Group #28379388 — subgroups

Subgroups — #28379388

Order 13952.

140 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 62 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×#16812770	C₂	1	1	yes	maximal, characteristic
2	D₁₀₉⋊#108	C₂	1	1	yes	maximal, characteristic
3	#20295955	C₂	1	2	yes	maximal, complemented
4	#20295954	C₂	1	2	yes	maximal, complemented
5	M₃₂×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	—	2	1	no	—
7	D₁₀₉⋊M₁₆	—	2	1	no	—
8	#16812770	—	2	1	no	—
9	#16812770	C₂²	1	1	yes	characteristic
10	#16812770	C₂²	1	2	yes	—
11	#16812770	C₂²	1	1	yes	characteristic
12	#16812769	—	2	1	no	—
13	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
14	#16812639	C₄	1	1	yes	characteristic
15	#16812638	C₄	1	2	yes	—
16	C₁₆×D₁₀₉	C₂²	1	2	yes	—
17	M₃₂×C₁₀₉	C₄	1	1	yes	characteristic
18	#13182039	—	2	1	no	—
19	#13182039	—	2	1	no	—
20	#13182039	—	2	2	no	—
21	#13182039	—	4	1	no	—
22	#13182038	—	4	1	no	—
23	D₁₀₉⋊C₈	—	4	1	no	—
24	C₂×#11724786	—	4	1	no	—
25	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
26	C₂×#11724784	Q₈	1	1	yes	characteristic
27	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₈×D₁₀₉	D₄	1	1	yes	characteristic
29	C₈×D₁₀₉	Q₈	1	1	yes	characteristic
30	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
31	#13182004	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₂²×D₁₀₉	Q₁₆	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
37	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₄×D₁₀₉	D₈	1	1	yes	characteristic
39	C₄×D₁₀₉	Q₁₆	1	1	yes	characteristic
40	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
41	#11724786	—	4	2	no	—
42	#11724786	—	4	1	no	—
43	#11724786	—	4	1	no	—
44	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
45	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	#11724784	C₄⋊C₄	1	1	yes	characteristic
47	#11724784	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₁₀₉	#107	1	1	yes	characteristic
49	C₂×D₁₀₉	—	4	1	no	—
50	C₂×D₁₀₉	#107	1	1	yes	characteristic
51	C₂×D₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₄×C₁₀₉	#107	1	1	yes	characteristic
53	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	Dic₁₀₉	#107	1	1	yes	characteristic
55	Dic₁₀₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₁₀₉	—	2	1	no	—
57	C₂×C₁₀₉	C₂×#107	1	1	yes	characteristic
58	D₁₀₉	—	2	1	no	—
59	D₁₀₉	—	2	1	no	—
60	#2161	—	109	1	no	maximal, Sylow
61	C₁₀₉	#2161	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₃₂	—	109	1	no	—
63	C₂×#108	—	109	1	no	—
64	C₂×#108	—	109	1	no	—
65	#368	—	109	2	no	—
66	#368	—	109	2	no	—
67	C₂×M₁₆	—	218	1	no	—
68	C₂×M₁₆	—	218	1	no	—
69	C₂²×C₈	—	109	1	no	—
70	M₃₂	—	109	1	no	—
71	M₃₂	—	109	2	no	—
72	M₃₂	#89137	1	1	yes	characteristic
73	C₂×C₁₆	—	109	2	no	—
74	#108	—	109	1	no	—
75	#108	—	109	2	no	—
76	#108	—	109	1	no	—
77	#108	—	218	1	no	—
78	#108	—	218	1	no	—
79	C₂²×C₄	—	109	1	no	—
80	M₁₆	—	436	1	no	—
81	M₁₆	—	436	1	no	—
82	M₁₆	—	218	1	no	—
83	M₁₆	—	218	1	no	—
84	M₁₆	—	218	2	no	—
85	C₂×C₈	—	109	1	no	—
86	C₂×C₈	—	109	1	no	—
87	C₂×C₈	—	109	1	no	—
88	C₂×C₈	—	109	1	no	—
89	C₂×C₈	—	109	1	no	—
90	C₂×C₈	—	436	1	no	—
91	C₂×C₈	—	436	1	no	—
92	C₂×C₈	C₂×#89137	1	1	yes	characteristic
93	C₁₆	—	109	2	no	—
94	C₁₆	C₂×#89137	1	2	yes	—
95	C₂³	—	109	1	no	—
96	C₂×C₄	—	109	1	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	C₄⋊#89137	1	1	yes	characteristic
102	C₈	—	109	1	no	—
103	C₈	—	109	1	no	—
104	C₈	—	436	2	no	—
105	C₈	—	436	1	no	—
106	C₈	—	436	1	no	—
107	C₈	C₄⋊#89137	1	1	yes	characteristic
108	C₈	C₂²×#89137	1	1	yes	characteristic
109	C₂²	—	109	1	no	—
110	C₂²	—	109	1	no	—
111	C₂²	—	436	1	no	—
112	C₂²	#16812768	1	1	yes	characteristic
113	C₄	—	109	1	no	—
114	C₄	—	109	1	no	—
115	C₄	#16812768	1	1	yes	characteristic
116	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	218	1	no	—
119	C₂	—	218	1	no	—
120	C₂	C₂×#16812768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812770

