Group #28378233 — subgroups

Subgroups — #28378233

Order 13952.

224 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 108 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	C₄×#13182038	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11724786	C₂	1	1	yes	maximal, characteristic
3	#11724786⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₈×#11724786	C₂	1	2	yes	maximal, complemented
5	#20295076	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182038	C₄	1	2	yes	—
7	C₂²×#11724786	C₄	1	2	yes	—
8	#16812754	C₄	1	2	yes	—
9	C₄×#11724786	C₄	1	2	yes	—
10	C₄×#11724786	C₂²	1	1	yes	characteristic
11	C₄×#11724786	C₂²	1	1	yes	characteristic
12	C₄×#11724786	C₂²	1	2	yes	—
13	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	#16812628	C₄	1	2	yes	—
15	C₂²⋊#11724784	C₄	1	1	yes	characteristic
16	C₈×Dic₁₀₉	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
18	#13182038	C₈	1	8	yes	complemented
19	C₂×#11724786	—	2	4	no	—
20	C₂×#11724786	C₂×C₄	1	2	yes	—
21	C₂×#11724786	C₂×C₄	1	4	yes	—
22	C₂×#11724786	C₂×C₄	1	2	yes	—
23	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×#11724784	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂³×C₁₀₉	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	—	2	2	no	—
37	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
39	C₂×Dic₁₀₉	C₂×C₈	1	4	yes	—
40	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	#11724786	—	2	2	no	—
43	#11724786	C₂×C₈	1	8	yes	complemented
44	#11724786	M₁₆	1	4	yes	complemented
45	C₈×C₁₀₉	—	2	2	no	—
46	#11724784	—	2	2	no	—
47	C₂²×C₁₀₉	—	2	1	no	—
48	C₂²×C₁₀₉	—	2	1	no	—
49	C₂²×C₁₀₉	C₄×C₈	1	2	yes	—
50	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
51	C₄×C₁₀₉	—	2	1	no	—
52	C₄×C₁₀₉	#131	1	2	yes	—
53	Dic₁₀₉	—	2	1	no	—
54	Dic₁₀₉	C₄×C₈	1	2	yes	—
55	Dic₁₀₉	C₂²×C₈	1	2	yes	—
56	Dic₁₀₉	C₂×M₁₆	1	2	yes	—
57	C₂×C₁₀₉	—	2	2	no	—
58	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
59	C₂×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
60	C₂×C₁₀₉	#405	1	1	yes	characteristic
61	C₈×M₁₆	—	109	1	no	maximal, Sylow
62	C₁₀₉	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
63	#431	—	109	1	no	—
64	C₄×M₁₆	—	109	1	no	—
65	C₂×C₄×C₈	—	109	1	no	—
66	#322	—	109	2	no	—
67	C₈²	—	109	2	no	—
68	C₂×M₁₆	—	109	2	no	—
69	C₂²×C₈	—	109	2	no	—
70	C₂×C₄²	—	109	1	no	—
71	C₄⋊C₈	—	109	2	no	—
72	C₂²⋊C₈	—	109	1	no	—
73	C₂²⋊C₈	#89137	1	1	yes	characteristic
74	#97	—	109	2	no	—
75	C₄×C₈	—	109	1	no	—
76	C₄×C₈	—	109	2	no	—
77	C₄×C₈	—	109	1	no	—
78	C₄×C₈	—	109	2	no	—
79	C₄×C₈	—	109	2	no	—
80	C₂²×C₄	—	109	1	no	—
81	C₂²×C₄	—	109	1	no	—
82	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
83	M₁₆	—	109	8	no	—
84	C₂×C₈	—	109	2	no	—
85	C₂×C₈	—	109	4	no	—
86	C₂×C₈	—	109	2	no	—
87	C₂×C₈	—	109	2	no	—
88	C₂×C₈	—	218	4	no	—
89	C₂×C₈	C₂×#89137	1	2	yes	—
90	C₄²	—	109	1	no	—
91	C₄²	—	109	1	no	—
92	C₄²	—	109	1	no	—
93	C₄²	—	109	1	no	—
94	C₂³	C₄×#89137	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	109	4	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	109	1	no	—
100	C₂×C₄	—	109	1	no	—
101	C₂×C₄	—	218	2	no	—
102	C₂×C₄	C₄×#89137	1	1	yes	characteristic
103	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	—	109	4	no	—
106	C₈	—	109	8	no	—
107	C₈	—	218	2	no	—
108	C₈	—	218	2	no	—
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	C₈×#89137	1	2	yes	—
112	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	109	2	no	—
115	C₄	—	109	2	no	—
116	C₄	—	109	2	no	—
117	C₄	—	218	1	no	—
118	C₄	#16812786	1	2	yes	—
119	C₂	—	2	2	no	—
120	C₂	C₂×C₈×#89137	1	1	yes	characteristic
121	C₂	M₁₆×#89137	1	1	yes	characteristic
122	C₂	#20295990	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13182038

