Group #34175905 — subgroups

Subgroups — #34175905

Order 20224.

132 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₄⋊(C₇₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
2	#23761472	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#2247	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₉⋊C₃₂	—	2	1	no	—
5	#19098316	—	2	1	no	—
6	C₂×Q₃₂×C₇₉	—	2	1	no	—
7	C₂×Q₃₂×C₇₉	C₄	1	2	yes	—
8	#494×C₇₉	—	2	1	no	—
9	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
10	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
11	C₇₉⋊C₃₂	—	4	1	no	—
12	C₂×Q₁₆×C₇₉	—	2	2	no	—
13	C₂×Q₁₆×C₇₉	—	4	1	no	—
14	C₄⋊Q₈×C₇₉	—	4	1	no	—
15	Q₃₂×C₇₉	—	4	2	no	—
16	Q₃₂×C₇₉	—	4	1	no	—
17	Q₃₂×C₇₉	—	4	1	no	—
18	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
20	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
21	C₂×C₇₉⋊C₈	—	8	1	no	—
22	C₂×Q₈×C₇₉	—	4	2	no	—
23	Q₁₆×C₇₉	—	4	2	no	—
24	Q₁₆×C₇₉	—	4	2	no	—
25	Q₁₆×C₇₉	—	8	2	no	—
26	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
28	C₄⋊C₄×C₇₉	—	8	1	no	—
29	C₄⋊C₄×C₇₉	—	8	1	no	—
30	C₄²×C₇₉	D₈	1	1	yes	characteristic
31	C₁₆×C₇₉	—	2	1	no	—
32	C₁₆×C₇₉	D₈	1	1	yes	characteristic
33	C₁₆×C₇₉	SD₁₆	1	1	yes	characteristic
34	Q₈×C₇₉	—	8	2	no	—
35	Q₈×C₇₉	—	8	2	no	—
36	C₂×C₄×C₇₉	—	8	2	no	—
37	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
38	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₉	SD₃₂	1	1	yes	characteristic
40	C₈×C₇₉	—	2	1	no	—
41	C₈×C₇₉	#100	1	1	yes	characteristic
42	C₈×C₇₉	#102	1	1	yes	characteristic
43	C₇₉⋊C₈	—	16	1	no	—
44	C₂²×C₇₉	#357	1	1	yes	characteristic
45	C₄×C₇₉	—	2	1	no	—
46	C₄×C₇₉	—	16	2	no	—
47	C₄×C₇₉	#331	1	1	yes	characteristic
48	C₄×C₇₉	#361	1	1	yes	characteristic
49	C₄×C₇₉	SD₆₄	1	1	yes	characteristic
50	C₄×C₇₉	Q₆₄	1	1	yes	characteristic
51	#7445	—	79	1	no	maximal, Sylow
52	C₂×C₇₉	#1361	1	1	yes	characteristic
53	C₂×C₇₉	#1416	1	1	yes	characteristic
54	C₂×C₇₉	#1420	1	1	yes	characteristic
55	#2247	D₇₉	1	1	yes	characteristic
56	C₄⋊C₃₂	—	79	1	no	—
57	#1371	—	79	1	no	—
58	C₇₉	#7445	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₃₂	—	2	1	no	—
60	C₂×Q₃₂	Dic₇₉	1	2	yes	—
61	#494	—	2	1	no	—
62	C₂×C₃₂	—	158	1	no	—
63	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
64	#335	—	158	1	no	—
65	C₂×Q₁₆	—	2	2	no	—
66	C₂×Q₁₆	—	4	1	no	—
67	C₄⋊Q₈	—	4	1	no	—
68	Q₃₂	—	4	2	no	—
69	Q₃₂	—	4	1	no	—
70	Q₃₂	—	4	1	no	—
71	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
72	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
73	C₄⋊C₈	—	316	1	no	—
74	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
75	C₃₂	—	316	1	no	—
76	C₂×Q₈	—	4	2	no	—
77	Q₁₆	—	4	2	no	—
78	Q₁₆	—	4	2	no	—
79	Q₁₆	—	8	2	no	—
80	C₂×C₈	—	632	1	no	—
81	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
82	C₂×C₈	#11929612	1	1	yes	characteristic
83	C₄⋊C₄	—	8	1	no	—
84	C₄⋊C₄	—	8	1	no	—
85	C₄²	#11929609	1	1	yes	characteristic
86	C₁₆	—	2	1	no	—
87	C₁₆	#11929609	1	1	yes	characteristic
88	C₁₆	#11929612	1	1	yes	characteristic
89	Q₈	—	8	2	no	—
90	Q₈	—	8	2	no	—
91	C₂×C₄	—	8	2	no	—
92	C₂×C₄	#14579082	1	1	yes	characteristic
93	C₂×C₄	#14579102	1	1	yes	characteristic
94	C₂×C₄	#14579105	1	1	yes	characteristic
95	C₈	—	2	1	no	—
96	C₈	—	1264	1	no	—
97	C₈	#14579078	1	1	yes	characteristic
98	C₈	#14579082	1	1	yes	characteristic
99	C₂²	#19098364	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	16	2	no	—
102	C₄	#19098307	1	1	yes	characteristic
103	C₄	#19098376	1	1	yes	characteristic
104	C₄	#19098400	1	1	yes	characteristic
105	C₄	#19098403	1	1	yes	characteristic
106	C₂	#23761446	1	1	yes	characteristic
107	C₂	#23761581	1	1	yes	characteristic
108	C₂	#23761593	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₇₉⋊C₃₂)

