Group #34187770 — subgroups

Subgroups — #34187770

Order 20224.

454 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 127 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	#23763538	C₂	1	1	yes	maximal, characteristic
2	#23762072	C₂	1	4	yes	maximal
3	#23761692	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19098616	C₂²	1	2	yes	—
6	#19098587	C₂²	1	2	yes	—
7	#19098328	—	2	2	no	—
8	#19098309	—	2	8	no	—
9	#19098285	—	2	2	no	—
10	C₂²×Q₁₆×C₇₉	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
12	#494×C₇₉	—	2	2	no	—
13	#494×C₇₉	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#11929608	—	2	2	no	—
16	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
17	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
18	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
19	C₄×C₇₉⋊C₈	—	2	2	no	—
20	C₂²×Q₈×C₇₉	—	2	2	no	—
21	C₂×Q₁₆×C₇₉	—	2	8	no	—
22	C₂×Q₁₆×C₇₉	—	4	8	no	—
23	C₂²×C₈×C₇₉	D₄	1	2	yes	—
24	C₄⋊Q₈×C₇₉	—	2	4	no	—
25	C₄⋊Q₈×C₇₉	—	2	2	no	—
26	C₄⋊Q₈×C₇₉	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
28	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₉	D₄	1	2	yes	—
31	#11929608	—	4	4	no	—
32	C₂×C₇₉⋊C₈	—	2	4	no	—
33	C₂×C₇₉⋊C₈	—	4	2	no	—
34	C₂×Q₈×C₇₉	—	2	4	no	—
35	C₂×Q₈×C₇₉	—	4	2	no	—
36	C₂×Q₈×C₇₉	—	4	8	no	—
37	C₂²×C₄×C₇₉	—	4	2	no	—
38	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	#41	1	2	yes	—
40	Q₁₆×C₇₉	—	4	16	no	—
41	C₂×C₈×C₇₉	—	2	4	no	—
42	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
43	C₄⋊C₄×C₇₉	—	2	8	no	—
44	C₄⋊C₄×C₇₉	—	4	4	no	—
45	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₇₉	C₂×D₄	1	2	yes	—
48	C₂³×C₇₉	#124	1	1	yes	characteristic
49	Q₈×C₇₉	—	4	8	no	—
50	Q₈×C₇₉	—	8	4	no	—
51	C₂×C₄×C₇₉	—	2	4	no	—
52	C₂×C₄×C₇₉	—	4	4	no	—
53	C₂×C₄×C₇₉	—	8	4	no	—
54	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
56	C₂×C₄×C₇₉	#121	1	2	yes	—
57	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
58	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
59	C₈×C₇₉	—	2	4	no	—
60	C₇₉⋊C₈	—	4	4	no	—
61	C₇₉⋊C₈	—	8	2	no	—
62	C₂²×C₇₉	—	2	2	no	—
63	C₂²×C₇₉	#390	1	1	yes	characteristic
64	C₂²×C₇₉	#490	1	2	yes	—
65	C₄×C₇₉	—	2	2	no	—
66	C₄×C₇₉	—	8	4	no	—
67	C₄×C₇₉	#418	1	2	yes	—
68	C₄×C₇₉	#473	1	4	yes	—
69	C₄×C₇₉	#496	1	2	yes	—
70	#9779	—	79	1	no	maximal, Sylow
71	C₂×C₇₉	—	2	2	no	—
72	C₂×C₇₉	#1961	1	1	yes	characteristic
73	C₂×C₇₉	#1979	1	2	yes	—
74	C₂×#494	D₇₉	1	1	yes	characteristic
75	#1837	—	79	1	no	—
76	#1546	—	79	4	no	—
77	#1452	—	79	1	no	—
78	C₇₉	#9779	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	—	2	2	no	—
80	C₂×C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
81	#494	—	2	2	no	—
82	#494	Dic₇₉	1	4	yes	—
83	#431	—	79	2	no	—
84	#423	—	79	2	no	—
85	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
86	#341	—	158	2	no	—
87	#332	—	158	8	no	—
88	#322	—	158	2	no	—
89	C₂²×Q₈	—	2	2	no	—
90	C₂×Q₁₆	—	2	8	no	—
91	C₂×Q₁₆	—	4	8	no	—
92	C₂×M₁₆	—	158	2	no	—
93	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
94	C₄⋊Q₈	—	2	4	no	—
95	C₄⋊Q₈	—	2	2	no	—
96	C₄⋊Q₈	C₂×Dic₇₉	1	4	yes	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
99	C₄⋊C₈	—	79	4	no	—
100	C₄⋊C₈	—	158	4	no	—
101	C₂²⋊C₈	—	158	2	no	—
102	C₄×C₈	—	158	2	no	—
103	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
104	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	8	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
110	C₂²×C₄	#11929623	1	2	yes	—
111	Q₁₆	—	4	16	no	—
112	M₁₆	—	316	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	316	2	no	—
115	C₂×C₈	—	158	4	no	—
116	C₂×C₈	C₂²⋊Dic₇₉	1	4	yes	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	4	4	no	—
119	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
120	C₄²	D₄×D₇₉	1	2	yes	—
121	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
122	C₂³	#14579148	1	1	yes	characteristic
123	Q₈	—	4	8	no	—
124	Q₈	—	8	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	8	4	no	—
128	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
129	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
130	C₂×C₄	#14579132	1	2	yes	—
131	C₂×C₄	#14579157	1	1	yes	characteristic
132	C₂×C₄	#14579187	1	4	yes	—
133	C₈	—	2	4	no	—
134	C₈	—	316	4	no	—
135	C₈	—	632	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#19098466	1	1	yes	characteristic
138	C₂²	#19098927	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	4	no	—
141	C₄	#19098568	1	2	yes	—
142	C₄	#19098835	1	4	yes	—
143	C₄	#19098950	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#23764225	1	1	yes	characteristic
146	C₂	#23764316	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763538

