Group #34181966 — subgroups

Subgroups — #34181966

Order 20224.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#23763572	C₂	1	1	yes	maximal, characteristic
2	#23761692	C₂	1	1	yes	maximal, characteristic
3	#23761258	C₂	1	4	yes	maximal
4	C₇₉×#1835	C₂	1	1	yes	maximal, characteristic
5	#19098616	C₂²	1	2	yes	—
6	C₂×#14579097	—	2	2	no	—
7	#19098587	C₂²	1	2	yes	—
8	#19098285	C₄	1	4	yes	complemented
9	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
10	#382×C₇₉	C₂²	1	2	yes	—
11	#340×C₇₉	—	2	2	no	—
12	C₂×#11929608	—	2	2	no	—
13	#14579097	—	4	4	no	—
14	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
15	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
16	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
17	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₇₉	D₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
20	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₉	Q₈	1	2	yes	—
23	#95×C₇₉	—	2	4	no	—
24	#11929608	—	4	4	no	—
25	C₂×C₇₉⋊C₈	—	2	4	no	—
26	C₂×C₇₉⋊C₈	—	4	2	no	—
27	C₂²×C₄×C₇₉	—	4	2	no	—
28	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	#41	1	2	yes	—
30	C₂×C₈×C₇₉	—	2	2	no	—
31	C₂×C₈×C₇₉	—	2	2	no	—
32	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
33	C₄⋊C₄×C₇₉	—	4	2	no	—
34	C₄⋊C₄×C₇₉	—	4	2	no	—
35	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₇₉	#41	1	2	yes	—
38	C₂³×C₇₉	#124	1	1	yes	characteristic
39	C₂×C₄×C₇₉	—	4	2	no	—
40	C₂×C₄×C₇₉	—	8	4	no	—
41	C₂×C₄×C₇₉	—	8	2	no	—
42	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	#117	1	2	yes	—
44	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
45	C₂×C₄×C₇₉	#125	1	2	yes	—
46	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×C₄×C₇₉	#137	1	2	yes	—
48	C₈×C₇₉	—	2	4	no	—
49	C₇₉⋊C₈	—	4	4	no	—
50	C₇₉⋊C₈	—	8	2	no	—
51	C₂²×C₇₉	—	2	2	no	—
52	C₂²×C₇₉	#382	1	1	yes	characteristic
53	C₂²×C₇₉	#489	1	2	yes	—
54	C₄×C₇₉	—	2	2	no	—
55	C₄×C₇₉	—	2	2	no	—
56	C₄×C₇₉	—	16	2	no	—
57	C₄×C₇₉	#428	1	2	yes	—
58	C₄×C₇₉	#490	1	1	yes	characteristic
59	C₄×C₇₉	#491	1	1	yes	characteristic
60	#8794	—	79	1	no	maximal, Sylow
61	C₂×C₇₉	—	2	2	no	—
62	C₂×C₇₉	#1833	1	2	yes	—
63	C₂×C₇₉	#1836	1	1	yes	characteristic
64	#1848	—	79	1	no	—
65	#1835	D₇₉	1	1	yes	characteristic
66	#1452	—	79	1	no	—
67	#1286	—	79	4	no	—
68	C₇₉	#8794	1	1	yes	characteristic, complemented, Sylow
69	#431	—	79	2	no	—
70	C₂×#108	—	158	2	no	—
71	#423	—	79	2	no	—
72	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
73	#382	C₂×D₇₉	1	2	yes	—
74	#340	—	2	2	no	—
75	#322	—	79	4	no	—
76	C₂×M₁₆	—	158	2	no	—
77	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
80	#108	—	316	4	no	—
81	C₄⋊C₈	—	158	4	no	—
82	C₄⋊C₈	—	158	2	no	—
83	C₂²⋊C₈	—	158	2	no	—
84	C₄×C₈	—	79	4	no	—
85	C₄×C₈	Dic₁₅₈	1	2	yes	—
86	C₄×C₈	C₄×D₇₉	1	2	yes	—
87	#95	—	2	4	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
90	C₂²×C₄	#11929623	1	2	yes	—
91	M₁₆	—	316	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	316	2	no	—
95	C₂×C₈	—	158	4	no	—
96	C₂×C₈	Dic₇₉⋊C₄	1	4	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
100	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
101	C₄²	#11929625	1	2	yes	—
102	C₂³	#14579148	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
107	C₂×C₄	#14579119	1	2	yes	—
108	C₂×C₄	#14579150	1	1	yes	characteristic
109	C₂×C₄	#14579152	1	2	yes	—
110	C₂×C₄	#14579186	1	2	yes	—
111	C₂×C₄	#14579191	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	316	4	no	—
114	C₈	—	632	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#19098428	1	1	yes	characteristic
117	C₂²	#19098922	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	#19098606	1	2	yes	—
122	C₄	#19098929	1	1	yes	characteristic
123	C₄	#19098933	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#23763523	1	2	yes	—
126	C₂	#23763533	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763572

