Group #23765429 — subgroups

Subgroups — #23765429

Order 10112.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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	#19099293	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098746	C₂	1	2	yes	maximal
3	#19098362	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098361	C₂	1	2	yes	maximal, complemented
5	#456×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#14579191	—	2	2	no	—
7	#14579160	—	2	2	no	—
8	#14579159	—	2	1	no	—
9	#14579159	C₂²	1	2	yes	—
10	C₄×Dic₁₅₈	—	2	1	no	—
11	#14579126	—	2	2	no	—
12	#14579088	—	2	2	no	—
13	#14579080	C₂²	1	2	yes	—
14	2₋¹⁺⁴×C₇₉	C₂²	1	1	yes	characteristic
15	#137×C₇₉	—	2	1	no	—
16	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
17	#104×C₇₉	—	2	1	no	—
18	#101×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₅₈	—	2	1	no	—
20	C₂×Dic₁₅₈	—	2	1	no	—
21	C₂×Dic₁₅₈	D₄	1	2	yes	—
22	#11929614	—	4	2	no	—
23	#11929612	—	4	2	no	—
24	#11929608	—	2	2	no	—
25	C₄⋊Dic₇₉	—	2	1	no	—
26	Dic₇₉⋊C₄	—	2	2	no	—
27	Dic₇₉⋊C₄	—	2	2	no	—
28	Dic₇₉⋊C₄	—	2	2	no	—
29	C₄×Dic₇₉	—	2	1	no	—
30	C₄×Dic₇₉	—	2	2	no	—
31	C₄×Dic₇₉	D₄	1	2	yes	—
32	#41×C₇₉	—	2	1	no	—
33	#41×C₇₉	—	2	2	no	—
34	#41×C₇₉	—	4	1	no	—
35	C₂×Q₈×C₇₉	—	4	1	no	—
36	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
38	Q₁₆×C₇₉	—	4	1	no	—
39	SD₁₆×C₇₉	—	4	1	no	—
40	M₁₆×C₇₉	—	2	1	no	—
41	C₄⋊C₄×C₇₉	—	2	2	no	—
42	C₄²×C₇₉	D₄	1	1	yes	characteristic
43	Q₈×C₇₉	—	2	2	no	—
44	Q₈×C₇₉	—	2	1	no	—
45	Q₈×C₇₉	—	4	1	no	—
46	Q₈×C₇₉	—	4	1	no	—
47	D₄×C₇₉	—	2	2	no	—
48	D₄×C₇₉	—	2	1	no	—
49	D₄×C₇₉	—	4	1	no	—
50	C₂×C₄×C₇₉	—	2	1	no	—
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	—	4	1	no	—
53	C₂×C₄×C₇₉	—	4	2	no	—
54	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	—	2	2	no	—
58	Dic₁₅₈	—	2	4	no	—
59	Dic₁₅₈	—	4	2	no	—
60	Dic₁₅₈	—	8	1	no	—
61	C₈×C₇₉	—	4	1	no	—
62	C₇₉⋊C₈	—	4	2	no	—
63	C₂²×C₇₉	—	4	1	no	—
64	C₂²×C₇₉	#120	1	1	yes	characteristic
65	C₄×C₇₉	—	2	2	no	—
66	C₄×C₇₉	—	2	1	no	—
67	C₄×C₇₉	—	2	2	no	—
68	C₄×C₇₉	—	4	1	no	—
69	C₄×C₇₉	—	4	1	no	—
70	Dic₇₉	—	2	4	no	—
71	Dic₇₉	—	4	2	no	—
72	C₂×C₇₉	—	2	1	no	—
73	C₂×C₇₉	—	8	1	no	—
74	C₂×C₇₉	B(4,𝔽₂)	1	1	yes	characteristic
75	#2205	—	79	1	no	maximal, Sylow
76	C₇₉	#2205	1	1	yes	characteristic, complemented, Sylow
77	Q₈²	—	79	1	no	—
78	#456	—	79	2	no	—
79	#456	D₇₉	1	1	yes	characteristic
80	#356	—	79	1	no	—
81	#356	—	79	2	no	—
82	2₋¹⁺⁴	C₂×D₇₉	1	1	yes	characteristic
83	#137	—	2	1	no	—
84	#137	—	158	2	no	—
85	C₄⋊Q₈	—	79	2	no	—
86	C₄⋊Q₈	—	158	2	no	—
87	C₄⋊Q₈	—	158	1	no	—
88	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
89	C₄×Q₈	—	158	2	no	—
90	C₄×Q₈	—	158	1	no	—
91	#104	—	2	1	no	—
92	#104	—	158	2	no	—
93	#101	—	79	2	no	—
94	#101	C₂×D₇₉	1	1	yes	characteristic
95	#41	—	2	2	no	—
96	#41	—	2	1	no	—
97	#41	—	4	1	no	—
98	C₂×Q₈	—	4	1	no	—
99	C₂×Q₈	—	79	2	no	—
100	C₂×Q₈	—	158	1	no	—
101	C₂×Q₈	—	158	1	no	—
102	C₂×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
103	C₂×Q₈	C₂²×D₇₉	1	1	yes	characteristic
104	Q₁₆	—	4	1	no	—
105	Q₁₆	—	316	2	no	—
106	SD₁₆	—	4	1	no	—
107	SD₁₆	—	316	2	no	—
108	M₁₆	—	2	1	no	—
109	M₁₆	—	158	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	158	2	no	—
112	C₄⋊C₄	—	158	1	no	—
113	C₄⋊C₄	—	158	2	no	—
114	C₄⋊C₄	—	158	2	no	—
115	C₄²	—	79	2	no	—
116	C₄²	—	158	2	no	—
117	C₄²	—	158	1	no	—
118	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
119	Q₈	—	2	1	no	—
120	Q₈	—	2	2	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	316	2	no	—
124	Q₈	—	632	1	no	—
125	Q₈	—	158	4	no	—
126	D₄	—	2	1	no	—
127	D₄	—	2	2	no	—
128	D₄	—	4	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	158	2	no	—
134	C₂×C₄	—	158	2	no	—
135	C₂×C₄	D₄×D₇₉	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
137	C₈	—	4	1	no	—
138	C₈	—	316	2	no	—
139	C₂²	—	4	1	no	—
140	C₂²	#14579129	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	2	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	316	2	no	—
147	C₄	—	158	4	no	—
148	C₂	—	2	1	no	—
149	C₂	—	8	1	no	—
150	C₂	#19098749	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099293

