Group #28378751 — subgroups

Subgroups — #28378751

Order 13952.

240 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 98 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×#16812754	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11724786	C₂	1	1	yes	maximal, characteristic
3	#20295892	C₂	1	4	yes	maximal
4	#20294871	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724786	C₂²	1	2	yes	—
6	C₂²×#11724786	C₂²	1	2	yes	—
7	#16812754	C₄	1	4	yes	complemented
8	C₄×#11724786	C₄	1	4	yes	complemented
9	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	#16812606	C₄	1	2	yes	—
12	#16812606	C₄	1	1	yes	characteristic
13	#95×C₁₀₉	C₄	1	1	yes	characteristic
14	C₂×#11724786	—	2	4	no	—
15	C₂×#11724786	—	2	4	no	—
16	C₂×#11724786	C₂×C₄	1	4	yes	—
17	C₂×#11724786	C₂×C₄	1	4	yes	—
18	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
19	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
21	C₄×Dic₁₀₉	—	2	4	no	—
22	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
23	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	—	2	2	no	—
28	C₂×C₄×C₁₀₉	—	2	2	no	—
29	C₂×C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
32	C₂×Dic₁₀₉	—	2	1	no	—
33	C₂×Dic₁₀₉	—	2	1	no	—
34	C₂×Dic₁₀₉	—	2	2	no	—
35	C₂×Dic₁₀₉	—	2	2	no	—
36	C₂×Dic₁₀₉	—	2	2	no	—
37	C₂×Dic₁₀₉	C₄²	1	2	yes	—
38	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₁₀₉	#41	1	4	yes	—
41	#11724786	—	2	4	no	—
42	#11724786	—	2	4	no	—
43	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₁₀₉	#117	1	2	yes	—
45	C₂²×C₁₀₉	#131	1	2	yes	—
46	C₂²×C₁₀₉	#131	1	1	yes	characteristic
47	C₂²×C₁₀₉	#131	1	1	yes	characteristic
48	C₄×C₁₀₉	—	2	2	no	—
49	C₄×C₁₀₉	—	4	2	no	—
50	Dic₁₀₉	—	2	2	no	—
51	Dic₁₀₉	—	4	2	no	—
52	Dic₁₀₉	#117	1	2	yes	—
53	Dic₁₀₉	#117	1	2	yes	—
54	C₂×C₁₀₉	#376	1	1	yes	characteristic
55	C₂×C₁₀₉	#405	1	1	yes	characteristic
56	C₂×C₁₀₉	#405	1	1	yes	characteristic
57	C₂×C₁₀₉	#433	1	2	yes	—
58	C₂×C₁₀₉	#433	1	2	yes	—
59	#1746	—	109	1	no	maximal, Sylow
60	C₁₀₉	#1746	1	1	yes	characteristic, complemented, Sylow
61	C₂×#97	—	109	1	no	—
62	C₂×C₄×C₈	—	109	1	no	—
63	#376	—	109	1	no	—
64	#336	—	109	4	no	—
65	C₂²×C₈	—	109	2	no	—
66	C₂²×C₈	—	109	2	no	—
67	C₂×C₄²	—	109	1	no	—
68	C₂×C₄²	—	109	2	no	—
69	#97	—	109	4	no	—
70	C₄×C₈	—	109	4	no	—
71	#95	—	109	2	no	—
72	#95	—	109	1	no	—
73	#95	#89137	1	1	yes	characteristic
74	C₂²×C₄	—	109	1	no	—
75	C₂²×C₄	—	109	1	no	—
76	C₂²×C₄	—	109	2	no	—
77	C₂²×C₄	C₂×#89137	1	2	yes	—
78	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
79	C₂×C₈	—	109	4	no	—
80	C₂×C₈	—	109	4	no	—
81	C₂×C₈	—	218	4	no	—
82	C₂×C₈	—	218	4	no	—
83	C₄²	—	109	2	no	—
84	C₄²	—	109	2	no	—
85	C₄²	—	218	4	no	—
86	C₂³	C₂²×#89137	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	109	1	no	—
92	C₂×C₄	—	109	1	no	—
93	C₂×C₄	—	109	4	no	—
94	C₂×C₄	—	109	2	no	—
95	C₂×C₄	—	218	2	no	—
96	C₂×C₄	—	218	2	no	—
97	C₂×C₄	—	218	2	no	—
98	C₂×C₄	—	218	1	no	—
99	C₂×C₄	—	218	1	no	—
100	C₂×C₄	C₄×#89137	1	2	yes	—
101	C₈	—	218	4	no	—
102	C₈	—	218	4	no	—
103	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
104	C₂²	#16812778	1	2	yes	—
105	C₂²	#16812786	1	2	yes	—
106	C₂²	#16812786	1	1	yes	characteristic
107	C₂²	#16812787	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	4	2	no	—
110	C₄	—	109	2	no	—
111	C₄	—	109	2	no	—
112	C₄	—	436	2	no	—
113	C₄	—	218	2	no	—
114	C₂	#20295962	1	1	yes	characteristic
115	C₂	#20295990	1	1	yes	characteristic
116	C₂	#20295991	1	1	yes	characteristic
117	C₂	#20296050	1	2	yes	—
118	C₂	#20296051	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812754

