Group #34188237 — subgroups

Subgroups — #34188237

Order 20224.

406 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 155 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	#23773914	C₂	1	1	yes	maximal, characteristic
2	#23763572	C₂	1	1	yes	maximal, characteristic
3	#23761856	C₂	1	4	yes	maximal
4	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19099120	C₂²	1	2	yes	—
6	#19098968	C₄	1	2	yes	—
7	#19098965	—	2	2	no	—
8	#19098961	C₄	1	2	yes	—
9	C₂×#14579097	—	2	2	no	—
10	#19098601	—	2	2	no	—
11	#19098587	C₂²	1	2	yes	—
12	#19098295	—	2	8	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	#322×C₇₉	—	2	2	no	—
16	C₂×#11929608	—	2	2	no	—
17	#14579161	C₂×C₄	1	4	yes	—
18	#14579153	—	2	2	no	—
19	#14579140	—	2	4	no	—
20	C₄×Dic₁₅₈	—	2	4	no	—
21	#14579120	—	2	2	no	—
22	#14579097	—	4	4	no	—
23	#14579095	—	2	4	no	—
24	#14579093	—	2	4	no	—
25	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
26	C₂²×C₈×C₇₉	—	2	2	no	—
27	C₂²×C₈×C₇₉	D₄	1	2	yes	—
28	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₇₉	—	2	2	no	—
30	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	D₄	1	2	yes	—
33	C₄×C₈×C₇₉	D₄	1	4	yes	—
34	C₂×Dic₁₅₈	—	2	4	no	—
35	#11929608	—	4	4	no	—
36	C₂×C₇₉⋊C₈	—	4	2	no	—
37	C₄⋊Dic₇₉	—	2	4	no	—
38	C₄⋊Dic₇₉	—	2	4	no	—
39	Dic₇₉⋊C₄	—	4	4	no	—
40	C₂²⋊Dic₇₉	—	4	2	no	—
41	C₄×Dic₇₉	—	4	2	no	—
42	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	#41	1	2	yes	—
44	C₂×C₈×C₇₉	—	2	4	no	—
45	C₂×C₈×C₇₉	—	4	4	no	—
46	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
49	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
50	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₇₉	C₂×D₄	1	2	yes	—
53	C₂³×C₇₉	#124	1	1	yes	characteristic
54	C₂×C₄×C₇₉	—	2	4	no	—
55	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₇₉	#121	1	2	yes	—
58	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
61	C₂×Dic₇₉	—	4	4	no	—
62	Dic₁₅₈	—	4	8	no	—
63	C₈×C₇₉	—	4	4	no	—
64	C₈×C₇₉	#103	1	4	yes	—
65	C₈×C₇₉	#103	1	4	yes	—
66	C₇₉⋊C₈	—	8	2	no	—
67	C₂²×C₇₉	—	2	2	no	—
68	C₂²×C₇₉	#390	1	1	yes	characteristic
69	C₂²×C₇₉	#487	1	2	yes	—
70	C₄×C₇₉	—	2	2	no	—
71	C₄×C₇₉	C₂×#103	1	2	yes	—
72	C₄×C₇₉	#444	1	4	yes	—
73	C₄×C₇₉	#492	1	1	yes	characteristic
74	C₄×C₇₉	#494	1	1	yes	characteristic
75	Dic₇₉	—	8	4	no	—
76	#9861	—	79	1	no	maximal, Sylow
77	C₂×C₇₉	—	2	2	no	—
78	C₂×C₇₉	#1956	1	1	yes	characteristic
79	C₂×C₇₉	#1975	1	2	yes	—
80	#3160	—	79	1	no	—
81	#1848	—	79	1	no	—
82	#1493	—	79	4	no	—
83	C₂×#322	D₇₉	1	1	yes	characteristic
84	C₇₉	#9861	1	1	yes	characteristic, complemented, Sylow
85	#533	—	79	2	no	—
86	#500	—	79	2	no	—
87	#499	—	158	2	no	—
88	#498	—	79	2	no	—
89	C₂×#108	—	158	2	no	—
90	#427	—	158	2	no	—
91	#423	—	79	2	no	—
92	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
93	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
94	Q₈⋊C₈	—	158	8	no	—
95	#322	—	2	2	no	—
96	C₂×M₁₆	—	158	2	no	—
97	C₂²×C₈	—	2	2	no	—
98	C₂²×C₈	D₃₁₆	1	2	yes	—
99	C₄⋊Q₈	—	79	4	no	—
100	#125	—	158	2	no	—
101	C₂²⋊Q₈	—	158	4	no	—
102	C₄×Q₈	—	158	4	no	—
103	#117	—	158	2	no	—
104	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
105	#108	—	316	4	no	—
106	#107	—	158	4	no	—
107	#106	—	158	4	no	—
108	C₄⋊C₈	—	158	4	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
111	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
112	C₄×C₈	D₃₁₆	1	4	yes	—
113	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
114	C₂×Q₈	—	158	4	no	—
115	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
116	C₂²×C₄	#11929624	1	2	yes	—
117	M₁₆	—	316	4	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	4	4	no	—
120	C₂×C₈	—	316	2	no	—
121	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
122	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
123	C₂×C₈	#11929613	1	4	yes	—
124	C₂×C₈	Dic₃₁₆	1	4	yes	—
125	C₄⋊C₄	—	316	4	no	—
126	C₄⋊C₄	—	158	4	no	—
127	C₄⋊C₄	—	158	4	no	—
128	C₂²⋊C₄	—	316	2	no	—
129	C₄²	—	316	2	no	—
130	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
131	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
132	C₄²	C₂×D₃₁₆	1	2	yes	—
133	C₂³	#14579149	1	1	yes	characteristic
134	Q₈	—	316	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	316	4	no	—
137	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
138	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
139	C₂×C₄	#14579135	1	2	yes	—
140	C₂×C₄	#14579158	1	1	yes	characteristic
141	C₂×C₄	C₂×#11929613	1	2	yes	—
142	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
143	C₈	—	4	4	no	—
144	C₈	—	632	2	no	—
145	C₈	#14579086	1	4	yes	—
146	C₈	#14579086	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#19098465	1	1	yes	characteristic
149	C₂²	#19098914	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	632	4	no	—
152	C₄	C₂×#14579086	1	2	yes	—
153	C₄	#19098686	1	4	yes	—
154	C₄	#19098936	1	1	yes	characteristic
155	C₄	#19098943	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#23764202	1	1	yes	characteristic
158	C₂	#23764293	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23773914

