Group #34208716 — subgroups

Subgroups — #34208716

Order 20224.

290 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂×#19098397	C₂	1	1	yes	maximal, characteristic
2	#23765184	C₂	1	1	yes	maximal, characteristic
3	#23761581	C₂	1	4	yes	maximal, complemented
4	C₂²×Q₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19098601	—	2	1	no	—
6	#19098397	—	2	2	no	—
7	C₂×C₇₉⋊C₃₂	C₂²	1	2	yes	—
8	#19098391	—	2	1	no	—
9	#19098389	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₇₉	—	2	1	no	—
11	C₂×Q₃₂×C₇₉	—	2	1	no	—
12	C₂×Q₃₂×C₇₉	—	2	2	no	—
13	C₂×Q₃₂×C₇₉	C₄	1	4	yes	complemented
14	C₂×Q₃₂×C₇₉	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	#14579120	—	4	1	no	—
17	#14579095	—	2	2	no	—
18	#14579093	—	4	1	no	—
19	C₇₉⋊C₃₂	—	2	2	no	—
20	C₂²×Q₈×C₇₉	—	4	1	no	—
21	C₂×Q₁₆×C₇₉	—	2	4	no	—
22	C₂×Q₁₆×C₇₉	—	2	2	no	—
23	C₂×Q₁₆×C₇₉	—	4	1	no	—
24	C₂×Q₁₆×C₇₉	—	4	2	no	—
25	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
26	Q₃₂×C₇₉	—	2	4	no	—
27	Q₃₂×C₇₉	—	2	2	no	—
28	Q₃₂×C₇₉	C₂×C₄	1	4	yes	—
29	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₇₉	—	4	2	no	—
34	C₂²⋊Dic₇₉	—	8	1	no	—
35	C₄×Dic₇₉	—	8	1	no	—
36	C₂×Q₈×C₇₉	—	4	4	no	—
37	C₂×Q₈×C₇₉	—	4	2	no	—
38	C₂×Q₈×C₇₉	—	8	1	no	—
39	C₂×Q₈×C₇₉	—	8	2	no	—
40	C₂²×C₄×C₇₉	—	8	1	no	—
41	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
42	Q₁₆×C₇₉	—	2	4	no	—
43	Q₁₆×C₇₉	—	4	4	no	—
44	Q₁₆×C₇₉	—	4	2	no	—
45	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
47	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
48	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₇₉	C₂²⋊C₄	1	2	yes	—
50	C₁₆×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₇₉	D₁₆	1	1	yes	characteristic
53	Q₈×C₇₉	—	4	4	no	—
54	Q₈×C₇₉	—	8	4	no	—
55	Q₈×C₇₉	—	8	2	no	—
56	C₂×C₄×C₇₉	—	8	4	no	—
57	C₂×C₄×C₇₉	—	8	2	no	—
58	C₂×C₄×C₇₉	#102	1	2	yes	—
59	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
60	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
61	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
62	C₂×Dic₇₉	—	8	2	no	—
63	C₈×C₇₉	#102	1	2	yes	—
64	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₇₉	—	2	2	no	—
67	C₂²×C₇₉	#357	1	2	yes	—
68	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₇₉	—	8	4	no	—
70	C₄×C₇₉	#357	1	2	yes	—
71	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
72	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
73	Dic₇₉	—	16	2	no	—
74	#13618	—	79	1	no	maximal, Sylow
75	C₂×C₇₉	—	2	2	no	—
76	C₂×C₇₉	C₂×#357	1	1	yes	characteristic
77	C₂×C₇₉	#2264	1	2	yes	—
78	C₂²×Q₃₂	D₇₉	1	1	yes	characteristic
