Group #28378664 — subgroups

Subgroups — #28378664

Order 13952.

180 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 58 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	#20296060	C₂	1	2	yes	maximal, complemented
2	#20295889	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295874	C₂	1	2	yes	maximal, complemented
4	C₈×#11724786	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295108	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	—	2	2	no	—
7	C₄⋊#11724786	C₂²	1	2	yes	—
8	#16812756	—	2	2	no	—
9	C₄×#11724786	C₂²	1	1	yes	characteristic
10	C₄×#11724786	C₂²	1	1	yes	characteristic
11	C₂×D₈₇₂	C₄	1	1	yes	characteristic
12	#16812661	C₂²	1	2	yes	—
13	#16812619	C₄	1	2	yes	—
14	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	#107×C₁₀₉	C₄	1	1	yes	characteristic
16	#13182039	—	4	4	no	—
17	C₂×#11724786	—	2	1	no	—
18	C₂×#11724786	—	2	2	no	—
19	C₂×#11724786	D₄	1	1	yes	characteristic
20	C₂×#11724786	D₄	1	1	yes	characteristic
21	C₂×D₄₃₆	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₁₀₉	—	2	2	no	—
23	D₈₇₂	—	2	2	no	—
24	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
25	#13182006	—	2	2	no	—
26	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂²×D₁₀₉	—	2	2	no	—
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
34	D₄₃₆	—	2	2	no	—
35	D₄₃₆	—	2	2	no	—
36	C₄×D₁₀₉	—	4	2	no	—
37	C₄×D₁₀₉	—	4	2	no	—
38	#11724786	—	2	2	no	—
39	#11724786	—	2	1	no	—
40	#11724786	—	4	2	no	—
41	#11724786	D₈	1	2	yes	complemented
42	C₈×C₁₀₉	M₁₆	1	2	yes	complemented
43	#11724784	—	2	1	no	—
44	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
45	C₂×D₁₀₉	—	4	2	no	—
46	C₂×D₁₀₉	—	4	2	no	—
47	C₂×D₁₀₉	—	4	2	no	—
48	C₄×C₁₀₉	—	4	2	no	—
49	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₁₀₉	#131	1	1	yes	characteristic
51	Dic₁₀₉	—	2	1	no	—
52	Dic₁₀₉	C₂×D₈	1	1	yes	characteristic
53	Dic₁₀₉	#135	1	1	yes	characteristic
54	C₂×C₁₀₉	#436	1	1	yes	characteristic
55	C₂×C₁₀₉	C₄×D₈	1	1	yes	characteristic
56	C₂×C₁₀₉	#443	1	1	yes	characteristic
57	D₁₀₉	—	8	2	no	—
58	#1589	—	109	1	no	maximal, Sylow
59	C₁₀₉	#1589	1	1	yes	characteristic, complemented, Sylow
60	C₄×D₈	—	109	1	no	—
61	#436	—	109	2	no	—
62	#335	—	109	1	no	—
63	D₄⋊C₈	—	109	2	no	—
64	C₈²	—	109	1	no	—
65	C₂×D₈	—	109	1	no	—
66	C₂×M₁₆	—	218	2	no	—
67	C₄×D₄	—	109	2	no	—
68	#107	#89137	1	1	yes	characteristic
69	C₄⋊C₈	—	109	2	no	—
70	#102	—	109	2	no	—
71	C₂²⋊C₈	—	218	2	no	—
72	C₄×C₈	—	109	1	no	—
73	C₄×C₈	—	109	1	no	—
74	C₄×C₈	—	109	1	no	—
75	C₂×D₄	—	109	2	no	—
76	C₂²×C₄	—	218	2	no	—
77	D₈	—	218	2	no	—
78	M₁₆	—	436	4	no	—
79	C₂×C₈	—	109	1	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	—	109	1	no	—
82	C₂×C₈	—	218	2	no	—
83	C₂×C₈	—	218	1	no	—
84	C₂×C₈	C₂×#89137	1	1	yes	characteristic
85	C₄⋊C₄	C₂×#89137	1	2	yes	—
86	C₂²⋊C₄	—	218	2	no	—
87	C₄²	—	109	1	no	—
88	C₂³	—	218	2	no	—
89	D₄	—	218	2	no	—
90	D₄	—	218	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	109	1	no	—
93	C₂×C₄	—	109	1	no	—
94	C₂×C₄	—	436	2	no	—
95	C₂×C₄	—	436	2	no	—
96	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
97	C₈	—	109	2	no	—
98	C₈	—	436	2	no	—
99	C₈	—	218	1	no	—
100	C₈	—	218	2	no	—
101	C₈	—	218	1	no	—
102	C₈	#13182039	1	2	yes	complemented
103	C₂²	—	436	2	no	—
104	C₂²	—	436	2	no	—
105	C₂²	—	436	2	no	—
106	C₂²	D₄×#89137	1	1	yes	characteristic
107	C₄	—	4	2	no	—
108	C₄	—	109	1	no	—
109	C₄	—	109	1	no	—
110	C₄	—	218	1	no	—
111	C₄	C₂×#13182039	1	1	yes	characteristic
112	C₄	#16812787	1	1	yes	characteristic
113	C₂	—	872	2	no	—
114	C₂	#20296060	1	1	yes	characteristic
115	C₂	D₈×#89137	1	1	yes	characteristic
116	C₂	#20296070	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296060