C₂

yes

maximal, characteristic

C₂×C₄×#11724786

C₂

yes

maximal, characteristic

#11724786⋊C₈

C₂

yes

maximal, complemented

C₈×#11724786

C₂

yes

maximal, complemented

#20295076

C₂

yes

maximal, characteristic

C₂×#13182038

C₄

yes

—

C₂²×#11724786

C₄

yes

—

#16812754

C₄

yes

—

C₄×#11724786

C₄

yes

—

C₄×#11724786

C₂²

yes

characteristic

C₄×#11724786

C₂²

yes

characteristic

C₄×#11724786

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812628

C₄

yes

—

C₂²⋊#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

—

C₂²⋊C₈×C₁₀₉

C₄

yes

characteristic

#13182038

C₈

yes

complemented

C₂×#11724786

—

C₂×#11724786

C₂×C₄

yes

—

C₂×#11724786

C₂×C₄

yes

—

C₂×#11724786

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄²

yes

characteristic

C₂×Dic₁₀₉

C₄²

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

#11724786

—

#11724786

C₂×C₈

yes

complemented

#11724786

M₁₆

yes

complemented

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×C₈

yes

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#131

yes

—

Dic₁₀₉

—

Dic₁₀₉

C₄×C₈

yes

—

Dic₁₀₉

C₂²×C₈

yes

—

Dic₁₀₉

C₂×M₁₆

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₉

C₄×M₁₆

yes

characteristic

C₂×C₁₀₉

#405

yes

characteristic

C₈×M₁₆

—

109

maximal, Sylow

C₁₀₉

C₈×M₁₆

yes

characteristic, complemented, Sylow

#431

—

109

—

C₄×M₁₆

—

109

—

C₂×C₄×C₈

—

109

—

#322

—

109

—

C₈²

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

#89137

yes

characteristic

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

M₁₆

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

C₄×#89137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

218

—

102

C₂×C₄

C₄×#89137

yes

characteristic

103

C₂×C₄

C₂²×#89137

yes

characteristic

104

C₈

—

105

C₈

—

109

—

106

C₈

—

109

—

107

C₈

—

218

—

108

C₈

—

218

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₈×#89137

yes

—

112

C₂²

C₂×C₄×#89137

yes

characteristic

113

C₄

—

114

C₄

—

109

—

115

C₄

—

109

—

116

C₄

—

109

—

117

C₄

—

218

—

118

C₄

#16812786

yes

—

119

C₂

—

120

C₂

C₂×C₈×#89137

yes

characteristic

121

C₂

M₁₆×#89137

yes

characteristic

122

C₂

#20295990

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Fitting	C₂²⋊C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄²
prime radical	C₂³×C₁₀₉	C₄²
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂²⋊C₈	#89137
hypocenter	C₁₀₉	C₈×M₁₆
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator

C₂×C₁₀₉

C₂×C₄×C₈

Fitting

C₂²⋊C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄²

prime radical

C₂³×C₁₀₉

C₄²

cosocle

C₄×Dic₁₀₉

C₂³

hypercenter

C₂²⋊C₈

#89137

hypocenter

C₁₀₉

C₈×M₁₆

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
3. #11724786⋊C₈	C₂	semidirect
4. C₈×#11724786	C₂	semidirect
18. #13182038	C₈	semidirect
43. #11724786	C₂×C₈	semidirect
44. #11724786	M₁₆	semidirect
62. C₁₀₉	C₈×M₁₆	semidirect

H ≅ G/N

type

3. #11724786⋊C₈

C₂

semidirect

4. C₈×#11724786

C₂

semidirect

18. #13182038

C₈

semidirect

43. #11724786

C₂×C₈

semidirect

44. #11724786

M₁₆

semidirect

62. C₁₀₉

C₈×M₁₆

semidirect

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

subgroup	index
C₄×#13182038	2
C₂×C₄×#11724786	2
#11724786⋊C₈	2
C₈×#11724786	2
#20295076	2
C₈×M₁₆	109

subgroup

index

C₈×M₁₆

109

Subgroups — #28378233

Conjugacy classes of subgroups

Complements

Maximal subgroups