Group #34209259 — subgroups

Subgroups — #34209259

Order 20224.

174 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 67 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	#23765679	C₂	1	1	yes	maximal, characteristic
2	#23761581	C₂	1	4	yes	maximal
3	#23761534	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#2247	C₂	1	1	yes	maximal, characteristic
5	#19098965	—	2	1	no	—
6	C₂×C₇₉⋊C₃₂	C₂²	1	2	yes	—
7	#19098391	—	2	1	no	—
8	#19098389	C₂²	1	2	yes	—
9	C₂×Q₃₂×C₇₉	—	2	1	no	—
10	C₂×Q₃₂×C₇₉	C₂²	1	2	yes	—
11	#494×C₇₉	—	2	1	no	—
12	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	#14579153	—	4	1	no	—
14	#14579095	—	2	2	no	—
15	#14579093	—	4	1	no	—
16	C₇₉⋊C₃₂	—	2	2	no	—
17	C₂×Q₁₆×C₇₉	—	2	2	no	—
18	C₂×Q₁₆×C₇₉	—	4	1	no	—
19	C₄⋊Q₈×C₇₉	—	4	1	no	—
20	Q₃₂×C₇₉	—	2	4	no	—
21	Q₃₂×C₇₉	—	4	2	no	—
22	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₄⋊Dic₇₉	—	4	2	no	—
26	Dic₇₉⋊C₄	—	8	2	no	—
27	C₂×Q₈×C₇₉	—	4	2	no	—
28	Q₁₆×C₇₉	—	4	2	no	—
29	Q₁₆×C₇₉	—	4	2	no	—
30	Q₁₆×C₇₉	—	8	2	no	—
31	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
32	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₇₉	—	8	2	no	—
34	C₄²×C₇₉	D₈	1	1	yes	characteristic
35	C₁₆×C₇₉	SD₁₆	1	2	yes	—
36	C₁₆×C₇₉	#41	1	2	yes	—
37	Q₈×C₇₉	—	8	2	no	—
38	Q₈×C₇₉	—	8	2	no	—
39	C₂×C₄×C₇₉	—	8	2	no	—
40	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
41	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
42	C₂×Dic₇₉	—	8	2	no	—
43	C₈×C₇₉	SD₃₂	1	2	yes	—
44	C₈×C₇₉	#124	1	1	yes	characteristic
45	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
47	C₄×C₇₉	—	2	2	no	—
48	C₄×C₇₉	—	16	2	no	—
49	C₄×C₇₉	#486	1	1	yes	characteristic
50	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
51	Dic₇₉	—	16	2	no	—
52	#13724	—	79	1	no	maximal, Sylow
53	C₂×C₇₉	#2240	1	1	yes	characteristic
54	C₂×C₇₉	#2264	1	2	yes	—
55	#2253	—	79	1	no	—
56	#2247	D₇₉	1	1	yes	characteristic
57	#1416	—	79	4	no	—
58	#1397	—	79	1	no	—
59	C₇₉	#13724	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₃₂	—	2	1	no	—
61	C₂×Q₃₂	C₂×D₇₉	1	2	yes	—
62	#499	—	158	1	no	—
63	#494	—	2	1	no	—
64	C₂×C₃₂	—	79	2	no	—
65	#367	—	158	1	no	—
66	#366	—	79	2	no	—
67	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
68	C₂×Q₁₆	—	2	2	no	—
69	C₂×Q₁₆	—	4	1	no	—
70	C₄⋊Q₈	—	4	1	no	—
71	#125	—	316	1	no	—
72	Q₃₂	—	2	4	no	—
73	Q₃₂	—	4	2	no	—
74	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
76	#107	—	158	2	no	—
77	#106	—	316	1	no	—
78	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
79	C₃₂	—	158	2	no	—
80	C₂×Q₈	—	4	2	no	—
81	Q₁₆	—	4	2	no	—
82	Q₁₆	—	4	2	no	—
83	Q₁₆	—	8	2	no	—
84	C₂×C₈	#11929609	1	1	yes	characteristic
85	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
86	C₄⋊C₄	—	8	2	no	—
87	C₄⋊C₄	—	316	2	no	—
88	C₄⋊C₄	—	632	2	no	—
89	C₄²	#11929609	1	1	yes	characteristic
90	C₁₆	#11929612	1	2	yes	—
91	C₁₆	#11929623	1	2	yes	—
92	Q₈	—	8	2	no	—
93	Q₈	—	8	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	632	2	no	—
96	C₂×C₄	#14579102	1	2	yes	—
97	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
98	C₈	#14579105	1	2	yes	—
99	C₈	#14579148	1	1	yes	characteristic
100	C₈	C₂×#11929612	1	1	yes	characteristic
101	C₂²	C₂×#14579102	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	16	2	no	—
104	C₄	—	1264	2	no	—
105	C₄	#19098909	1	1	yes	characteristic
106	C₄	C₂×#14579105	1	1	yes	characteristic
107	C₂	#23765623	1	1	yes	characteristic
108	C₂	#23765718	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765679

