Group #34202919 — subgroups

Subgroups — #34202919

Order 20224.

306 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 87 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	#23765675	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765623	C₂	1	2	yes	maximal, complemented
3	#23762989	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#2246	C₂	1	1	yes	maximal, characteristic
6	#19098968	C₂²	1	2	yes	—
7	#19098935	—	2	2	no	—
8	#19098909	—	2	2	no	—
9	#19098389	—	2	2	no	—
10	#19098364	—	2	4	no	—
11	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×D₁₆×C₇₉	—	2	2	no	—
14	#493×C₇₉	C₂²	1	2	yes	—
15	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#11929612	—	4	4	no	—
17	#14579161	—	2	2	no	—
18	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
19	#14579095	—	2	2	no	—
20	#14579095	—	2	2	no	—
21	#14579082	—	4	4	no	—
22	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
24	C₂×D₈×C₇₉	—	2	2	no	—
25	C₂×D₈×C₇₉	—	2	2	no	—
26	#127×C₇₉	—	2	2	no	—
27	D₁₆×C₇₉	—	4	4	no	—
28	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
29	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₅₈	—	4	2	no	—
31	#11929612	—	8	8	no	—
32	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
33	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
34	C₄⋊Dic₇₉	—	4	2	no	—
35	C₄⋊Dic₇₉	—	4	2	no	—
36	C₇₉⋊C₁₆	—	2	2	no	—
37	C₂×D₄×C₇₉	—	4	2	no	—
38	C₂×D₄×C₇₉	—	4	2	no	—
39	C₂×D₄×C₇₉	—	4	2	no	—
40	D₈×C₇₉	—	4	4	no	—
41	D₈×C₇₉	—	4	4	no	—
42	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₇₉	SD₁₆	1	4	yes	complemented
46	C₂³×C₇₉	—	8	2	no	—
47	D₄×C₇₉	—	8	2	no	—
48	D₄×C₇₉	—	8	2	no	—
49	D₄×C₇₉	—	8	4	no	—
50	D₄×C₇₉	—	8	4	no	—
51	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
54	C₂×Dic₇₉	—	8	2	no	—
55	Dic₁₅₈	—	8	4	no	—
56	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
57	C₈×C₇₉	#135	1	2	yes	—
58	C₇₉⋊C₈	—	2	2	no	—
59	C₇₉⋊C₈	D₁₆	1	4	yes	complemented
60	C₂²×C₇₉	—	16	2	no	—
61	C₂²×C₇₉	—	16	2	no	—
62	C₂²×C₇₉	—	16	2	no	—
63	C₂²×C₇₉	#493	1	1	yes	characteristic
64	C₄×C₇₉	#492	1	1	yes	characteristic
65	C₄×C₇₉	#495	1	1	yes	characteristic
66	C₄×C₇₉	C₂×D₁₆	1	2	yes	—
67	C₄×C₇₉	#508	1	2	yes	—
68	Dic₇₉	—	16	2	no	—
69	#12503	—	79	1	no	maximal, Sylow
70	C₂×C₇₉	—	32	2	no	—
71	C₂×C₇₉	#1706	1	1	yes	characteristic
72	C₂×C₇₉	#2246	1	1	yes	characteristic
73	C₂×C₇₉	#2249	1	1	yes	characteristic
74	#2252	—	79	1	no	—
75	#2246	D₇₉	1	1	yes	characteristic
76	#2240	—	79	2	no	—
77	#1706	—	79	2	no	—
78	C₈×C₁₆	—	79	1	no	—
79	C₇₉	#12503	1	1	yes	characteristic, complemented, Sylow
80	C₂×D₁₆	—	2	2	no	—
81	#500	—	79	2	no	—
82	#493	C₂×D₇₉	1	2	yes	—
83	#492	—	158	2	no	—
84	#486	—	158	2	no	—
85	#366	—	158	2	no	—
86	#357	—	158	4	no	—
87	C₄×C₁₆	—	79	1	no	—
88	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
89	C₈²	—	79	1	no	—
90	C₂×SD₁₆	—	316	4	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	—	2	2	no	—
93	C₄⋊Q₈	—	158	2	no	—
94	#127	—	2	2	no	—
95	D₁₆	—	4	4	no	—
96	C₂×C₁₆	—	79	2	no	—
97	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
98	#107	—	158	2	no	—
99	#107	—	158	2	no	—
100	#102	—	316	4	no	—
101	C₄×C₈	—	79	1	no	—
102	C₄×C₈	—	79	1	no	—
103	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	C₂×Q₈	—	316	2	no	—
105	C₂×D₄	—	4	2	no	—
106	C₂×D₄	—	4	2	no	—
107	C₂×D₄	—	4	2	no	—
108	SD₁₆	—	632	8	no	—
109	D₈	—	4	4	no	—
110	D₈	—	4	4	no	—
111	C₂×C₈	—	79	2	no	—
112	C₂×C₈	—	79	2	no	—
113	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
114	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
115	C₄⋊C₄	—	316	2	no	—
116	C₄⋊C₄	—	316	2	no	—
117	C₄²	D₄×D₇₉	1	1	yes	characteristic
118	C₁₆	—	158	2	no	—
119	C₁₆	#11929612	1	4	yes	complemented
120	C₂³	—	8	2	no	—
121	Q₈	—	632	4	no	—
122	D₄	—	8	2	no	—
123	D₄	—	8	2	no	—
124	D₄	—	8	4	no	—
125	D₄	—	8	4	no	—
126	C₂×C₄	—	632	2	no	—
127	C₂×C₄	#14579157	1	1	yes	characteristic
128	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
129	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
130	C₈	—	79	4	no	—
131	C₈	—	158	2	no	—
132	C₈	C₂×#11929612	1	2	yes	—
133	C₈	#14579180	1	2	yes	—
134	C₂²	—	16	2	no	—
135	C₂²	—	16	2	no	—
136	C₂²	—	16	2	no	—
137	C₂²	#19098938	1	1	yes	characteristic
138	C₄	—	1264	2	no	—
139	C₄	#19098935	1	1	yes	characteristic
140	C₄	#19098946	1	1	yes	characteristic
141	C₄	D₁₆×D₇₉	1	2	yes	—
142	C₄	#19098992	1	2	yes	—
143	C₂	—	32	2	no	—
144	C₂	#23762989	1	1	yes	characteristic
145	C₂	#23765648	1	1	yes	characteristic
146	C₂	#23765658	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765675

