Group #28378747 — subgroups

Subgroups — #28378747

Order 13952.

296 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 142 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×#11724786	C₂	1	2	yes	maximal
2	#20295892	C₂	1	4	yes	maximal
3	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11724786	C₂²	1	4	yes	—
5	C₄×#11724786	C₄	1	8	yes	complemented
6	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₉	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×#11724786	—	2	8	no	—
12	C₂×#11724786	C₂×C₄	1	8	yes	—
13	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
14	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
16	C₄×Dic₁₀₉	—	2	2	no	—
17	C₄×Dic₁₀₉	—	2	4	no	—
18	C₄×Dic₁₀₉	C₈	1	8	yes	complemented
19	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₁₀₉	—	2	2	no	—
23	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	—	2	2	no	—
25	C₂×C₄×C₁₀₉	—	2	2	no	—
26	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
27	C₂×C₄×C₁₀₉	C₂×C₈	1	4	yes	—
28	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
29	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
30	C₂×Dic₁₀₉	—	2	2	no	—
31	C₂×Dic₁₀₉	—	2	2	no	—
32	C₂×Dic₁₀₉	C₄²	1	2	yes	—
33	C₂×Dic₁₀₉	C₂×C₈	1	4	yes	—
34	C₂×Dic₁₀₉	C₂×C₈	1	4	yes	—
35	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
36	C₂×Dic₁₀₉	#41	1	4	yes	—
37	#11724786	—	2	8	no	—
38	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
39	C₂²×C₁₀₉	#117	1	2	yes	—
40	C₂²×C₁₀₉	C₂²×C₈	1	2	yes	—
41	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
42	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
43	C₄×C₁₀₉	—	2	4	no	—
44	C₄×C₁₀₉	#97	1	4	yes	—
45	Dic₁₀₉	—	2	4	no	—
46	Dic₁₀₉	C₄×C₈	1	4	yes	—
47	Dic₁₀₉	#117	1	4	yes	—
48	C₂×C₁₀₉	#376	1	1	yes	characteristic
49	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
50	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
51	C₂×C₁₀₉	#431	1	2	yes	—
52	C₂×C₁₀₉	#431	1	2	yes	—
53	#1744	—	109	1	no	maximal, Sylow
54	C₁₀₉	#1744	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₄×C₈	—	109	2	no	—
56	C₄³	—	109	1	no	—
57	#336	—	109	4	no	—
58	C₂²×C₈	—	109	4	no	—
59	C₂×C₄²	—	109	2	no	—
60	C₂×C₄²	—	109	1	no	—
61	C₂×C₄²	—	109	2	no	—
62	C₂×C₄²	—	109	1	no	—
63	C₂×C₄²	#89137	1	1	yes	characteristic
64	C₄×C₈	—	109	8	no	—
65	C₂²×C₄	—	109	1	no	—
66	C₂²×C₄	—	109	1	no	—
67	C₂²×C₄	—	109	2	no	—
68	C₂²×C₄	C₂×#89137	1	2	yes	—
69	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
70	C₂×C₈	—	109	8	no	—
71	C₂×C₈	—	218	8	no	—
72	C₄²	—	2	2	no	—
73	C₄²	—	109	4	no	—
74	C₄²	—	109	8	no	—
75	C₄²	—	218	4	no	—
76	C₄²	—	218	2	no	—
77	C₂³	C₂²×#89137	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	109	2	no	—
81	C₂×C₄	—	109	2	no	—
82	C₂×C₄	—	109	4	no	—
83	C₂×C₄	—	109	4	no	—
84	C₂×C₄	—	109	4	no	—
85	C₂×C₄	—	218	2	no	—
86	C₂×C₄	—	218	2	no	—
87	C₂×C₄	C₄×#89137	1	2	yes	—
88	C₂×C₄	D₁₀₉⋊C₈	1	4	yes	—
89	C₂×C₄	#13182038	1	2	yes	—
90	C₂×C₄	#13182039	1	2	yes	—
91	C₈	—	218	8	no	—
92	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
93	C₂²	#16812778	1	2	yes	—
94	C₂²	C₂×D₁₀₉⋊C₈	1	2	yes	—
95	C₂²	C₂×#13182038	1	1	yes	characteristic
96	C₂²	C₂×#13182039	1	1	yes	characteristic
97	C₄	—	2	4	no	—
98	C₄	—	109	4	no	—
99	C₄	—	109	4	no	—
100	C₄	—	218	4	no	—
101	C₄	#16812754	1	4	yes	—
102	C₂	#20295962	1	1	yes	characteristic
103	C₂	C₄×D₁₀₉⋊C₈	1	1	yes	characteristic
104	C₂	C₂×#16812754	1	1	yes	characteristic
105	C₂	#20296043	1	2	yes	—
106	C₂	#20296045	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#11724786

