Group #34176145 — subgroups

Subgroups — #34176145

Order 20224.

138 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 25 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	#23762999	C₂	1	1	yes	maximal, characteristic
2	#23761565	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1410	C₂	1	1	yes	maximal, characteristic
4	#19098961	C₄	1	1	yes	characteristic, complemented
5	#19098915	—	2	1	no	—
6	#19098909	—	2	1	no	—
7	#19098356	—	2	1	no	—
8	C₈×C₇₉⋊C₈	C₄	1	1	yes	characteristic, complemented
9	#492×C₇₉	C₂²	1	1	yes	characteristic
10	#356×C₇₉	—	2	1	no	—
11	#14579161	—	2	1	no	—
12	#14579093	—	4	1	no	—
13	#14579084	—	4	2	no	—
14	#14579082	—	4	2	no	—
15	#14579076	—	4	1	no	—
16	C₄×C₇₉⋊C₈	—	2	1	no	—
17	C₂×SD₁₆×C₇₉	—	4	1	no	—
18	C₄⋊Q₈×C₇₉	D₄	1	1	yes	characteristic
19	#127×C₇₉	D₄	1	1	yes	characteristic
20	#101×C₇₉	—	4	1	no	—
21	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₅₈	—	4	1	no	—
23	C₂×C₇₉⋊C₈	—	2	2	no	—
24	C₄⋊Dic₇₉	—	4	1	no	—
25	C₄⋊Dic₇₉	—	4	1	no	—
26	C₂²⋊Dic₇₉	—	8	1	no	—
27	C₂×Q₈×C₇₉	—	2	1	no	—
28	C₂×D₄×C₇₉	—	2	1	no	—
29	C₂×D₄×C₇₉	—	4	1	no	—
30	SD₁₆×C₇₉	—	8	2	no	—
31	M₁₆×C₇₉	—	8	1	no	—
32	C₂×C₈×C₇₉	—	2	1	no	—
33	C₄⋊C₄×C₇₉	—	4	1	no	—
34	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂³×C₇₉	—	4	1	no	—
36	Q₈×C₇₉	—	8	1	no	—
37	D₄×C₇₉	—	8	1	no	—
38	D₄×C₇₉	—	8	2	no	—
39	C₂×C₄×C₇₉	—	2	1	no	—
40	C₂×C₄×C₇₉	—	4	1	no	—
41	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
42	C₂×Dic₇₉	—	8	1	no	—
43	C₂×Dic₇₉	—	16	1	no	—
44	Dic₁₅₈	—	8	2	no	—
45	C₈×C₇₉	—	2	2	no	—
46	C₈×C₇₉	—	16	1	no	—
47	C₇₉⋊C₈	—	4	2	no	—
48	C₂²×C₇₉	—	8	1	no	—
49	C₂²×C₇₉	—	16	1	no	—
50	C₂²×C₇₉	#353	1	1	yes	characteristic
51	C₄×C₇₉	—	2	2	no	—
52	C₄×C₇₉	—	2	1	no	—
53	C₄×C₇₉	—	8	1	no	—
54	Dic₇₉	—	16	1	no	—
55	Dic₇₉	—	32	1	no	—
56	#7536	—	79	1	no	maximal, Sylow
57	C₂×C₇₉	—	2	1	no	—
58	C₂×C₇₉	—	16	1	no	—
59	C₂×C₇₉	#1409	1	1	yes	characteristic
60	#1708	—	79	1	no	—
61	#1410	D₇₉	1	1	yes	characteristic
62	#1409	—	79	1	no	—
63	C₇₉	#7536	1	1	yes	characteristic, complemented, Sylow
64	#498	—	79	1	no	—
65	#492	C₂×D₇₉	1	1	yes	characteristic
66	#487	—	158	1	no	—
67	#486	—	158	1	no	—
68	#356	—	2	1	no	—
69	#353	—	158	1	no	—
70	C₈²	—	79	1	no	—
71	C₂×SD₁₆	—	4	1	no	—
72	C₄⋊Q₈	—	158	1	no	—
73	C₄⋊Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
74	#127	D₃₁₆	1	1	yes	characteristic
75	#106	—	316	1	no	—
76	#103	—	316	2	no	—
77	#102	—	316	2	no	—
78	#101	—	4	1	no	—
79	#99	—	316	1	no	—
80	C₄×C₈	—	158	1	no	—
81	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	316	1	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂×D₄	—	4	1	no	—
86	SD₁₆	—	8	2	no	—
87	M₁₆	—	8	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	158	2	no	—
90	C₄⋊C₄	—	4	1	no	—
91	C₄⋊C₄	—	316	1	no	—
92	C₄⋊C₄	—	316	1	no	—
93	C₂²⋊C₄	—	632	1	no	—
94	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
95	C₂³	—	4	1	no	—
96	Q₈	—	8	1	no	—
97	Q₈	—	632	2	no	—
98	D₄	—	8	1	no	—
99	D₄	—	8	2	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	632	1	no	—
103	C₂×C₄	—	1264	1	no	—
104	C₂×C₄	#14579075	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	16	1	no	—
107	C₈	—	316	2	no	—
108	C₂²	—	8	1	no	—
109	C₂²	—	16	1	no	—
110	C₂²	#19098355	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	8	1	no	—
114	C₄	—	2528	1	no	—
115	C₄	—	1264	1	no	—
116	C₂	—	2	1	no	—
117	C₂	—	16	1	no	—
118	C₂	#23761564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23762999

