Group #28379372 — subgroups

Subgroups — C₂×#20295949 #28379372

Order 13952.

168 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 82 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×#16812773	C₂	1	2	yes	maximal
2	#20295949	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
4	#16812773	—	2	2	no	—
5	#16812773	C₂²	1	4	yes	—
6	C₂×#13182032	—	2	2	no	—
7	C₂²×#11724784	C₄	1	1	yes	characteristic
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	C₄×#11724784	C₄	1	2	yes	—
10	C₄×#11724784	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
12	#13182032	—	2	4	no	—
13	C₂×#11724784	C₈	1	2	yes	—
14	C₂×#11724784	C₈	1	2	yes	—
15	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
16	C₂×#11724784	C₂×C₄	1	2	yes	—
17	C₂×#11724784	D₄	1	1	yes	characteristic
18	C₂×#11724784	D₄	1	1	yes	characteristic
19	C₂×#11724784	Q₈	1	1	yes	characteristic
20	C₂×#11724784	Q₈	1	1	yes	characteristic
21	C₂×#11724784	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	—	2	1	no	—
23	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₀₉	—	2	1	no	—
25	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂³×C₁₀₉	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	—	2	4	no	—
28	C₂×C₄×C₁₀₉	—	2	2	no	—
29	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	#11724784	C₄⋊C₄	1	1	yes	characteristic
36	#11724784	C₄⋊C₄	1	1	yes	characteristic
37	#11724784	C₂×C₈	1	4	yes	—
38	#11724784	C₂×D₄	1	1	yes	characteristic
39	#11724784	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₁₀₉	—	2	2	no	—
41	C₂²×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
42	C₂²×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
43	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₁₀₉	—	2	4	no	—
45	C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
46	C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₁₀₉	—	2	1	no	—
50	C₂×C₁₀₉	—	2	1	no	—
51	C₂×C₁₀₉	#364	1	2	yes	complemented
52	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
53	C₂×#364	—	109	1	no	maximal, Sylow
54	C₁₀₉	C₂×#364	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	—	109	2	no	—
56	C₂×C₄×C₈	—	109	1	no	—
57	#364	—	109	4	no	—
58	C₂²×C₈	—	109	1	no	—
59	C₂²×C₈	—	109	1	no	—
60	C₂×C₄²	#89137	1	1	yes	characteristic
61	M₃₂	—	109	4	no	—
62	M₃₂	—	218	2	no	—
63	C₂×C₁₆	—	218	2	no	—
64	C₄×C₈	—	109	2	no	—
65	C₄×C₈	—	109	2	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
68	C₂×C₈	—	109	1	no	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	109	1	no	—
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	109	2	no	—
73	C₂×C₈	—	109	2	no	—
74	C₂×C₈	—	109	1	no	—
75	C₂×C₈	—	109	2	no	—
76	C₂×C₈	—	109	1	no	—
77	C₄²	—	2	1	no	—
78	C₄²	C₂×#89137	1	2	yes	—
79	C₁₆	—	218	4	no	—
80	C₂³	#13182038	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
84	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
85	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
86	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
87	C₂×C₄	#13182038	1	1	yes	characteristic
88	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
89	C₈	—	109	1	no	—
90	C₈	—	109	1	no	—
91	C₈	—	109	1	no	—
92	C₈	—	109	1	no	—
93	C₈	—	109	4	no	—
94	C₂²	—	2	2	no	—
95	C₂²	#16812766	1	1	yes	characteristic
96	C₂²	#16812766	1	1	yes	characteristic
97	C₂²	C₂×#13182038	1	1	yes	characteristic
98	C₄	—	2	4	no	—
99	C₄	#16812766	1	1	yes	characteristic
100	C₄	#16812766	1	1	yes	characteristic
101	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
102	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#20295949	1	2	yes	direct product factor
106	C₂	C₂×#16812766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812773

C₂

yes

maximal

#20295949

C₂

yes

maximal, direct product factor

C₂×C₄×#11724784

C₂

yes

maximal, characteristic

#16812773

—

#16812773

C₂²

yes

—

C₂×#13182032

—

C₂²×#11724784

C₄

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

—

C₄×#11724784

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₄

yes

characteristic

#13182032

—

C₂×#11724784

C₈

yes

—

C₂×#11724784

C₈

yes

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

D₄

yes

characteristic

C₂×#11724784

D₄

yes

characteristic

C₂×#11724784

Q₈

yes

characteristic

C₂×#11724784

Q₈

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

M₁₆

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

M₁₆

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

#11724784

C₂×C₈

yes

—

#11724784

C₂×D₄

yes

characteristic

#11724784

C₂×Q₈

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄⋊C₈

yes

characteristic

C₂²×C₁₀₉

C₄⋊C₈

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊C₈

yes

characteristic

C₄×C₁₀₉

C₄⋊C₈

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#364

yes

complemented

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

characteristic

C₂×#364

—

109

maximal, Sylow

C₁₀₉

C₂×#364

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂×C₄×C₈

—

109

—

#364

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

#89137

yes

characteristic

M₃₂

—

109

—

M₃₂

—

218

—

C₂×C₁₆

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄²

—

C₄²

C₂×#89137

yes

—

C₁₆

—

218

—

C₂³

#13182038

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

#13182038

yes

characteristic

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

#16812766

yes

characteristic

100

C₄

#16812766

yes

characteristic

101

C₄

C₂×C₄⋊#89137

yes

characteristic

102

C₄

C₂×D₁₀₉⋊C₈

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

#20295949

yes

direct product factor

106

C₂

C₂×#16812766

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×C₄²×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₄⋊C₈
prime radical	C₂³×C₁₀₉	M₁₆
cosocle	C₂×#11724784	C₂³
hypercenter	C₂×C₄²	#89137
hypocenter	C₁₀₉	C₂×#364
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂×C₄

C₄⋊#89137

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂×C₄²×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₄⋊C₈

prime radical

C₂³×C₁₀₉

M₁₆

C₂³

C₂×C₄²

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
2. #20295949	C₂	direct (N × H)
51. C₂×C₁₀₉	#364	semidirect
54. C₁₀₉	C₂×#364	semidirect
105. C₂	#20295949	direct (N × H)

H ≅ G/N

type

2. #20295949

C₂

direct (N × H)

51. C₂×C₁₀₉

#364

semidirect

54. C₁₀₉

C₂×#364

semidirect

105. C₂

#20295949

direct (N × H)

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

subgroup	index
C₂×#16812773	2
#20295949	2
C₂×C₄×#11724784	2
C₂×#364	109

subgroup

index

109

Subgroups — C₂×#20295949 #28379372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295949 #28379372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295949 #28379372