Group #18087224 — subgroups

Subgroups — #18087224

Order 4864.

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	#14562426	C₂	1	3	yes	maximal, complemented
2	#14562304	C₂	1	3	yes	maximal
3	C₁₉×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10597320	C₂²	1	1	yes	characteristic
5	#11926597	C₂²	1	3	yes	—
6	#11926296	—	2	3	no	—
7	#11926294	—	2	3	no	—
8	#11926287	—	2	3	no	—
9	#11926287	—	2	3	no	—
10	C₁₉×#423	C₂²	1	3	yes	—
11	C₂²×D₇₆	—	2	3	no	—
12	C₂×M₁₆×C₁₉	—	2	3	no	—
13	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
14	#10597320	—	2	3	no	—
15	#10597320	Q₈	1	2	yes	—
16	C₁₉×C₄⋊C₈	—	2	3	no	—
17	C₁₉×C₄⋊C₈	—	2	3	no	—
18	C₂²⋊(C₁₉⋊C₈)	—	2	3	no	—
19	C₄⋊(C₁₉⋊C₈)	D₄	1	3	yes	—
20	C₄⋊(C₁₉⋊C₈)	D₄	1	3	yes	—
21	C₄×C₁₉⋊C₈	—	2	3	no	—
22	C₂³×D₁₉	—	4	1	no	—
23	C₂²×C₄×C₁₉	#41	1	3	yes	—
24	C₂×D₇₆	—	2	6	no	—
25	C₂×D₇₆	—	4	3	no	—
26	C₂×D₇₆	—	8	3	no	—
27	C₂×D₇₆	—	8	3	no	—
28	M₁₆×C₁₉	—	8	3	no	—
29	C₂×C₈×C₁₉	—	4	3	no	—
30	C₄²×C₁₉	C₂×D₄	1	3	yes	—
31	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₁₉⋊C₈	—	2	3	no	—
33	C₂×C₁₉⋊C₈	—	2	3	no	—
34	#10612	—	19	1	no	maximal, Sylow
35	C₂³×C₁₉	#126	1	1	yes	characteristic
36	C₂²×D₁₉	—	4	2	no	—
37	C₂²×D₁₉	—	8	6	no	—
38	C₂×C₄×C₁₉	—	2	3	no	—
39	C₂×C₄×C₁₉	—	2	3	no	—
40	C₂×C₄×C₁₉	#121	1	3	yes	—
41	C₂×C₄×C₁₉	C₂²⋊Q₈	1	3	yes	—
42	D₇₆	—	8	6	no	—
43	D₇₆	—	8	3	no	—
44	D₇₆	—	8	3	no	—
45	C₈×C₁₉	—	8	3	no	—
46	C₁₉⋊C₈	—	4	3	no	—
47	C₁₉⋊C₈	—	4	3	no	—
48	#1574	—	19	3	no	—
49	#1545	—	19	3	no	—
50	#1309	D₁₉	1	1	yes	characteristic, complemented
51	C₂²×C₁₉	—	2	3	no	—
52	C₂²×C₁₉	#396	1	1	yes	characteristic
53	C₂×D₁₉	—	8	6	no	—
54	C₂×D₁₉	—	16	4	no	—
55	C₄×C₁₉	—	2	3	no	—
56	C₄×C₁₉	#471	1	3	yes	—
57	C₄×C₁₉	#472	1	3	yes	—
58	C₂×#127	—	19	1	no	—
59	#431	—	19	3	no	—
60	#423	C₂×D₁₉	1	3	yes	—
61	#335	—	38	3	no	—
62	#334	—	38	3	no	—
63	#331	—	38	3	no	—
64	#331	—	38	3	no	—
65	C₂×C₁₉	—	4	1	no	—
66	C₂×C₁₉	#2063	1	3	yes	—
67	D₁₉	—	16	2	no	—
68	C₂²×D₄	—	38	3	no	—
69	C₂×M₁₆	—	2	3	no	—
70	#127	—	19	2	no	—
71	#127	—	38	3	no	—
72	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
73	C₄⋊C₈	—	2	3	no	—
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	19	3	no	—
76	C₄⋊C₈	—	19	3	no	—
77	C₂²⋊C₈	—	38	3	no	—
78	C₄×C₈	—	38	3	no	—
79	C₁₉	#10612	1	1	yes	characteristic, complemented, Sylow
80	C₂⁴	—	76	1	no	—
81	C₂×D₄	—	38	6	no	—
82	C₂×D₄	—	76	3	no	—
83	C₂×D₄	—	152	3	no	—
84	C₂×D₄	—	152	3	no	—
85	C₂²×C₄	#64541	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	4	3	no	—
88	C₂×C₈	—	38	3	no	—
89	C₂×C₈	—	38	3	no	—
90	C₄²	D₄×D₁₉	1	3	yes	—
91	C₄²	Q₈×D₁₉	1	1	yes	characteristic
92	C₂³	—	76	2	no	—
93	C₂³	—	152	6	no	—
94	C₂³	#10597322	1	1	yes	characteristic
95	D₄	—	152	6	no	—
96	D₄	—	152	3	no	—
97	D₄	—	152	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#10597330	1	3	yes	—
101	C₂×C₄	#10597342	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	76	3	no	—
104	C₈	—	76	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	304	4	no	—
107	C₂²	—	152	6	no	—
108	C₂²	#11926468	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#11926810	1	3	yes	—
111	C₄	#11926813	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	304	2	no	—
114	C₂	#14564996	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14562426

