Group #34182041 — subgroups

Subgroups — #34182041

Order 20224.

422 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 203 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	#23763557	C₂	1	1	yes	maximal, characteristic
2	#23763527	C₂	1	4	yes	maximal
3	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579093	—	2	2	no	—
6	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
7	#19098433	C₂²	1	2	yes	—
8	#19098325	—	2	8	no	—
9	C₈×C₇₉⋊C₈	C₄	1	4	yes	complemented
10	C₂×#106×C₇₉	—	2	2	no	—
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	#384×C₇₉	C₂²	1	2	yes	—
13	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
14	C₂×C₄⋊Dic₇₉	—	2	4	no	—
15	C₂×C₄⋊Dic₇₉	—	2	2	no	—
16	#14579093	—	4	4	no	—
17	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₇₉⋊C₈	Q₈	1	4	yes	—
19	C₂²×C₈×C₇₉	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
22	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
23	#106×C₇₉	—	4	4	no	—
24	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₉	Q₈	1	2	yes	—
26	C₂²×Dic₇₉	—	4	2	no	—
27	C₂×C₇₉⋊C₈	—	2	8	no	—
28	C₂×C₇₉⋊C₈	C₄⋊C₄	1	8	yes	—
29	C₄⋊Dic₇₉	—	4	4	no	—
30	C₄⋊Dic₇₉	—	4	8	no	—
31	C₂²×C₄×C₇₉	—	4	2	no	—
32	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
34	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
36	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
37	C₄⋊C₄×C₇₉	—	4	4	no	—
38	C₄⋊C₄×C₇₉	—	4	8	no	—
39	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
41	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
42	C₂³×C₇₉	#127	1	1	yes	characteristic
43	C₂×C₄×C₇₉	—	8	2	no	—
44	C₂×C₄×C₇₉	—	8	4	no	—
45	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
49	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
50	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄×C₇₉	#135	1	8	yes	—
52	C₂×Dic₇₉	—	8	2	no	—
53	C₂×Dic₇₉	—	8	4	no	—
54	C₈×C₇₉	#106	1	8	yes	—
55	C₇₉⋊C₈	—	2	8	no	—
56	C₂²×C₇₉	#384	1	1	yes	characteristic
57	C₂²×C₇₉	#492	1	1	yes	characteristic
58	C₂²×C₇₉	#492	1	1	yes	characteristic
59	C₂²×C₇₉	#495	1	4	yes	—
60	C₄×C₇₉	—	16	2	no	—
61	C₄×C₇₉	C₂×#106	1	2	yes	—
62	C₄×C₇₉	#427	1	4	yes	—
63	C₄×C₇₉	#498	1	2	yes	—
64	C₄×C₇₉	#499	1	4	yes	—
65	Dic₇₉	—	16	2	no	—
66	#8815	—	79	1	no	maximal, Sylow
67	C₂×C₇₉	#1708	1	2	yes	—
68	C₂×C₇₉	#1708	1	2	yes	—
69	C₂×C₇₉	#1844	1	1	yes	characteristic
70	C₂×C₇₉	#1846	1	2	yes	—
71	#1844	—	79	1	no	—
72	#1844	D₇₉	1	1	yes	characteristic
73	#1834	—	79	4	no	—
74	C₂×C₈²	—	79	1	no	—
75	C₇₉	#8815	1	1	yes	characteristic, complemented, Sylow
76	C₂×#106	—	2	2	no	—
77	C₂×#106	—	158	2	no	—
78	C₂×C₄×C₈	—	79	2	no	—
79	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
80	#384	—	79	2	no	—
81	#384	C₂×D₇₉	1	2	yes	—
82	#340	—	158	8	no	—
83	C₈²	—	79	4	no	—
84	C₂²×C₈	—	79	4	no	—
85	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
86	C₂×C₄⋊C₄	—	2	4	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄⋊C₄	—	158	4	no	—
89	C₂×C₄⋊C₄	—	158	2	no	—
90	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
91	#106	—	4	4	no	—
92	#106	—	316	4	no	—
93	C₄×C₈	—	79	4	no	—
94	C₄×C₈	—	79	4	no	—
95	C₄×C₈	Dic₁₅₈	1	2	yes	—
96	C₄×C₈	C₄×D₇₉	1	2	yes	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	316	2	no	—
99	C₂²×C₄	D₄×D₇₉	1	2	yes	—
100	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
101	C₂×C₈	—	79	8	no	—
102	C₂×C₈	—	158	8	no	—
103	C₂×C₈	Dic₇₉⋊C₄	1	4	yes	—
104	C₂×C₈	#11929611	1	4	yes	—
105	C₂×C₈	#11929612	1	4	yes	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	8	no	—
108	C₄⋊C₄	—	316	4	no	—
109	C₄⋊C₄	—	316	8	no	—
110	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
111	C₄²	Q₈×D₇₉	1	2	yes	—
112	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
113	C₂³	#14579157	1	1	yes	characteristic
114	C₂×C₄	—	8	2	no	—
115	C₂×C₄	—	8	4	no	—
116	C₂×C₄	—	632	2	no	—
117	C₂×C₄	—	632	4	no	—
118	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
119	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
120	C₂×C₄	#14579159	1	1	yes	characteristic
121	C₂×C₄	#14579160	1	2	yes	—
122	C₂×C₄	C₂×#11929611	1	2	yes	—
123	C₂×C₄	C₂×#11929612	1	2	yes	—
124	C₂×C₄	#14579182	1	8	yes	—
125	C₈	—	158	8	no	—
126	C₈	#14579092	1	8	yes	—
127	C₂²	#19098432	1	1	yes	characteristic
128	C₂²	#19098935	1	1	yes	characteristic
129	C₂²	#19098937	1	1	yes	characteristic
130	C₂²	#19098944	1	4	yes	—
131	C₄	—	16	2	no	—
132	C₄	—	1264	2	no	—
133	C₄	C₂×#14579092	1	2	yes	—
134	C₄	#19098598	1	4	yes	—
135	C₄	#19098959	1	2	yes	—
136	C₄	#19098963	1	4	yes	—
137	C₂	#23762999	1	2	yes	—
138	C₂	#23763002	1	2	yes	—
139	C₂	#23763556	1	1	yes	characteristic
140	C₂	#23763561	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763557