C₂

yes

maximal, characteristic

#23761581

C₂

yes

maximal

#23761534

C₂

yes

maximal, characteristic

C₇₉×#2247

C₂

yes

maximal, characteristic

#19098965

—

C₂×C₇₉⋊C₃₂

C₂²

yes

—

#19098391

—

#19098389

C₂²

yes

—

C₂×Q₃₂×C₇₉

—

C₂×Q₃₂×C₇₉

C₂²

yes

—

#494×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

—

—

—

C₇₉⋊C₃₂

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

—

Q₃₂×C₇₉

—

Q₃₂×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

SD₁₆

yes

—

C₁₆×C₇₉

#41

yes

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

SD₃₂

yes

—

C₈×C₇₉

#124

yes

characteristic

C₈×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#486

yes

characteristic

C₄×C₇₉

C₂×SD₃₂

yes

characteristic

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

#2253

—

#2247

D₇₉

yes

characteristic

#1416

—

#1397

—

C₇₉

#13724

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₇₉

yes

—

#499

—

158

—

#494

—

C₂×C₃₂

—

#367

—

158

—

#366

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

#125

—

316

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

#107

—

158

—

#106

—

316

—

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₃₂

—

158

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#11929609

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

632

—

C₄²

#11929609

yes

characteristic

C₁₆

#11929612

yes

—

C₁₆

#11929623

yes

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

632

—

C₂×C₄

#14579102

yes

—

C₂×C₄

C₂×#11929609

yes

characteristic

C₈

#14579105

yes

—

C₈

#14579148

yes

characteristic

100

C₈

C₂×#11929612

yes

characteristic

101

C₂²

C₂×#14579102

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

1264

—

105

C₄

#19098909

yes

characteristic

106

C₄

C₂×#14579105

yes

characteristic

107

C₂

#23765623

yes

characteristic

108

C₂

#23765718

yes

—

	subgroup	G/H
center	C₂²	C₂×#14579102
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#2247	C₂
socle, prime radical	C₂²×C₇₉	C₂×D₁₆
hypercenter	#2247	D₇₉
hypocenter	C₇₉	#13724
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

subgroup

G/H

center

C₂²

C₂×#14579102

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₇₉×#2247

C₂

socle, prime radical

C₂²×C₇₉

C₂×D₁₆

D₇₉

C₇₉

G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

N	H ≅ G/N	type
59. C₇₉	#13724	semidirect

H ≅ G/N

type

59. C₇₉

#13724

semidirect

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

subgroup	index
#23765679	2
#23761581	2
#23761534	2
C₇₉×#2247	2
#13724	79

subgroup

index

C₇₉×#2247

Subgroups — #34209259

Conjugacy classes of subgroups

Complements

Maximal subgroups