Group #23766142 — subgroups

Subgroups — #23766142

Order 10112.

444 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 237 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	#19098462	C₂	1	2	yes	maximal
2	#19098462	C₂	1	4	yes	maximal
3	#19098449	C₂	1	4	yes	maximal
4	#19098433	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×Dic₇₉	C₂	1	2	yes	maximal
6	#19098409	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₇₉	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₇₉	C₂²	1	4	yes	—
11	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₇₉	C₂²	1	4	yes	—
13	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
14	C₇₉⋊#95	C₂²	1	2	yes	—
15	C₇₉⋊#95	C₂²	1	2	yes	—
16	C₇₉⋊#95	C₂²	1	4	yes	—
17	C₇₉⋊#95	C₂²	1	4	yes	—
18	#125×C₇₉	C₄	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₇₉	C₂²	1	4	yes	—
21	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
22	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₇₉	C₂³	1	4	yes	—
25	C₂²×Dic₇₉	C₂³	1	2	yes	—
26	C₄⋊Dic₇₉	—	2	4	no	—
27	C₄⋊Dic₇₉	—	4	4	no	—
28	C₄⋊Dic₇₉	—	4	4	no	—
29	C₄×Dic₇₉	—	2	2	no	—
30	C₄×Dic₇₉	—	4	4	no	—
31	C₄×Dic₇₉	Q₈	1	4	yes	—
32	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₇₉	C₂³	1	4	yes	—
34	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₇₉	C₂×C₄	1	16	yes	—
36	C₄⋊C₄×C₇₉	C₂×C₄	1	8	yes	—
37	C₄²×C₇₉	C₂×C₄	1	4	yes	—
38	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₇₉	—	2	4	no	—
40	C₂×C₄×C₇₉	—	2	2	no	—
41	C₂×C₄×C₇₉	—	2	2	no	—
42	C₂×C₄×C₇₉	—	2	4	no	—
43	C₂×C₄×C₇₉	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₇₉	C₂²×C₄	1	8	yes	—
45	C₂×C₄×C₇₉	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₇₉	#41	1	2	yes	—
48	C₂×Dic₇₉	—	2	2	no	—
49	C₂×Dic₇₉	—	2	2	no	—
50	C₂×Dic₇₉	—	2	1	no	—
51	C₂×Dic₇₉	—	2	1	no	—
52	C₂×Dic₇₉	—	2	4	no	—
53	C₂×Dic₇₉	—	2	4	no	—
54	C₂×Dic₇₉	—	2	2	no	—
55	C₂×Dic₇₉	—	2	4	no	—
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	C₂×Q₈	1	4	yes	—
58	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
59	C₂²×C₇₉	C₂²×Q₈	1	1	yes	characteristic
60	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
61	C₂²×C₇₉	H(5,𝔽₂)	1	2	yes	—
62	C₂²×C₇₉	2₋¹⁺⁴	1	2	yes	—
63	C₄×C₇₉	—	2	8	no	—
64	C₄×C₇₉	—	2	4	no	—
65	C₄×C₇₉	C₄×Q₈	1	4	yes	—
66	Dic₇₉	—	2	4	no	—
67	Dic₇₉	—	4	1	no	—
68	Dic₇₉	—	4	4	no	—
69	Dic₇₉	—	4	1	no	—
70	C₂×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₂×C₇₉	#520	1	2	yes	—
72	C₂×C₇₉	#540	1	1	yes	characteristic
73	C₂×C₇₉	#541	1	1	yes	characteristic
74	C₂×C₇₉	#544	1	2	yes	—
75	#2330	—	79	1	no	maximal, Sylow
76	C₇₉	#2330	1	1	yes	characteristic, complemented, Sylow
77	C₂×#125	D₇₉	1	1	yes	characteristic
78	#389	—	79	4	no	—
79	#389	—	79	2	no	—
80	#387	—	79	4	no	—
81	#384	—	79	1	no	—
82	C₄×C₄⋊C₄	—	79	2	no	—
83	#376	—	79	1	no	—
84	#125	Dic₇₉	1	8	yes	complemented
85	C₂×C₄⋊C₄	—	79	4	no	—
86	C₂×C₄⋊C₄	—	79	4	no	—
87	C₂×C₄⋊C₄	—	79	2	no	—
88	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
89	C₂×C₄⋊C₄	C₂×D₇₉	1	4	yes	—
90	C₂×C₄²	—	79	4	no	—
91	C₂×C₄²	—	79	1	no	—
92	C₂×C₄²	—	79	1	no	—
93	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
94	#95	—	79	2	no	—
95	#95	—	79	2	no	—
96	#95	—	79	4	no	—
97	#95	—	79	4	no	—
98	C₂²×C₄	—	79	2	no	—
99	C₂²×C₄	—	79	1	no	—
100	C₂²×C₄	—	79	1	no	—
101	C₂²×C₄	—	79	4	no	—
102	C₂²×C₄	C₂²×D₇₉	1	4	yes	—
103	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
104	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
105	C₄⋊C₄	—	316	4	no	—
106	C₄⋊C₄	—	316	4	no	—
107	C₄⋊C₄	—	158	4	no	—
108	C₄⋊C₄	C₂×Dic₇₉	1	16	yes	—
109	C₄⋊C₄	C₂×Dic₇₉	1	8	yes	—
110	C₄²	—	316	4	no	—
111	C₄²	—	79	4	no	—
112	C₄²	—	158	2	no	—
113	C₄²	C₂×Dic₇₉	1	4	yes	—
114	C₂³	C₂³×D₇₉	1	1	yes	characteristic
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	79	4	no	—
120	C₂×C₄	—	158	1	no	—
121	C₂×C₄	—	158	1	no	—
122	C₂×C₄	—	158	4	no	—
123	C₂×C₄	—	158	4	no	—
124	C₂×C₄	—	158	2	no	—
125	C₂×C₄	—	158	4	no	—
126	C₂×C₄	—	158	2	no	—
127	C₂×C₄	—	158	2	no	—
128	C₂×C₄	—	158	2	no	—
129	C₂×C₄	C₂²×Dic₇₉	1	4	yes	—
130	C₂×C₄	C₂²×Dic₇₉	1	8	yes	—
131	C₂×C₄	C₂²×Dic₇₉	1	2	yes	—
132	C₂×C₄	Q₈×D₇₉	1	2	yes	—
133	C₂×C₄	#11929625	1	2	yes	—
134	C₂²	C₂³×Dic₇₉	1	1	yes	characteristic
135	C₂²	C₂×Q₈×D₇₉	1	1	yes	characteristic
136	C₂²	C₂×#11929625	1	1	yes	characteristic
137	C₂²	#14579207	1	2	yes	—
138	C₂²	#14579208	1	2	yes	—
139	C₄	—	2	8	no	—
140	C₄	—	2	4	no	—
141	C₄	—	316	1	no	—
142	C₄	—	316	4	no	—
143	C₄	—	316	1	no	—
144	C₄	—	158	4	no	—
145	C₄	Q₈×Dic₇₉	1	4	yes	—
146	C₂	C₂×Q₈×Dic₇₉	1	1	yes	characteristic
147	C₂	#19099046	1	2	yes	—
148	C₂	#19099169	1	1	yes	characteristic
149	C₂	#19099176	1	1	yes	characteristic
150	C₂	#19099193	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19098462

