Group #34207953 — subgroups

Subgroups — #34207953

Order 20224.

416 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 81 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	#23763614	C₂	1	3	yes	maximal
2	#23761527	C₂	1	3	yes	maximal
3	C₇₉×#2811	C₂	1	1	yes	maximal, characteristic
4	#19099041	C₂²	1	1	yes	characteristic
5	#19098603	—	2	3	no	—
6	C₂×#14579080	C₂²	1	3	yes	—
7	#19098323	—	2	3	no	—
8	#395×C₇₉	—	2	3	no	—
9	#391×C₇₉	—	2	3	no	—
10	#391×C₇₉	C₂²	1	3	yes	—
11	C₂×#11929608	—	2	3	no	—
12	Q₈×Dic₇₉	—	4	1	no	—
13	Q₈×Dic₇₉	—	4	1	no	—
14	#14579120	—	2	3	no	—
15	#14579094	—	4	3	no	—
16	#14579092	—	4	3	no	—
17	#14579080	—	2	3	no	—
18	#14579080	Q₈	1	6	yes	complemented
19	C₂²×Q₈×C₇₉	—	2	1	no	—
20	C₂²×Q₈×C₇₉	—	2	1	no	—
21	C₂²×Q₈×C₇₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₇₉	—	2	3	no	—
23	C₂×C₄⋊C₄×C₇₉	D₄	1	3	yes	—
24	C₂×C₄²×C₇₉	D₄	1	3	yes	—
25	#95×C₇₉	—	2	3	no	—
26	#95×C₇₉	—	2	6	no	—
27	#11929608	—	2	6	no	—
28	#11929608	—	4	3	no	—
29	C₂×C₇₉⋊C₈	—	4	3	no	—
30	C₄⋊Dic₇₉	—	2	3	no	—
31	C₄⋊Dic₇₉	—	2	3	no	—
32	C₂²⋊Dic₇₉	—	4	1	no	—
33	C₄×Dic₇₉	—	4	3	no	—
34	C₂×Q₈×C₇₉	—	2	4	no	—
35	C₂×Q₈×C₇₉	—	4	1	no	—
36	C₂×Q₈×C₇₉	—	4	1	no	—
37	C₂×Q₈×C₇₉	—	8	1	no	—
38	C₂×Q₈×C₇₉	—	8	1	no	—
39	C₂×Q₈×C₇₉	—	8	2	no	—
40	C₂×Q₈×C₇₉	—	8	1	no	—
41	C₂×Q₈×C₇₉	C₂×Q₈	1	3	yes	—
42	C₂×Q₈×C₇₉	#41	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	—	2	3	no	—
44	C₂²×C₄×C₇₉	—	2	3	no	—
45	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
46	C₄⋊C₄×C₇₉	—	4	3	no	—
47	C₄⋊C₄×C₇₉	—	8	3	no	—
48	C₄²×C₇₉	—	4	3	no	—
49	C₂³×C₇₉	#120	1	1	yes	characteristic
50	Q₈×C₇₉	—	4	3	no	—
51	Q₈×C₇₉	—	4	1	no	—
52	Q₈×C₇₉	—	8	4	no	—
53	Q₈×C₇₉	—	8	4	no	—
54	C₂×C₄×C₇₉	—	2	3	no	—
55	C₂×C₄×C₇₉	—	2	3	no	—
56	C₂×C₄×C₇₉	—	2	6	no	—
57	C₂×C₄×C₇₉	—	4	3	no	—
58	C₂×C₄×C₇₉	—	4	3	no	—
59	C₂×C₄×C₇₉	—	4	6	no	—
60	C₂×C₄×C₇₉	—	4	3	no	—
61	C₂×C₄×C₇₉	C₂²⋊Q₈	1	3	yes	—
62	C₂×C₄×C₇₉	C₄⋊Q₈	1	3	yes	—
63	C₂×Dic₇₉	—	4	1	no	—
64	C₂×Dic₇₉	—	4	1	no	—
65	C₇₉⋊C₈	—	4	6	no	—
66	C₂²×C₇₉	—	2	1	no	—
67	C₂²×C₇₉	—	2	1	no	—
68	C₂²×C₇₉	#395	1	1	yes	characteristic
69	C₂²×C₇₉	B(4,𝔽₂)	1	1	yes	characteristic
70	C₂²×C₇₉	#458	1	1	yes	characteristic
71	C₄×C₇₉	—	2	3	no	—
72	C₄×C₇₉	—	2	3	no	—
73	C₄×C₇₉	—	4	6	no	—
74	C₄×C₇₉	—	8	3	no	—
75	Dic₇₉	—	8	1	no	—
76	Dic₇₉	—	8	1	no	—
77	#13464	—	79	1	no	maximal, Sylow
78	C₂×C₇₉	—	2	1	no	—
