Group #34204635 — subgroups

Subgroups — #34204635

Order 20224.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#23765166	C₂	1	1	yes	maximal, characteristic
2	#23763155	C₂	1	1	yes	maximal, characteristic
3	#23761488	C₂	1	2	yes	maximal
4	#23761487	C₂	1	2	yes	maximal
5	C₇₉×#2151	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	2	yes	—
7	#19098518	C₂²	1	2	yes	—
8	#19098384	—	2	2	no	—
9	C₇₉⋊#338	C₄	1	1	yes	characteristic
10	#19098320	C₄	1	1	yes	characteristic
11	#19098319	C₄	1	2	yes	—
12	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	#405×C₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₉	—	2	2	no	—
16	C₂×#11929608	C₂×C₄	1	2	yes	—
17	#14579120	C₂×C₄	1	2	yes	—
18	#14579101	—	2	2	no	—
19	#14579101	—	2	2	no	—
20	C₂×C₇₉⋊C₁₆	—	2	2	no	—
21	#14579071	—	2	2	no	—
22	C₈×Dic₇₉	—	2	2	no	—
23	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
25	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₇₉	—	2	2	no	—
27	C₂×C₁₆×C₇₉	—	2	1	no	—
28	C₂×C₁₆×C₇₉	—	2	1	no	—
29	C₂×C₁₆×C₇₉	—	2	2	no	—
30	#97×C₇₉	Q₈	1	2	yes	—
31	C₄×C₈×C₇₉	D₄	1	2	yes	—
32	#11929608	—	2	2	no	—
33	#11929608	—	2	2	no	—
34	C₂×C₇₉⋊C₈	—	2	2	no	—
35	C₄⋊Dic₇₉	—	2	2	no	—
36	C₂²⋊Dic₇₉	—	2	2	no	—
37	C₄×Dic₇₉	—	2	2	no	—
38	C₇₉⋊C₁₆	—	4	2	no	—
39	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₇₉	—	2	2	no	—
41	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₇₉	#41	1	2	yes	—
45	C₂×C₈×C₇₉	#41	1	2	yes	—
46	C₄⋊C₄×C₇₉	—	2	1	no	—
47	C₂²⋊C₄×C₇₉	—	2	1	no	—
48	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
49	C₁₆×C₇₉	—	4	1	no	—
50	C₁₆×C₇₉	—	4	1	no	—
51	C₂³×C₇₉	#131	1	1	yes	characteristic
52	C₂×C₄×C₇₉	—	2	2	no	—
53	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#117	1	2	yes	—
55	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#131	1	2	yes	—
57	C₂×Dic₇₉	—	4	2	no	—
58	C₈×C₇₉	—	2	2	no	—
59	C₈×C₇₉	#124	1	1	yes	characteristic
60	C₈×C₇₉	#124	1	1	yes	characteristic
61	C₈×C₇₉	#125	1	2	yes	—
62	C₇₉⋊C₈	—	4	2	no	—
63	C₂²×C₇₉	—	4	1	no	—
64	C₂²×C₇₉	#424	1	1	yes	characteristic
65	C₂²×C₇₉	#433	1	2	yes	—
66	C₄×C₇₉	—	4	2	no	—
67	C₄×C₇₉	#382	1	1	yes	characteristic
68	C₄×C₇₉	#424	1	1	yes	characteristic
69	C₄×C₇₉	#433	1	2	yes	—
70	Dic₇₉	—	8	2	no	—
71	#12885	—	79	1	no	maximal, Sylow
72	C₂×C₇₉	—	2	2	no	—
73	C₂×C₇₉	—	2	1	no	—
74	C₂×C₇₉	#1832	1	1	yes	characteristic
75	#2153	—	79	1	no	—
76	#2151	D₇₉	1	1	yes	characteristic
77	#1747	—	79	1	no	—
78	#1379	—	79	2	no	—
79	#1378	—	79	2	no	—
80	C₇₉	#12885	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	79	2	no	—
82	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
83	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
84	#405	—	79	2	no	—
85	#405	C₂×D₇₉	1	1	yes	characteristic
86	#364	—	158	2	no	—
87	C₄⋊C₁₆	—	2	2	no	—
88	#338	—	79	1	no	—
89	H(3,ℤ/4)	—	79	1	no	—
90	H(3,ℤ/4)	—	79	2	no	—
91	C₂×M₁₆	—	79	2	no	—
92	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
94	#117	—	79	2	no	—
95	#117	C₄×D₇₉	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	158	2	no	—
98	M₃₂	—	158	2	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	158	2	no	—
103	#97	—	158	2	no	—
104	#97	Dic₁₅₈	1	2	yes	—
105	C₄×C₈	—	158	2	no	—
106	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	158	2	no	—
110	M₁₆	—	158	2	no	—
111	C₂×C₈	—	158	2	no	—
112	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
113	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
114	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
115	C₂×C₈	#11929623	1	2	yes	—
116	C₂×C₈	#11929625	1	2	yes	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	158	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	158	2	no	—
121	C₄²	—	158	2	no	—
122	C₄²	Dic₇₉⋊C₄	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₁₆	—	316	2	no	—
126	C₂³	#14579167	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	316	2	no	—
129	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
130	C₂×C₄	#14579119	1	2	yes	—
131	C₂×C₄	#14579167	1	1	yes	characteristic
132	C₂×C₄	#14579169	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	316	2	no	—
135	C₈	#14579148	1	1	yes	characteristic
136	C₈	#14579150	1	1	yes	characteristic
137	C₈	#14579152	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#19098589	1	1	yes	characteristic
140	C₂²	#19098623	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	632	2	no	—
143	C₄	#19098428	1	1	yes	characteristic
144	C₄	#19098591	1	1	yes	characteristic
145	C₄	#19098623	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#23763519	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765166

