Group #23772045 — subgroups

Subgroups — #23772045

Order 10112.

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

normal subgroups only

This group has 400 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	C₄×#11929624	C₂	1	1	yes	maximal, characteristic
2	#19098625	C₂	1	4	yes	maximal, complemented
3	#19098614	C₂	1	4	yes	maximal, complemented
4	#19098591	C₂	1	2	yes	maximal, complemented
5	#19098531	C₂	1	2	yes	maximal
6	C₂×C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929624	C₂²	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₁₅₈	C₄	1	2	yes	—
11	C₄×D₃₁₆	C₄	1	2	yes	—
12	C₄×C₂²⋊D₇₉	C₄	1	4	yes	—
13	#14579120	C₂²	1	1	yes	characteristic
14	#14579118	C₂²	1	2	yes	—
15	C₄²×D₇₉	C₂²	1	2	yes	—
16	Dic₇₉⋊C₈	C₂²	1	8	yes	—
17	(C₂×D₇₉)⋊C₈	C₂²	1	8	yes	—
18	C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
19	C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
20	C₂×M₁₆×C₇₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
22	#97×C₇₉	C₂²	1	2	yes	—
23	C₄×C₈×C₇₉	C₂²	1	2	yes	—
24	#11929624	—	2	4	no	—
25	C₂×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
26	C₂×D₃₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₇₉	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₇₉	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₇₉	C₂³	1	2	yes	—
30	C₂×C₇₉⋊C₈	—	2	4	no	—
31	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
32	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
33	Dic₇₉⋊C₄	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₇₉	C₂×C₄	1	2	yes	—
35	(C₂×D₇₉)⋊C₄	C₂×C₄	1	4	yes	—
36	C₄×Dic₇₉	C₂×C₄	1	4	yes	—
37	C₄×Dic₇₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
40	M₁₆×C₇₉	—	2	4	no	—
41	C₂×C₈×C₇₉	C₂³	1	4	yes	—
42	C₄²×C₇₉	C₂×C₄	1	2	yes	—
43	C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	C₄²×C₇₉	C₂³	1	1	yes	characteristic
45	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₇₉	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₇₉	—	2	4	no	—
48	C₂×C₄×C₇₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₇₉	#41	1	4	yes	—
52	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
53	C₂²⋊D₇₉	—	2	4	no	—
54	C₂×Dic₇₉	C₂²×C₄	1	2	yes	—
55	C₂×Dic₇₉	C₂²×C₄	1	4	yes	—
56	D₃₁₆	—	2	2	no	—
57	C₄×D₇₉	—	2	8	no	—
58	C₄×D₇₉	—	2	4	no	—
59	Dic₁₅₈	—	2	2	no	—
60	C₈×C₇₉	—	2	4	no	—
61	C₇₉⋊C₈	—	2	4	no	—
62	C₂²×C₇₉	—	2	2	no	—
63	C₂²×C₇₉	#117	1	2	yes	—
64	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₇₉	—	2	2	no	—