79	C₂×C₇₉	—	2	1	no	—
80	C₂×C₇₉	#2062	1	1	yes	characteristic
81	C₂×C₇₉	#2202	1	1	yes	characteristic
82	C₂×C₇₉	#2205	1	1	yes	characteristic
83	#2811	D₇₉	1	1	yes	characteristic
84	#1857	—	79	3	no	—
85	#1394	—	79	3	no	—
86	C₇₉	#13464	1	1	yes	characteristic, complemented, Sylow
87	#519	—	79	1	no	—
88	#428	—	158	3	no	—
89	C₂×#101	—	79	3	no	—
90	#395	—	2	3	no	—
91	#391	—	2	3	no	—
92	#391	C₂×D₇₉	1	3	yes	—
93	#339	—	158	3	no	—
94	C₂²×Q₈	—	2	1	no	—
95	C₂²×Q₈	—	2	1	no	—
96	C₂²×Q₈	C₂²×D₇₉	1	1	yes	characteristic
97	C₂×M₁₆	—	158	3	no	—
98	C₄×Q₈	—	316	1	no	—
99	C₄×Q₈	—	316	1	no	—
100	#117	—	158	3	no	—
101	C₂×C₄⋊C₄	—	2	3	no	—
102	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	3	yes	—
103	C₂×C₄²	C₂²⋊D₇₉	1	3	yes	—
104	#107	—	316	3	no	—
105	#106	—	316	3	no	—
106	#101	—	79	6	no	—
107	#101	—	158	3	no	—
108	#95	—	2	3	no	—
109	#95	—	2	6	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	4	1	no	—
112	C₂×Q₈	—	4	1	no	—
113	C₂×Q₈	—	8	1	no	—
114	C₂×Q₈	—	8	1	no	—
115	C₂×Q₈	—	8	2	no	—
116	C₂×Q₈	—	8	1	no	—
117	C₂×Q₈	Q₈×D₇₉	1	3	yes	—
118	C₂×Q₈	#11929625	1	1	yes	characteristic
119	C₂²×C₄	—	2	3	no	—
120	C₂²×C₄	—	2	3	no	—
121	C₂²×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
122	M₁₆	—	316	3	no	—
123	M₁₆	—	158	6	no	—
124	C₂×C₈	—	316	3	no	—
125	C₄⋊C₄	—	4	3	no	—
126	C₄⋊C₄	—	8	3	no	—
127	C₄⋊C₄	—	158	3	no	—
128	C₄⋊C₄	—	158	3	no	—
129	C₂²⋊C₄	—	316	1	no	—
130	C₄²	—	4	3	no	—
131	C₄²	—	316	3	no	—
132	C₂³	#14579130	1	1	yes	characteristic
133	Q₈	—	4	3	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	8	4	no	—
136	Q₈	—	8	4	no	—
137	C₂×C₄	—	2	3	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	4	3	no	—
141	C₂×C₄	—	4	3	no	—
142	C₂×C₄	—	4	6	no	—
143	C₂×C₄	—	4	3	no	—
144	C₂×C₄	—	316	1	no	—
145	C₂×C₄	—	316	1	no	—
146	C₂×C₄	#14579137	1	3	yes	—
147	C₂×C₄	#14579159	1	3	yes	—
148	C₈	—	316	6	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#19098482	1	1	yes	characteristic
152	C₂²	#19098750	1	1	yes	characteristic
153	C₂²	#19098753	1	1	yes	characteristic
154	C₄	—	2	3	no	—
155	C₄	—	2	3	no	—
156	C₄	—	4	6	no	—
157	C₄	—	8	3	no	—
158	C₄	—	632	1	no	—
159	C₄	—	632	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	#23764752	1	1	yes	characteristic
163	C₂	#23765417	1	1	yes	characteristic
164	C₂	#23765430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763614