C₂

yes

maximal, characteristic

#23763155

C₂

yes

maximal, characteristic

#23761488

C₂

yes

maximal

#23761487

C₂

yes

maximal

C₇₉×#2151

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

—

#19098518

C₂²

yes

—

#19098384

—

C₇₉⋊#338

C₄

yes

characteristic

#19098320

C₄

yes

characteristic

#19098319

C₄

yes

—

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

#405×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₂×#11929608

C₂×C₄

yes

—

#14579120

C₂×C₄

yes

—

#14579101

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

#14579071

—

C₈×Dic₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#117×C₇₉

C₂×C₄

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

#97×C₇₉

Q₈

yes

—

C₄×C₈×C₇₉

D₄

yes

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₄⋊C₄

yes

—

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

#131

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

#131

yes

characteristic

C₂×C₄×C₇₉

#131

yes

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#424

yes

characteristic

C₂²×C₇₉

#433

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#12885

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

D₇₉

yes

—

—

—

C₇₉

#12885

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₉

yes

characteristic

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

#405

—

#405

C₂×D₇₉

yes

characteristic

#364

—

158

—

C₄⋊C₁₆

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₇₉

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

#117

—

#117

C₄×D₇₉

yes

characteristic

M₃₂

—

M₃₂

—

158

—

M₃₂

—

158

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

158

—

103

#97

—

158

—

104

#97

Dic₁₅₈

yes

—

105

C₄×C₈

—

158

—

106

C₄×C₈

C₂²⋊D₇₉

yes

—

107

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

108

M₁₆

—

109

M₁₆

—

158

—

110

M₁₆

—

158

—

111

C₂×C₈

—

158

—

112

C₂×C₈

Dic₇₉⋊C₄

yes

—

113

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

114

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

115

C₂×C₈

#11929623

yes

—

116

C₂×C₈

#11929625

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

158

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

158

—

121

C₄²

—

158

—

122

C₄²

Dic₇₉⋊C₄

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

316

—

126

C₂³

#14579167

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

316

—

129

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

130

C₂×C₄

#14579119

yes

—

131

C₂×C₄

#14579167

yes

characteristic

132

C₂×C₄

#14579169

yes

—

133

C₈

—

134

C₈

—

316

—

135

C₈

#14579148

yes

characteristic

136

C₈

#14579150

yes

characteristic

137

C₈

#14579152

yes

—

138

C₂²

—

139

C₂²

#19098589

yes

characteristic

140

C₂²

#19098623

yes

—

141

C₄

—

142

C₄

—

632

—

143

C₄

#19098428

yes

characteristic

144

C₄

#19098591

yes

characteristic

145

C₄

#19098623

yes

—

146

C₂

—

147

C₂

—

148

C₂

#23763519

yes

characteristic

	subgroup	G/H
center	C₈	#14579150
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2151	C₂
socle	C₂×C₇₉	#1832
prime radical	C₂³×C₇₉	#131
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2151	D₇₉
hypocenter	C₇₉	#12885
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

subgroup

G/H

center

C₈

#14579150

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2151

C₂

C₂×C₇₉

C₂³×C₇₉

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

N	H ≅ G/N	type
80. C₇₉	#12885	semidirect

H ≅ G/N

type

80. C₇₉

#12885

semidirect

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

subgroup	index
#23765166	2
#23763155	2
#23761488	2
#23761487	2
C₇₉×#2151	2
#12885	79

subgroup

index

C₇₉×#2151

Subgroups — #34204635

Conjugacy classes of subgroups

Complements

Maximal subgroups