66	C₂×D₇₉	—	2	4	no	—
67	C₄×C₇₉	—	2	2	no	—
68	C₄×C₇₉	#117	1	2	yes	—
69	C₄×C₇₉	#131	1	4	yes	—
70	C₄×C₇₉	C₂×#41	1	2	yes	—
71	Dic₇₉	—	2	2	no	—
72	Dic₇₉	—	2	4	no	—
73	C₂×C₇₉	—	2	2	no	—
74	C₂×C₇₉	C₂×#117	1	1	yes	characteristic
75	C₂×C₇₉	C₂×#131	1	2	yes	—
76	D₇₉	—	4	2	no	—
77	#2922	—	79	1	no	maximal, Sylow
78	C₇₉	#2922	1	1	yes	characteristic, complemented, Sylow
79	C₄×#41	—	79	1	no	—
80	#433	—	79	4	no	—
81	#431	—	79	4	no	—
82	#424	—	79	2	no	—
83	#408	—	79	2	no	—
84	C₄×M₁₆	D₇₉	1	1	yes	characteristic, complemented
85	C₂×C₄×C₈	—	79	1	no	—
86	C₂×#41	—	79	1	no	—
87	C₂×M₁₆	C₂×D₇₉	1	2	yes	—
88	C₂²×C₈	—	79	2	no	—
89	C₄×Q₈	—	79	2	no	—
90	C₄×D₄	—	79	4	no	—
91	C₄×D₄	—	79	2	no	—
92	#117	—	79	1	no	—
93	#117	—	79	2	no	—
94	C₂×C₄²	—	79	2	no	—
95	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
96	C₄⋊C₈	—	79	8	no	—
97	C₂²⋊C₈	—	79	8	no	—
98	#97	C₂×D₇₉	1	2	yes	—
99	C₄×C₈	—	79	2	no	—
100	C₄×C₈	—	79	2	no	—
101	C₄×C₈	C₂×D₇₉	1	2	yes	—
102	#41	—	158	4	no	—
103	C₂×Q₈	—	79	1	no	—
104	C₂×D₄	—	79	2	no	—
105	C₂×D₄	—	79	1	no	—
106	C₂²×C₄	—	79	4	no	—
107	C₂²×C₄	—	79	2	no	—
108	C₂²×C₄	C₄×D₇₉	1	2	yes	—
109	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	79	4	no	—
112	C₂×C₈	—	158	4	no	—
113	C₂×C₈	C₂²×D₇₉	1	4	yes	—
114	C₄⋊C₄	—	79	4	no	—
115	C₄⋊C₄	—	79	2	no	—
116	C₂²⋊C₄	—	79	2	no	—
117	C₂²⋊C₄	—	79	4	no	—
118	C₄²	—	79	4	no	—
119	C₄²	—	79	2	no	—
120	C₄²	C₄×D₇₉	1	2	yes	—
121	C₄²	C₂²×D₇₉	1	1	yes	characteristic
122	C₄²	C₂²×D₇₉	1	1	yes	characteristic
123	C₂³	—	79	2	no	—
124	C₂³	C₂×C₄×D₇₉	1	1	yes	characteristic
125	Q₈	—	158	2	no	—
126	D₄	—	158	4	no	—
127	D₄	—	158	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	79	4	no	—
130	C₂×C₄	—	79	2	no	—
131	C₂×C₄	—	158	4	no	—
132	C₂×C₄	—	158	8	no	—
133	C₂×C₄	C₂×C₄×D₇₉	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₇₉	1	2	yes	—
136	C₂×C₄	#11929624	1	4	yes	—
137	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	158	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	158	2	no	—
142	C₂²	—	158	4	no	—
143	C₂²	#14579118	1	2	yes	—
144	C₂²	C₂²×C₄×D₇₉	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	158	2	no	—
147	C₄	—	158	4	no	—
148	C₄	#14579118	1	2	yes	—
149	C₄	#14579169	1	4	yes	—
150	C₄	C₂×#11929624	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	316	2	no	—
153	C₂	C₂×#14579118	1	1	yes	characteristic
154	C₂	C₂×#14579169	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11929624