C₂

yes

maximal, complemented

#20295889

C₂

yes

maximal, characteristic, complemented

#20295874

C₂

yes

maximal, complemented

C₈×#11724786

C₂

yes

maximal, characteristic, complemented

#20295108

C₂

yes

maximal, characteristic

C₂×#13182039

—

C₄⋊#11724786

C₂²

yes

—

#16812756

—

C₄×#11724786

C₂²

yes

characteristic

C₄×#11724786

C₂²

yes

characteristic

C₂×D₈₇₂

C₄

yes

characteristic

#16812661

C₂²

yes

—

#16812619

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

C₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11724786

D₄

yes

characteristic

C₂×D₄₃₆

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

—

D₈₇₂

—

C₂×#11724784

C₂×C₄

yes

characteristic

#13182006

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

#41

yes

characteristic

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

—

—

—

D₈

yes

complemented

C₈×C₁₀₉

M₁₆

yes

complemented

#11724784

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

#131

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂×D₈

yes

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

C₄×D₈

yes

characteristic

C₂×C₁₀₉

#443

yes

characteristic

D₁₀₉

—

#1589

—

109

maximal, Sylow

C₁₀₉

#1589

yes

characteristic, complemented, Sylow

C₄×D₈

—

109

—

#436

—

109

—

#335

—

109

—

D₄⋊C₈

—

109

—

C₈²

—

109

—

C₂×D₈

—

109

—

C₂×M₁₆

—

218

—

C₄×D₄

—

109

—

#107

#89137

yes

characteristic

C₄⋊C₈

—

109

—

#102

—

109

—

C₂²⋊C₈

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

218

—

D₈

—

218

—

M₁₆

—

436

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

C₂×#89137

yes

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₂³

—

218

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

109

—

C₈

—

436

—

C₈

—

218

—

100

C₈

—

218

—

101

C₈

—

218

—

102

C₈

#13182039

yes

complemented

103

C₂²

—

436

—

104

C₂²

—

436

—

105

C₂²

—

436

—

106

C₂²

D₄×#89137

yes

characteristic

107

C₄

—

108

C₄

—

109

—

109

C₄

—

109

—

110

C₄

—

218

—

111

C₄

C₂×#13182039

yes

characteristic

112

C₄

#16812787

yes

characteristic

113

C₂

—

872

—

114

C₂

#20296060

yes

characteristic

115

C₂

D₈×#89137

yes

characteristic

116

C₂

#20296070

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	#107×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₄×D₄
prime radical	C₂×D₈₇₂	C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	#107	#89137
hypocenter	C₁₀₉	#1589
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

#107×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₄×D₄

prime radical

C₂×D₈₇₂

C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
1. #20296060	C₂	semidirect
2. #20295889	C₂	semidirect
3. #20295874	C₂	semidirect
4. C₈×#11724786	C₂	semidirect
41. #11724786	D₈	semidirect
42. C₈×C₁₀₉	M₁₆	semidirect
59. C₁₀₉	#1589	semidirect
102. C₈	#13182039	semidirect

H ≅ G/N

type

1. #20296060

C₂

semidirect

2. #20295889

C₂

semidirect

3. #20295874

C₂

semidirect

4. C₈×#11724786

C₂

semidirect

41. #11724786

D₈

semidirect

42. C₈×C₁₀₉

M₁₆

semidirect

59. C₁₀₉

#1589

semidirect

102. C₈

#13182039

semidirect

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

subgroup	index
#20296060	2
#20295889	2
#20295874	2
C₈×#11724786	2
#20295108	2
#1589	109

subgroup

index

109

Subgroups — #28378664

Conjugacy classes of subgroups

Complements

Maximal subgroups