Group #34187662 — subgroups

Subgroups — #34187662

Order 20224.

370 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 119 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	#23763531	C₂	1	1	yes	maximal, characteristic
2	#23761964	C₂	1	4	yes	maximal
3	#23761692	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#3160	C₂	1	1	yes	maximal, characteristic
5	#19098616	C₂²	1	2	yes	—
6	#19098429	C₂²	1	2	yes	—
7	#19098326	—	2	2	no	—
8	#19098300	—	2	8	no	—
9	#19098285	—	2	2	no	—
10	#533×C₇₉	C₂²	1	2	yes	—
11	#500×C₇₉	C₄	1	2	yes	complemented
12	#499×C₇₉	—	2	2	no	—
13	#498×C₇₉	C₄	1	2	yes	complemented
14	#427×C₇₉	—	2	2	no	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇₉	—	2	2	no	—
17	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
19	C₄×C₇₉⋊C₈	—	2	2	no	—
20	C₇₉⋊#95	—	2	4	no	—
21	C₂²×C₈×C₇₉	D₄	1	2	yes	—
22	C₄⋊Q₈×C₇₉	C₂×C₄	1	4	yes	—
23	#125×C₇₉	—	2	2	no	—
24	C₂²⋊Q₈×C₇₉	—	2	4	no	—
25	C₄×Q₈×C₇₉	—	2	4	no	—
26	#117×C₇₉	—	2	2	no	—
27	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
28	#107×C₇₉	—	2	4	no	—
29	#106×C₇₉	—	2	4	no	—
30	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	D₄	1	2	yes	—
33	C₂²×Dic₇₉	—	4	2	no	—
34	C₂×C₇₉⋊C₈	—	2	4	no	—
35	C₄⋊Dic₇₉	—	4	4	no	—
36	C₂×Q₈×C₇₉	—	2	4	no	—
37	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	—	2	4	no	—
40	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
42	C₄⋊C₄×C₇₉	—	2	4	no	—
43	C₄⋊C₄×C₇₉	—	2	4	no	—
44	C₄⋊C₄×C₇₉	—	4	4	no	—
45	C₂²⋊C₄×C₇₉	—	4	2	no	—
46	C₄²×C₇₉	—	4	2	no	—
47	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₇₉	#41	1	2	yes	—
50	C₂³×C₇₉	#127	1	1	yes	characteristic
51	Q₈×C₇₉	—	8	4	no	—
52	C₂×C₄×C₇₉	—	4	2	no	—
53	C₂×C₄×C₇₉	—	4	4	no	—
54	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
56	C₂×C₄×C₇₉	#121	1	2	yes	—
57	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#137	1	4	yes	—
59	C₂×Dic₇₉	—	8	4	no	—
60	C₂×Dic₇₉	—	8	2	no	—
61	C₈×C₇₉	—	2	2	no	—
62	C₈×C₇₉	—	2	2	no	—
63	C₇₉⋊C₈	—	4	4	no	—
64	C₂²×C₇₉	—	2	2	no	—
65	C₂²×C₇₉	#390	1	1	yes	characteristic
66	C₂²×C₇₉	#497	1	2	yes	—
67	C₄×C₇₉	—	2	2	no	—
68	C₄×C₇₉	—	2	2	no	—
69	C₄×C₇₉	—	8	4	no	—
70	C₄×C₇₉	#419	1	2	yes	—
71	C₄×C₇₉	#491	1	2	yes	—
72	Dic₇₉	—	16	2	no	—
73	#9759	—	79	1	no	maximal, Sylow
74	C₂×C₇₉	—	2	2	no	—
75	C₂×C₇₉	#1960	1	1	yes	characteristic
76	C₂×C₇₉	#1976	1	2	yes	—
77	#3160	D₇₉	1	1	yes	characteristic
78	#1835	—	79	1	no	—
79	#1518	—	79	4	no	—
80	#1452	—	79	1	no	—
81	C₇₉	#9759	1	1	yes	characteristic, complemented, Sylow
82	#533	C₂×D₇₉	1	2	yes	—
83	#500	Dic₇₉	1	2	yes	complemented
84	#499	—	2	2	no	—
85	#498	Dic₇₉	1	2	yes	complemented
86	#431	—	79	2	no	—
87	#427	—	2	2	no	—
88	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
89	#382	—	79	2	no	—
90	#340	—	158	2	no	—
91	#328	—	158	8	no	—
92	#322	—	158	2	no	—
93	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
94	C₄⋊Q₈	C₂×Dic₇₉	1	4	yes	—
95	#125	—	2	2	no	—
96	C₂²⋊Q₈	—	2	4	no	—
97	C₄×Q₈	—	2	4	no	—
98	#117	—	2	2	no	—
99	C₂×C₄⋊C₄	—	158	2	no	—
100	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
101	#107	—	2	4	no	—
102	#106	—	2	4	no	—
103	C₄⋊C₈	—	158	2	no	—
104	C₂²⋊C₈	—	79	4	no	—
105	C₄×C₈	—	158	2	no	—
106	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
109	#95	—	158	4	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂²×C₄	—	316	2	no	—
112	C₂²×C₄	D₄×D₇₉	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	158	4	no	—
116	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
117	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	4	4	no	—
121	C₄⋊C₄	—	316	4	no	—
122	C₂²⋊C₄	—	4	2	no	—
123	C₄²	—	4	2	no	—
124	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
125	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
126	C₄²	#11929623	1	2	yes	—
127	C₂³	#14579157	1	1	yes	characteristic
128	Q₈	—	8	4	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	632	4	no	—
132	C₂×C₄	—	632	2	no	—
133	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
134	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
135	C₂×C₄	#14579132	1	2	yes	—
136	C₂×C₄	#14579148	1	1	yes	characteristic
137	C₂×C₄	#14579192	1	4	yes	—
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	316	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#19098466	1	1	yes	characteristic
143	C₂²	#19098955	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	8	4	no	—
147	C₄	—	1264	2	no	—
148	C₄	#19098572	1	2	yes	—
149	C₄	#19098931	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#23764221	1	1	yes	characteristic
152	C₂	#23764299	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763531