C₂

yes

maximal, characteristic

#23761472

C₂

yes

maximal, characteristic

C₇₉×#2247

C₂

yes

maximal, characteristic

C₂×C₇₉⋊C₃₂

—

#19098316

—

C₂×Q₃₂×C₇₉

—

C₂×Q₃₂×C₇₉

C₄

yes

—

#494×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

C₇₉⋊C₃₂

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

—

Q₃₂×C₇₉

—

Q₃₂×C₇₉

—

Q₃₂×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

SD₁₆

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

SD₁₆

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#102

yes

characteristic

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

SD₃₂

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₇₉⋊C₈

—

C₂²×C₇₉

#357

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

SD₆₄

yes

characteristic

C₄×C₇₉

Q₆₄

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

C₄⋊C₃₂

—

#1371

—

C₇₉

#7445

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

Dic₇₉

yes

—

#494

—

C₂×C₃₂

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

#335

—

158

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₄⋊C₈

—

316

—

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₃₂

—

316

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

632

—

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

C₂×C₈

#11929612

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

#11929609

yes

characteristic

C₁₆

—

C₁₆

yes

C₁₆

yes

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

1264

—

C₈

yes

C₈

yes

C₂²

yes

100

C₄

—

101

C₄

—

102

C₄

#19098307

yes

characteristic

103

C₄

#19098376

yes

characteristic

104

C₄

#19098400

yes

characteristic

105

C₄

#19098403

yes

characteristic

106

C₂

#23761446

yes

characteristic

107

C₂

#23761581

yes

characteristic

108

C₂

#23761593

yes

characteristic

	subgroup	G/H
center	C₂²	#19098364
commutator	C₂×C₁₆×C₇₉	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₇₉
Fitting	C₇₉×#2247	C₂
socle, prime radical	C₂²×C₇₉	#357
cosocle	C₄×C₁₆×C₇₉	C₂²
hypercenter	#2247	D₇₉
hypocenter	C₇₉	#7445
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

subgroup

G/H

center

C₂²

#19098364

commutator

C₂×C₁₆×C₇₉

C₂×C₄

Frattini

C₄×C₁₆

C₂×D₇₉

Fitting

C₇₉×#2247

C₂

socle, prime radical

C₂²×C₇₉

#357

cosocle

C₄×C₁₆×C₇₉

C₂²

D₇₉

C₇₉

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

UCS

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

N	H ≅ G/N	type
58. C₇₉	#7445	semidirect

H ≅ G/N

type

58. C₇₉

#7445

semidirect

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

subgroup	index
C₄⋊(C₇₉⋊C₃₂)	2
#23761472	2
C₇₉×#2247	2
#7445	79

subgroup

index

C₄⋊(C₇₉⋊C₃₂)

#23761472

C₇₉×#2247

#7445

Subgroups — #34175905

Conjugacy classes of subgroups

Complements

Maximal subgroups