Group #28364608 — subgroups

Subgroups — #28364608

Order 13952.

158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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	#20295075	C₂	1	1	yes	maximal, characteristic
2	#20295049	C₂	1	1	yes	maximal, characteristic
3	#20294796	C₂	1	2	yes	maximal
4	#20294785	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂×#13182013	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812635	—	2	2	no	—
10	#16812628	—	2	2	no	—
11	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
12	#117×C₁₀₉	C₂²	1	2	yes	—
13	C₂²×Dic₁₀₉	—	2	1	no	—
14	#13182013	—	2	2	no	—
15	#13182013	—	2	2	no	—
16	#13182013	—	2	2	no	—
17	C₂×#11724784	—	2	1	no	—
18	C₂×#11724784	—	2	2	no	—
19	C₄×Dic₁₀₉	—	2	1	no	—
20	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
21	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₀₉	—	2	2	no	—
24	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₀₉	—	2	2	no	—
26	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
27	C₄²×C₁₀₉	—	2	2	no	—
28	C₂³×C₁₀₉	#41	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	—	4	2	no	—
30	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	#41	1	2	yes	—
33	C₂×C₄×C₁₀₉	#41	1	2	yes	—
34	C₂×Dic₁₀₉	—	2	1	no	—
35	C₂×Dic₁₀₉	—	2	1	no	—
36	C₂×Dic₁₀₉	—	4	2	no	—
37	C₈×C₁₀₉	—	2	2	no	—
38	#11724784	—	4	1	no	—
39	#11724784	—	4	2	no	—
40	C₂²×C₁₀₉	—	4	1	no	—
41	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₀₉	#126	1	2	yes	—
43	C₄×C₁₀₉	—	8	2	no	—
44	C₄×C₁₀₉	#123	1	1	yes	characteristic
45	C₄×C₁₀₉	#124	1	1	yes	characteristic
46	C₄×C₁₀₉	#125	1	2	yes	—
47	Dic₁₀₉	—	2	2	no	—
48	Dic₁₀₉	—	4	1	no	—
49	C₂×C₁₀₉	—	2	1	no	—
50	C₂×C₁₀₉	—	2	2	no	—
51	C₂×C₁₀₉	#400	1	1	yes	characteristic
52	#2102	—	109	1	no	maximal, Sylow
53	C₁₀₉	#2102	1	1	yes	characteristic, complemented, Sylow
54	#430	—	109	1	no	—
55	#423	—	109	1	no	—
56	#344	—	109	2	no	—
57	#339	—	109	2	no	—
58	H(3,ℤ/4)	D₁₀₉	1	1	yes	characteristic
59	C₂×M₁₆	—	109	2	no	—
60	C₂×M₁₆	—	109	1	no	—
61	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
62	#117	C₂×D₁₀₉	1	2	yes	—
63	C₂×C₄²	—	109	1	no	—
64	#108	—	218	2	no	—
65	C₄⋊C₈	—	218	2	no	—
66	C₂²×C₄	—	218	1	no	—
67	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	218	2	no	—
70	M₁₆	—	218	2	no	—
71	M₁₆	—	218	2	no	—
72	C₂×C₈	—	218	1	no	—
73	C₂×C₈	—	218	2	no	—
74	C₂×C₈	D₄₃₆	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	109	1	no	—
79	C₄²	—	109	1	no	—
80	C₄²	—	218	1	no	—
81	C₂³	#13182028	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	436	2	no	—
84	C₂×C₄	—	218	1	no	—
85	C₂×C₄	—	218	1	no	—
86	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
88	C₂×C₄	#13182028	1	2	yes	—
89	C₂×C₄	#13182029	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	436	1	no	—
92	C₈	—	436	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#16812679	1	1	yes	characteristic
95	C₂²	#16812692	1	2	yes	—
96	C₄	—	8	2	no	—
97	C₄	—	436	1	no	—
98	C₄	—	218	2	no	—
99	C₄	#16812683	1	1	yes	characteristic
100	C₄	#16812687	1	1	yes	characteristic
101	C₄	#16812689	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#20294966	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295075

C₂

yes

maximal, characteristic

#20295049

C₂

yes

maximal, characteristic

#20294796

C₂

yes

maximal

#20294785

C₂

yes

maximal

H(3,ℤ/4)×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂×#13182013

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812635

—

#16812628

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

—

—

—

—

—

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

#41

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₉

#126

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#400

yes

characteristic

#2102

—

109

maximal, Sylow

C₁₀₉

#2102

yes

characteristic, complemented, Sylow

#430

—

109

—

#423

—

109

—

#344

—

109

—

#339

—

109

—

H(3,ℤ/4)

D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

#117

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

#108

—

218

—

C₄⋊C₈

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

D₄₃₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₂³

#13182028

yes

characteristic

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

C₂×C₄

#13182028

yes

—

C₂×C₄

#13182029

yes

—

C₈

—

C₈

—

436

—

C₈

—

436

—

C₂²

—

C₂²

#16812679

yes

characteristic

C₂²

#16812692

yes

—

C₄

—

C₄

—

436

—

C₄

—

218

—

C₄

#16812683

yes

characteristic

100

C₄

#16812687

yes

characteristic

101

C₄

#16812689

yes

—

102

C₂

—

103

C₂

—

104

C₂

#20294966

yes

characteristic

	subgroup	G/H
center	C₄	#16812687
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	H(3,ℤ/4)×C₁₀₉	C₂
socle	C₂×C₁₀₉	#400
prime radical	C₂³×C₁₀₉	#41
hypercenter	H(3,ℤ/4)	D₁₀₉
hypocenter	C₁₀₉	#2102
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#16812687

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

H(3,ℤ/4)×C₁₀₉

C₂

C₂×C₁₀₉

C₂³×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
53. C₁₀₉	#2102	semidirect

H ≅ G/N

type

53. C₁₀₉

#2102

semidirect

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

subgroup	index
#20295075	2
#20295049	2
#20294796	2
#20294785	2
H(3,ℤ/4)×C₁₀₉	2
#2102	109

subgroup

index

H(3,ℤ/4)×C₁₀₉

109

Subgroups — #28364608

Conjugacy classes of subgroups

Complements

Maximal subgroups