C₂

yes

maximal, characteristic

#23763527

C₂

yes

maximal

C₂×C₈×C₇₉⋊C₈

C₂

yes

maximal, characteristic

C₇₉×#1844

C₂

yes

maximal, characteristic

C₂×#14579093

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

#19098433

C₂²

yes

—

#19098325

—

C₈×C₇₉⋊C₈

C₄

yes

complemented

C₂×#106×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#384×C₇₉

C₂²

yes

—

C₂²×C₇₉⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579093

—

C₄×C₇₉⋊C₈

C₂×C₄

yes

—

C₄×C₇₉⋊C₈

Q₈

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#106×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

#106

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#106

yes

—

C₄×C₇₉

#427

yes

—

C₄×C₇₉

#498

yes

—

C₄×C₇₉

#499

yes

—

Dic₇₉

—

#8815

—

maximal, Sylow

C₂×C₇₉

#1708

yes

—

C₂×C₇₉

#1708

yes

—

C₂×C₇₉

#1844

yes

characteristic

C₂×C₇₉

#1846

yes

—

#1844

—

#1844

D₇₉

yes

characteristic

#1834

—

C₂×C₈²

—

C₇₉

#8815

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×#106

—

158

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#384

—

#384

C₂×D₇₉

yes

—

#340

—

158

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₇₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

#106

—

#106

—

316

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

C₄×D₇₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

316

—

C₂²×C₄

D₄×D₇₉

yes

—

100

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

158

—

103

C₂×C₈

Dic₇₉⋊C₄

yes

—

104

C₂×C₈

#11929611

yes

—

105

C₂×C₈

#11929612

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

316

—

109

C₄⋊C₄

—

316

—

110

C₄²

C₂×C₄×D₇₉

yes

characteristic

111

C₄²

Q₈×D₇₉

yes

—

112

C₄²

C₂×Dic₁₅₈

yes

characteristic

113

C₂³

#14579157

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

632

—

117

C₂×C₄

—

632

—

118

C₂×C₄

C₄⋊C₄×D₇₉

yes

—

119

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

120

C₂×C₄

#14579159

yes

characteristic

121

C₂×C₄

#14579160

yes

—

122

C₂×C₄

C₂×#11929611

yes

—

123

C₂×C₄

C₂×#11929612

yes

—

124

C₂×C₄

#14579182

yes

—

125

C₈

—

158

—

126

C₈

#14579092

yes

—

127

C₂²

#19098432

yes

characteristic

128

C₂²

#19098935

yes

characteristic

129

C₂²

#19098937

yes

characteristic

130

C₂²

#19098944

yes

—

131

C₄

—

132

C₄

—

1264

—

133

C₄

C₂×#14579092

yes

—

134

C₄

#19098598

yes

—

135

C₄

#19098959

yes

—

136

C₄

#19098963

yes

—

137

C₂

#23762999

yes

—

138

C₂

#23763002

yes

—

139

C₂

#23763556

yes

characteristic

140

C₂

#23763561

yes

—

	subgroup	G/H
center	C₂³	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1844	C₂
socle, prime radical	C₂³×C₇₉	#127
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1844	D₇₉
hypocenter	C₇₉	#8815
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14579157

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1844

C₂

socle, prime radical

C₂³×C₇₉

#127

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
9. C₈×C₇₉⋊C₈	C₄	semidirect
75. C₇₉	#8815	semidirect

H ≅ G/N

type

9. C₈×C₇₉⋊C₈

C₄

semidirect

75. C₇₉

#8815

semidirect

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

subgroup	index
#23763557	2
#23763527	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#1844	2
#8815	79

subgroup

index

#23763557

#23763527

C₂×C₈×C₇₉⋊C₈

C₇₉×#1844

#8815

Subgroups — #34182041

Conjugacy classes of subgroups

Complements

Maximal subgroups