Group #34202628 — subgroups

Subgroups — #34202628

Order 20224.

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 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
2	#23762794	C₂	1	2	yes	maximal
3	#23761472	C₂	1	1	yes	maximal, characteristic
4	#23761454	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19098968	C₂²	1	2	yes	—
7	#19098773	—	2	2	no	—
8	#19098389	—	2	1	no	—
9	#19098389	—	2	1	no	—
10	#19098316	C₂²	1	2	yes	—
11	#19098308	—	2	2	no	—
12	C₈×Q₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₇₉	—	2	1	no	—
14	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₇₉	—	2	1	no	—
16	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#11929614	—	4	2	no	—
18	#14579161	—	2	2	no	—
19	#14579095	—	2	2	no	—
20	#14579095	—	2	2	no	—
21	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
22	#14579084	—	4	2	no	—
23	C₄×Q₈×C₇₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₉	—	2	1	no	—
25	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₉	—	2	1	no	—
28	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	—	2	1	no	—
30	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₅₈	—	4	2	no	—
32	#11929614	—	8	4	no	—
33	C₂×C₇₉⋊C₈	—	4	2	no	—
34	C₄⋊Dic₇₉	—	4	2	no	—
35	C₄⋊Dic₇₉	—	4	2	no	—
36	C₂×Q₈×C₇₉	D₈	1	1	yes	characteristic
37	C₂×C₈×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₇₉	—	2	1	no	—
41	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
42	C₄²×C₇₉	—	2	1	no	—
43	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₉	—	2	1	no	—
45	C₁₆×C₇₉	—	4	1	no	—
46	C₁₆×C₇₉	Q₁₆	1	2	yes	complemented
47	Q₈×C₇₉	—	2	1	no	—
48	Q₈×C₇₉	D₁₆	1	2	yes	—
49	C₂×C₄×C₇₉	—	2	1	no	—
50	C₂×C₄×C₇₉	—	2	1	no	—
51	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
54	C₂×Dic₇₉	—	8	2	no	—
55	Dic₁₅₈	—	8	2	no	—
56	Dic₁₅₈	—	8	2	no	—
57	C₈×C₇₉	—	2	1	no	—
58	C₈×C₇₉	—	4	1	no	—
59	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₇₉	#137	1	1	yes	characteristic
61	C₇₉⋊C₈	—	8	2	no	—
62	C₂²×C₇₉	#466	1	1	yes	characteristic
63	C₄×C₇₉	—	2	2	no	—
64	C₄×C₇₉	—	2	1	no	—
65	C₄×C₇₉	—	4	1	no	—
66	C₄×C₇₉	#462	1	1	yes	characteristic
67	C₄×C₇₉	#473	1	1	yes	characteristic
68	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₇₉	#508	1	1	yes	characteristic
70	Dic₇₉	—	16	2	no	—
71	#12457	—	79	1	no	maximal, Sylow
72	C₂×C₇₉	#1674	1	1	yes	characteristic
73	C₂×C₇₉	#2215	1	1	yes	characteristic
74	C₂×C₇₉	#2222	1	1	yes	characteristic
75	#2252	—	79	1	no	—
76	Q₈×C₁₆	D₇₉	1	1	yes	characteristic
77	#1674	—	79	2	no	—
78	#1371	—	79	1	no	—
79	#1364	—	79	2	no	—
80	C₇₉	#12457	1	1	yes	characteristic, complemented, Sylow
81	#500	—	79	2	no	—
82	#462	—	158	2	no	—
83	C₈×Q₈	C₂×D₇₉	1	1	yes	characteristic
84	#366	—	158	1	no	—
85	#366	—	158	1	no	—
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
90	#335	—	79	2	no	—
91	#332	—	158	2	no	—
92	C₂×Q₁₆	—	316	2	no	—
93	C₄⋊Q₈	—	158	2	no	—
94	C₄×Q₈	D₃₁₆	1	1	yes	characteristic
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
98	#107	—	158	2	no	—
99	#107	—	158	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	158	2	no	—
102	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
103	#103	—	316	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
106	C₂×Q₈	—	316	2	no	—
107	C₂×Q₈	D₆₃₂	1	1	yes	characteristic
108	Q₁₆	—	632	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	316	2	no	—
111	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
112	C₂×C₈	#11929624	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	316	2	no	—
115	C₄⋊C₄	—	316	2	no	—
116	C₄⋊C₄	D₆₃₂	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	#11929614	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	632	2	no	—
124	Q₈	—	632	2	no	—
125	Q₈	D₁₂₆₄	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	632	2	no	—
129	C₂×C₄	#14579135	1	1	yes	characteristic
130	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
131	C₂×C₄	#14579183	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	632	2	no	—
135	C₈	C₂×#11929614	1	1	yes	characteristic
136	C₈	#14579191	1	1	yes	characteristic
137	C₂²	#19098799	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	4	1	no	—
141	C₄	—	1264	2	no	—
142	C₄	#19098773	1	1	yes	characteristic
143	C₄	#19098838	1	1	yes	characteristic
144	C₄	C₂×D₁₂₆₄	1	1	yes	characteristic
145	C₄	#19098995	1	1	yes	characteristic
146	C₂	#23762794	1	1	yes	characteristic
147	C₂	#23765486	1	1	yes	characteristic
148	C₂	#23765525	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765675

