Group #28379237 — subgroups

Subgroups — C₄×#16812774 #28379237

Order 13952.

196 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 110 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×#16812774	C₂	1	2	yes	maximal
2	#20295915	C₂	1	2	yes	maximal, complemented
3	C₄×#13182032	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#16812774	C₄	1	8	yes	direct product factor
6	C₂×#13182032	C₂²	1	4	yes	—
7	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
8	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₄²×D₁₀₉	C₄	1	1	yes	characteristic
10	C₄×#11724784	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
14	#13182032	C₂×C₄	1	8	yes	complemented
15	C₂×C₄×D₁₀₉	C₈	1	2	yes	—
16	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
18	C₂×#11724784	C₂³	1	1	yes	characteristic
19	C₈×D₁₀₉	C₈	1	4	yes	—
20	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₀₉	C₈	1	2	yes	—
22	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
30	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
31	C₄×D₁₀₉	—	2	4	no	—
32	C₄×D₁₀₉	C₄²	1	2	yes	—
33	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
34	C₈×C₁₀₉	C₄²	1	2	yes	—
35	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
36	#11724784	C₂×C₈	1	2	yes	—
37	#11724784	C₂²×C₄	1	2	yes	—
38	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
39	C₂×D₁₀₉	—	2	2	no	—
40	C₂×D₁₀₉	C₄×C₈	1	2	yes	—
41	C₄×C₁₀₉	M₃₂	1	4	yes	complemented
42	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
43	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
44	Dic₁₀₉	—	2	2	no	—
45	Dic₁₀₉	C₄×C₈	1	2	yes	—
46	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
47	C₂×C₁₀₉	C₂×M₃₂	1	2	yes	—
48	D₁₀₉	—	2	2	no	—
49	C₄×M₃₂	—	109	1	no	maximal, Sylow
50	C₁₀₉	C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
51	C₂×M₃₂	—	109	2	no	—
52	C₂×C₄×C₈	—	109	1	no	—
53	#346	—	109	2	no	—
54	C₄×C₁₆	—	109	2	no	—
55	C₂²×C₈	—	109	1	no	—
56	C₂²×C₈	—	109	1	no	—
57	C₂×C₄²	—	109	1	no	—
58	M₃₂	—	109	8	no	—
59	C₂×C₁₆	—	109	4	no	—
60	C₄×C₈	—	109	1	no	—
61	C₄×C₈	—	109	1	no	—
62	C₄×C₈	—	109	1	no	—
63	C₄×C₈	#89137	1	1	yes	characteristic
64	C₂²×C₄	—	109	1	no	—
65	C₂²×C₄	—	109	2	no	—
66	C₂×C₈	—	109	1	no	—
67	C₂×C₈	—	109	1	no	—
68	C₂×C₈	—	109	4	no	—
69	C₂×C₈	—	109	4	no	—
70	C₂×C₈	C₂×#89137	1	1	yes	characteristic
71	C₂×C₈	C₂×#89137	1	1	yes	characteristic
72	C₄²	—	109	2	no	—
73	C₄²	—	109	1	no	—
74	C₄²	C₂×#89137	1	1	yes	characteristic
75	C₁₆	—	109	8	no	—
76	C₂³	—	109	1	no	—
77	C₂×C₄	—	109	1	no	—
78	C₂×C₄	—	109	2	no	—
79	C₂×C₄	—	109	2	no	—
80	C₂×C₄	—	109	2	no	—
81	C₂×C₄	—	218	4	no	—
82	C₂×C₄	D₁₀₉⋊C₈	1	2	yes	—
83	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
84	C₈	—	109	2	no	—
85	C₈	—	109	2	no	—
86	C₈	C₄×#89137	1	2	yes	—
87	C₈	D₁₀₉⋊C₈	1	2	yes	—
88	C₂²	—	109	2	no	—
89	C₂²	—	218	2	no	—
90	C₂²	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
91	C₄	—	109	2	no	—
92	C₄	—	218	2	no	—
93	C₄	#16812774	1	4	yes	direct product factor
94	C₄	C₂×C₄×#89137	1	1	yes	characteristic
95	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
96	C₂	—	218	2	no	—
97	C₂	C₄×D₁₀₉⋊C₈	1	1	yes	characteristic
98	C₂	C₂×#16812774	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812774

C₂

yes

maximal

#20295915

C₂

yes

maximal, complemented

C₄×#13182032

C₂

yes

maximal, complemented

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#16812774

C₄

yes

direct product factor

C₂×#13182032

C₂²

yes

—

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₄²×D₁₀₉

C₄

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₄

yes

characteristic

#13182032

C₂×C₄

yes

complemented

C₂×C₄×D₁₀₉

C₈

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₈×D₁₀₉

C₈

yes

—

C₈×D₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₈

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₄²

yes

—

C₄×D₁₀₉

C₂×C₈

yes

—

C₈×C₁₀₉

C₄²

yes

—

C₈×C₁₀₉

C₂×C₈

yes

—

#11724784

C₂×C₈

yes

—

#11724784

C₂²×C₄

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄×C₈

yes

—

C₄×C₁₀₉

M₃₂

yes

complemented

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₄×C₈

yes

—

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

—

D₁₀₉

—

C₄×M₃₂

—

109

maximal, Sylow

C₁₀₉

C₄×M₃₂

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂×C₄×C₈

—

109

—

#346

—

109

—

C₄×C₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

M₃₂

—

109

—

C₂×C₁₆

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₁₆

—

109

—

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

D₁₀₉⋊C₈

yes

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

C₄×#89137

yes

—

C₈

D₁₀₉⋊C₈

yes

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

C₂×D₁₀₉⋊C₈

yes

characteristic

C₄

—

109

—

C₄

—

218

—

C₄

#16812774

yes

direct product factor

C₄

C₂×C₄×#89137

yes

characteristic

C₄

C₂×D₁₀₉⋊C₈

yes

characteristic

C₂

—

218

—

C₂

C₄×D₁₀₉⋊C₈

yes

characteristic

C₂

C₂×#16812774

yes

—

	subgroup	G/H
center	C₄²	C₂×#89137
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²×C₈
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₂×#11724784	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	C₄×M₃₂
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₄²

C₂×#89137

commutator

C₂×C₁₀₉

C₂×C₄×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂²×C₈

prime radical

C₂²×D₁₀₉

C₂×C₈

C₂³

C₄×C₈

C₁₀₉

C₄×M₃₂

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
2. #20295915	C₂	semidirect
3. C₄×#13182032	C₂	semidirect
5. #16812774	C₄	direct (N × H)
14. #13182032	C₂×C₄	semidirect
41. C₄×C₁₀₉	M₃₂	semidirect
50. C₁₀₉	C₄×M₃₂	semidirect
93. C₄	#16812774	direct (N × H)

H ≅ G/N

type

2. #20295915

C₂

semidirect

3. C₄×#13182032

C₂

semidirect

5. #16812774

C₄

direct (N × H)

14. #13182032

C₂×C₄

semidirect

41. C₄×C₁₀₉

M₃₂

semidirect

50. C₁₀₉

C₄×M₃₂

semidirect

93. C₄

#16812774

direct (N × H)

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

subgroup	index
C₂×#16812774	2
#20295915	2
C₄×#13182032	2
C₄×C₈×D₁₀₉	2
C₄×M₃₂	109

subgroup

index

C₂×#16812774

#20295915

C₄×#13182032

C₄×C₈×D₁₀₉

C₄×M₃₂

109

Subgroups — C₄×#16812774 #28379237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#16812774 #28379237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16812774 #28379237