C₂

yes

maximal, characteristic

#23761964

C₂

yes

maximal

#23761692

C₂

yes

maximal, characteristic

C₇₉×#3160

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

—

#19098429

C₂²

yes

—

#19098326

—

#19098300

—

#19098285

—

#533×C₇₉

C₂²

yes

—

#500×C₇₉

C₄

yes

complemented

#499×C₇₉

—

#498×C₇₉

C₄

yes

complemented

#427×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

D₄

yes

—

C₄×C₇₉⋊C₈

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

#125×C₇₉

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#106×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

#127

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#497

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#419

yes

—

C₄×C₇₉

#491

yes

—

Dic₇₉

—

#9759

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1960

yes

characteristic

C₂×C₇₉

#1976

yes

—

#3160

D₇₉

yes

—

—

—

C₇₉

#9759

yes

characteristic, complemented, Sylow

#533

C₂×D₇₉

yes

—

#500

Dic₇₉

yes

complemented

#499

—

#498

Dic₇₉

yes

complemented

#431

—

#427

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#382

—

#340

—

158

—

#328

—

158

—

#322

—

158

—

C₂²×C₈

C₂²⋊D₇₉

yes

—

C₄⋊Q₈

C₂×Dic₇₉

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

158

—

100

C₂×C₄²

C₂²×D₇₉

yes

characteristic

101

#107

—

102

#106

—

103

C₄⋊C₈

—

158

—

104

C₂²⋊C₈

—

105

C₄×C₈

—

158

—

106

C₄×C₈

C₂×Dic₇₉

yes

characteristic

107

C₄×C₈

C₂×Dic₇₉

yes

characteristic

108

C₄×C₈

C₂²⋊D₇₉

yes

—

109

#95

—

158

—

110

C₂×Q₈

—

111

C₂²×C₄

—

316

—

112

C₂²×C₄

D₄×D₇₉

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

158

—

116

C₂×C₈

C₂²⋊Dic₇₉

yes

—

117

C₂×C₈

C₂²⋊Dic₇₉

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

316

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

C₂²×Dic₇₉

yes

characteristic

125

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

126

C₄²

#11929623

yes

—

127

C₂³

#14579157

yes

characteristic

128

Q₈

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

632

—

132

C₂×C₄

—

632

—

133

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

134

C₂×C₄

D₄×Dic₇₉

yes

—

135

C₂×C₄

#14579132

yes

—

136

C₂×C₄

#14579148

yes

characteristic

137

C₂×C₄

#14579192

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

316

—

141

C₂²

—

142

C₂²

#19098466

yes

characteristic

143

C₂²

#19098955

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

1264

—

148

C₄

#19098572

yes

—

149

C₄

#19098931

yes

—

150

C₂

—

151

C₂

#23764221

yes

characteristic

152

C₂

#23764299

yes

—

	subgroup	G/H
center	C₂²	#19098466
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3160	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂³×C₇₉	#127
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3160	D₇₉
hypocenter	C₇₉	#9759
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂²

#19098466

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#3160

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
11. #500×C₇₉	C₄	semidirect
13. #498×C₇₉	C₄	semidirect
81. C₇₉	#9759	semidirect
83. #500	Dic₇₉	semidirect
85. #498	Dic₇₉	semidirect

H ≅ G/N

type

11. #500×C₇₉

C₄

semidirect

13. #498×C₇₉

C₄

semidirect

81. C₇₉

#9759

semidirect

83. #500

Dic₇₉

semidirect

85. #498

Dic₇₉

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23763531	2
#23761964	2
#23761692	2
C₇₉×#3160	2
#9759	79

subgroup

index

C₇₉×#3160

Subgroups — #34187662

Conjugacy classes of subgroups

Complements

Maximal subgroups