₉

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂²×Dic₇₉

—

#11929608

C₂×C₄

yes

complemented

C₂×C₇₉⋊C₈

C₂×C₄

yes

complemented

C₂×C₇₉⋊C₈

C₂³

yes

—

C₂×C₇₉⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

characteristic

C₄⋊Dic₇₉

C₂³

yes

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

#41×C₇₉

C₂×C₄

yes

—

#41×C₇₉

Q₈

yes

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×D₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

C₂³

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

—

Q₈×C₇₉

C₄⋊C₄

yes

—

D₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₇₉⋊C₈

C₂²×C₄

yes

—

C₇₉⋊C₈

C₂²×C₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₄⋊C₄

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₄×C₇₉

C₂³×C₄

yes

characteristic

C₄×C₇₉

C₂²×Q₈

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂²×C₄⋊C₄

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×#131

—

#520

—

#520

D₇₉

yes

—

—

—

—

C₂×#41

C₂×D₇₉

yes

characteristic

#131

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

158

—

C₄×D₄

—

C₄×D₄

—

158

—

#117

—

#117

C₂×D₇₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

#107

—

#107

—

#107

—

100

#106

—

101

#106

—

102

#41

Dic₁₅₈

yes

—

103

#41

C₄×D₇₉

yes

—

104

C₂×Q₈

C₂²⋊D₇₉

yes

characteristic

105

C₂×D₄

C₂²⋊D₇₉

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

158

—

108

C₂²×C₄

C₂²×D₇₉

yes

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

158

—

117

C₄⋊C₄

C₂²×D₇₉

yes

characteristic

118

C₄⋊C₄

C₂²×D₇₉

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

158

—

121

C₄²

—

122

C₄²

—

158

—

123

C₂³

C₂×C₂²⋊D₇₉

yes

—

124

Q₈

Dic₇₉⋊C₄

yes

—

125

D₄

Dic₇₉⋊C₄

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

632

—

130

C₂×C₄

—

158

—

131

C₂×C₄

—

158

—

132

C₂×C₄

C₂×C₄×D₇₉

yes

—

133

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

134

C₂×C₄

C₂×Dic₁₅₈

yes

—

135

C₂×C₄

C₂³×D₇₉

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×Dic₇₉⋊C₄

yes

—

141

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

316

—

145

C₄

—

316

—

146

C₄

C₂×Dic₇₉⋊C₄

yes

—

147

C₄

C₂²×C₄×D₇₉

yes

characteristic

148

C₄

C₂²×Dic₁₅₈

yes

characteristic

149

C₂

—

150

C₂

C₂²×Dic₇₉⋊C₄

yes

characteristic

151

C₂

#19099366

yes

characteristic

152

C₂

#19099378

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#520×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂×#41×C₇₉	C₂²
cosocle	C₂×C₄×C₇₉	C₂⁴
hypercenter	#520	D₇₉
hypocenter	C₇₉	#2913
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₉

commutator

C₄×C₇₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#520×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂×#41×C₇₉

C₂²

cosocle

C₂×C₄×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

N	H ≅ G/N	type
3. #19098606	C₂	semidirect
4. #19098603	C₂	semidirect
5. #19098599	C₂	semidirect
6. C₂×#14579094	C₂	semidirect
8. #14579168	C₄	semidirect
26. #11929608	C₂×C₄	semidirect
27. C₂×C₇₉⋊C₈	C₂×C₄	semidirect
77. C₇₉	#2913	semidirect

H ≅ G/N

type

3. #19098606

C₂

semidirect

4. #19098603

C₂

semidirect

5. #19098599

C₂

semidirect

6. C₂×#14579094

C₂

semidirect

8. #14579168

C₄

semidirect

26. #11929608

C₂×C₄

semidirect

27. C₂×C₇₉⋊C₈

C₂×C₄

semidirect

77. C₇₉

#2913

semidirect

Subgroups — #23771998

Conjugacy classes of subgroups

Complements