Group #34202616 — subgroups

Subgroups — #34202616

Order 20224.

206 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 71 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	#23762799	C₂	1	2	yes	maximal
3	#23761474	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761457	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2169	C₂	1	1	yes	maximal, characteristic
6	#19098968	C₂²	1	2	yes	—
7	#19098853	—	2	2	no	—
8	#19098389	—	2	1	no	—
9	#19098389	—	2	1	no	—
10	#19098316	C₂²	1	2	yes	—
11	#19098311	—	2	2	no	—
12	C₂×M₃₂×C₇₉	—	2	1	no	—
13	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²⋊C₁₆×C₇₉	—	2	1	no	—
16	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	#14579161	—	2	2	no	—
18	#14579095	—	2	2	no	—
19	#14579095	—	2	2	no	—
20	#14579092	—	4	2	no	—
21	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
22	#14579082	—	4	2	no	—
23	C₂²×C₈×C₇₉	—	2	1	no	—
24	C₄×D₄×C₇₉	D₄	1	1	yes	characteristic
25	M₃₂×C₇₉	—	4	2	no	—
26	C₂×C₁₆×C₇₉	—	2	1	no	—
27	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₂²⋊C₈×C₇₉	—	2	1	no	—
31	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₅₈	—	4	2	no	—
33	C₂×C₇₉⋊C₈	—	4	2	no	—
34	C₄⋊Dic₇₉	—	4	2	no	—
35	C₄⋊Dic₇₉	—	4	2	no	—
36	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₂×C₈×C₇₉	—	4	1	no	—
40	C₂×C₈×C₇₉	—	4	1	no	—
41	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
43	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
44	C₂²⋊C₄×C₇₉	—	2	1	no	—
45	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₇₉	—	2	1	no	—
47	C₁₆×C₇₉	—	4	1	no	—
48	C₁₆×C₇₉	SD₁₆	1	2	yes	—
49	C₂³×C₇₉	—	2	1	no	—
50	D₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	—	4	1	no	—
53	C₂×C₄×C₇₉	—	4	1	no	—
54	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
57	C₂×Dic₇₉	—	8	2	no	—
58	Dic₁₅₈	—	16	2	no	—
59	C₈×C₇₉	—	2	1	no	—
60	C₈×C₇₉	—	4	1	no	—
61	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₈×C₇₉	#137	1	1	yes	characteristic
63	C₇₉⋊C₈	—	8	2	no	—
64	C₂²×C₇₉	—	4	1	no	—
65	C₂²×C₇₉	—	4	1	no	—
66	C₂²×C₇₉	—	4	1	no	—
67	C₂²×C₇₉	#466	1	1	yes	characteristic
68	C₄×C₇₉	—	2	1	no	—
69	C₄×C₇₉	—	4	1	no	—
70	C₄×C₇₉	#461	1	1	yes	characteristic
71	C₄×C₇₉	#473	1	1	yes	characteristic
72	C₄×C₇₉	#510	1	2	yes	—
73	Dic₇₉	—	16	2	no	—
74	#12455	—	79	1	no	maximal, Sylow
75	C₂×C₇₉	—	8	1	no	—
76	C₂×C₇₉	#1673	1	1	yes	characteristic
77	C₂×C₇₉	#2219	1	1	yes	characteristic
78	C₂×C₇₉	#2221	1	1	yes	characteristic
79	#2252	—	79	1	no	—
80	#2169	D₇₉	1	1	yes	characteristic
81	#1675	—	79	2	no	—
82	#1372	—	79	1	no	—
83	#1365	—	79	2	no	—
84	C₇₉	#12455	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	2	1	no	—
86	#500	—	79	2	no	—
87	#476	—	158	2	no	—
88	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
89	#366	—	158	1	no	—
90	#366	—	158	1	no	—
91	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
92	C₂²⋊C₁₆	—	2	1	no	—
93	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
94	#335	—	79	2	no	—
95	#333	—	158	2	no	—
96	C₂²×C₈	—	2	1	no	—
97	C₄⋊Q₈	—	158	2	no	—
98	C₄×D₄	D₃₁₆	1	1	yes	characteristic
99	M₃₂	—	4	2	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
103	#107	—	158	2	no	—
104	#107	—	158	2	no	—
105	#106	—	316	2	no	—
106	C₄⋊C₈	—	158	2	no	—
107	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
108	#102	—	316	2	no	—
109	C₂²⋊C₈	—	2	1	no	—
110	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
111	C₂×Q₈	—	316	2	no	—
112	C₂×D₄	D₆₃₂	1	1	yes	characteristic
113	C₂²×C₄	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	316	2	no	—
118	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
119	C₂×C₈	#11929624	1	1	yes	characteristic
120	C₄⋊C₄	—	316	2	no	—
121	C₄⋊C₄	—	316	2	no	—
122	C₄⋊C₄	D₆₃₂	1	1	yes	characteristic
123	C₂²⋊C₄	—	2	1	no	—
124	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	4	1	no	—
127	C₁₆	#11929612	1	2	yes	—
128	C₂³	—	2	1	no	—
129	Q₈	—	1264	2	no	—
130	D₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	632	2	no	—
135	C₂×C₄	#14579135	1	1	yes	characteristic
136	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
137	C₂×C₄	#14579183	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	632	2	no	—
141	C₈	C₂×#11929612	1	1	yes	characteristic
142	C₈	#14579191	1	1	yes	characteristic
143	C₂²	—	4	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	#19098799	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	1264	2	no	—
150	C₄	#19098768	1	1	yes	characteristic
151	C₄	#19098838	1	1	yes	characteristic
152	C₄	#19099006	1	2	yes	—
153	C₂	—	8	1	no	—
154	C₂	#23762789	1	1	yes	characteristic
155	C₂	#23765508	1	1	yes	characteristic
156	C₂	#23765520	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765675

