Group #28360915 — subgroups

Subgroups — #28360915

Order 13952.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×C₄⋊#11724784	C₂	1	2	yes	maximal
2	#384×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11724784	—	2	2	no	—
4	C₄⋊#11724784	—	2	2	no	—
5	C₄⋊#11724784	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
7	C₂×C₄⋊C₄×C₁₀₉	C₄	1	2	yes	—
8	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#11724784	—	2	4	no	—
10	C₂×#11724784	—	4	2	no	—
11	C₂×#11724784	—	4	2	no	—
12	C₂²×C₄×C₁₀₉	—	2	2	no	—
13	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
15	C₄⋊C₄×C₁₀₉	—	4	2	no	—
16	C₄⋊C₄×C₁₀₉	—	4	2	no	—
17	C₄⋊C₄×C₁₀₉	—	4	2	no	—
18	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
20	C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
21	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₉	—	2	1	no	—
23	C₂×C₄×C₁₀₉	—	2	1	no	—
24	C₂×C₄×C₁₀₉	—	4	2	no	—
25	C₂×C₄×C₁₀₉	—	4	4	no	—
26	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	D₈	1	2	yes	—
31	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
32	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
33	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
34	#11724784	—	4	4	no	—
35	C₂²×C₁₀₉	#95	1	1	yes	characteristic
36	C₂²×C₁₀₉	#100	1	1	yes	characteristic
37	C₂²×C₁₀₉	#101	1	1	yes	characteristic
38	C₂²×C₁₀₉	#102	1	2	yes	—
39	C₂²×C₁₀₉	#103	1	2	yes	—
40	C₄×C₁₀₉	—	2	1	no	—
41	C₄×C₁₀₉	—	2	1	no	—
42	C₄×C₁₀₉	—	8	2	no	—
43	C₄×C₁₀₉	#102	1	2	yes	—
44	C₄×C₁₀₉	#103	1	2	yes	—
45	C₄×C₁₀₉	#106	1	2	yes	—
46	C₄×C₁₀₉	#107	1	2	yes	—
47	C₂×C₁₀₉	#331	1	1	yes	characteristic
48	C₂×C₁₀₉	#332	1	2	yes	—
49	C₂×C₁₀₉	#333	1	1	yes	characteristic
50	C₂×C₁₀₉	#340	1	2	yes	—
51	C₂×C₁₀₉	#343	1	1	yes	characteristic
52	#1296	—	109	1	no	maximal, Sylow
53	C₁₀₉	#1296	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₄⋊C₈	—	109	2	no	—
55	#384	D₁₀₉	1	1	yes	characteristic
56	C₂²×C₈	—	218	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₁₀₉	1	2	yes	—
59	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
60	C₄⋊C₈	—	109	4	no	—
61	C₄⋊C₈	—	218	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
65	C₂×C₈	—	436	2	no	—
66	C₂×C₈	—	436	2	no	—
67	C₂×C₈	—	218	4	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	Dic₂₁₈	1	1	yes	characteristic
72	C₄²	C₄×D₁₀₉	1	2	yes	—
73	C₄²	D₄₃₆	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
80	C₂×C₄	#13182006	1	2	yes	—
81	C₂×C₄	#13182008	1	2	yes	—
82	C₂×C₄	C₄⋊Dic₁₀₉	1	1	yes	characteristic
83	C₂×C₄	#13182014	1	2	yes	—
84	C₂×C₄	#13182016	1	2	yes	—
85	C₂×C₄	#13182017	1	2	yes	—
86	C₂×C₄	#13182019	1	2	yes	—
87	C₈	—	436	4	no	—
88	C₂²	#16812606	1	1	yes	characteristic
89	C₂²	#16812616	1	1	yes	characteristic
90	C₂²	#16812618	1	1	yes	characteristic
91	C₂²	#16812620	1	2	yes	—
92	C₂²	#16812623	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#16812619	1	2	yes	—
97	C₄	#16812622	1	2	yes	—
98	C₄	#16812630	1	2	yes	—
99	C₄	#16812632	1	2	yes	—
100	C₂	#20294770	1	1	yes	characteristic
101	C₂	#20294772	1	2	yes	—
102	C₂	#20294775	1	1	yes	characteristic
103	C₂	#20294788	1	2	yes	—
104	C₂	#20294795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11724784

C₂

yes

maximal

#384×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

—

C₄⋊#11724784

—

C₄⋊#11724784

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

—

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

Q₈

yes

characteristic

C₂³×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

D₈

yes

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

Q₁₆

yes

—

#11724784

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

—

C₂²×C₁₀₉

#103

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#102

yes

—

C₄×C₁₀₉

#103

yes

—

C₄×C₁₀₉

#106

yes

—

C₄×C₁₀₉

#107

yes

—

C₂×C₁₀₉

#331

yes

characteristic

C₂×C₁₀₉

#332

yes

—

C₂×C₁₀₉

#333

yes

characteristic

C₂×C₁₀₉

#340

yes

—

C₂×C₁₀₉

#343

yes

characteristic

#1296

—

109

maximal, Sylow

C₁₀₉

#1296

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

#384

D₁₀₉

yes

characteristic

C₂²×C₈

—

218

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂×C₈

—

436

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₂₁₈

yes

characteristic

C₄²

C₄×D₁₀₉

yes

—

C₄²

D₄₃₆

yes

characteristic

C₂³

C₂²⋊Dic₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₀₉

yes

characteristic

C₂×C₄

#13182006

yes

—

C₂×C₄

#13182008

yes

—

C₂×C₄

C₄⋊Dic₁₀₉

yes

characteristic

C₂×C₄

#13182014

yes

—

C₂×C₄

#13182016

yes

—

C₂×C₄

#13182017

yes

—

C₂×C₄

#13182019

yes

—

C₈

—

436

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#16812623

yes

—

C₄

—

C₄

—

C₄

—

C₄

#16812619

yes

—

C₄

#16812622

yes

—

C₄

#16812630

yes

—

C₄

#16812632

yes

—

100

C₂

#20294770

yes

characteristic

101

C₂

#20294772

yes

—

102

C₂

#20294775

yes

characteristic

103

C₂

#20294788

yes

—

104

C₂

#20294795

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₁₀₉
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₂×C₄²	C₂×D₁₀₉
Fitting	#384×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₀₉	C₂²
hypercenter	#384	D₁₀₉
hypocenter	C₁₀₉	#1296
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₁₀₉

commutator

C₂×C₄×C₁₀₉

C₄²

Frattini

C₂×C₄²

C₂×D₁₀₉

Fitting

#384×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂²⋊C₄

cosocle

C₂×C₄²×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊#11724784	C₄	semidirect
53. C₁₀₉	#1296	semidirect

H ≅ G/N

type

5. C₄⋊#11724784

C₄

semidirect

53. C₁₀₉

#1296

semidirect

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

subgroup	index
C₂×C₄⋊#11724784	2
#384×C₁₀₉	2
#1296	109

subgroup

index

C₂×C₄⋊#11724784

#384×C₁₀₉

#1296

109

Subgroups — #28360915

Conjugacy classes of subgroups

Complements

Maximal subgroups