Group #34182022 — subgroups

Subgroups — #34182022

Order 20224.

402 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 211 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	#23763552	C₂	1	2	yes	maximal
2	C₂×#19098314	C₂	1	4	yes	maximal
3	C₇₉×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
5	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
6	#19098314	—	2	8	no	—
7	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
8	C₄³×C₇₉	C₂²	1	1	yes	characteristic
9	#336×C₇₉	C₄	1	4	yes	—
10	C₂²×C₇₉⋊C₈	—	2	4	no	—
11	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
12	C₄⋊(C₇₉⋊C₈)	—	4	8	no	—
13	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₇₉	D₄	1	2	yes	—
16	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
17	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₇₉	D₄	1	4	yes	—
19	C₄×C₈×C₇₉	Q₈	1	4	yes	—
20	C₂×C₇₉⋊C₈	—	4	4	no	—
21	C₂×C₇₉⋊C₈	—	4	8	no	—
22	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
23	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₈×C₇₉	—	2	8	no	—
27	C₂×C₈×C₇₉	C₄⋊C₄	1	8	yes	—
28	C₄²×C₇₉	—	2	4	no	—
29	C₄²×C₇₉	—	4	2	no	—
30	C₄²×C₇₉	SD₁₆	1	8	yes	—
31	C₄²×C₇₉	C₂×D₄	1	2	yes	—
32	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
33	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₉	—	2	4	no	—
35	C₂×C₄×C₇₉	—	2	4	no	—
36	C₂×C₄×C₇₉	—	2	2	no	—
37	C₂×C₄×C₇₉	#106	1	8	yes	—
38	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
40	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
41	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₄×C₇₉	#131	1	4	yes	—
43	C₂×C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
44	C₈×C₇₉	—	2	8	no	—
45	C₇₉⋊C₈	—	8	4	no	—
46	C₂²×C₇₉	#384	1	1	yes	characteristic
47	C₂²×C₇₉	#424	1	2	yes	—
48	C₂²×C₇₉	C₂×#106	1	2	yes	—
49	C₂²×C₇₉	#430	1	2	yes	—
50	C₄×C₇₉	—	2	8	no	—
51	C₄×C₇₉	—	2	2	no	—
52	C₄×C₇₉	#424	1	4	yes	—
53	C₄×C₇₉	#492	1	2	yes	—
54	C₄×C₇₉	#498	1	2	yes	—
55	#8809	—	79	1	no	maximal, Sylow
56	C₂×C₇₉	#1570	1	4	yes	—
57	C₂×C₇₉	#1843	1	1	yes	characteristic
58	C₂×C₇₉	#1844	1	1	yes	characteristic
59	C₂×C₇₉	#1849	1	1	yes	characteristic
60	#1842	—	79	2	no	—
61	#1744	D₇₉	1	1	yes	characteristic
62	C₂×#334	—	79	4	no	—
63	C₇₉	#8809	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊C₈	—	79	4	no	—
65	C₂×C₄⋊C₈	—	158	4	no	—
66	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
67	C₄³	C₂×D₇₉	1	1	yes	characteristic
68	#336	Dic₇₉	1	4	yes	—
69	#334	—	158	8	no	—
70	C₂²×C₈	—	158	4	no	—
71	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
72	C₂×C₄²	Dic₁₅₈	1	2	yes	—
73	C₂×C₄²	D₃₁₆	1	2	yes	—
74	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
75	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
76	C₄⋊C₈	—	316	8	no	—
77	C₄⋊C₈	—	158	8	no	—
78	C₄×C₈	Dic₁₅₈	1	4	yes	—
79	C₄×C₈	D₃₁₆	1	4	yes	—
80	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
81	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
83	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
84	C₂×C₈	—	2	8	no	—
85	C₂×C₈	—	316	4	no	—
86	C₂×C₈	—	316	8	no	—
87	C₂×C₈	C₄⋊Dic₇₉	1	8	yes	—
88	C₄²	—	2	4	no	—
89	C₄²	—	4	2	no	—
90	C₄²	#11929613	1	8	yes	—
91	C₄²	C₂×D₃₁₆	1	2	yes	—
92	C₄²	C₂×Dic₁₅₈	1	2	yes	—
93	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	#14579093	1	8	yes	—
98	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
99	C₂×C₄	#14579158	1	1	yes	characteristic
100	C₂×C₄	#14579161	1	2	yes	—
101	C₂×C₄	#14579161	1	1	yes	characteristic
102	C₂×C₄	#14579168	1	4	yes	—
103	C₂×C₄	C₂×#11929613	1	4	yes	—
104	C₈	—	2	8	no	—
105	C₈	—	632	4	no	—
106	C₂²	#19098433	1	1	yes	characteristic
107	C₂²	#19098590	1	2	yes	—
108	C₂²	C₂×#14579093	1	2	yes	—
109	C₂²	#19098612	1	2	yes	—
110	C₄	—	2	2	no	—
111	C₄	—	2	8	no	—
112	C₄	#19098590	1	4	yes	—
113	C₄	#19098936	1	2	yes	—
114	C₄	#19098961	1	2	yes	—
115	C₂	#23762171	1	4	yes	—
116	C₂	#23763554	1	1	yes	characteristic
117	C₂	#23763557	1	1	yes	characteristic
118	C₂	#23763575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763552

