Group #34182083 — subgroups

Subgroups — #34182083

Order 20224.

338 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 187 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	2	yes	maximal
2	#23762159	C₂	1	4	yes	maximal
3	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579097	—	2	4	no	—
5	#19098587	C₂²	1	4	yes	—
6	#19098316	—	2	8	no	—
7	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
9	#322×C₇₉	C₄	1	4	yes	—
10	C₂×#11929608	—	2	4	no	—
11	#14579097	—	4	8	no	—
12	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
13	C₂²×C₈×C₇₉	D₄	1	2	yes	—
14	C₂²×C₈×C₇₉	Q₈	1	4	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	D₄	1	4	yes	—
19	C₄×C₈×C₇₉	Q₈	1	2	yes	—
20	#11929608	—	4	8	no	—
21	C₂×C₇₉⋊C₈	—	4	4	no	—
22	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
24	C₂×C₈×C₇₉	—	4	4	no	—
25	C₂×C₈×C₇₉	C₄⋊C₄	1	8	yes	—
26	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
27	C₂×C₈×C₇₉	D₈	1	4	yes	—
28	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
29	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₇₉	C₂×D₄	1	2	yes	—
31	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
32	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
33	C₂×C₄×C₇₉	—	2	4	no	—
34	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
37	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
38	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
39	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
40	C₈×C₇₉	—	2	8	no	—
41	C₈×C₇₉	#107	1	8	yes	—
42	C₇₉⋊C₈	—	8	4	no	—
43	C₂²×C₇₉	—	2	2	no	—
44	C₂²×C₇₉	#384	1	1	yes	characteristic
45	C₂²×C₇₉	#500	1	2	yes	—
46	C₄×C₇₉	—	2	2	no	—
47	C₄×C₇₉	C₂×#107	1	2	yes	—
48	C₄×C₇₉	#430	1	4	yes	—
49	C₄×C₇₉	#493	1	1	yes	characteristic
50	C₄×C₇₉	#494	1	1	yes	characteristic
51	#8826	—	79	1	no	maximal, Sylow
52	C₂×C₇₉	—	2	2	no	—
53	C₂×C₇₉	#1845	1	1	yes	characteristic
54	C₂×C₇₉	#1849	1	2	yes	—
55	#1848	—	79	2	no	—
56	#1567	—	79	4	no	—
57	C₂×#322	D₇₉	1	1	yes	characteristic
58	C₇₉	#8826	1	1	yes	characteristic, complemented, Sylow
59	C₂×#108	—	158	4	no	—
60	#423	—	79	4	no	—
61	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
63	#335	—	158	8	no	—
64	#322	Dic₇₉	1	4	yes	—
65	C₂×M₁₆	—	158	4	no	—
66	C₂²×C₈	Dic₁₅₈	1	4	yes	—
67	C₂²×C₈	D₃₁₆	1	2	yes	—
68	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
69	#108	—	316	8	no	—
70	C₄⋊C₈	—	158	8	no	—
71	C₄×C₈	Dic₁₅₈	1	2	yes	—
72	C₄×C₈	D₃₁₆	1	4	yes	—
73	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
74	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
75	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
77	M₁₆	—	316	8	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	—	316	4	no	—
80	C₂×C₈	C₄⋊Dic₇₉	1	8	yes	—
81	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
82	C₂×C₈	D₆₃₂	1	4	yes	—
83	C₂×C₈	Dic₃₁₆	1	4	yes	—
84	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
85	C₄²	C₂×D₃₁₆	1	2	yes	—
86	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
87	C₂³	#14579161	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
91	C₂×C₄	#14579158	1	1	yes	characteristic
92	C₂×C₄	#14579161	1	2	yes	—
93	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
94	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	—	632	4	no	—
97	C₈	#14579095	1	8	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	#19098433	1	1	yes	characteristic
100	C₂²	#19098968	1	2	yes	—
101	C₄	—	2	2	no	—
102	C₄	C₂×#14579095	1	2	yes	—
103	C₄	#19098612	1	4	yes	—
104	C₄	#19098940	1	1	yes	characteristic
105	C₄	#19098943	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	#23763560	1	1	yes	characteristic
108	C₂	#23763575	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763572

C₂

yes

maximal

#23762159

C₂

yes

maximal

C₂×#322×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579097

—

#19098587

C₂²

yes

—

#19098316

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#322×C₇₉

C₄

yes

—

C₂×#11929608

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

Q₈

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

#107

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#384

yes

characteristic

C₂²×C₇₉

#500

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#107

yes

—

C₄×C₇₉

#430

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

#1845

yes

characteristic

C₂×C₇₉

#1849

yes

—

#1848

—

#1567

—

C₂×#322

D₇₉

yes

characteristic

C₇₉

#8826

yes

characteristic, complemented, Sylow

C₂×#108

—

158

—

#423

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄×C₈

C₂×D₇₉

yes

—

#335

—

158

—

#322

Dic₇₉

yes

—

C₂×M₁₆

—

158

—

C₂²×C₈

Dic₁₅₈

yes

—

C₂²×C₈

D₃₁₆

yes

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

#108

—

316

—

C₄⋊C₈

—

158

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

D₃₁₆

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₅₈

yes

—

M₁₆

—

316

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

D₆₃₂

yes

—

C₂×C₈

Dic₃₁₆

yes

—

C₄²

C₂²×Dic₇₉

yes

characteristic

C₄²

C₂×D₃₁₆

yes

—

C₄²

C₂×Dic₁₅₈

yes

characteristic

C₂³

#14579161

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

—

C₂×C₄

#14579158

yes

characteristic

C₂×C₄

#14579161

yes

—

C₂×C₄

C₂×D₆₃₂

yes

—

C₂×C₄

C₂×Dic₃₁₆

yes

—

C₈

—

C₈

—

632

—

C₈

#14579095

yes

—

C₂²

—

C₂²

#19098433

yes

characteristic

100

C₂²

#19098968

yes

—

101

C₄

—

102

C₄

C₂×#14579095

yes

—

103

C₄

#19098612

yes

—

104

C₄

#19098940

yes

characteristic

105

C₄

#19098943

yes

characteristic

106

C₂

—

107

C₂

#23763560

yes

characteristic

108

C₂

#23763575

yes

—

	subgroup	G/H
center	C₂×C₄	#14579158
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#322×C₇₉	C₂
socle	C₂²×C₇₉	#384
prime radical	C₂³×C₇₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#8826
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14579158

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×#322×C₇₉

C₂

socle

C₂²×C₇₉

#384

prime radical

C₂³×C₇₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
58. C₇₉	#8826	semidirect

H ≅ G/N

type

58. C₇₉

#8826

semidirect

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

subgroup	index
#23763572	2
#23762159	2
C₂×#322×C₇₉	2
#8826	79

subgroup

index

#23763572

#23762159

C₂×#322×C₇₉

#8826

Subgroups — #34182083

Conjugacy classes of subgroups

Complements

Maximal subgroups