Group #28378753 — subgroups

Subgroups — #28378753

Order 13952.

200 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 78 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	#20295989	C₂	1	2	yes	maximal
2	#20295895	C₂	1	1	yes	maximal, characteristic
3	#20295894	C₂	1	1	yes	maximal, characteristic
4	#20295893	C₂	1	2	yes	maximal
5	#20294980	C₂	1	1	yes	maximal, characteristic
6	D₁₀₉⋊M₁₆	C₂²	1	2	yes	—
7	#16812778	C₂²	1	2	yes	—
8	#16812753	C₄	1	4	yes	complemented
9	C₈×#89137	C₄	1	4	yes	complemented
10	C₂×#13182012	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
12	#16812656	C₂²	1	1	yes	characteristic
13	#16812610	C₄	1	1	yes	characteristic, complemented
14	#16812609	C₄	1	1	yes	characteristic, complemented
15	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic, complemented
16	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
17	#13182039	—	2	2	no	—
18	#13182038	—	2	2	no	—
19	D₁₀₉⋊C₈	C₂×C₄	1	4	yes	—
20	C₄⋊#89137	—	2	2	no	—
21	C₂²⋊#89137	—	2	2	no	—
22	C₄×#89137	C₂×C₄	1	4	yes	—
23	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
24	#13182012	—	2	2	no	—
25	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
26	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
27	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
28	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
29	#13182008	—	2	1	no	—
30	#13182006	—	2	1	no	—
31	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₁₀₉	#41	1	1	yes	characteristic
36	C₂×#89137	—	2	4	no	—
37	C₂×C₄×C₁₀₉	—	2	1	no	—
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₁₀₉	—	2	1	no	—
40	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
41	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
42	C₄×D₁₀₉	#41	1	2	yes	—
43	#11724786	—	2	4	no	—
44	C₈×C₁₀₉	—	2	1	no	—
45	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
46	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
47	#11724784	—	2	1	no	—
48	#11724784	C₄²	1	2	yes	—
49	C₂²×C₁₀₉	#117	1	1	yes	characteristic
50	C₂×D₁₀₉	—	4	1	no	—
51	C₂×D₁₀₉	#117	1	2	yes	—
52	#89137	—	4	4	no	—
53	C₄×C₁₀₉	—	4	1	no	—
54	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₁₀₉	#117	1	1	yes	characteristic
56	Dic₁₀₉	—	4	1	no	—
57	Dic₁₀₉	#117	1	2	yes	—
58	C₂×C₁₀₉	—	2	1	no	—
59	C₂×C₁₀₉	#376	1	1	yes	characteristic
60	D₁₀₉	—	2	2	no	—
61	#1747	—	109	1	no	maximal, Sylow
62	C₁₀₉	#1747	1	1	yes	characteristic, complemented, Sylow
63	#405	—	109	2	no	—
64	#405	—	109	1	no	—
65	#338	—	109	1	no	—
66	H(3,ℤ/4)	—	109	2	no	—
67	H(3,ℤ/4)	—	109	1	no	—
68	C₂×M₁₆	—	109	2	no	—
69	C₂×M₁₆	—	109	1	no	—
70	C₂²×C₈	—	109	1	no	—
71	#117	—	109	2	no	—
72	#117	—	109	1	no	—
73	#97	—	109	4	no	—
74	#97	—	109	1	no	—
75	#97	—	109	1	no	—
76	C₄×C₈	—	109	4	no	—
77	C₄×C₈	—	109	1	no	—
78	C₄×C₈	#89137	1	1	yes	characteristic, complemented
79	C₂²×C₄	—	109	1	no	—
80	M₁₆	—	218	2	no	—
81	M₁₆	—	218	2	no	—
82	M₁₆	—	218	2	no	—
83	C₂×C₈	—	109	1	no	—
84	C₂×C₈	—	109	2	no	—
85	C₂×C₈	—	109	2	no	—
86	C₂×C₈	—	109	4	no	—
87	C₂×C₈	—	109	1	no	—
88	C₂×C₈	C₂×#89137	1	1	yes	characteristic
89	C₂×C₈	C₂×#89137	1	1	yes	characteristic
90	C₄⋊C₄	—	218	1	no	—
91	C₄⋊C₄	—	218	2	no	—
92	C₂²⋊C₄	—	218	1	no	—
93	C₂²⋊C₄	—	218	2	no	—
94	C₄²	—	109	4	no	—
95	C₄²	—	109	1	no	—
96	C₄²	C₂×#89137	1	1	yes	characteristic
97	C₂³	—	109	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	109	1	no	—
100	C₂×C₄	—	109	2	no	—
101	C₂×C₄	—	109	2	no	—
102	C₂×C₄	—	218	4	no	—
103	C₂×C₄	—	218	1	no	—
104	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	109	2	no	—
107	C₈	—	218	4	no	—
108	C₈	—	218	1	no	—
109	C₈	C₄×#89137	1	1	yes	characteristic
110	C₈	C₄×#89137	1	1	yes	characteristic
111	C₂²	—	109	2	no	—
112	C₂²	—	436	1	no	—
113	C₂²	#16812778	1	1	yes	characteristic
114	C₄	—	4	1	no	—
115	C₄	—	109	2	no	—
116	C₄	—	436	1	no	—
117	C₄	—	436	4	no	—
118	C₄	C₂×C₄×#89137	1	1	yes	characteristic
119	C₄	#16812778	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	218	2	no	—
122	C₂	#20295962	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295989