C₂

yes

maximal, complemented

#14562304

C₂

yes

maximal

C₁₉×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#10597320

C₂²

yes

characteristic

#11926597

C₂²

yes

—

—

—

—

—

C₁₉×#423

C₂²

yes

—

C₂²×D₇₆

—

C₂×M₁₆×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

#10597320

—

#10597320

Q₈

yes

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₂²⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

D₄

yes

—

C₄⋊(C₁₉⋊C₈)

D₄

yes

—

C₄×C₁₉⋊C₈

—

C₂³×D₁₉

—

C₂²×C₄×C₁₉

#41

yes

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

C₂×Q₈

yes

characteristic

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

D₇₆

—

D₇₆

—

D₇₆

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1574

—

#1545

—

#1309

D₁₉

yes

characteristic, complemented

C₂²×C₁₉

—

C₂²×C₁₉

#396

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#471

yes

—

C₄×C₁₉

#472

yes

—

C₂×#127

—

#431

—

#423

C₂×D₁₉

yes

—

—

—

—

—

C₂×C₁₉

—

C₂×C₁₉

#2063

yes

—

D₁₉

—

C₂²×D₄

—

C₂×M₁₆

—

#127

—

#127

—

C₂×C₄²

C₂²×D₁₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₁₉

#10612

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

152

—

C₂×D₄

—

152

—

C₂²×C₄

#64541

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

D₄×D₁₉

yes

—

C₄²

Q₈×D₁₉

yes

characteristic

C₂³

—

C₂³

—

152

—

C₂³

#10597322

yes

characteristic

D₄

—

152

—

D₄

—

152

—

D₄

—

152

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#10597330

yes

—

101

C₂×C₄

#10597342

yes

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₂²

—

106

C₂²

—

304

—

107

C₂²

—

152

—

108

C₂²

#11926468

yes

characteristic

109

C₄

—

110

C₄

#11926810

yes

—

111

C₄

#11926813

yes

—

112

C₂

—

113

C₂

—

304

—

114

C₂

#14564996

yes

—

	subgroup	G/H
center	C₂²	#11926468
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1309	C₂
socle	C₂²×C₁₉	#396
prime radical	C₂×#10597320	C₂²
hypercenter	#1309	D₁₉
hypocenter	C₁₉	#10612
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#11926468

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1309

C₂

C₂²×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #14562426	C₂	semidirect
3. C₁₉×#1309	C₂	semidirect
50. #1309	D₁₉	semidirect
79. C₁₉	#10612	semidirect

H ≅ G/N

type

1. #14562426

C₂

semidirect

3. C₁₉×#1309

C₂

semidirect

50. #1309

D₁₉

semidirect

79. C₁₉

#10612

semidirect

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

subgroup	index
#14562426	2
#14562304	2
C₁₉×#1309	2
#10612	19

subgroup

index

#14562426

#14562304

C₁₉×#1309

#10612

Subgroups — #18087224

Conjugacy classes of subgroups

Complements

Maximal subgroups