C₂

yes

maximal

#20295892

C₂

yes

maximal

C₄²×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724786

C₂²

yes

—

C₄×#11724786

C₄

yes

complemented

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₄

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₄

yes

characteristic

C₂×#11724786

—

C₂×#11724786

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂³

yes

characteristic

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₈

yes

complemented

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

#41

yes

—

#11724786

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂²×C₁₀₉

#117

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

—

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#97

yes

—

Dic₁₀₉

—

Dic₁₀₉

C₄×C₈

yes

—

Dic₁₀₉

#117

yes

—

C₂×C₁₀₉

#376

yes

characteristic

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₉

C₂×#97

yes

characteristic

C₂×C₁₀₉

#431

yes

—

C₂×C₁₀₉

#431

yes

—

#1744

—

109

maximal, Sylow

C₁₀₉

#1744

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

109

—

C₄³

—

109

—

#336

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

#89137

yes

characteristic

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₄²

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

—

218

—

C₂³

C₂²×#89137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₄×#89137

yes

—

C₂×C₄

D₁₀₉⋊C₈

yes

—

C₂×C₄

#13182038

yes

—

C₂×C₄

#13182039

yes

—

C₈

—

218

—

C₂²

C₂×C₄×#89137

yes

characteristic

C₂²

#16812778

yes

—

C₂²

C₂×D₁₀₉⋊C₈

yes

—

C₂²

yes

C₂²

yes

C₄

—

C₄

—

109

—

C₄

—

109

—

100

C₄

—

218

—

101

C₄

#16812754

yes

—

102

C₂

#20295962

yes

characteristic

103

C₂

C₄×D₁₀₉⋊C₈

yes

characteristic

104

C₂

C₂×#16812754

yes

characteristic

105

C₂

#20296043

yes

—

106

C₂

#20296045

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂×#89137
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Frattini	C₂³	C₂²×#89137
Fitting	C₂×C₄²×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×Dic₁₀₉	C₂³
hypercenter	C₂×C₄²	#89137
hypocenter	C₁₀₉	#1744
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂²×C₄

C₂×#89137

commutator

C₂×C₁₀₉

C₂×C₄×C₈

Frattini

C₂³

C₂²×#89137

Fitting

C₂×C₄²×C₁₀₉

C₄

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×Dic₁₀₉

C₂³

C₂×C₄²

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
5. C₄×#11724786	C₄	semidirect
18. C₄×Dic₁₀₉	C₈	semidirect
54. C₁₀₉	#1744	semidirect

H ≅ G/N

type

5. C₄×#11724786

C₄

semidirect

18. C₄×Dic₁₀₉

C₈

semidirect

54. C₁₀₉

#1744

semidirect

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

subgroup	index
C₂×C₄×#11724786	2
#20295892	2
C₄²×Dic₁₀₉	2
#1744	109

subgroup

index

C₂×C₄×#11724786

#20295892

C₄²×Dic₁₀₉

#1744

109

Subgroups — #28378747

Conjugacy classes of subgroups

Complements

Maximal subgroups