C₂

yes

maximal, characteristic

#23761692

C₂

yes

maximal, characteristic

#23761258

C₂

yes

maximal

C₇₉×#1835

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

—

C₂×#14579097

—

#19098587

C₂²

yes

—

#19098285

C₄

yes

complemented

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#382×C₇₉

C₂²

yes

—

#340×C₇₉

—

C₂×#11929608

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

#95×C₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

#125

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#382

yes

characteristic

C₂²×C₇₉

#489

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#428

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

#1833

yes

—

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

#1452

—

#1286

—

C₇₉

#8794

yes

characteristic, complemented, Sylow

#431

—

C₂×#108

—

158

—

#423

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#382

C₂×D₇₉

yes

—

#340

—

#322

—

C₂×M₁₆

—

158

—

C₂²×C₈

C₂²⋊D₇₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

#108

—

316

—

C₄⋊C₈

—

158

—

C₄⋊C₈

—

158

—

C₂²⋊C₈

—

158

—

C₄×C₈

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

C₄×D₇₉

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

C₂²×C₄

#11929623

yes

—

M₁₆

—

316

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

—

158

—

C₂×C₈

Dic₇₉⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×D₇₉

yes

characteristic

100

C₄²

C₂×Dic₁₅₈

yes

characteristic

101

C₄²

#11929625

yes

—

102

C₂³

#14579148

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

107

C₂×C₄

#14579119

yes

—

108

C₂×C₄

#14579150

yes

characteristic

109

C₂×C₄

#14579152

yes

—

110

C₂×C₄

#14579186

yes

—

111

C₂×C₄

#14579191

yes

—

112

C₈

—

113

C₈

—

316

—

114

C₈

—

632

—

115

C₂²

—

116

C₂²

#19098428

yes

characteristic

117

C₂²

#19098922

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#19098606

yes

—

122

C₄

#19098929

yes

characteristic

123

C₄

#19098933

yes

characteristic

124

C₂

—

125

C₂

#23763523

yes

—

126

C₂

#23763533

yes

characteristic

	subgroup	G/H
center	C₂²	#19098428
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1835	C₂
socle	C₂²×C₇₉	#382
prime radical	C₂³×C₇₉	#124
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1835	D₇₉
hypocenter	C₇₉	#8794
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#19098428

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1835

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
8. #19098285	C₄	semidirect
68. C₇₉	#8794	semidirect

H ≅ G/N

type

8. #19098285

C₄

semidirect

68. C₇₉

#8794

semidirect

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

subgroup	index
#23763572	2
#23761692	2
#23761258	2
C₇₉×#1835	2
#8794	79

subgroup

index

C₇₉×#1835

Subgroups — #34181966

Conjugacy classes of subgroups

Complements

Maximal subgroups