C₂

yes

maximal, characteristic, complemented

#23762799

C₂

yes

maximal

#23761474

C₂

yes

maximal, characteristic, complemented

#23761457

C₂

yes

maximal, complemented

C₇₉×#2169

C₂

yes

maximal, characteristic

#19098968

C₂²

yes

—

—

—

—

C₂²

yes

—

#19098311

—

C₂×M₃₂×C₇₉

—

C₈×D₄×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

—

—

—

—

C₄⋊(C₇₉⋊C₈)

—

#14579082

—

C₂²×C₈×C₇₉

—

C₄×D₄×C₇₉

D₄

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

D₄

yes

characteristic

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×D₄×C₇₉

D₈

yes

characteristic

C₂²×C₄×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

D₈

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

SD₁₆

yes

—

C₂³×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

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

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×SD₁₆

yes

characteristic

C₈×C₇₉

#137

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#466

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#12455

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

—

—

—

C₇₉

#12455

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

#500

—

#476

—

158

—

C₈×D₄

C₂×D₇₉

yes

characteristic

#366

—

158

—

#366

—

158

—

C₄⋊C₁₆

C₂×D₇₉

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

#335

—

#333

—

158

—

C₂²×C₈

—

C₄⋊Q₈

—

158

—

C₄×D₄

D₃₁₆

yes

characteristic

M₃₂

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

102

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

103

#107

—

158

—

104

#107

—

158

—

105

#106

—

316

—

106

C₄⋊C₈

—

158

—

107

C₄⋊C₈

D₃₁₆

yes

characteristic

108

#102

—

316

—

109

C₂²⋊C₈

—

110

C₄×C₈

C₂²×D₇₉

yes

characteristic

111

C₂×Q₈

—

316

—

112

C₂×D₄

D₆₃₂

yes

characteristic

113

C₂²×C₄

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

316

—

118

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

119

C₂×C₈

#11929624

yes

characteristic

120

C₄⋊C₄

—

316

—

121

C₄⋊C₄

—

316

—

122

C₄⋊C₄

D₆₃₂

yes

characteristic

123

C₂²⋊C₄

—

124

C₄²

C₂×D₃₁₆

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₁₆

#11929612

yes

—

128

C₂³

—

129

Q₈

—

1264

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

632

—

135

C₂×C₄

#14579135

yes

characteristic

136

C₂×C₄

C₂×D₆₃₂

yes

characteristic

137

C₂×C₄

#14579183

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

632

—

141

C₈

C₂×#11929612

yes

characteristic

142

C₈

#14579191

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#19098799

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

1264

—

150

C₄

#19098768

yes

characteristic

151

C₄

#19098838

yes

characteristic

152

C₄

#19099006

yes

—

153

C₂

—

154

C₂

#23762789

yes

characteristic

155

C₂

#23765508

yes

characteristic

156

C₂

#23765520

yes

characteristic

	subgroup	G/H
center	C₂²	#19098799
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2169	C₂
socle	C₂²×C₇₉	#466
prime radical	C₂×D₄×C₇₉	D₈
hypercenter	#2169	D₇₉
hypocenter	C₇₉	#12455
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

subgroup

G/H

center

C₂²

#19098799

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#2169

C₂

socle

C₂²×C₇₉

#466

prime radical

C₂×D₄×C₇₉

D₈

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

N	H ≅ G/N	type
1. #23765675	C₂	semidirect
3. #23761474	C₂	semidirect
4. #23761457	C₂	semidirect
84. C₇₉	#12455	semidirect

H ≅ G/N

type

1. #23765675

C₂

semidirect

3. #23761474

C₂

semidirect

4. #23761457

C₂

semidirect

84. C₇₉

#12455

semidirect

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

subgroup	index
#23765675	2
#23762799	2
#23761474	2
#23761457	2
C₇₉×#2169	2
#12455	79

subgroup

index

C₇₉×#2169

Subgroups — #34202616

Conjugacy classes of subgroups

Complements

Maximal subgroups