C₂

yes

maximal, characteristic

D₁₀₉⋊#108

C₂

yes

maximal, characteristic

#20295955

C₂

yes

maximal, complemented

#20295954

C₂

yes

maximal, complemented

M₃₂×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182039

—

D₁₀₉⋊M₁₆

—

#16812770

—

#16812770

C₂²

yes

characteristic

#16812770

C₂²

yes

—

#16812770

C₂²

yes

characteristic

#16812769

—

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812639

C₄

yes

characteristic

#16812638

C₄

yes

—

C₁₆×D₁₀₉

C₂²

yes

—

M₃₂×C₁₀₉

C₄

yes

—

—

—

—

—

D₁₀₉⋊C₈

—

C₂×#11724786

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×#11724784

Q₈

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

Q₈

yes

characteristic

C₈×D₁₀₉

C₂³

yes

characteristic

#13182004

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

Q₁₆

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

D₈

yes

characteristic

C₄×D₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

D₈

yes

characteristic

C₄×D₁₀₉

Q₁₆

yes

characteristic

C₄×D₁₀₉

C₂×D₄

yes

—

—

—

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

C₈×C₁₀₉

C₂²×C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

#11724784

C₂×Q₈

yes

characteristic

C₂²×C₁₀₉

#107

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

#107

yes

characteristic

C₂×D₁₀₉

C₂×Q₁₆

yes

characteristic

C₄×C₁₀₉

#107

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

Dic₁₀₉

#107

yes

characteristic

Dic₁₀₉

C₂×D₈

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#107

yes

characteristic

D₁₀₉

—

D₁₀₉

—

#2161

—

109

maximal, Sylow

C₁₀₉

#2161

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂×#108

—

109

—

C₂×#108

—

109

—

#368

—

109

—

#368

—

109

—

C₂×M₁₆

—

218

—

C₂×M₁₆

—

218

—

C₂²×C₈

—

109

—

M₃₂

—

109

—

M₃₂

—

109

—

M₃₂

#89137

yes

characteristic

C₂×C₁₆

—

109

—

#108

—

109

—

#108

—

109

—

#108

—

109

—

#108

—

218

—

#108

—

218

—

C₂²×C₄

—

109

—

M₁₆

—

436

—

M₁₆

—

436

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

436

—

C₂×C₈

—

436

—

C₂×C₈

C₂×#89137

yes

characteristic

C₁₆

—

109

—

C₁₆

C₂×#89137

yes

—

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

C₄⋊#89137

yes

characteristic

102

C₈

—

109

—

103

C₈

—

109

—

104

C₈

—

436

—

105

C₈

—

436

—

106

C₈

—

436

—

107

C₈

C₄⋊#89137

yes

characteristic

108

C₈

C₂²×#89137

yes

characteristic

109

C₂²

—

109

—

110

C₂²

—

109

—

111

C₂²

—

436

—

112

C₂²

#16812768

yes

characteristic

113

C₄

—

109

—

114

C₄

—

109

—

115

C₄

#16812768

yes

characteristic

116

C₄

C₂×C₄⋊#89137

yes

characteristic

117

C₂

—

118

C₂

—

218

—

119

C₂

—

218

—

120

C₂

C₂×#16812768

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#16812768
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₈	C₂²×#89137
Fitting	M₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×#107
prime radical	C₂²×D₁₀₉	Q₁₆
cosocle	C₈×D₁₀₉	C₂³
hypercenter	M₃₂	#89137
hypocenter	C₁₀₉	#2161
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

subgroup

G/H

center

C₂

C₂×#16812768

commutator

C₈×C₁₀₉

C₂²×C₄

Frattini

C₈

C₂²×#89137

Fitting

M₃₂×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₂×#107

prime radical

C₂²×D₁₀₉

Q₁₆

cosocle

C₈×D₁₀₉

C₂³

M₃₂

C₁₀₉

G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

N	H ≅ G/N	type
3. #20295955	C₂	semidirect
4. #20295954	C₂	semidirect
61. C₁₀₉	#2161	semidirect

H ≅ G/N

type

3. #20295955

C₂

semidirect

4. #20295954

C₂

semidirect

61. C₁₀₉

#2161

semidirect

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

subgroup	index
C₂×#16812770	2
D₁₀₉⋊#108	2
#20295955	2
#20295954	2
M₃₂×D₁₀₉	2
#2161	109

subgroup

index

D₁₀₉⋊#108

M₃₂×D₁₀₉

109

Subgroups — #28379388

Conjugacy classes of subgroups

Complements

Maximal subgroups