Group #28364784 — subgroups

Subgroups — C₂×#20294844 #28364784

Order 13952.

184 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×#13182004	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294844	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
7	C₂×#13182004	—	2	2	no	—
8	C₂×#13182004	C₂²	1	2	yes	—
9	C₈×Dic₁₀₉	C₄	1	2	yes	—
10	C₈×Dic₁₀₉	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₀₉	—	2	2	no	—
13	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×Dic₁₀₉	C₈	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	C₈	1	1	yes	characteristic
16	C₂×#11724784	—	2	2	no	—
17	C₂×#11724784	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₀₉	C₈	1	2	yes	—
19	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
20	#13182004	—	2	2	no	—
21	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
24	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
25	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₁₆×C₁₀₉	—	2	2	no	—
27	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
31	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	—	2	2	no	—
33	C₂×Dic₁₀₉	C₁₆	1	4	yes	complemented
34	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
35	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
36	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
37	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
39	#11724784	—	2	2	no	—
40	C₂²×C₁₀₉	C₄⋊C₈	1	2	yes	—
41	C₂²×C₁₀₉	C₂×C₁₆	1	2	yes	—
42	C₂²×C₁₀₉	M₃₂	1	2	yes	—
43	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
45	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
47	Dic₁₀₉	—	2	2	no	—
48	Dic₁₀₉	C₂×C₁₆	1	4	yes	complemented
49	C₂×C₁₀₉	C₄⋊C₁₆	1	4	yes	complemented
50	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
52	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
53	C₂×C₄⋊C₁₆	—	109	1	no	maximal, Sylow
54	C₁₀₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₁₆	—	109	1	no	—
56	C₂²×C₁₆	D₁₀₉	1	1	yes	characteristic
57	C₂×C₄×C₈	—	109	1	no	—
58	C₄⋊C₁₆	—	109	4	no	—
59	C₂²×C₈	—	109	1	no	—
60	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
61	C₂×C₄²	—	109	1	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	109	2	no	—
64	C₂×C₁₆	—	218	2	no	—
65	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	—
66	C₄×C₈	—	109	2	no	—
67	C₄×C₈	—	109	2	no	—
68	C₂²×C₄	—	109	1	no	—
69	C₂²×C₄	—	109	1	no	—
70	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
71	C₂×C₈	—	109	2	no	—
72	C₂×C₈	—	218	2	no	—
73	C₂×C₈	Dic₂₁₈	1	2	yes	—
74	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
75	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
76	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
77	C₄²	—	109	2	no	—
78	C₄²	—	109	2	no	—
79	C₁₆	—	2	2	no	—
80	C₁₆	—	218	2	no	—
81	C₂³	C₈×D₁₀₉	1	1	yes	characteristic
82	C₂×C₄	—	109	4	no	—
83	C₂×C₄	—	109	2	no	—
84	C₂×C₄	—	109	2	no	—
85	C₂×C₄	—	218	2	no	—
86	C₂×C₄	#13182008	1	2	yes	—
87	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
88	C₂×C₄	#13182012	1	2	yes	—
89	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
90	C₈	—	218	2	no	—
91	C₈	#13182008	1	2	yes	—
92	C₈	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
93	C₈	C₂×Dic₂₁₈	1	1	yes	characteristic
94	C₂²	#16812628	1	2	yes	—
95	C₂²	C₁₆×D₁₀₉	1	2	yes	—
96	C₂²	#16812638	1	2	yes	—
97	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
98	C₄	—	109	4	no	—
99	C₄	—	218	2	no	—
100	C₄	#16812628	1	2	yes	—
101	C₄	C₂×#13182008	1	1	yes	characteristic
102	C₄	C₂×#13182012	1	1	yes	characteristic
103	C₂	#20294844	1	4	yes	direct product factor
104	C₂	C₂×#16812628	1	1	yes	characteristic
105	C₂	C₂×C₁₆×D₁₀₉	1	1	yes	characteristic
106	C₂	C₂×#16812638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13182004