C₂

yes

maximal, characteristic, complemented

#23765623

C₂

yes

maximal, complemented

#23762989

C₂

yes

maximal, complemented

C₁₆×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₇₉×#2246

C₂

yes

maximal, characteristic

#19098968

C₂²

yes

—

—

—

—

—

C₄×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×D₁₆×C₇₉

—

#493×C₇₉

C₂²

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×#11929612

—

#14579161

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

#14579095

—

#14579095

—

#14579082

—

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

#127×C₇₉

—

D₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

#11929612

—

C₂×C₇₉⋊C₈

D₈

yes

—

C₂×C₇₉⋊C₈

D₈

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

SD₁₆

yes

complemented

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

#135

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

D₁₆

yes

complemented

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×D₁₆

yes

—

C₄×C₇₉

#508

yes

—

Dic₇₉

—

#12503

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

#2240

—

#1706

—

C₈×C₁₆

—

C₇₉

#12503

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

#500

—

#493

C₂×D₇₉

yes

—

#492

—

158

—

#486

—

158

—

#366

—

158

—

#357

—

158

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₈²

—

C₂×SD₁₆

—

316

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

158

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₉

yes

—

#107

—

158

—

#107

—

158

—

100

#102

—

316

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂²×D₇₉

yes

characteristic

104

C₂×Q₈

—

316

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

SD₁₆

—

632

—

109

D₈

—

110

D₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₇₉

yes

characteristic

114

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

115

C₄⋊C₄

—

316

—

116

C₄⋊C₄

—

316

—

117

C₄²

D₄×D₇₉

yes

characteristic

118

C₁₆

—

158

—

119

C₁₆

#11929612

yes

complemented

120

C₂³

—

121

Q₈

—

632

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

632

—

127

C₂×C₄

#14579157

yes

characteristic

128

C₂×C₄

D₈×D₇₉

yes

characteristic

129

C₂×C₄

D₈×D₇₉

yes

characteristic

130

C₈

—

131

C₈

—

158

—

132

C₈

C₂×#11929612

yes

—

133

C₈

#14579180

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#19098938

yes

characteristic

138

C₄

—

1264

—

139

C₄

#19098935

yes

characteristic

140

C₄

#19098946

yes

characteristic

141

C₄

D₁₆×D₇₉

yes

—

142

C₄

#19098992

yes

—

143

C₂

—

144

C₂

#23762989

yes

characteristic

145

C₂

#23765648

yes

characteristic

146

C₂

#23765658

yes

characteristic

	subgroup	G/H
center	C₂²	#19098938
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting, prime radical	C₇₉×#2246	C₂
socle	C₂²×C₇₉	#493
hypercenter	#2246	D₇₉
hypocenter	C₇₉	#12503
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#19098938

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting, prime radical

C₇₉×#2246

C₂

C₂²×C₇₉

D₇₉

C₇₉

G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #23765675	C₂	semidirect
2. #23765623	C₂	semidirect
3. #23762989	C₂	semidirect
4. C₁₆×C₇₉⋊C₈	C₂	semidirect
45. C₁₆×C₇₉	SD₁₆	semidirect
59. C₇₉⋊C₈	D₁₆	semidirect
79. C₇₉	#12503	semidirect
119. C₁₆	#11929612	semidirect

H ≅ G/N

type

1. #23765675

C₂

semidirect

2. #23765623

C₂

semidirect

3. #23762989

C₂

semidirect

4. C₁₆×C₇₉⋊C₈

C₂

semidirect

45. C₁₆×C₇₉

SD₁₆

semidirect

59. C₇₉⋊C₈

D₁₆

semidirect

79. C₇₉

#12503

semidirect

119. C₁₆

#11929612

semidirect

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

subgroup	index
#23765675	2
#23765623	2
#23762989	2
C₁₆×C₇₉⋊C₈	2
C₇₉×#2246	2
#12503	79

subgroup

index

#23765675

#23765623

#23762989

C₁₆×C₇₉⋊C₈

C₇₉×#2246

#12503

Subgroups — #34202919

Conjugacy classes of subgroups

Complements

Maximal subgroups