Group #23773803 — subgroups

Subgroups — #23773803

Order 10112.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#19099113	C₂	1	1	yes	maximal, characteristic
2	#19098933	C₂	1	2	yes	maximal, complemented
3	#19098929	C₂	1	2	yes	maximal, complemented
4	#19098920	C₂	1	4	yes	maximal, complemented
5	#19098562	C₂	1	4	yes	maximal, complemented
6	C₄×#11929608	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#125×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929624	C₂²	1	2	yes	—
9	C₂×#11929608	C₂²	1	2	yes	—
10	#14579161	C₂²	1	1	yes	characteristic
11	#14579158	C₂²	1	1	yes	characteristic
12	#14579149	C₂²	1	2	yes	—
13	#14579135	—	2	2	no	—
14	C₄×D₃₁₆	—	2	2	no	—
15	#14579084	C₂²	1	8	yes	—
16	#14579081	C₂²	1	8	yes	—
17	#14579072	C₂²	1	2	yes	—
18	C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
19	#125×C₇₉	—	2	2	no	—
20	#125×C₇₉	C₂²	1	2	yes	—
21	#125×C₇₉	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
25	#11929624	—	2	8	no	—
26	C₂×Dic₁₅₈	C₂³	1	2	yes	—
27	C₂×D₃₁₆	—	2	2	no	—
28	C₂×D₃₁₆	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₇₉	—	2	2	no	—
30	C₂×C₄×D₇₉	—	2	2	no	—
31	#11929608	—	2	4	no	—
32	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
33	C₄⋊Dic₇₉	—	2	2	no	—
34	(C₂×D₇₉)⋊C₄	—	2	4	no	—
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	4	no	—
39	C₄⋊C₄×C₇₉	—	2	4	no	—
40	C₄⋊C₄×C₇₉	—	2	2	no	—
41	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
42	C₄²×C₇₉	D₄	1	2	yes	—
43	C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	C₄²×C₇₉	C₂³	1	1	yes	characteristic
45	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	2	4	no	—
48	C₂×C₄×C₇₉	—	4	2	no	—
49	C₂×C₄×C₇₉	—	4	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	1	yes	characteristic
53	C₂×C₄×C₇₉	#41	1	4	yes	—
54	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
55	C₂²⋊D₇₉	—	4	4	no	—
56	C₂×Dic₇₉	—	2	2	no	—
57	D₃₁₆	—	2	4	no	—
58	D₃₁₆	—	4	4	no	—
59	C₄×D₇₉	—	4	4	no	—
60	Dic₁₅₈	—	2	4	no	—
61	C₇₉⋊C₈	—	2	4	no	—
62	C₂²×C₇₉	—	2	2	no	—
63	C₂²×C₇₉	#124	1	2	yes	—
64	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₇₉	—	4	2	no	—
66	C₂×D₇₉	—	4	4	no	—
67	C₄×C₇₉	—	2	2	no	—
68	C₄×C₇₉	—	2	2	no	—
69	C₄×C₇₉	—	4	4	no	—
70	C₄×C₇₉	#124	1	2	yes	—
71	C₄×C₇₉	C₂×#41	1	2	yes	—
72	Dic₇₉	—	4	2	no	—
73	C₂×C₇₉	—	2	2	no	—
74	C₂×C₇₉	C₂×#124	1	1	yes	characteristic
75	C₂×C₇₉	#575	1	2	yes	—
76	D₇₉	—	8	2	no	—
77	#3142	—	79	1	no	maximal, Sylow
78	C₇₉	#3142	1	1	yes	characteristic, complemented, Sylow
79	#532	—	79	1	no	—
80	C₂×#125	D₇₉	1	1	yes	characteristic, complemented
81	#491	—	79	2	no	—
82	#490	—	79	2	no	—
83	#488	—	79	4	no	—
84	#417	—	79	4	no	—
85	C₄×M₁₆	—	79	1	no	—
86	C₂×#41	—	79	2	no	—
87	C₂×M₁₆	—	79	2	no	—
88	C₄⋊Q₈	—	79	1	no	—
89	#127	—	79	1	no	—
90	#125	—	2	2	no	—
91	#125	C₂×D₇₉	1	2	yes	—
92	#125	C₂×D₇₉	1	2	yes	—
93	#124	—	79	2	no	—
94	#121	—	158	2	no	—
95	C₄×D₄	—	158	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
98	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
99	#103	—	79	8	no	—
100	#102	—	79	8	no	—
101	#97	—	79	2	no	—
102	C₄×C₈	—	79	2	no	—
103	#41	—	158	8	no	—
104	C₂×Q₈	—	79	2	no	—
105	C₂×D₄	—	79	2	no	—
106	C₂×D₄	—	158	2	no	—
107	C₂×D₄	—	158	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	158	2	no	—
110	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
112	M₁₆	—	158	4	no	—
113	C₂×C₈	—	79	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	158	2	no	—
118	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
119	C₂²⋊C₄	—	158	4	no	—
120	C₄²	C₂²⋊D₇₉	1	2	yes	—
121	C₄²	C₂²×D₇₉	1	1	yes	characteristic
122	C₄²	C₂²×D₇₉	1	1	yes	characteristic
123	C₂³	—	158	2	no	—
124	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
125	Q₈	—	158	4	no	—
126	D₄	—	316	4	no	—
127	D₄	—	316	4	no	—
128	D₄	—	158	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	316	4	no	—
133	C₂×C₄	—	158	2	no	—
134	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
137	C₂×C₄	#11929625	1	4	yes	—
138	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
139	C₈	—	158	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	316	2	no	—
142	C₂²	—	316	4	no	—
143	C₂²	#14579150	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	316	2	no	—
149	C₄	#14579150	1	2	yes	—
150	C₄	C₂×#11929625	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	632	2	no	—
153	C₂	C₂×#14579150	1	1	yes	characteristic
154	C₂	#19099361	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19099113