C₂

yes

maximal

#19098462

C₂

yes

maximal

#19098449

C₂

yes

maximal

#19098433

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₇₉

C₂

yes

maximal

#19098409

C₂

yes

maximal, characteristic

C₂×#125×C₇₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

C₂×C₄×Dic₇₉

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

#125×C₇₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

C₂³

yes

characteristic

C₂²×Dic₇₉

C₂³

yes

characteristic

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

Q₈

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

C₂×C₄

yes

—

C₂³×C₇₉

C₂⁴

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×Q₈

yes

—

C₂²×C₇₉

C₂³×C₄

yes

characteristic

C₂²×C₇₉

C₂²×Q₈

yes

characteristic

C₂²×C₇₉

C₂×#41

yes

characteristic

C₂²×C₇₉

H(5,𝔽₂)

yes

—

C₂²×C₇₉

2₋¹⁺⁴

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×Q₈

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₇₉

#520

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

#2330

—

maximal, Sylow

C₇₉

#2330

yes

characteristic, complemented, Sylow

C₂×#125

D₇₉

yes

—

—

—

—

C₄×C₄⋊C₄

—

#376

—

#125

Dic₇₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₉

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₇₉

yes

—

103

C₂²×C₄

C₂²×D₇₉

yes

—

104

C₂²×C₄

C₂²×D₇₉

yes

characteristic

105

C₄⋊C₄

—

316

—

106

C₄⋊C₄

—

316

—

107

C₄⋊C₄

—

158

—

108

C₄⋊C₄

C₂×Dic₇₉

yes

—

109

C₄⋊C₄

C₂×Dic₇₉

yes

—

110

C₄²

—

316

—

111

C₄²

—

112

C₄²

—

158

—

113

C₄²

C₂×Dic₇₉

yes

—

114

C₂³

C₂³×D₇₉

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

158

—

121

C₂×C₄

—

158

—

122

C₂×C₄

—

158

—

123

C₂×C₄

—

158

—

124

C₂×C₄

—

158

—

125

C₂×C₄

—

158

—

126

C₂×C₄

—

158

—

127

C₂×C₄

—

158

—

128

C₂×C₄

—

158

—

129

C₂×C₄

C₂²×Dic₇₉

yes

—

130

C₂×C₄

C₂²×Dic₇₉

yes

—

131

C₂×C₄

C₂²×Dic₇₉

yes

—

132

C₂×C₄

Q₈×D₇₉

yes

—

133

C₂×C₄

#11929625

yes

—

134

C₂²

C₂³×Dic₇₉

yes

characteristic

135

C₂²

C₂×Q₈×D₇₉

yes

characteristic

136

C₂²

C₂×#11929625

yes

characteristic

137

C₂²

#14579207

yes

—

138

C₂²

#14579208

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

316

—

142

C₄

—

316

—

143

C₄

—

316

—

144

C₄

—

158

—

145

C₄

Q₈×Dic₇₉

yes

—

146

C₂

C₂×Q₈×Dic₇₉

yes

characteristic

147

C₂

#19099046

yes

—

148

C₂

#19099169

yes

characteristic

149

C₂

#19099176

yes

characteristic

150

C₂

#19099193

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂³×C₄
Fitting	C₂×#125×C₇₉	C₂
socle, prime radical, cosocle	C₂³×C₇₉	C₂⁴
hypercenter	C₂×#125	D₇₉
hypocenter	C₇₉	#2330
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator

C₂²×C₇₉

C₂³×C₄

Fitting

C₂×#125×C₇₉

C₂

socle, prime radical, cosocle

C₂³×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
18. #125×C₇₉	C₄	semidirect
76. C₇₉	#2330	semidirect
84. #125	Dic₇₉	semidirect

H ≅ G/N

type

18. #125×C₇₉

C₄

semidirect

76. C₇₉

#2330

semidirect

84. #125

Dic₇₉

semidirect

Subgroups — #23766142

Conjugacy classes of subgroups

Complements