Group #34193319 — subgroups

Subgroups — #34193319

Order 20224.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#23763560	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×#14579095	—	2	6	no	—
6	C₂×#14579083	—	2	6	no	—
7	#19098433	C₂²	1	3	yes	—
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	6	no	—
14	#14579095	—	4	12	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	12	no	—
27	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
28	C₂×C₈×C₇₉	—	2	6	no	—
29	C₂×C₈×C₇₉	—	2	6	no	—
30	C₂×C₈×C₇₉	D₈	1	12	yes	—
31	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₇₉	#41	1	3	yes	—
33	C₂³×C₇₉	#127	1	1	yes	characteristic
34	C₂²×D₇₉	—	4	2	no	—
35	C₂²×D₇₉	—	8	3	no	—
36	C₂²×D₇₉	—	8	3	no	—
37	C₂×C₄×C₇₉	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₇₉	#123	1	3	yes	—
39	C₂×C₄×C₇₉	C₂×D₈	1	6	yes	—
40	C₂×C₄×C₇₉	#137	1	6	yes	—
41	C₂×Dic₇₉	—	8	3	no	—
42	C₂×Dic₇₉	—	8	6	no	—
43	D₃₁₆	—	4	12	no	—
44	D₃₁₆	—	8	6	no	—
45	C₈×C₇₉	—	2	12	no	—
46	C₂²×C₇₉	#399	1	1	yes	characteristic
47	C₂²×C₇₉	#493	1	3	yes	—
48	C₂²×C₇₉	#497	1	3	yes	—
49	C₂×D₇₉	—	8	6	no	—
50	C₂×D₇₉	—	16	1	no	—
51	C₂×D₇₉	—	16	3	no	—
52	C₄×C₇₉	#474	1	6	yes	—
53	C₄×C₇₉	#480	1	6	yes	—
54	Dic₇₉	—	16	3	no	—
55	#10673	—	79	1	no	maximal, Sylow
56	C₂×C₇₉	#1712	1	1	yes	characteristic
57	C₂×C₇₉	#1722	1	3	yes	—
58	C₂×C₇₉	#2093	1	3	yes	—
59	D₇₉	—	16	2	no	—
60	#1957	—	79	3	no	—
61	#1845	—	79	3	no	—
62	#1273	D₇₉	1	1	yes	characteristic, complemented
63	C₇₉	#10673	1	1	yes	characteristic, complemented, Sylow
64	C₂×#127	—	79	1	no	—
65	C₂×#107	—	158	6	no	—
66	C₂×#102	—	158	6	no	—
67	C₂×C₄×C₈	C₂×D₇₉	1	3	yes	—
68	#384	—	79	3	no	—
69	C₂²×D₄	—	158	3	no	—
70	C₂²×C₈	D₃₁₆	1	6	yes	—
71	#127	—	79	2	no	—
72	#127	—	158	3	no	—
73	C₂×C₄⋊C₄	—	158	3	no	—
74	C₂×C₄⋊C₄	—	158	6	no	—
75	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
76	#107	—	316	12	no	—
77	#102	—	316	12	no	—
78	C₄×C₈	—	2	6	no	—
79	C₂⁴	—	316	1	no	—
80	C₂×D₄	—	316	3	no	—
81	C₂×D₄	—	316	6	no	—
82	C₂×D₄	—	316	6	no	—
83	C₂×D₄	—	158	6	no	—
84	C₂²×C₄	—	316	3	no	—
85	C₂²×C₄	C₂×D₃₁₆	1	3	yes	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	D₆₃₂	1	12	yes	—
89	C₄⋊C₄	—	316	6	no	—
90	C₄⋊C₄	—	316	12	no	—
91	C₄²	Q₈×D₇₉	1	1	yes	characteristic
92	C₄²	#11929623	1	3	yes	—
93	C₂³	—	316	2	no	—
94	C₂³	—	632	3	no	—
95	C₂³	—	632	3	no	—
96	C₂³	#14579158	1	1	yes	characteristic
97	D₄	—	316	12	no	—
98	D₄	—	632	6	no	—
99	C₂×C₄	—	632	3	no	—
100	C₂×C₄	—	632	6	no	—
101	C₂×C₄	#14579141	1	3	yes	—
102	C₂×C₄	#14579145	1	3	yes	—
103	C₂×C₄	C₂×D₆₃₂	1	6	yes	—
104	C₂×C₄	#14579194	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	632	6	no	—
107	C₂²	—	1264	1	no	—
108	C₂²	—	1264	3	no	—
109	C₂²	#19098500	1	1	yes	characteristic
110	C₂²	#19098940	1	3	yes	—
111	C₂²	#19098957	1	3	yes	—
112	C₄	—	1264	3	no	—
113	C₄	#19098844	1	6	yes	—
114	C₄	#19098877	1	6	yes	—
115	C₂	—	1264	2	no	—
116	C₂	#23763015	1	1	yes	characteristic
117	C₂	#23763071	1	3	yes	—
118	C₂	#23764927	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764206

C₂

yes

maximal

#23763560

C₂

yes

maximal, complemented

C₇₉×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#14579158

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂²×D₃₁₆

—

#14579158

—

#14579158

Q₈

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579095

—

#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

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

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

C₂×D₈

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₂²×C₇₉

#399

yes

characteristic

C₂²×C₇₉

#493

yes

—

C₂²×C₇₉

#497

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

#474

yes

—

C₄×C₇₉

#480

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#2093

yes

—

D₇₉

—

#1957

—

#1845

—

#1273

D₇₉

yes

characteristic, complemented

C₇₉

#10673

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#107

—

158

—

C₂×#102

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

#384

—

C₂²×D₄

—

158

—

C₂²×C₈

D₃₁₆

yes

—

#127

—

#127

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

#107

—

316

—

#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

D₆₃₂

yes

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

316

—

C₄²

Q₈×D₇₉

yes

characteristic

C₄²

#11929623

yes

—

C₂³

—

316

—

C₂³

—

632

—

C₂³

—

632

—

C₂³

#14579158

yes

characteristic

D₄

—

316

—

D₄

—

632

—

C₂×C₄

—

632

—

100

C₂×C₄

—

632

—

101

C₂×C₄

#14579141

yes

—

102

C₂×C₄

#14579145

yes

—

103

C₂×C₄

C₂×D₆₃₂

yes

—

104

C₂×C₄

#14579194

yes

—

105

C₈

—

106

C₂²

—

632

—

107

C₂²

—

1264

—

108

C₂²

—

1264

—

109

C₂²

#19098500

yes

characteristic

110

C₂²

#19098940

yes

—

111

C₂²

#19098957

yes

—

112

C₄

—

1264

—

113

C₄

#19098844

yes

—

114

C₄

#19098877

yes

—

115

C₂

—

1264

—

116

C₂

#23763015

yes

characteristic

117

C₂

#23763071

yes

—

118

C₂

#23764927

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	#10673
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. #23763560	C₂	semidirect
3. C₇₉×#1273	C₂	semidirect
62. #1273	D₇₉	semidirect
63. C₇₉	#10673	semidirect

H ≅ G/N

type

2. #23763560

C₂

semidirect

3. C₇₉×#1273

C₂

semidirect

62. #1273

D₇₉

semidirect

63. C₇₉

#10673

semidirect

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

subgroup	index
#23764206	2
#23763560	2
C₇₉×#1273	2
#10673	79

subgroup

index

#23764206

#23763560

C₇₉×#1273

#10673

Subgroups — #34193319

Conjugacy classes of subgroups

Complements

Maximal subgroups