C₂

yes

maximal

#23761527

C₂

yes

maximal

C₇₉×#2811

C₂

yes

maximal, characteristic

#19099041

C₂²

yes

characteristic

#19098603

—

C₂×#14579080

C₂²

yes

—

#19098323

—

#395×C₇₉

—

#391×C₇₉

—

#391×C₇₉

C₂²

yes

—

C₂×#11929608

—

Q₈×Dic₇₉

—

Q₈×Dic₇₉

—

—

—

—

—

Q₈

yes

complemented

C₂²×Q₈×C₇₉

—

C₂²×Q₈×C₇₉

—

C₂²×Q₈×C₇₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

—

#95×C₇₉

—

#95×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂×Q₈

yes

—

C₂×Q₈×C₇₉

#41

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

#120

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

Dic₇₉

—

#13464

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

#1857

—

#1394

—

C₇₉

#13464

yes

characteristic, complemented, Sylow

#519

—

#428

—

158

—

—

—

—

C₂×D₇₉

yes

—

#339

—

158

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂²×D₇₉

yes

characteristic

C₂×M₁₆

—

158

—

C₄×Q₈

—

316

—

C₄×Q₈

—

316

—

100

#117

—

158

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂²⋊D₇₉

yes

—

103

C₂×C₄²

C₂²⋊D₇₉

yes

—

104

#107

—

316

—

105

#106

—

316

—

106

#101

—

107

#101

—

158

—

108

#95

—

109

#95

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

Q₈×D₇₉

yes

—

118

C₂×Q₈

#11929625

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

122

M₁₆

—

316

—

123

M₁₆

—

158

—

124

C₂×C₈

—

316

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

158

—

128

C₄⋊C₄

—

158

—

129

C₂²⋊C₄

—

316

—

130

C₄²

—

131

C₄²

—

316

—

132

C₂³

#14579130

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

316

—

145

C₂×C₄

—

316

—

146

C₂×C₄

#14579137

yes

—

147

C₂×C₄

#14579159

yes

—

148

C₈

—

316

—

149

C₂²

—

150

C₂²

—

151

C₂²

#19098482

yes

characteristic

152

C₂²

#19098750

yes

characteristic

153

C₂²

#19098753

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

632

—

159

C₄

—

632

—

160

C₂

—

161

C₂

—

162

C₂

#23764752

yes

characteristic

163

C₂

#23765417

yes

characteristic

164

C₂

#23765430

yes

characteristic

	subgroup	G/H
center	C₂²	#19098750
commutator, cosocle	C₂²×Q₈×C₇₉	C₂³
Frattini	C₂²×Q₈	C₂²×D₇₉
Fitting	C₇₉×#2811	C₂
socle	C₂²×C₇₉	B(4,𝔽₂)
prime radical	C₂³×C₇₉	#120
hypercenter	#2811	D₇₉
hypocenter	C₇₉	#13464
LCS	G ▷ C₂²×Q₈×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #2811

subgroup

G/H

center

C₂²

#19098750

commutator, cosocle

C₂²×Q₈×C₇₉

C₂³

Frattini

C₂²×Q₈

C₂²×D₇₉

Fitting

C₇₉×#2811

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂²×Q₈×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #2811

N	H ≅ G/N	type
18. #14579080	Q₈	semidirect
86. C₇₉	#13464	semidirect

H ≅ G/N

type

18. #14579080

Q₈

semidirect

86. C₇₉

#13464

semidirect

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

subgroup	index
#23763614	2
#23761527	2
C₇₉×#2811	2
#13464	79

subgroup

index

#23763614

#23761527

C₇₉×#2811

#13464

Subgroups — #34207953

Conjugacy classes of subgroups

Complements

Maximal subgroups