C₂

yes

maximal, characteristic

C₂×C₄×#11724786

C₂

yes

maximal, characteristic

#20295892

C₂

yes

maximal

#20294871

C₂

yes

maximal, characteristic

C₂²×#11724786

C₂²

yes

—

C₂²×#11724786

C₂²

yes

—

#16812754

C₄

yes

complemented

C₄×#11724786

C₄

yes

complemented

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812606

C₄

yes

—

#16812606

C₄

yes

characteristic

#95×C₁₀₉

C₄

yes

—

—

C₂×C₄

yes

—

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

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

#41

yes

—

#11724786

—

#11724786

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂²×C₁₀₉

#117

yes

—

C₂²×C₁₀₉

#131

yes

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#117

yes

—

Dic₁₀₉

#117

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

C₂×C₁₀₉

#433

yes

—

#1746

—

109

maximal, Sylow

C₁₀₉

#1746

yes

characteristic, complemented, Sylow

C₂×#97

—

109

—

C₂×C₄×C₈

—

109

—

#376

—

109

—

#336

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

#95

—

109

—

#95

—

109

—

#95

#89137

yes

characteristic

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

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₂³

C₂²×#89137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

100

C₂×C₄

C₄×#89137

yes

—

101

C₈

—

218

—

102

C₈

—

218

—

103

C₂²

C₂×C₄×#89137

yes

characteristic

104

C₂²

#16812778

yes

—

105

C₂²

#16812786

yes

—

106

C₂²

#16812786

yes

characteristic

107

C₂²

#16812787

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

109

—

111

C₄

—

109

—

112

C₄

—

436

—

113

C₄

—

218

—

114

C₂

#20295962

yes

characteristic

115

C₂

#20295990

yes

characteristic

116

C₂

#20295991

yes

characteristic

117

C₂

#20296050

yes

—

118

C₂

#20296051

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#89137
commutator	C₂²×C₁₀₉	C₂×C₄²
Fitting	#95×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×Dic₁₀₉	C₂³
hypercenter	#95	#89137
hypocenter	C₁₀₉	#1746
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×#89137

commutator

C₂²×C₁₀₉

C₂×C₄²

Fitting

#95×C₁₀₉

C₄

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×Dic₁₀₉

C₂³

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
7. #16812754	C₄	semidirect
8. C₄×#11724786	C₄	semidirect
60. C₁₀₉	#1746	semidirect

H ≅ G/N

type

7. #16812754

C₄

semidirect

8. C₄×#11724786

C₄

semidirect

60. C₁₀₉

#1746

semidirect

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

subgroup	index
C₂×#16812754	2
C₂×C₄×#11724786	2
#20295892	2
#20294871	2
#1746	109

subgroup

index

109

Subgroups — #28378751

Conjugacy classes of subgroups

Complements

Maximal subgroups