C₂

yes

maximal

#20295895

C₂

yes

maximal, characteristic

#20295894

C₂

yes

maximal, characteristic

#20295893

C₂

yes

maximal

#20294980

C₂

yes

maximal, characteristic

D₁₀₉⋊M₁₆

C₂²

yes

—

#16812778

C₂²

yes

—

#16812753

C₄

yes

complemented

C₈×#89137

C₄

yes

complemented

C₂×#13182012

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812656

C₂²

yes

characteristic

#16812610

C₄

yes

characteristic, complemented

#16812609

C₄

yes

characteristic, complemented

C₈×Dic₁₀₉

C₄

yes

characteristic, complemented

C₄×C₈×C₁₀₉

C₄

yes

characteristic, complemented

#13182039

—

#13182038

—

D₁₀₉⋊C₈

C₂×C₄

yes

—

C₄⋊#89137

—

C₂²⋊#89137

—

C₄×#89137

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

#13182012

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

—

C₈×D₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182006

—

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

C₂²×D₁₀₉

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

#41

yes

characteristic

C₄×D₁₀₉

C₂²×C₄

yes

—

C₄×D₁₀₉

#41

yes

—

#11724786

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄²

yes

characteristic

C₈×C₁₀₉

C₄²

yes

characteristic

#11724784

—

#11724784

C₄²

yes

—

C₂²×C₁₀₉

#117

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

#117

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

#117

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

#117

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#376

yes

characteristic

D₁₀₉

—

#1747

—

109

maximal, Sylow

C₁₀₉

#1747

yes

characteristic, complemented, Sylow

#405

—

109

—

#405

—

109

—

#338

—

109

—

H(3,ℤ/4)

—

109

—

H(3,ℤ/4)

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

#117

—

109

—

#97

—

109

—

#97

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic, complemented

C₂²×C₄

—

109

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

218

—

103

C₂×C₄

—

218

—

104

C₂×C₄

C₂²×#89137

yes

characteristic

105

C₈

—

106

C₈

—

109

—

107

C₈

—

218

—

108

C₈

—

218

—

109

C₈

C₄×#89137

yes

characteristic

110

C₈

C₄×#89137

yes

characteristic

111

C₂²

—

109

—

112

C₂²

—

436

—

113

C₂²

#16812778

yes

characteristic

114

C₄

—

115

C₄

—

109

—

116

C₄

—

436

—

117

C₄

—

436

—

118

C₄

C₂×C₄×#89137

yes

characteristic

119

C₄

#16812778

yes

characteristic

120

C₂

—

121

C₂

—

218

—

122

C₂

#20295962

yes

characteristic

	subgroup	G/H
center	C₈	C₄×#89137
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂×C₁₀₉	#376
prime radical	C₂²×D₁₀₉	#41
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#1747
LCS	G ▷ C₄×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₈ ◁ C₄×C₈

subgroup

G/H

center

C₈

C₄×#89137

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

C₂×C₁₀₉

C₂²×D₁₀₉

C₂×C₄×D₁₀₉

C₂³

C₄×C₈

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₈ ◁ C₄×C₈

N	H ≅ G/N	type
8. #16812753	C₄	semidirect
9. C₈×#89137	C₄	semidirect
13. #16812610	C₄	semidirect
14. #16812609	C₄	semidirect
15. C₈×Dic₁₀₉	C₄	semidirect
16. C₄×C₈×C₁₀₉	C₄	semidirect
62. C₁₀₉	#1747	semidirect
78. C₄×C₈	#89137	semidirect

H ≅ G/N

type

8. #16812753

C₄

semidirect

9. C₈×#89137

C₄

semidirect

13. #16812610

C₄

semidirect

14. #16812609

C₄

semidirect

15. C₈×Dic₁₀₉

C₄

semidirect

16. C₄×C₈×C₁₀₉

C₄

semidirect

62. C₁₀₉

#1747

semidirect

78. C₄×C₈

#89137

semidirect

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

subgroup	index
#20295989	2
#20295895	2
#20295894	2
#20295893	2
#20294980	2
#1747	109

subgroup

index

109

Subgroups — #28378753

Conjugacy classes of subgroups

Complements

Maximal subgroups