Group #34187768 — subgroups

Subgroups — #34187768

Order 20224.

454 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 119 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×#19098943	C₂	1	1	yes	maximal, characteristic
2	#23762073	C₂	1	4	yes	maximal
3	#23761692	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1837	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₆	—	2	2	no	—
6	C₂×#14579161	C₂²	1	2	yes	—
7	#19098943	—	2	2	no	—
8	#19098943	C₄	1	4	yes	—
9	#19098616	C₂²	1	2	yes	—
10	#19098310	—	2	8	no	—
11	#19098285	—	2	2	no	—
12	#423×C₇₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#341×C₇₉	—	2	2	no	—
15	C₂²×Dic₁₅₈	—	2	2	no	—
16	C₂×Dic₃₁₆	—	2	8	no	—
17	C₂×Dic₃₁₆	—	4	8	no	—
18	#14579161	—	2	4	no	—
19	#14579161	—	2	2	no	—
20	#14579161	C₂×C₄	1	4	yes	—
21	C₂×C₄⋊Dic₇₉	—	2	4	no	—
22	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
23	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄×C₇₉⋊C₈	—	2	2	no	—
25	C₂×M₁₆×C₇₉	—	2	2	no	—
26	C₂²×C₈×C₇₉	D₄	1	2	yes	—
27	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₇₉	—	2	4	no	—
29	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₉	D₄	1	2	yes	—
31	C₂×Dic₁₅₈	—	2	4	no	—
32	C₂×Dic₁₅₈	—	4	2	no	—
33	C₂×Dic₁₅₈	—	4	8	no	—
34	C₂²×Dic₇₉	—	4	2	no	—
35	Dic₃₁₆	—	4	16	no	—
36	C₂×C₇₉⋊C₈	—	2	4	no	—
37	C₄⋊Dic₇₉	—	2	8	no	—
38	C₄⋊Dic₇₉	—	4	4	no	—
39	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	#41	1	2	yes	—
41	M₁₆×C₇₉	—	4	4	no	—
42	C₂×C₈×C₇₉	—	2	4	no	—
43	C₂×C₈×C₇₉	—	4	2	no	—
44	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
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	C₂×C₄×C₇₉	—	2	4	no	—
50	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
52	C₂×C₄×C₇₉	#121	1	2	yes	—
53	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
54	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
55	C₂×Dic₇₉	—	4	4	no	—
56	C₂×Dic₇₉	—	8	4	no	—
57	Dic₁₅₈	—	4	8	no	—
58	Dic₁₅₈	—	8	4	no	—
59	C₈×C₇₉	—	2	4	no	—
60	C₈×C₇₉	—	8	2	no	—
61	C₇₉⋊C₈	—	4	4	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	#418	1	2	yes	—
67	C₄×C₇₉	#473	1	4	yes	—
68	C₄×C₇₉	#496	1	2	yes	—
69	Dic₇₉	—	8	4	no	—
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	—	79	1	no	—
75	#1837	D₇₉	1	1	yes	characteristic
76	#1546	—	79	4	no	—
77	#1452	—	79	1	no	—
78	C₇₉	#9779	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	—	158	2	no	—
80	C₂×C₄⋊Q₈	—	79	2	no	—
81	#494	—	79	4	no	—
82	#494	—	158	2	no	—
83	#431	—	79	2	no	—
84	#423	C₂×D₇₉	1	2	yes	—
85	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
86	#341	—	2	2	no	—
87	#332	—	158	8	no	—
88	#322	—	158	2	no	—
89	C₂²×Q₈	—	158	2	no	—
90	C₂×Q₁₆	—	316	8	no	—
91	C₂×Q₁₆	—	158	8	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
94	C₄⋊Q₈	—	79	4	no	—
95	C₄⋊Q₈	—	158	4	no	—
96	C₄⋊Q₈	—	158	2	no	—
97	C₂×C₄⋊C₄	—	158	4	no	—
98	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	79	4	no	—
101	C₂²⋊C₈	—	158	2	no	—
102	C₄×C₈	—	158	2	no	—
103	C₄×C₈	C₄×D₇₉	1	2	yes	—
104	C₄×C₈	D₃₁₆	1	2	yes	—
105	C₂×Q₈	—	316	2	no	—
106	C₂×Q₈	—	316	8	no	—
107	C₂×Q₈	—	158	4	no	—
108	C₂²×C₄	—	316	2	no	—
109	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
110	C₂²×C₄	#11929625	1	2	yes	—
111	Q₁₆	—	316	16	no	—
112	M₁₆	—	4	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	158	4	no	—
116	C₂×C₈	(C₂×D₇₉)⋊C₄	1	4	yes	—
117	C₄⋊C₄	—	316	4	no	—
118	C₄⋊C₄	—	158	8	no	—
119	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
120	C₄²	D₄×D₇₉	1	2	yes	—
121	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
122	C₂³	#14579150	1	1	yes	characteristic
123	Q₈	—	316	8	no	—
124	Q₈	—	632	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	316	4	no	—
127	C₂×C₄	—	632	4	no	—
128	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
129	C₂×C₄	#14579123	1	2	yes	—
130	C₂×C₄	#14579136	1	2	yes	—
131	C₂×C₄	#14579157	1	1	yes	characteristic
132	C₂×C₄	#14579186	1	4	yes	—
133	C₈	—	2	4	no	—
134	C₈	—	8	2	no	—
135	C₈	—	316	4	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#19098464	1	1	yes	characteristic
138	C₂²	#19098929	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	632	4	no	—
141	C₄	#19098567	1	2	yes	—
142	C₄	#19098839	1	4	yes	—
143	C₄	#19098953	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#23764223	1	1	yes	characteristic
146	C₂	#23764313	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098943

