Group #34182080 — subgroups

Subgroups — #34182080

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	#23762156	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	#19098314	—	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	SD₁₆	1	8	yes	—
28	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₇₉	C₂×D₄	1	2	yes	—
30	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
31	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
32	C₂×C₄×C₇₉	—	2	4	no	—
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	#127	1	1	yes	characteristic
36	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
38	C₈×C₇₉	—	2	8	no	—
39	C₈×C₇₉	#106	1	8	yes	—
40	C₇₉⋊C₈	—	8	4	no	—
41	C₂²×C₇₉	—	2	2	no	—
42	C₂²×C₇₉	#384	1	1	yes	characteristic
43	C₂²×C₇₉	#498	1	2	yes	—
44	C₄×C₇₉	—	2	2	no	—
45	C₄×C₇₉	C₂×#106	1	2	yes	—
46	C₄×C₇₉	#430	1	4	yes	—
47	C₄×C₇₉	#492	1	2	yes	—
48	#8825	—	79	1	no	maximal, Sylow
49	C₂×C₇₉	—	2	2	no	—
50	C₂×C₇₉	#1844	1	1	yes	characteristic
51	C₂×C₇₉	#1849	1	2	yes	—
52	#1848	—	79	2	no	—
53	#1566	—	79	4	no	—
54	C₂×#322	D₇₉	1	1	yes	characteristic
55	C₇₉	#8825	1	1	yes	characteristic, complemented, Sylow
56	C₂×#108	—	158	4	no	—
57	#423	—	79	4	no	—
58	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
60	#334	—	158	8	no	—
61	#322	Dic₇₉	1	4	yes	—
62	C₂×M₁₆	—	158	4	no	—
63	C₂²×C₈	Dic₁₅₈	1	4	yes	—
64	C₂²×C₈	D₃₁₆	1	2	yes	—
65	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
66	#108	—	316	8	no	—
67	C₄⋊C₈	—	158	8	no	—
68	C₄×C₈	Dic₁₅₈	1	2	yes	—
69	C₄×C₈	D₃₁₆	1	4	yes	—
70	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
71	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
72	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
73	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
74	M₁₆	—	316	8	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	316	4	no	—
77	C₂×C₈	C₄⋊Dic₇₉	1	8	yes	—
78	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
79	C₂×C₈	#11929613	1	8	yes	—
80	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
81	C₄²	C₂×D₃₁₆	1	2	yes	—
82	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
83	C₂³	#14579161	1	1	yes	characteristic
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
86	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
87	C₂×C₄	#14579158	1	1	yes	characteristic
88	C₂×C₄	#14579161	1	2	yes	—
89	C₂×C₄	C₂×#11929613	1	4	yes	—
90	C₈	—	2	8	no	—
91	C₈	—	632	4	no	—
92	C₈	#14579093	1	8	yes	—
93	C₂²	—	2	2	no	—
94	C₂²	#19098433	1	1	yes	characteristic
95	C₂²	#19098961	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	C₂×#14579093	1	2	yes	—
98	C₄	#19098612	1	4	yes	—
99	C₄	#19098936	1	2	yes	—
100	C₂	—	2	2	no	—
101	C₂	#23763557	1	1	yes	characteristic
102	C₂	#23763575	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763572

C₂

yes

maximal

#23762156

C₂

yes

maximal

C₂×#322×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579097

—

#19098587

C₂²

yes

—

#19098314

—

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

SD₁₆

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×SD₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

#106

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#384

yes

characteristic

C₂²×C₇₉

#498

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#106

yes

—

C₄×C₇₉

#430

yes

—

C₄×C₇₉

#492

yes

—

#8825

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1844

yes

characteristic

C₂×C₇₉

#1849

yes

—

#1848

—

#1566

—

C₂×#322

D₇₉

yes

characteristic

C₇₉

#8825

yes

characteristic, complemented, Sylow

C₂×#108

—

158

—

#423

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄×C₈

C₂×D₇₉

yes

—

#334

—

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

#11929613

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×#11929613

yes

—

C₈

—

C₈

—

632

—

C₈

#14579093

yes

—

C₂²

—

C₂²

#19098433

yes

characteristic

C₂²

#19098961

yes

—

C₄

—

C₄

C₂×#14579093

yes

—

C₄

#19098612

yes

—

C₄

#19098936

yes

—

100

C₂

—

101

C₂

#23763557

yes

characteristic

102

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	#8825
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
55. C₇₉	#8825	semidirect

H ≅ G/N

type

55. C₇₉

#8825

semidirect

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

subgroup	index
#23763572	2
#23762156	2
C₂×#322×C₇₉	2
#8825	79

subgroup

index

#23763572

#23762156

C₂×#322×C₇₉

#8825

Subgroups — #34182080

Conjugacy classes of subgroups

Complements

Maximal subgroups