C₂

yes

maximal, characteristic

#19098933

C₂

yes

maximal, complemented

#19098929

C₂

yes

maximal, complemented

#19098920

C₂

yes

maximal, complemented

#19098562

C₂

yes

maximal, complemented

C₄×#11929608

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929624

C₂²

yes

—

C₂×#11929608

C₂²

yes

—

#14579161

C₂²

yes

characteristic

#14579158

C₂²

yes

characteristic

#14579149

C₂²

yes

—

#14579135

—

C₄×D₃₁₆

—

#14579084

C₂²

yes

—

#14579081

C₂²

yes

—

#14579072

C₂²

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

—

#125×C₇₉

—

#125×C₇₉

C₂²

yes

—

#125×C₇₉

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#11929624

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂³

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

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

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#124

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#124

yes

—

C₄×C₇₉

C₂×#41

yes

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×#124

yes

characteristic

C₂×C₇₉

#575

yes

—

D₇₉

—

#3142

—

maximal, Sylow

C₇₉

#3142

yes

characteristic, complemented, Sylow

#532

—

C₂×#125

D₇₉

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#125

—

#125

C₂×D₇₉

yes

—

#125

C₂×D₇₉

yes

—

#124

—

#121

—

158

—

C₄×D₄

—

158

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#103

—

100

#102

—

101

#97

—

102

C₄×C₈

—

103

#41

—

158

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

158

—

107

C₂×D₄

—

158

—

108

C₂²×C₄

—

109

C₂²×C₄

—

158

—

110

C₂²×C₄

C₂²⋊D₇₉

yes

—

111

C₂²×C₄

C₂²×D₇₉

yes

characteristic

112

M₁₆

—

158

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

158

—

118

C₄⋊C₄

C₂²×D₇₉

yes

—

119

C₂²⋊C₄

—

158

—

120

C₄²

C₂²⋊D₇₉

yes

—

121

C₄²

C₂²×D₇₉

yes

characteristic

122

C₄²

C₂²×D₇₉

yes

characteristic

123

C₂³

—

158

—

124

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

125

Q₈

—

158

—

126

D₄

—

316

—

127

D₄

—

316

—

128

D₄

—

158

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

316

—

133

C₂×C₄

—

158

—

134

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

135

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

136

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

137

C₂×C₄

#11929625

yes

—

138

C₂×C₄

C₂³×D₇₉

yes

characteristic

139

C₈

—

158

—

140

C₂²

—

141

C₂²

—

316

—

142

C₂²

—

316

—

143

C₂²

#14579150

yes

—

144

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

316

—

149

C₄

#14579150

yes

—

150

C₄

C₂×#11929625

yes

—

151

C₂

—

152

C₂

—

632

—

153

C₂

C₂×#14579150

yes

characteristic

154

C₂

#19099361

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂×#125×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	#19099113	C₂
hypercenter	C₂×#125	D₇₉
hypocenter	C₇₉	#3142
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂×#125×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

C₂

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #19098933	C₂	semidirect
3. #19098929	C₂	semidirect
4. #19098920	C₂	semidirect
5. #19098562	C₂	semidirect
6. C₄×#11929608	C₂	semidirect
7. C₂×#125×C₇₉	C₂	semidirect
78. C₇₉	#3142	semidirect
80. C₂×#125	D₇₉	semidirect

H ≅ G/N

type

2. #19098933

C₂

semidirect

3. #19098929

C₂

semidirect

4. #19098920

C₂

semidirect

5. #19098562

C₂

semidirect

6. C₄×#11929608

C₂

semidirect

7. C₂×#125×C₇₉

C₂

semidirect

78. C₇₉

#3142

semidirect

80. C₂×#125

D₇₉

semidirect

Subgroups — #23773803

Conjugacy classes of subgroups

Complements