Group #28379418 — subgroups

Subgroups — #28379418

Order 13952.

162 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 66 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, complemented
2	C₄⋊#13182032	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295921	C₂	1	2	yes	maximal, complemented
4	C₄×#13182032	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295091	C₂	1	1	yes	maximal, characteristic
6	#16812774	—	2	4	no	—
7	C₂×#13182032	C₂²	1	2	yes	—
8	C₂×#13182032	C₂²	1	1	yes	characteristic
9	C₂×#13182032	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
11	C₄×D₄₃₆	C₄	1	1	yes	characteristic
12	#16812611	C₄	1	2	yes	—
13	C₄×#11724784	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
15	#13182032	—	2	2	no	—
16	#13182032	—	2	1	no	—
17	#13182032	D₄	1	2	yes	complemented
18	C₂×D₄₃₆	C₈	1	1	yes	characteristic
19	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
20	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
21	C₂×#11724784	C₂³	1	1	yes	characteristic
22	C₈×D₁₀₉	—	2	2	no	—
23	C₈×D₁₀₉	—	2	2	no	—
24	C₄⋊Dic₁₀₉	C₈	1	1	yes	characteristic
25	#13182006	C₈	1	2	yes	—
26	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₁₀₉	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
33	D₄₃₆	—	2	2	no	—
34	C₄×D₁₀₉	—	2	2	no	—
35	C₄×D₁₀₉	—	2	2	no	—
36	C₈×C₁₀₉	—	2	2	no	—
37	#11724784	—	2	1	no	—
38	#11724784	C₂×D₄	1	1	yes	characteristic
39	#11724784	#41	1	1	yes	characteristic
40	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
41	C₂×D₁₀₉	—	2	2	no	—
42	C₂×D₁₀₉	—	2	2	no	—
43	C₂×D₁₀₉	—	2	2	no	—
44	C₄×C₁₀₉	—	2	1	no	—
45	C₄×C₁₀₉	M₃₂	1	2	yes	complemented
46	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
47	C₄×C₁₀₉	#131	1	1	yes	characteristic
48	Dic₁₀₉	—	2	2	no	—
49	C₂×C₁₀₉	C₈×D₄	1	1	yes	characteristic
50	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
51	C₂×C₁₀₉	#504	1	1	yes	characteristic
52	D₁₀₉	—	4	2	no	—
53	#2169	—	109	1	no	maximal, Sylow
54	C₁₀₉	#2169	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	—	109	2	no	—
56	C₈×D₄	—	109	1	no	—
57	C₄⋊C₁₆	—	109	1	no	—
58	C₂²⋊C₁₆	—	109	2	no	—
59	C₄×C₁₆	—	109	1	no	—
60	C₂²×C₈	—	109	2	no	—
61	C₄×D₄	—	109	1	no	—
62	M₃₂	—	218	4	no	—
63	C₂×C₁₆	—	109	1	no	—
64	C₂×C₁₆	—	109	1	no	—
65	C₂×C₁₆	—	109	2	no	—
66	C₄⋊C₈	#89137	1	1	yes	characteristic
67	C₂²⋊C₈	—	109	2	no	—
68	C₄×C₈	—	109	1	no	—
69	C₂×D₄	—	109	1	no	—
70	C₂²×C₄	—	109	2	no	—
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	—	218	2	no	—
74	C₂×C₈	—	218	2	no	—
75	C₂×C₈	C₂×#89137	1	2	yes	—
76	C₄⋊C₄	—	109	1	no	—
77	C₂²⋊C₄	—	109	2	no	—
78	C₄²	C₂×#89137	1	1	yes	characteristic
79	C₁₆	—	109	2	no	—
80	C₁₆	—	218	1	no	—
81	C₁₆	—	218	2	no	—
82	C₂³	—	109	2	no	—
83	D₄	—	218	2	no	—
84	C₂×C₄	—	109	2	no	—
85	C₂×C₄	—	218	2	no	—
86	C₂×C₄	—	218	2	no	—
87	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
88	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
89	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	109	1	no	—
92	C₈	—	109	1	no	—
93	C₈	—	218	1	no	—
94	C₂²	—	218	2	no	—
95	C₂²	—	218	2	no	—
96	C₂²	—	218	2	no	—
97	C₂²	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	218	2	no	—
100	C₄	#16812774	1	2	yes	complemented
101	C₄	D₄×#89137	1	1	yes	characteristic
102	C₄	#16812787	1	1	yes	characteristic
103	C₂	—	436	2	no	—
104	C₂	#20296054	1	1	yes	characteristic
105	C₂	C₂×#16812774	1	1	yes	characteristic
106	C₂	#20296085	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812774

