Group #34193344 — subgroups

Subgroups — #34193344

Order 20224.

478 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 141 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	#23764206	C₂	1	3	yes	maximal
2	#23763528	C₂	1	3	yes	maximal, complemented
3	C₇₉×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14579158	C₂²	1	1	yes	characteristic
5	C₂×#14579083	—	2	6	no	—
6	#19098433	C₂²	1	3	yes	—
7	#19098326	—	2	6	no	—
8	C₂×C₄×C₈×C₇₉	C₂²	1	3	yes	—
9	C₂²×D₃₁₆	—	2	3	no	—
10	#14579158	—	2	3	no	—
11	#14579158	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₇₉	—	2	3	no	—
13	C₂×C₄⋊Dic₇₉	—	2	3	no	—
14	C₂×C₄⋊Dic₇₉	—	2	3	no	—
15	#14579083	—	4	12	no	—
16	C₂²×C₈×C₇₉	D₄	1	6	yes	—
17	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₇₉	—	2	6	no	—
19	C₂³×D₇₉	—	4	1	no	—
20	C₂×D₃₁₆	—	2	6	no	—
21	C₂×D₃₁₆	—	4	3	no	—
22	C₂×D₃₁₆	—	4	6	no	—
23	C₂×D₃₁₆	—	4	6	no	—
24	C₂²×Dic₇₉	—	4	3	no	—
25	C₄⋊Dic₇₉	—	4	6	no	—
26	C₄⋊Dic₇₉	—	4	6	no	—
27	C₄⋊Dic₇₉	—	4	6	no	—
28	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
29	C₂×C₈×C₇₉	—	2	6	no	—
30	C₂×C₈×C₇₉	—	2	6	no	—
31	C₂×C₈×C₇₉	—	2	6	no	—
32	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₇₉	#41	1	3	yes	—
34	C₂³×C₇₉	#127	1	1	yes	characteristic
35	C₂²×D₇₉	—	4	2	no	—
36	C₂²×D₇₉	—	8	3	no	—
37	C₂²×D₇₉	—	8	3	no	—
38	C₂×C₄×C₇₉	C₂²⋊Q₈	1	3	yes	—
39	C₂×C₄×C₇₉	#123	1	3	yes	—
40	C₂×C₄×C₇₉	#135	1	6	yes	—
41	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	6	yes	—
42	C₂×Dic₇₉	—	8	3	no	—
43	C₂×Dic₇₉	—	8	3	no	—
44	C₂×Dic₇₉	—	8	3	no	—
45	D₃₁₆	—	4	12	no	—
46	D₃₁₆	—	8	6	no	—
47	C₈×C₇₉	—	4	6	no	—
48	C₂²×C₇₉	#399	1	1	yes	characteristic
49	C₂²×C₇₉	#495	1	3	yes	—
50	C₂²×C₇₉	#496	1	3	yes	—
51	C₂×D₇₉	—	8	6	no	—
52	C₂×D₇₉	—	16	1	no	—
53	C₂×D₇₉	—	16	3	no	—
54	C₄×C₇₉	#478	1	6	yes	—
55	C₄×C₇₉	#482	1	6	yes	—
56	Dic₇₉	—	16	3	no	—
57	#10678	—	79	1	no	maximal, Sylow
58	C₂×C₇₉	#1711	1	1	yes	characteristic
59	C₂×C₇₉	#1717	1	3	yes	—
60	C₂×C₇₉	#2092	1	3	yes	—
61	D₇₉	—	16	2	no	—
62	#1957	—	79	3	no	—
63	#1834	—	79	3	no	—
64	#1273	D₇₉	1	1	yes	characteristic, complemented
65	C₇₉	#10678	1	1	yes	characteristic, complemented, Sylow
66	C₂×#127	—	79	1	no	—
67	C₂×#102	—	158	6	no	—
68	C₂×C₄×C₈	C₂×D₇₉	1	3	yes	—
69	#384	—	79	3	no	—
70	#340	—	158	6	no	—
71	C₂²×D₄	—	158	3	no	—
72	C₂²×C₈	D₃₁₆	1	6	yes	—
73	#127	—	79	2	no	—
74	#127	—	158	3	no	—
75	C₂×C₄⋊C₄	—	158	3	no	—
76	C₂×C₄⋊C₄	—	158	3	no	—
77	C₂×C₄⋊C₄	—	158	3	no	—
78	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
79	#102	—	316	12	no	—
80	C₄×C₈	—	2	6	no	—
81	C₂⁴	—	316	1	no	—
82	C₂×D₄	—	316	3	no	—
83	C₂×D₄	—	316	6	no	—
84	C₂×D₄	—	316	6	no	—
85	C₂×D₄	—	158	6	no	—
86	C₂²×C₄	—	316	3	no	—
87	C₂²×C₄	C₂×D₃₁₆	1	3	yes	—
88	C₂×C₈	—	2	6	no	—
89	C₂×C₈	—	2	6	no	—
90	C₂×C₈	—	2	6	no	—
91	C₄⋊C₄	—	316	6	no	—
92	C₄⋊C₄	—	316	6	no	—
93	C₄⋊C₄	—	316	6	no	—
94	C₄²	Q₈×D₇₉	1	1	yes	characteristic
95	C₄²	#11929623	1	3	yes	—
96	C₂³	—	316	2	no	—
97	C₂³	—	632	3	no	—
98	C₂³	—	632	3	no	—
99	C₂³	#14579158	1	1	yes	characteristic
100	D₄	—	316	12	no	—
101	D₄	—	632	6	no	—
102	C₂×C₄	—	632	3	no	—
103	C₂×C₄	—	632	3	no	—
104	C₂×C₄	—	632	3	no	—
105	C₂×C₄	#14579141	1	3	yes	—
106	C₂×C₄	#14579145	1	3	yes	—
107	C₂×C₄	#14579183	1	6	yes	—
108	C₂×C₄	#14579189	1	6	yes	—
109	C₈	—	4	6	no	—
110	C₂²	—	632	6	no	—
111	C₂²	—	1264	1	no	—
112	C₂²	—	1264	3	no	—
113	C₂²	#19098500	1	1	yes	characteristic
114	C₂²	#19098947	1	3	yes	—
115	C₂²	#19098952	1	3	yes	—
116	C₄	—	1264	3	no	—
117	C₄	#19098864	1	6	yes	—
118	C₄	#19098888	1	6	yes	—
119	C₂	—	1264	2	no	—
120	C₂	#23763012	1	1	yes	characteristic
121	C₂	#23763039	1	3	yes	—
122	C₂	#23764920	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764206