79	C₂×M₆₄	—	79	1	no	—
80	#2158	—	79	1	no	—
81	#1416	—	79	4	no	—
82	C₇₉	#13618	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₁₆	—	2	1	no	—
84	C₂×Q₃₂	—	2	1	no	—
85	C₂×Q₃₂	—	2	2	no	—
86	C₂×Q₃₂	Dic₇₉	1	4	yes	complemented
87	C₂×Q₃₂	C₂×D₇₉	1	2	yes	—
88	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
89	#427	—	158	1	no	—
90	M₆₄	—	158	2	no	—
91	C₂×C₃₂	—	79	2	no	—
92	#367	—	158	1	no	—
93	#366	—	79	2	no	—
94	C₂²×Q₈	—	4	1	no	—
95	C₂×Q₁₆	—	2	4	no	—
96	C₂×Q₁₆	—	2	2	no	—
97	C₂×Q₁₆	—	4	1	no	—
98	C₂×Q₁₆	—	4	2	no	—
99	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
100	#117	—	316	1	no	—
101	Q₃₂	—	2	4	no	—
102	Q₃₂	—	2	2	no	—
103	Q₃₂	C₂×Dic₇₉	1	4	yes	—
104	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
105	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
106	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
107	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
108	#107	—	158	2	no	—
109	#106	—	316	1	no	—
110	C₃₂	—	158	2	no	—
111	C₂×Q₈	—	4	4	no	—
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	8	1	no	—
114	C₂×Q₈	—	8	2	no	—
115	C₂²×C₄	—	8	1	no	—
116	C₂²×C₄	#11929609	1	1	yes	characteristic
117	Q₁₆	—	2	4	no	—
118	Q₁₆	—	4	4	no	—
119	Q₁₆	—	4	2	no	—
120	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
121	C₂×C₈	#11929609	1	1	yes	characteristic
122	C₂×C₈	#11929612	1	2	yes	—
123	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
124	C₄⋊C₄	—	316	2	no	—
125	C₂²⋊C₄	—	632	1	no	—
126	C₄²	—	632	1	no	—
127	C₁₆	C₂²⋊Dic₇₉	1	2	yes	—
128	C₁₆	C₂²×Dic₇₉	1	1	yes	characteristic
129	C₁₆	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
130	C₂³	#14579102	1	1	yes	characteristic
131	Q₈	—	4	4	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	8	4	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	8	4	no	—
136	C₂×C₄	—	632	2	no	—
137	C₂×C₄	#14579082	1	2	yes	—
138	C₂×C₄	#14579102	1	1	yes	characteristic
139	C₂×C₄	#14579105	1	2	yes	—
140	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
141	C₈	#14579082	1	2	yes	—
142	C₈	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
143	C₈	C₂×#11929612	1	1	yes	characteristic
144	C₂²	—	2	2	no	—
145	C₂²	#19098364	1	2	yes	—
146	C₂²	C₂×#14579102	1	1	yes	characteristic
147	C₄	—	8	4	no	—
148	C₄	—	1264	2	no	—
149	C₄	#19098364	1	2	yes	—
150	C₄	C₂×#14579082	1	1	yes	characteristic
151	C₄	C₂×#14579105	1	1	yes	characteristic
152	C₂	—	2	2	no	—
153	C₂	C₂×#19098364	1	1	yes	characteristic
154	C₂	#23765718	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098397

