Group #34182043 — subgroups

Subgroups — #34182043

Order 20224.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#23763557	C₂	1	2	yes	maximal
2	#23763528	C₂	1	4	yes	maximal
3	C₂×C₈²×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579093	—	2	4	no	—
5	#19098433	C₂²	1	4	yes	—
6	#19098326	—	2	8	no	—
7	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
9	C₈²×C₇₉	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₇₉	—	2	8	no	—
11	C₂×C₄⋊Dic₇₉	—	2	4	no	—
12	#14579093	—	4	8	no	—
13	C₂²×C₈×C₇₉	D₄	1	4	yes	—
14	C₂²×C₈×C₇₉	D₄	1	2	yes	—
15	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₇₉	Q₈	1	4	yes	—
19	C₄×C₈×C₇₉	Q₈	1	2	yes	—
20	C₂²×Dic₇₉	—	4	4	no	—
21	C₄⋊Dic₇₉	—	4	8	no	—
22	C₄⋊Dic₇₉	—	4	16	no	—
23	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
25	C₂×C₈×C₇₉	—	2	8	no	—
26	C₂×C₈×C₇₉	C₄⋊C₄	1	8	yes	—
27	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
28	C₂×C₈×C₇₉	SD₁₆	1	8	yes	—
29	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
31	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
32	C₂³×C₇₉	#127	1	1	yes	characteristic
33	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
36	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
38	C₂×C₄×C₇₉	#135	1	8	yes	—
39	C₂×Dic₇₉	—	8	4	no	—
40	C₂×Dic₇₉	—	8	8	no	—
41	C₈×C₇₉	—	2	8	no	—
42	C₈×C₇₉	#106	1	8	yes	—
43	C₂²×C₇₉	#384	1	1	yes	characteristic
44	C₂²×C₇₉	#492	1	2	yes	—
45	C₂²×C₇₉	#495	1	4	yes	—
46	C₄×C₇₉	C₂×#106	1	2	yes	—
47	C₄×C₇₉	#427	1	4	yes	—
48	C₄×C₇₉	#498	1	2	yes	—
49	C₄×C₇₉	#499	1	4	yes	—
50	Dic₇₉	—	16	4	no	—
51	#8815	—	79	1	no	maximal, Sylow
52	C₂×C₇₉	#1708	1	4	yes	—
53	C₂×C₇₉	#1844	1	1	yes	characteristic
54	C₂×C₇₉	#1846	1	2	yes	—
55	#1844	—	79	2	no	—
56	#1834	—	79	4	no	—
57	C₂×C₈²	D₇₉	1	1	yes	characteristic
58	C₇₉	#8815	1	1	yes	characteristic, complemented, Sylow
59	C₂×#106	—	158	4	no	—
60	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
61	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
62	#384	—	79	4	no	—
63	#340	—	158	8	no	—
64	C₈²	Dic₇₉	1	4	yes	complemented
65	C₂²×C₈	D₃₁₆	1	4	yes	—
66	C₂²×C₈	D₃₁₆	1	2	yes	—
67	C₂×C₄⋊C₄	—	158	8	no	—
68	C₂×C₄⋊C₄	—	158	4	no	—
69	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
70	#106	—	316	8	no	—
71	C₄×C₈	Dic₁₅₈	1	4	yes	—
72	C₄×C₈	Dic₁₅₈	1	2	yes	—
73	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
74	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
75	C₂²×C₄	—	316	4	no	—
76	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	C₄⋊Dic₇₉	1	8	yes	—
80	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
81	C₂×C₈	#11929613	1	8	yes	—
82	C₄⋊C₄	—	316	8	no	—
83	C₄⋊C₄	—	316	16	no	—
84	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
85	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
86	C₄²	C₂×Dic₁₅₈	1	2	yes	—
87	C₂³	#14579158	1	1	yes	characteristic
88	C₂×C₄	—	632	4	no	—
89	C₂×C₄	—	632	8	no	—
90	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
92	C₂×C₄	#14579161	1	1	yes	characteristic
93	C₂×C₄	#14579161	1	2	yes	—
94	C₂×C₄	C₂×#11929613	1	4	yes	—
95	C₂×C₄	#14579183	1	8	yes	—
96	C₈	—	2	8	no	—
97	C₈	#14579093	1	8	yes	—
98	C₂²	#19098433	1	1	yes	characteristic
99	C₂²	#19098936	1	2	yes	—
100	C₂²	#19098947	1	4	yes	—
101	C₄	—	1264	4	no	—
102	C₄	C₂×#14579093	1	2	yes	—
103	C₄	#19098601	1	4	yes	—
104	C₄	#19098961	1	2	yes	—
105	C₄	#19098965	1	4	yes	—
106	C₂	#23763001	1	4	yes	—
107	C₂	#23763557	1	1	yes	characteristic
108	C₂	#23763564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763557