C₂

yes

maximal, characteristic

#23761565

C₂

yes

maximal, characteristic

C₇₉×#1410

C₂

yes

maximal, characteristic

#19098961

C₄

yes

characteristic, complemented

#19098915

—

#19098909

—

#19098356

—

C₈×C₇₉⋊C₈

C₄

yes

characteristic, complemented

#492×C₇₉

C₂²

yes

characteristic

#356×C₇₉

—

—

—

—

—

—

C₄×C₇₉⋊C₈

—

C₂×SD₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

D₄

yes

characteristic

#127×C₇₉

D₄

yes

characteristic

#101×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂×Dic₁₅₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

SD₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₂³×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#99

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#353

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

Dic₇₉

—

#7536

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

#1409

—

C₇₉

#7536

yes

characteristic, complemented, Sylow

#498

—

#492

C₂×D₇₉

yes

characteristic

#487

—

158

—

#486

—

158

—

#356

—

#353

—

158

—

C₈²

—

C₂×SD₁₆

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

C₂²⋊D₇₉

yes

characteristic

#127

D₃₁₆

yes

characteristic

#106

—

316

—

#103

—

316

—

#102

—

316

—

#101

—

#99

—

316

—

C₄×C₈

—

158

—

C₄×C₈

C₄×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

316

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

316

—

C₂²⋊C₄

—

632

—

C₄²

(C₂×D₇₉)⋊C₄

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

632

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

632

—

103

C₂×C₄

—

1264

—

104

C₂×C₄

#14579075

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

316

—

108

C₂²

—

109

C₂²

—

110

C₂²

#19098355

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

2528

—

115

C₄

—

1264

—

116

C₂

—

117

C₂

—

118

C₂

#23761564

yes

characteristic

	subgroup	G/H
center	C₂	#23761564
commutator	C₄×C₈×C₇₉	C₂×C₄
Frattini	#492	C₂×D₇₉
Fitting	C₇₉×#1410	C₂
socle	C₂×C₇₉	#1409
prime radical	#127×C₇₉	D₄
cosocle	#492×C₇₉	C₂²
hypercenter	#1410	D₇₉
hypocenter	C₇₉	#7536
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #492 ◁ #1410

subgroup

G/H

center

C₂

#23761564

commutator

C₄×C₈×C₇₉

C₂×C₄

Frattini

#492

C₂×D₇₉

Fitting

C₇₉×#1410

C₂

socle

C₂×C₇₉

#1409

prime radical

#127×C₇₉

D₄

cosocle

#492×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₄×C₈×C₇₉ ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #492 ◁ #1410

N	H ≅ G/N	type
4. #19098961	C₄	semidirect
8. C₈×C₇₉⋊C₈	C₄	semidirect
63. C₇₉	#7536	semidirect

H ≅ G/N

type

4. #19098961

C₄

semidirect

8. C₈×C₇₉⋊C₈

C₄

semidirect

63. C₇₉

#7536

semidirect

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

subgroup	index
#23762999	2
#23761565	2
C₇₉×#1410	2
#7536	79

subgroup

index

#23762999

#23761565

C₇₉×#1410

#7536

Subgroups — #34176145

Conjugacy classes of subgroups

Complements

Maximal subgroups