C₂

yes

maximal, characteristic

#23762073

C₂

yes

maximal

#23761692

C₂

yes

maximal, characteristic

C₇₉×#1837

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₆

—

C₂×#14579161

C₂²

yes

—

#19098943

—

#19098943

C₄

yes

—

#19098616

C₂²

yes

—

#19098310

—

#19098285

—

#423×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#341×C₇₉

—

C₂²×Dic₁₅₈

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

#14579161

—

#14579161

—

#14579161

C₂×C₄

yes

—

C₂×C₄⋊Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

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

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#490

yes

—

C₄×C₇₉

—

C₄×C₇₉

#418

yes

—

C₄×C₇₉

#473

yes

—

C₄×C₇₉

#496

yes

—

Dic₇₉

—

#9779

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1961

yes

characteristic

C₂×C₇₉

#1979

yes

—

C₂×#494

—

#1837

D₇₉

yes

characteristic

#1546

—

#1452

—

C₇₉

#9779

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

158

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

158

—

#431

—

#423

C₂×D₇₉

yes

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#341

—

#332

—

158

—

#322

—

158

—

C₂²×Q₈

—

158

—

C₂×Q₁₆

—

316

—

C₂×Q₁₆

—

158

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₇₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

158

—

102

C₄×C₈

—

158

—

103

C₄×C₈

C₄×D₇₉

yes

—

104

C₄×C₈

D₃₁₆

yes

—

105

C₂×Q₈

—

316

—

106

C₂×Q₈

—

316

—

107

C₂×Q₈

—

158

—

108

C₂²×C₄

—

316

—

109

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

110

C₂²×C₄

#11929625

yes

—

111

Q₁₆

—

316

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

158

—

116

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

117

C₄⋊C₄

—

316

—

118

C₄⋊C₄

—

158

—

119

C₄²

C₂×C₄×D₇₉

yes

characteristic

120

C₄²

D₄×D₇₉

yes

—

121

C₄²

C₂×D₃₁₆

yes

characteristic

122

C₂³

#14579150

yes

characteristic

123

Q₈

—

316

—

124

Q₈

—

632

—

125

C₂×C₄

—

126

C₂×C₄

—

316

—

127

C₂×C₄

—

632

—

128

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

129

C₂×C₄

#14579123

yes

—

130

C₂×C₄

#14579136

yes

—

131

C₂×C₄

#14579157

yes

characteristic

132

C₂×C₄

#14579186

yes

—

133

C₈

—

134

C₈

—

135

C₈

—

316

—

136

C₂²

—

137

C₂²

#19098464

yes

characteristic

138

C₂²

#19098929

yes

—

139

C₄

—

140

C₄

—

632

—

141

C₄

#19098567

yes

—

142

C₄

#19098839

yes

—

143

C₄

#19098953

yes

—

144

C₂

—

145

C₂

#23764223

yes

characteristic

146

C₂

#23764313

yes

—

	subgroup	G/H
center	C₂²	#19098464
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1837	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂³×C₇₉	#124
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1837	D₇₉
hypocenter	C₇₉	#9779
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#19098464

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1837

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

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
C₂×#19098943	2
#23762073	2
#23761692	2
C₇₉×#1837	2
#9779	79

subgroup

index

C₇₉×#1837

Subgroups — #34187768

Conjugacy classes of subgroups

Complements

Maximal subgroups