C₂

yes

maximal

#23763528

C₂

yes

maximal, complemented

C₇₉×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#14579158

C₂²

yes

characteristic

C₂×#14579083

—

#19098433

C₂²

yes

—

#19098326

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂²×D₃₁₆

—

#14579158

—

#14579158

Q₈

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579083

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

—

C₂³×D₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

#127

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₂²×C₇₉

#399

yes

characteristic

C₂²×C₇₉

#495

yes

—

C₂²×C₇₉

#496

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

#478

yes

—

C₄×C₇₉

#482

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#2092

yes

—

D₇₉

—

#1957

—

#1834

—

#1273

D₇₉

yes

characteristic, complemented

C₇₉

#10678

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#102

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

#384

—

#340

—

158

—

C₂²×D₄

—

158

—

C₂²×C₈

D₃₁₆

yes

—

#127

—

#127

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

#102

—

316

—

C₄×C₈

—

C₂⁴

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

158

—

C₂²×C₄

—

316

—

C₂²×C₄

C₂×D₃₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

316

—

C₄²

Q₈×D₇₉

yes

characteristic

C₄²

#11929623

yes

—

C₂³

—

316

—

C₂³

—

632

—

C₂³

—

632

—

C₂³

#14579158

yes

characteristic

100

D₄

—

316

—

101

D₄

—

632

—

102

C₂×C₄

—

632

—

103

C₂×C₄

—

632

—

104

C₂×C₄

—

632

—

105

C₂×C₄

#14579141

yes

—

106

C₂×C₄

#14579145

yes

—

107

C₂×C₄

#14579183

yes

—

108

C₂×C₄

#14579189

yes

—

109

C₈

—

110

C₂²

—

632

—

111

C₂²

—

1264

—

112

C₂²

—

1264

—

113

C₂²

#19098500

yes

characteristic

114

C₂²

#19098947

yes

—

115

C₂²

#19098952

yes

—

116

C₄

—

1264

—

117

C₄

#19098864

yes

—

118

C₄

#19098888

yes

—

119

C₂

—

1264

—

120

C₂

#23763012

yes

characteristic

121

C₂

#23763039

yes

—

122

C₂

#23764920

yes

—

	subgroup	G/H
center	C₂³	#14579158
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1273	C₂
socle	C₂³×C₇₉	#127
prime radical	C₂×#14579158	C₂²
hypercenter	#1273	D₇₉
hypocenter	C₇₉	#10678
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#14579158

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1273

C₂

C₂³×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #23763528	C₂	semidirect
3. C₇₉×#1273	C₂	semidirect
64. #1273	D₇₉	semidirect
65. C₇₉	#10678	semidirect

H ≅ G/N

type

2. #23763528

C₂

semidirect

3. C₇₉×#1273

C₂

semidirect

64. #1273

D₇₉

semidirect

65. C₇₉

#10678

semidirect

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

subgroup	index
#23764206	2
#23763528	2
C₇₉×#1273	2
#10678	79

subgroup

index

#23764206

#23763528

C₇₉×#1273

#10678

Subgroups — #34193344

Conjugacy classes of subgroups

Complements

Maximal subgroups