C₂

yes

maximal, characteristic

#23765184

C₂

yes

maximal, characteristic

#23761581

C₂

yes

maximal, complemented

C₂²×Q₃₂×C₇₉

C₂

yes

maximal, characteristic

#19098601

—

#19098397

—

C₂×C₇₉⋊C₃₂

C₂²

yes

—

#19098391

—

#19098389

C₂²

yes

—

C₂²×Q₁₆×C₇₉

—

C₂×Q₃₂×C₇₉

—

C₂×Q₃₂×C₇₉

—

C₂×Q₃₂×C₇₉

C₄

yes

complemented

C₂×Q₃₂×C₇₉

C₂²

yes

—

C₂²×C₁₆×C₇₉

C₂²

yes

—

—

—

C₇₉⋊C₃₂

—

C₂²×Q₈×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

Q₃₂×C₇₉

—

Q₃₂×C₇₉

—

Q₃₂×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

—

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₈

yes

characteristic

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

C₂²⋊C₄

yes

—

C₁₆×C₇₉

C₂²×C₄

yes

characteristic

C₁₆×C₇₉

C₂×D₄

yes

characteristic

C₂³×C₇₉

D₁₆

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

SD₃₂

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

#102

yes

—

C₈×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

#357

yes

—

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#357

yes

—

C₄×C₇₉

C₂×#102

yes

characteristic

C₄×C₇₉

C₂×SD₃₂

yes

characteristic

Dic₇₉

—

#13618

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₂×#357

yes

characteristic

C₂×C₇₉

#2264

yes

—

C₂²×Q₃₂

D₇₉

yes

characteristic

C₂×M₆₄

—

#2158

—

#1416

—

C₇₉

#13618

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

Dic₇₉

yes

complemented

C₂×Q₃₂

C₂×D₇₉

yes

—

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

#427

—

158

—

M₆₄

—

158

—

C₂×C₃₂

—

#367

—

158

—

#366

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

100

#117

—

316

—

101

Q₃₂

—

102

Q₃₂

—

103

Q₃₂

C₂×Dic₇₉

yes

—

104

C₂×C₁₆

C₂×Dic₇₉

yes

—

105

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

106

C₂×C₁₆

C₂²⋊D₇₉

yes

—

107

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

108

#107

—

158

—

109

#106

—

316

—

110

C₃₂

—

158

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂²×C₄

—

116

C₂²×C₄

#11929609

yes

characteristic

117

Q₁₆

—

118

Q₁₆

—

119

Q₁₆

—

120

C₂×C₈

C₂²⋊Dic₇₉

yes

—

121

C₂×C₈

#11929609

yes

characteristic

122

C₂×C₈

#11929612

yes

—

123

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

124

C₄⋊C₄

—

316

—

125

C₂²⋊C₄

—

632

—

126

C₄²

—

632

—

127

C₁₆

C₂²⋊Dic₇₉

yes

—

128

C₁₆

C₂²×Dic₇₉

yes

characteristic

129

C₁₆

C₂×C₂²⋊D₇₉

yes

characteristic

130

C₂³

#14579102

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

632

—

137

C₂×C₄

#14579082

yes

—

138

C₂×C₄

#14579102

yes

characteristic

139

C₂×C₄

#14579105

yes

—

140

C₂×C₄

C₂×#11929609

yes

characteristic

141

C₈

#14579082

yes

—

142

C₈

C₂×C₂²⋊Dic₇₉

yes

characteristic

143

C₈

C₂×#11929612

yes

characteristic

144

C₂²

—

145

C₂²

#19098364

yes

—

146

C₂²

C₂×#14579102

yes

characteristic

147

C₄

—

148

C₄

—

1264

—

149

C₄

#19098364

yes

—

150

C₄

C₂×#14579082

yes

characteristic

151

C₄

C₂×#14579105

yes

characteristic

152

C₂

—

153

C₂

C₂×#19098364

yes

characteristic

154

C₂

#23765718

yes

—

	subgroup	G/H
center	C₂²	C₂×#14579102
commutator	C₁₆×C₇₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₂²×Q₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₁₆
prime radical	C₂³×C₇₉	D₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	C₂²×Q₃₂	D₇₉
hypocenter	C₇₉	#13618
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂²

C₂×#14579102

commutator

C₁₆×C₇₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₂²×Q₃₂×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₁₆

prime radical

C₂³×C₇₉

D₁₆

cosocle

C₂×C₁₆×C₇₉

C₂³

hypercenter

C₂²×Q₃₂

D₇₉

hypocenter

C₇₉

#13618

LCS

G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
3. #23761581	C₂	semidirect
13. C₂×Q₃₂×C₇₉	C₄	semidirect
82. C₇₉	#13618	semidirect
86. C₂×Q₃₂	Dic₇₉	semidirect

H ≅ G/N

type

3. #23761581

C₂

semidirect

13. C₂×Q₃₂×C₇₉

C₄

semidirect

82. C₇₉

#13618

semidirect

86. C₂×Q₃₂

Dic₇₉

semidirect

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

subgroup	index
C₂×#19098397	2
#23765184	2
#23761581	2
C₂²×Q₃₂×C₇₉	2
#13618	79

subgroup

index

C₂²×Q₃₂×C₇₉

Subgroups — #34208716

Conjugacy classes of subgroups

Complements

Maximal subgroups