Group #34192951 — subgroups

Subgroups — #34192951

Order 20224.

328 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 79 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	#23762191	C₂	1	3	yes	maximal, complemented
2	#23762069	C₂	1	3	yes	maximal
3	C₇₉×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14579158	C₂²	1	1	yes	characteristic
5	#19098616	C₂²	1	3	yes	—
6	#19098315	—	2	3	no	—
7	#19098313	—	2	3	no	—
8	#19098306	—	2	3	no	—
9	#19098306	—	2	3	no	—
10	#423×C₇₉	C₂²	1	3	yes	—
11	C₂²×D₃₁₆	—	2	3	no	—
12	#14579158	—	2	3	no	—
13	#14579158	Q₈	1	2	yes	—
14	C₄⋊(C₇₉⋊C₈)	D₄	1	3	yes	—
15	C₄⋊(C₇₉⋊C₈)	D₄	1	3	yes	—
16	C₂²⋊(C₇₉⋊C₈)	—	2	3	no	—
17	C₄×C₇₉⋊C₈	—	2	3	no	—
18	C₂×M₁₆×C₇₉	—	2	3	no	—
19	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₇₉	—	2	3	no	—
21	C₄⋊C₈×C₇₉	—	2	3	no	—
22	C₂³×D₇₉	—	4	1	no	—
23	C₂×D₃₁₆	—	2	6	no	—
24	C₂×D₃₁₆	—	4	3	no	—
25	C₂×D₃₁₆	—	8	3	no	—
26	C₂×D₃₁₆	—	8	3	no	—
27	C₂×C₇₉⋊C₈	—	2	3	no	—
28	C₂×C₇₉⋊C₈	—	2	3	no	—
29	C₂²×C₄×C₇₉	#41	1	3	yes	—
30	M₁₆×C₇₉	—	8	3	no	—
31	C₂×C₈×C₇₉	—	4	3	no	—
32	C₄²×C₇₉	C₂×D₄	1	3	yes	—
33	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
34	C₂³×C₇₉	#126	1	1	yes	characteristic
35	C₂²×D₇₉	—	4	2	no	—
36	C₂²×D₇₉	—	8	6	no	—
37	C₂×C₄×C₇₉	—	2	3	no	—
38	C₂×C₄×C₇₉	—	2	3	no	—
39	C₂×C₄×C₇₉	#121	1	3	yes	—
40	C₂×C₄×C₇₉	C₂²⋊Q₈	1	3	yes	—
41	D₃₁₆	—	8	6	no	—
42	D₃₁₆	—	8	3	no	—
43	D₃₁₆	—	8	3	no	—
44	C₈×C₇₉	—	8	3	no	—
45	C₇₉⋊C₈	—	4	3	no	—
46	C₇₉⋊C₈	—	4	3	no	—
47	C₂²×C₇₉	—	2	3	no	—
48	C₂²×C₇₉	#396	1	1	yes	characteristic
49	C₂×D₇₉	—	8	6	no	—
50	C₂×D₇₉	—	16	4	no	—
51	C₄×C₇₉	—	2	3	no	—
52	C₄×C₇₉	#471	1	3	yes	—
53	C₄×C₇₉	#472	1	3	yes	—
54	#10611	—	79	1	no	maximal, Sylow
55	C₂×C₇₉	—	4	1	no	—
56	C₂×C₇₉	#2063	1	3	yes	—
57	D₇₉	—	16	2	no	—
58	#1574	—	79	3	no	—
59	#1545	—	79	3	no	—
60	#1309	D₇₉	1	1	yes	characteristic, complemented
61	C₇₉	#10611	1	1	yes	characteristic, complemented, Sylow
62	C₂×#127	—	79	1	no	—
63	#431	—	79	3	no	—
64	#423	C₂×D₇₉	1	3	yes	—
65	#335	—	158	3	no	—
66	#334	—	158	3	no	—
67	#331	—	158	3	no	—
68	#331	—	158	3	no	—
69	C₂²×D₄	—	158	3	no	—
70	C₂×M₁₆	—	2	3	no	—
71	#127	—	79	2	no	—
72	#127	—	158	3	no	—
73	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	2	3	no	—
76	C₄⋊C₈	—	79	3	no	—
77	C₄⋊C₈	—	79	3	no	—
78	C₂²⋊C₈	—	158	3	no	—
79	C₄×C₈	—	158	3	no	—
80	C₂⁴	—	316	1	no	—
81	C₂×D₄	—	316	3	no	—
82	C₂×D₄	—	632	3	no	—
83	C₂×D₄	—	632	3	no	—
84	C₂×D₄	—	158	6	no	—
85	C₂²×C₄	#11929624	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	4	3	no	—
88	C₂×C₈	—	158	3	no	—
89	C₂×C₈	—	158	3	no	—
90	C₄²	D₄×D₇₉	1	3	yes	—
91	C₄²	Q₈×D₇₉	1	1	yes	characteristic
92	C₂³	—	316	2	no	—
93	C₂³	—	632	6	no	—
94	C₂³	#14579155	1	1	yes	characteristic
95	D₄	—	632	6	no	—
96	D₄	—	632	3	no	—
97	D₄	—	632	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#14579134	1	3	yes	—
101	C₂×C₄	#14579139	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	316	3	no	—
104	C₈	—	316	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	632	6	no	—
107	C₂²	—	1264	4	no	—
108	C₂²	#19098487	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#19098829	1	3	yes	—
111	C₄	#19098832	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	1264	2	no	—
114	C₂	#23764761	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23762191