C₂

yes

maximal, characteristic

#19098625

C₂

yes

maximal, complemented

#19098614

C₂

yes

maximal, complemented

#19098591

C₂

yes

maximal, complemented

#19098531

C₂

yes

maximal

C₂×C₄×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929624

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

—

C₄×Dic₁₅₈

C₄

yes

—

C₄×D₃₁₆

C₄

yes

—

C₄×C₂²⋊D₇₉

C₄

yes

—

#14579120

C₂²

yes

characteristic

#14579118

C₂²

yes

—

C₄²×D₇₉

C₂²

yes

—

Dic₇₉⋊C₈

C₂²

yes

—

(C₂×D₇₉)⋊C₈

C₂²

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

—

C₂×M₁₆×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#97×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

—

#11929624

—

C₂×Dic₁₅₈

C₂×C₄

yes

characteristic

C₂×D₃₁₆

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₇₉

C₂×C₄

yes

—

C₂×C₄×D₇₉

C₂×C₄

yes

—

C₂×C₄×D₇₉

C₂³

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

Dic₇₉⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₇₉

C₂×C₄

yes

—

(C₂×D₇₉)⋊C₄

C₂×C₄

yes

—

C₄×Dic₇₉

C₂×C₄

yes

—

C₄×Dic₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂²×C₄

yes

characteristic

C₂²×D₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

C₂²×C₄

yes

—

C₂×Dic₇₉

C₂²×C₄

yes

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#117

yes

—

C₂²×C₇₉

C₂³×C₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#117

yes

—

C₄×C₇₉

#131

yes

—

C₄×C₇₉

C₂×#41

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×#117

yes

characteristic

C₂×C₇₉

C₂×#131

yes

—

D₇₉

—

#2922

—

maximal, Sylow

C₇₉

#2922

yes

characteristic, complemented, Sylow

C₄×#41

—

—

—

—

—

C₄×M₁₆

D₇₉

yes

characteristic, complemented

C₂×C₄×C₈

—

C₂×#41

—

C₂×M₁₆

C₂×D₇₉

yes

—

C₂²×C₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

C₂×D₇₉

yes

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂×D₇₉

yes

—

102

#41

—

158

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₇₉

yes

—

109

C₂²×C₄

C₂²×D₇₉

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

158

—

113

C₂×C₈

C₂²×D₇₉

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₇₉

yes

—

121

C₄²

C₂²×D₇₉

yes

characteristic

122

C₄²

C₂²×D₇₉

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₄×D₇₉

yes

characteristic

125

Q₈

—

158

—

126

D₄

—

158

—

127

D₄

—

158

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

158

—

132

C₂×C₄

—

158

—

133

C₂×C₄

C₂×C₄×D₇₉

yes

—

134

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

135

C₂×C₄

C₂×C₄×D₇₉

yes

—

136

C₂×C₄

#11929624

yes

—

137

C₂×C₄

C₂³×D₇₉

yes

characteristic

138

C₈

—

139

C₈

—

158

—

140

C₂²

—

141

C₂²

—

158

—

142

C₂²

—

158

—

143

C₂²

#14579118

yes

—

144

C₂²

C₂²×C₄×D₇₉

yes

characteristic

145

C₄

—

146

C₄

—

158

—

147

C₄

—

158

—

148

C₄

#14579118

yes

—

149

C₄

#14579169

yes

—

150

C₄

C₂×#11929624

yes

—

151

C₂

—

152

C₂

—

316

—

153

C₂

C₂×#14579118

yes

characteristic

154

C₂

C₂×#14579169

yes

—

	subgroup	G/H
center	C₄²	C₂²×D₇₉
commutator, socle	C₂²×C₇₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₄×M₁₆×C₇₉	C₂
prime radical	C₂×#11929624	C₂²
cosocle	C₂×C₄×C₇₉	C₂⁴
hypercenter	C₄×M₁₆	D₇₉
hypocenter	C₇₉	#2922
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²×D₇₉

commutator, socle

C₂²×C₇₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₄×M₁₆×C₇₉

C₂

prime radical

C₂×#11929624

C₂²

cosocle

C₂×C₄×C₇₉

C₂⁴

hypercenter

C₄×M₁₆

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #19098625	C₂	semidirect
3. #19098614	C₂	semidirect
4. #19098591	C₂	semidirect
6. C₂×C₄×C₇₉⋊C₈	C₂	semidirect
7. C₄×M₁₆×C₇₉	C₂	semidirect
78. C₇₉	#2922	semidirect
84. C₄×M₁₆	D₇₉	semidirect

H ≅ G/N

type

2. #19098625

C₂

semidirect

3. #19098614

C₂

semidirect

4. #19098591

C₂

semidirect

6. C₂×C₄×C₇₉⋊C₈

C₂

semidirect

7. C₄×M₁₆×C₇₉

C₂

semidirect

78. C₇₉

#2922

semidirect

84. C₄×M₁₆

D₇₉

semidirect

Subgroups — #23772045

Conjugacy classes of subgroups

Complements