C₂

yes

maximal, characteristic

#23762072

C₂

yes

maximal

#23761692

C₂

yes

maximal, characteristic

C₂×#494×C₇₉

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

—

#19098587

C₂²

yes

—

#19098328

—

#19098309

—

#19098285

—

C₂²×Q₁₆×C₇₉

—

C₂×C₄⋊Q₈×C₇₉

C₂²

yes

—

#494×C₇₉

—

#494×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂²×Q₈×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

#124

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

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

AGL(1,ℤ/8)

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#490

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#418

yes

—

C₄×C₇₉

#473

yes

—

C₄×C₇₉

#496

yes

—

#9779

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1961

yes

characteristic

C₂×C₇₉

#1979

yes

—

C₂×#494

D₇₉

yes

—

—

—

C₇₉

#9779

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₇₉

yes

—

#494

—

#494

Dic₇₉

yes

—

#431

—

#423

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#341

—

158

—

#332

—

158

—

#322

—

158

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

158

—

C₂²×C₈

C₂²⋊D₇₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×Dic₇₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

158

—

101

C₂²⋊C₈

—

158

—

102

C₄×C₈

—

158

—

103

C₄×C₈

C₂×Dic₇₉

yes

—

104

C₄×C₈

C₂²⋊D₇₉

yes

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

110

C₂²×C₄

#11929623

yes

—

111

Q₁₆

—

112

M₁₆

—

316

—

113

C₂×C₈

—

114

C₂×C₈

—

316

—

115

C₂×C₈

—

158

—

116

C₂×C₈

C₂²⋊Dic₇₉

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

C₂²×Dic₇₉

yes

characteristic

120

C₄²

D₄×D₇₉

yes

—

121

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

122

C₂³

#14579148

yes

characteristic

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

129

C₂×C₄

D₄×Dic₇₉

yes

—

130

C₂×C₄

#14579132

yes

—

131

C₂×C₄

#14579157

yes

characteristic

132

C₂×C₄

#14579187

yes

—

133

C₈

—

134

C₈

—

316

—

135

C₈

—

632

—

136

C₂²

—

137

C₂²

#19098466

yes

characteristic

138

C₂²

#19098927

yes

—

139

C₄

—

140

C₄

—

141

C₄

#19098568

yes

—

142

C₄

#19098835

yes

—

143

C₄

#19098950

yes

—

144

C₂

—

145

C₂

#23764225

yes

characteristic

146

C₂

#23764316

yes

—

	subgroup	G/H
center	C₂²	#19098466
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#494×C₇₉	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂³×C₇₉	#124
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#494	D₇₉
hypocenter	C₇₉	#9779
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#19098466

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×#494×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
78. C₇₉	#9779	semidirect

H ≅ G/N

type

78. C₇₉

#9779

semidirect

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

subgroup	index
#23763538	2
#23762072	2
#23761692	2
C₂×#494×C₇₉	2
#9779	79

subgroup

index

C₂×#494×C₇₉

Subgroups — #34187770

Conjugacy classes of subgroups

Complements

Maximal subgroups