C₂

yes

maximal, complemented

#23762069

C₂

yes

maximal

C₇₉×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#14579158

C₂²

yes

characteristic

#19098616

C₂²

yes

—

—

—

—

—

#423×C₇₉

C₂²

yes

—

C₂²×D₃₁₆

—

#14579158

—

#14579158

Q₈

yes

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂×M₁₆×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₂³×D₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

#41

yes

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

#126

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#396

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#471

yes

—

C₄×C₇₉

#472

yes

—

#10611

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#2063

yes

—

D₇₉

—

#1574

—

#1545

—

#1309

D₇₉

yes

characteristic, complemented

C₇₉

#10611

yes

characteristic, complemented, Sylow

C₂×#127

—

#431

—

#423

C₂×D₇₉

yes

—

#335

—

158

—

#334

—

158

—

#331

—

158

—

#331

—

158

—

C₂²×D₄

—

158

—

C₂×M₁₆

—

#127

—

#127

—

158

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

158

—

C₄×C₈

—

158

—

C₂⁴

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

632

—

C₂×D₄

—

632

—

C₂×D₄

—

158

—

C₂²×C₄

#11929624

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₄²

D₄×D₇₉

yes

—

C₄²

Q₈×D₇₉

yes

characteristic

C₂³

—

316

—

C₂³

—

632

—

C₂³

#14579155

yes

characteristic

D₄

—

632

—

D₄

—

632

—

D₄

—

632

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#14579134

yes

—

101

C₂×C₄

#14579139

yes

—

102

C₈

—

103

C₈

—

316

—

104

C₈

—

316

—

105

C₂²

—

106

C₂²

—

632

—

107

C₂²

—

1264

—

108

C₂²

#19098487

yes

characteristic

109

C₄

—

110

C₄

#19098829

yes

—

111

C₄

#19098832

yes

—

112

C₂

—

113

C₂

—

1264

—

114

C₂

#23764761

yes

—

	subgroup	G/H
center	C₂²	#19098487
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1309	C₂
socle	C₂²×C₇₉	#396
prime radical	C₂×#14579158	C₂²
hypercenter	#1309	D₇₉
hypocenter	C₇₉	#10611
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#19098487

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1309

C₂

C₂²×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #23762191	C₂	semidirect
3. C₇₉×#1309	C₂	semidirect
60. #1309	D₇₉	semidirect
61. C₇₉	#10611	semidirect

H ≅ G/N

type

1. #23762191

C₂

semidirect

3. C₇₉×#1309

C₂

semidirect

60. #1309

D₇₉

semidirect

61. C₇₉

#10611

semidirect

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

subgroup	index
#23762191	2
#23762069	2
C₇₉×#1309	2
#10611	79

subgroup

index

#23762191

#23762069

C₇₉×#1309

#10611

Subgroups — #34192951

Conjugacy classes of subgroups

Complements

Maximal subgroups