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294844

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×#13182004

—

C₂×#13182004

C₂²

yes

—

C₈×Dic₁₀₉

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₈

yes

characteristic

C₂²×Dic₁₀₉

C₈

yes

characteristic

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₈

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

#13182004

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₁₆

yes

complemented

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

C₂×Q₈

yes

characteristic

#11724784

—

C₂²×C₁₀₉

C₄⋊C₈

yes

—

C₂²×C₁₀₉

C₂×C₁₆

yes

—

C₂²×C₁₀₉

M₃₂

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

C₄⋊C₈

yes

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂×C₁₆

yes

complemented

C₂×C₁₀₉

C₄⋊C₁₆

yes

complemented

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₀₉

C₂²×C₁₆

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

C₂×C₄⋊C₁₆

—

109

maximal, Sylow

C₁₀₉

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

109

—

C₂²×C₁₆

D₁₀₉

yes

characteristic

C₂×C₄×C₈

—

109

—

C₄⋊C₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄²

—

109

—

C₂×C₁₆

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

C₂×C₁₆

C₂×D₁₀₉

yes

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄²

—

109

—

C₄²

—

109

—

C₁₆

—

C₁₆

—

218

—

C₂³

C₈×D₁₀₉

yes

characteristic

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

#13182008

yes

—

C₂×C₄

C₈×D₁₀₉

yes

characteristic

C₂×C₄

#13182012

yes

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₈

—

218

—

C₈

#13182008

yes

—

C₈

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₈

C₂×Dic₂₁₈

yes

characteristic

C₂²

#16812628

yes

—

C₂²

C₁₆×D₁₀₉

yes

—

C₂²

#16812638

yes

—

C₂²

C₂×C₈×D₁₀₉

yes

characteristic

C₄

—

109

—

C₄

—

218

—

100

C₄

#16812628

yes

—

101

C₄

C₂×#13182008

yes

characteristic

102

C₄

C₂×#13182012

yes

characteristic

103

C₂

#20294844

yes

direct product factor

104

C₂

C₂×#16812628

yes

characteristic

105

C₂

C₂×C₁₆×D₁₀₉

yes

characteristic

106

C₂

C₂×#16812638

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	C₂²×C₁₆×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₂²×C₁₆	D₁₀₉
hypocenter	C₁₀₉	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²×C₈

C₂×D₁₀₉

commutator

C₂×C₁₀₉

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

C₂²×C₁₆×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×C₈

cosocle

C₂×C₈×C₁₀₉

C₂³

hypercenter

C₂²×C₁₆

D₁₀₉

hypocenter

C₁₀₉

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. #20294844	C₂	direct (N × H)
33. C₂×Dic₁₀₉	C₁₆	semidirect
48. Dic₁₀₉	C₂×C₁₆	semidirect
49. C₂×C₁₀₉	C₄⋊C₁₆	semidirect
54. C₁₀₉	C₂×C₄⋊C₁₆	semidirect
103. C₂	#20294844	direct (N × H)

H ≅ G/N

type

3. #20294844

C₂

direct (N × H)

33. C₂×Dic₁₀₉

C₁₆

semidirect

48. Dic₁₀₉

C₂×C₁₆

semidirect

49. C₂×C₁₀₉

C₄⋊C₁₆

semidirect

54. C₁₀₉

C₂×C₄⋊C₁₆

semidirect

103. C₂

#20294844

direct (N × H)

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

subgroup	index
C₂²×#13182004	2
C₂×C₈×Dic₁₀₉	2
#20294844	2
C₂²×C₁₆×C₁₀₉	2
C₂×C₄⋊C₁₆	109

subgroup

index

C₂²×#13182004

C₂×C₈×Dic₁₀₉

#20294844

C₂²×C₁₆×C₁₀₉

C₂×C₄⋊C₁₆

109

Subgroups — C₂×#20294844 #28364784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20294844 #28364784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20294844 #28364784