C₂

yes

maximal

C₂×#19098314

C₂

yes

maximal

C₇₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#19098314

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

#336×C₇₉

C₄

yes

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

Q₈

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#106

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#384

yes

characteristic

C₂²×C₇₉

#424

yes

—

C₂²×C₇₉

C₂×#106

yes

—

C₂²×C₇₉

#430

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#424

yes

—

C₄×C₇₉

#492

yes

—

C₄×C₇₉

#498

yes

—

#8809

—

maximal, Sylow

C₂×C₇₉

#1570

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

C₂×#334

—

C₇₉

#8809

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

C₄³

C₂×D₇₉

yes

characteristic

#336

Dic₇₉

yes

—

#334

—

158

—

C₂²×C₈

—

158

—

C₂²×C₈

C₂×Dic₇₉

yes

—

C₂×C₄²

Dic₁₅₈

yes

—

C₂×C₄²

D₃₁₆

yes

—

C₂×C₄²

C₂×Dic₇₉

yes

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

316

—

C₄⋊C₈

—

158

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

D₃₁₆

yes

—

C₂²×C₄

C₄⋊Dic₇₉

yes

—

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₅₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

—

316

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₄²

—

C₄²

—

C₄²

#11929613

yes

—

C₄²

C₂×D₃₁₆

yes

—

C₄²

C₂×Dic₁₅₈

yes

—

C₂³

C₂×C₄⋊Dic₇₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14579093

yes

—

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

—

C₂×C₄

#14579158

yes

characteristic

100

C₂×C₄

#14579161

yes

—

101

C₂×C₄

#14579161

yes

characteristic

102

C₂×C₄

#14579168

yes

—

103

C₂×C₄

C₂×#11929613

yes

—

104

C₈

—

105

C₈

—

632

—

106

C₂²

#19098433

yes

characteristic

107

C₂²

#19098590

yes

—

108

C₂²

C₂×#14579093

yes

—

109

C₂²

#19098612

yes

—

110

C₄

—

111

C₄

—

112

C₄

#19098590

yes

—

113

C₄

#19098936

yes

—

114

C₄

#19098961

yes

—

115

C₂

#23762171

yes

—

116

C₂

#23763554

yes

characteristic

117

C₂

#23763557

yes

characteristic

118

C₂

#23763575

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1744	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1744	D₇₉
hypocenter	C₇₉	#8809
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₃₁₆

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1744

C₂

socle, prime radical

C₂³×C₇₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
63. C₇₉	#8809	semidirect

H ≅ G/N

type

63. C₇₉

#8809

semidirect

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

subgroup	index
#23763552	2
C₂×#19098314	2
C₇₉×#1744	2
#8809	79

subgroup

index

#23763552

C₂×#19098314

C₇₉×#1744

#8809

Subgroups — #34182022

Conjugacy classes of subgroups

Complements

Maximal subgroups