Group #28378244 — subgroups

Subgroups — #28378244

Order 13952.

208 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 92 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×#13182039	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#11724786⋊C₈	C₂	1	2	yes	maximal, complemented
4	#20295867	C₂	1	2	yes	maximal, complemented
5	#20295078	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	C₄	1	2	yes	—
7	C₂×D₁₀₉⋊C₈	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	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊#11724784	C₄	1	1	yes	characteristic
15	#16812611	C₄	1	2	yes	—
16	C₈×Dic₁₀₉	C₂²	1	2	yes	—
17	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
18	#13182039	—	2	4	no	—
19	D₁₀₉⋊C₈	—	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	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×#11724784	C₂×C₄	1	2	yes	—
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	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₁₀₉	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₄×D₁₀₉	—	2	1	no	—
40	C₄×D₁₀₉	—	2	1	no	—
41	C₄×D₁₀₉	—	2	2	no	—
42	C₄×D₁₀₉	M₁₆	1	4	yes	—
43	#11724786	—	2	2	no	—
44	#11724786	—	2	4	no	—
45	#11724786	M₁₆	1	4	yes	complemented
46	C₈×C₁₀₉	—	2	2	no	—
47	#11724784	—	2	2	no	—
48	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
49	C₂×D₁₀₉	—	2	1	no	—
50	C₂×D₁₀₉	—	2	1	no	—
51	C₂×D₁₀₉	#97	1	2	yes	—
52	C₄×C₁₀₉	—	2	1	no	—
53	C₄×C₁₀₉	#97	1	2	yes	—
54	C₄×C₁₀₉	#131	1	1	yes	characteristic
55	C₄×C₁₀₉	#131	1	1	yes	characteristic
56	Dic₁₀₉	—	2	1	no	—
57	Dic₁₀₉	C₂×M₁₆	1	2	yes	—
58	Dic₁₀₉	C₂×M₁₆	1	2	yes	—
59	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
60	C₂×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
61	C₂×C₁₀₉	#405	1	1	yes	characteristic
62	D₁₀₉	—	2	2	no	—
63	#1451	—	109	1	no	maximal, Sylow
64	C₁₀₉	#1451	1	1	yes	characteristic, complemented, Sylow
65	#431	—	109	1	no	—
66	C₄×M₁₆	—	109	1	no	—
67	C₂×C₄×C₈	—	109	1	no	—
68	#322	—	109	2	no	—
69	#322	—	109	2	no	—
70	C₂×M₁₆	—	109	2	no	—
71	C₂²×C₈	—	109	2	no	—
72	C₂×C₄²	—	109	1	no	—
73	C₄⋊C₈	—	109	1	no	—
74	C₄⋊C₈	#89137	1	1	yes	characteristic
75	C₂²⋊C₈	—	109	2	no	—
76	#97	—	109	2	no	—
77	C₄×C₈	—	109	1	no	—
78	C₄×C₈	—	109	1	no	—
79	C₄×C₈	—	109	2	no	—
80	C₄×C₈	—	109	2	no	—
81	C₄×C₈	—	109	2	no	—
82	C₂²×C₄	—	109	1	no	—
83	C₂²×C₄	—	109	1	no	—
84	C₂²×C₄	—	109	1	no	—
85	M₁₆	—	218	4	no	—
86	C₂×C₈	—	109	2	no	—
87	C₂×C₈	—	109	2	no	—
88	C₂×C₈	—	109	4	no	—
89	C₂×C₈	—	109	2	no	—
90	C₂×C₈	—	218	4	no	—
91	C₂×C₈	C₂×#89137	1	2	yes	—
92	C₄²	—	109	1	no	—
93	C₄²	—	109	1	no	—
94	C₄²	—	109	1	no	—
95	C₄²	C₂×#89137	1	1	yes	characteristic
96	C₂³	—	109	1	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	109	4	no	—
100	C₂×C₄	—	109	1	no	—
101	C₂×C₄	—	218	2	no	—
102	C₂×C₄	—	218	1	no	—
103	C₂×C₄	—	218	1	no	—
104	C₂×C₄	C₄×#89137	1	1	yes	characteristic
105	C₂×C₄	C₄×#89137	1	1	yes	characteristic
106	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	109	4	no	—
109	C₈	—	218	2	no	—
110	C₈	—	218	4	no	—
111	C₈	—	218	2	no	—
112	C₂²	—	109	2	no	—
113	C₂²	—	218	1	no	—
114	C₂²	—	218	1	no	—
115	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	109	2	no	—
118	C₄	—	109	2	no	—
119	C₄	—	218	1	no	—
120	C₄	#16812753	1	2	yes	—
121	C₄	#16812786	1	1	yes	characteristic
122	C₄	#16812787	1	1	yes	characteristic
123	C₂	—	218	2	no	—
124	C₂	C₂×#16812753	1	1	yes	characteristic
125	C₂	M₁₆×#89137	1	1	yes	characteristic
126	C₂	#20295991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13182039