C₂

yes

maximal, characteristic

#23763572

C₂

yes

maximal, characteristic

#23761856

C₂

yes

maximal

C₂×#322×C₇₉

C₂

yes

maximal, characteristic

#19099120

C₂²

yes

—

#19098968

C₄

yes

—

#19098965

—

#19098961

C₄

yes

—

C₂×#14579097

—

#19098601

—

#19098587

C₂²

yes

—

#19098295

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#322×C₇₉

—

C₂×#11929608

—

#14579161

C₂×C₄

yes

—

#14579153

—

#14579140

—

C₄×Dic₁₅₈

—

—

—

—

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

#103

yes

—

C₈×C₇₉

#103

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#487

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#103

yes

—

C₄×C₇₉

#444

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

#9861

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1956

yes

characteristic

C₂×C₇₉

#1975

yes

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#9861

yes

characteristic, complemented, Sylow

#533

—

#500

—

#499

—

158

—

#498

—

C₂×#108

—

158

—

#427

—

158

—

#423

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

Q₈⋊C₈

—

158

—

#322

—

C₂×M₁₆

—

158

—

C₂²×C₈

—

C₂²×C₈

D₃₁₆

yes

—

C₄⋊Q₈

—

100

#125

—

158

—

101

C₂²⋊Q₈

—

158

—

102

C₄×Q₈

—

158

—

103

#117

—

158

—

104

C₂×C₄²

C₂²×D₇₉

yes

characteristic

105

#108

—

316

—

106

#107

—

158

—

107

#106

—

158

—

108

C₄⋊C₈

—

158

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₇₉

yes

characteristic

111

C₄×C₈

C₄×D₇₉

yes

characteristic

112

C₄×C₈

D₃₁₆

yes

—

113

C₄×C₈

C₂²⋊D₇₉

yes

—

114

C₂×Q₈

—

158

—

115

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

116

C₂²×C₄

#11929624

yes

—

117

M₁₆

—

316

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

316

—

121

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

122

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

123

C₂×C₈

#11929613

yes

—

124

C₂×C₈

Dic₃₁₆

yes

—

125

C₄⋊C₄

—

316

—

126

C₄⋊C₄

—

158

—

127

C₄⋊C₄

—

158

—

128

C₂²⋊C₄

—

316

—

129

C₄²

—

316

—

130

C₄²

C₂×C₄×D₇₉

yes

characteristic

131

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

132

C₄²

C₂×D₃₁₆

yes

—

133

C₂³

#14579149

yes

characteristic

134

Q₈

—

316

—

135

C₂×C₄

—

136

C₂×C₄

—

316

—

137

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

138

C₂×C₄

C₄×D₃₁₆

yes

—

139

C₂×C₄

#14579135

yes

—

140

C₂×C₄

#14579158

yes

characteristic

141

C₂×C₄

C₂×#11929613

yes

—

142

C₂×C₄

C₂×Dic₃₁₆

yes

—

143

C₈

—

144

C₈

—

632

—

145

C₈

#14579086

yes

—

146

C₈

#14579086

yes

—

147

C₂²

—

148

C₂²

#19098465

yes

characteristic

149

C₂²

#19098914

yes

—

150

C₄

—

151

C₄

—

632

—

152

C₄

C₂×#14579086

yes

—

153

C₄

#19098686

yes

—

154

C₄

#19098936

yes

characteristic

155

C₄

#19098943

yes

characteristic

156

C₂

—

157

C₂

#23764202

yes

characteristic

158

C₂

#23764293

yes

—

	subgroup	G/H
center	C₂×C₄	#14579158
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#322×C₇₉	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂³×C₇₉	#124
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#9861
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14579158

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×#322×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
84. C₇₉	#9861	semidirect

H ≅ G/N

type

84. C₇₉

#9861

semidirect

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

subgroup	index
#23773914	2
#23763572	2
#23761856	2
C₂×#322×C₇₉	2
#9861	79

subgroup

index

C₂×#322×C₇₉

Subgroups — #34188237

Conjugacy classes of subgroups

Complements

Maximal subgroups