C₂

yes

maximal, complemented

C₄⋊#13182032

C₂

yes

maximal, characteristic, complemented

#20295921

C₂

yes

maximal, complemented

C₄×#13182032

C₂

yes

maximal, characteristic, complemented

#20295091

C₂

yes

maximal, characteristic

#16812774

—

C₂×#13182032

C₂²

yes

—

C₂×#13182032

C₂²

yes

characteristic

C₂×#13182032

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

—

C₄×D₄₃₆

C₄

yes

characteristic

#16812611

C₄

yes

—

C₄×#11724784

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₄

yes

—

—

D₄

yes

complemented

C₂×D₄₃₆

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄⋊Dic₁₀₉

C₈

yes

characteristic

#13182006

C₈

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

C₂×D₄

yes

yes

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

M₃₂

yes

complemented

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

#131

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

C₈×D₄

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

C₂×C₁₀₉

#504

yes

characteristic

D₁₀₉

—

#2169

—

109

maximal, Sylow

C₁₀₉

#2169

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₈×D₄

—

109

—

C₄⋊C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

C₄×C₁₆

—

109

—

C₂²×C₈

—

109

—

C₄×D₄

—

109

—

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₄⋊C₈

#89137

yes

characteristic

C₂²⋊C₈

—

109

—

C₄×C₈

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₁₆

—

109

—

C₁₆

—

218

—

C₁₆

—

218

—

C₂³

—

109

—

D₄

—

218

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×D₁₀₉⋊C₈

yes

characteristic

C₄

—

C₄

—

218

—

100

C₄

#16812774

yes

complemented

101

C₄

D₄×#89137

yes

characteristic

102

C₄

#16812787

yes

characteristic

103

C₂

—

436

—

104

C₂

#20296054

yes

characteristic

105

C₂

C₂×#16812774

yes

characteristic

106

C₂

#20296085

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₄⋊C₈×C₁₀₉	C₄
prime radical	C₂×D₄₃₆	C₈
cosocle	C₂×#11724784	C₂³
hypercenter	C₄⋊C₈	#89137
hypocenter	C₁₀₉	#2169
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₄⋊C₈×C₁₀₉

C₄

prime radical

C₂×D₄₃₆

C₈

C₂³

C₄⋊C₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
1. C₂×#16812774	C₂	semidirect
2. C₄⋊#13182032	C₂	semidirect
3. #20295921	C₂	semidirect
4. C₄×#13182032	C₂	semidirect
17. #13182032	D₄	semidirect
45. C₄×C₁₀₉	M₃₂	semidirect
54. C₁₀₉	#2169	semidirect
100. C₄	#16812774	semidirect

H ≅ G/N

type

1. C₂×#16812774

C₂

semidirect

2. C₄⋊#13182032

C₂

semidirect

3. #20295921

C₂

semidirect

4. C₄×#13182032

C₂

semidirect

17. #13182032

D₄

semidirect

45. C₄×C₁₀₉

M₃₂

semidirect

54. C₁₀₉

#2169

semidirect

100. C₄

#16812774

semidirect

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

subgroup	index
C₂×#16812774	2
C₄⋊#13182032	2
#20295921	2
C₄×#13182032	2
#20295091	2
#2169	109

subgroup

index

109

Subgroups — #28379418

Conjugacy classes of subgroups

Complements

Maximal subgroups