C₂

yes

maximal

#23763528

C₂

yes

maximal

C₂×C₈²×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579093

—

#19098433

C₂²

yes

—

#19098326

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₈²×C₇₉

C₄

yes

complemented

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579093

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#106

yes

—

C₂²×C₇₉

#384

yes

characteristic

C₂²×C₇₉

#492

yes

—

C₂²×C₇₉

#495

yes

—

C₄×C₇₉

C₂×#106

yes

—

C₄×C₇₉

#427

yes

—

C₄×C₇₉

#498

yes

—

C₄×C₇₉

#499

yes

—

Dic₇₉

—

#8815

—

maximal, Sylow

C₂×C₇₉

#1708

yes

—

C₂×C₇₉

#1844

yes

characteristic

C₂×C₇₉

#1846

yes

—

#1844

—

#1834

—

C₂×C₈²

D₇₉

yes

characteristic

C₇₉

#8815

yes

characteristic, complemented, Sylow

C₂×#106

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#384

—

#340

—

158

—

C₈²

Dic₇₉

yes

complemented

C₂²×C₈

D₃₁₆

yes

—

C₂²×C₈

D₃₁₆

yes

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

#106

—

316

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₂²×C₄

—

316

—

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

C₂²×C₄

C₂×D₃₁₆

yes

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

#11929613

yes

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

316

—

C₄²

C₂²×Dic₇₉

yes

characteristic

C₄²

C₂×Dic₁₅₈

yes

characteristic

C₄²

C₂×Dic₁₅₈

yes

—

C₂³

#14579158

yes

characteristic

C₂×C₄

—

632

—

C₂×C₄

—

632

—

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

—

C₂×C₄

#14579161

yes

characteristic

C₂×C₄

#14579161

yes

—

C₂×C₄

C₂×#11929613

yes

—

C₂×C₄

#14579183

yes

—

C₈

—

C₈

#14579093

yes

—

C₂²

#19098433

yes

characteristic

C₂²

#19098936

yes

—

100

C₂²

#19098947

yes

—

101

C₄

—

1264

—

102

C₄

C₂×#14579093

yes

—

103

C₄

#19098601

yes

—

104

C₄

#19098961

yes

—

105

C₄

#19098965

yes

—

106

C₂

#23763001

yes

—

107

C₂

#23763557

yes

characteristic

108

C₂

#23763564

yes

—

	subgroup	G/H
center	C₂³	#14579158
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×C₈²×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#127
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×C₈²	D₇₉
hypocenter	C₇₉	#8815
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#14579158

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×C₈²×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

#127

cosocle

C₂×C₄²×C₇₉

C₂³

hypercenter

C₂×C₈²

D₇₉

hypocenter

C₇₉

#8815

LCS

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₇₉	C₄	semidirect
58. C₇₉	#8815	semidirect
64. C₈²	Dic₇₉	semidirect

H ≅ G/N

type

9. C₈²×C₇₉

C₄

semidirect

58. C₇₉

#8815

semidirect

64. C₈²

Dic₇₉

semidirect

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

subgroup	index
#23763557	2
#23763528	2
C₂×C₈²×C₇₉	2
#8815	79

subgroup

index

#23763557

#23763528

C₂×C₈²×C₇₉

#8815

Subgroups — #34182043

Conjugacy classes of subgroups

Complements

Maximal subgroups