C₂

yes

maximal, characteristic

C₄×D₁₀₉⋊C₈

C₂

yes

maximal, characteristic

#11724786⋊C₈

C₂

yes

maximal, complemented

#20295867

C₂

yes

maximal, complemented

#20295078

C₂

yes

maximal, characteristic

C₂×#13182039

C₄

yes

—

C₂×D₁₀₉⋊C₈

C₄

yes

—

#16812754

C₄

yes

—

C₄×#11724786

C₄

yes

—

C₄×#11724786

C₂²

yes

characteristic

C₄×#11724786

C₂²

yes

characteristic

C₄×#11724786

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

characteristic

#16812611

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₄⋊C₈×C₁₀₉

C₄

yes

characteristic

#13182039

—

D₁₀₉⋊C₈

—

C₂×#11724786

C₂×C₄

yes

—

C₂×#11724786

C₂×C₄

yes

—

C₂×#11724786

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄²

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

M₁₆

yes

—

#11724786

—

#11724786

—

#11724786

M₁₆

yes

complemented

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#97

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#97

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

C₂×M₁₆

yes

—

Dic₁₀₉

C₂×M₁₆

yes

—

C₂×C₁₀₉

C₂×#97

yes

characteristic

C₂×C₁₀₉

C₄×M₁₆

yes

characteristic

C₂×C₁₀₉

#405

yes

characteristic

D₁₀₉

—

#1451

—

109

maximal, Sylow

C₁₀₉

#1451

yes

characteristic, complemented, Sylow

#431

—

109

—

C₄×M₁₆

—

109

—

C₂×C₄×C₈

—

109

—

#322

—

109

—

#322

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

#89137

yes

characteristic

C₂²⋊C₈

—

109

—

#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

—

109

—

M₁₆

—

218

—

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

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

218

—

102

C₂×C₄

—

218

—

103

C₂×C₄

—

218

—

104

C₂×C₄

C₄×#89137

yes

characteristic

105

C₂×C₄

C₄×#89137

yes

characteristic

106

C₂×C₄

C₂²×#89137

yes

characteristic

107

C₈

—

108

C₈

—

109

—

109

C₈

—

218

—

110

C₈

—

218

—

111

C₈

—

218

—

112

C₂²

—

109

—

113

C₂²

—

218

—

114

C₂²

—

218

—

115

C₂²

C₂×C₄×#89137

yes

characteristic

116

C₄

—

117

C₄

—

109

—

118

C₄

—

109

—

119

C₄

—

218

—

120

C₄

#16812753

yes

—

121

C₄

#16812786

yes

characteristic

122

C₄

#16812787

yes

characteristic

123

C₂

—

218

—

124

C₂

C₂×#16812753

yes

characteristic

125

C₂

M₁₆×#89137

yes

characteristic

126

C₂

#20295991

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂×C₄²
Fitting	C₄⋊C₈×C₁₀₉	C₄
prime radical	C₂²×D₁₀₉	C₄²
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₄⋊C₈	#89137
hypocenter	C₁₀₉	#1451
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂×C₄²

Fitting

C₄⋊C₈×C₁₀₉

C₄

prime radical

C₂²×D₁₀₉

C₄²

cosocle

C₄×Dic₁₀₉

C₂³

C₄⋊C₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
3. #11724786⋊C₈	C₂	semidirect
4. #20295867	C₂	semidirect
45. #11724786	M₁₆	semidirect
64. C₁₀₉	#1451	semidirect

H ≅ G/N

type

3. #11724786⋊C₈

C₂

semidirect

4. #20295867

C₂

semidirect

45. #11724786

M₁₆

semidirect

64. C₁₀₉

#1451

semidirect

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

subgroup	index
C₄×#13182039	2
C₄×D₁₀₉⋊C₈	2
#11724786⋊C₈	2
#20295867	2
#20295078	2
#1451	109

subgroup

index

C₄×D₁₀₉⋊C₈

109

Subgroups — #28378244

Conjugacy classes of subgroups

Complements

Maximal subgroups