C₂

yes

maximal, characteristic, complemented

#19098746

C₂

yes

maximal

#19098362

C₂

yes

maximal, characteristic, complemented

#19098361

C₂

yes

maximal, complemented

#456×C₇₉

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×Dic₁₅₈

—

#14579126

—

#14579088

—

#14579080

C₂²

yes

—

2₋¹⁺⁴×C₇₉

C₂²

yes

characteristic

#137×C₇₉

—

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#104×C₇₉

—

#101×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

—

#11929614

—

#11929612

—

#11929608

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

D₄

yes

—

#41×C₇₉

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×Q₈×C₇₉

C₂³

yes

characteristic

Q₁₆×C₇₉

—

SD₁₆×C₇₉

—

M₁₆×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

D₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×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

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

Q₈²

—

#456

—

#456

D₇₉

yes

characteristic

#356

—

#356

—

2₋¹⁺⁴

C₂×D₇₉

yes

characteristic

#137

—

#137

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

C₄×Q₈

—

158

—

C₄×Q₈

—

158

—

#104

—

#104

—

158

—

#101

—

#101

C₂×D₇₉

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

158

—

101

C₂×Q₈

—

158

—

102

C₂×Q₈

C₂²⋊D₇₉

yes

characteristic

103

C₂×Q₈

C₂²×D₇₉

yes

characteristic

104

Q₁₆

—

105

Q₁₆

—

316

—

106

SD₁₆

—

107

SD₁₆

—

316

—

108

M₁₆

—

109

M₁₆

—

158

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

158

—

112

C₄⋊C₄

—

158

—

113

C₄⋊C₄

—

158

—

114

C₄⋊C₄

—

158

—

115

C₄²

—

116

C₄²

—

158

—

117

C₄²

—

158

—

118

C₄²

C₂²⋊D₇₉

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

316

—

124

Q₈

—

632

—

125

Q₈

—

158

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

158

—

134

C₂×C₄

—

158

—

135

C₂×C₄

D₄×D₇₉

yes

—

136

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

137

C₈

—

138

C₈

—

316

—

139

C₂²

—

140

C₂²

#14579129

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

316

—

147

C₄

—

158

—

148

C₂

—

149

C₂

—

150

C₂

#19098749

yes

characteristic

	subgroup	G/H
center	C₂	#19098749
commutator, cosocle	C₂×Q₈×C₇₉	C₂³
Frattini	C₂×Q₈	C₂²×D₇₉
Fitting	#456×C₇₉	C₂
socle	C₂×C₇₉	B(4,𝔽₂)
prime radical	2₋¹⁺⁴×C₇₉	C₂²
hypercenter	#456	D₇₉
hypocenter	C₇₉	#2205
LCS	G ▷ C₂×Q₈×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #456

subgroup

G/H

center

C₂

#19098749

commutator, cosocle

C₂×Q₈×C₇₉

C₂³

Frattini

C₂×Q₈

C₂²×D₇₉

Fitting

#456×C₇₉

C₂

socle

C₂×C₇₉

B(4,𝔽₂)

prime radical

2₋¹⁺⁴×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×Q₈×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #456

N	H ≅ G/N	type
1. #19099293	C₂	semidirect
3. #19098362	C₂	semidirect
4. #19098361	C₂	semidirect
76. C₇₉	#2205	semidirect

H ≅ G/N

type

1. #19099293

C₂

semidirect

3. #19098362

C₂

semidirect

4. #19098361

C₂

semidirect

76. C₇₉

#2205

semidirect

Subgroups — #23765429

Conjugacy classes of subgroups

Complements