C₂

yes

maximal, characteristic

#23762794

C₂

yes

maximal

#23761472

C₂

yes

maximal, characteristic

#23761454

C₂

yes

maximal

Q₈×C₁₆×C₇₉

C₂

yes

maximal, characteristic

#19098968

C₂²

yes

—

—

—

—

C₂²

yes

—

#19098308

—

C₈×Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₄⋊C₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

—

—

—

—

C₄⋊(C₇₉⋊C₈)

—

#14579084

—

C₄×Q₈×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

#11929614

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₈

yes

characteristic

C₄²×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

Q₁₆

yes

complemented

Q₈×C₇₉

—

Q₈×C₇₉

D₁₆

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×Q₁₆

yes

characteristic

C₈×C₇₉

#137

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

#466

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

#508

yes

characteristic

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

Q₈×C₁₆

D₇₉

yes

—

—

—

C₇₉

#12457

yes

characteristic, complemented, Sylow

#500

—

#462

—

158

—

C₈×Q₈

C₂×D₇₉

yes

characteristic

#366

—

158

—

#366

—

158

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₇₉

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

#335

—

#332

—

158

—

C₂×Q₁₆

—

316

—

C₄⋊Q₈

—

158

—

C₄×Q₈

D₃₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

#107

—

158

—

#107

—

158

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

158

—

102

C₄⋊C₈

D₃₁₆

yes

characteristic

103

#103

—

316

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₇₉

yes

characteristic

106

C₂×Q₈

—

316

—

107

C₂×Q₈

D₆₃₂

yes

characteristic

108

Q₁₆

—

632

—

109

C₂×C₈

—

110

C₂×C₈

—

316

—

111

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

112

C₂×C₈

#11929624

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

316

—

115

C₄⋊C₄

—

316

—

116

C₄⋊C₄

D₆₃₂

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₃₁₆

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#11929614

yes

complemented

122

Q₈

—

123

Q₈

—

632

—

124

Q₈

—

632

—

125

Q₈

D₁₂₆₄

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

632

—

129

C₂×C₄

#14579135

yes

characteristic

130

C₂×C₄

C₂×D₆₃₂

yes

characteristic

131

C₂×C₄

#14579183

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

632

—

135

C₈

C₂×#11929614

yes

characteristic

136

C₈

#14579191

yes

characteristic

137

C₂²

#19098799

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

1264

—

142

C₄

#19098773

yes

characteristic

143

C₄

#19098838

yes

characteristic

144

C₄

C₂×D₁₂₆₄

yes

characteristic

145

C₄

#19098995

yes

characteristic

146

C₂

#23762794

yes

characteristic

147

C₂

#23765486

yes

characteristic

148

C₂

#23765525

yes

characteristic

	subgroup	G/H
center	C₂²	#19098799
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	Q₈×C₁₆×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	#466
hypercenter	Q₈×C₁₆	D₇₉
hypocenter	C₇₉	#12457
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#19098799

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

Q₈×C₁₆×C₇₉

C₂

socle, prime radical

C₂²×C₇₉

#466

hypercenter

Q₈×C₁₆

D₇₉

hypocenter

C₇₉

#12457

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
46. C₁₆×C₇₉	Q₁₆	semidirect
80. C₇₉	#12457	semidirect
121. C₁₆	#11929614	semidirect

H ≅ G/N

type

46. C₁₆×C₇₉

Q₁₆

semidirect

80. C₇₉

#12457

semidirect

121. C₁₆

#11929614

semidirect

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

subgroup	index
#23765675	2
#23762794	2
#23761472	2
#23761454	2
Q₈×C₁₆×C₇₉	2
#12457	79

subgroup

index

Q₈×C₁₆×C₇₉

Subgroups — #34202628

Conjugacy classes of subgroups

Complements

Maximal subgroups