Group #28378974 — subgroups

Subgroups — #28378974

Order 13952.

212 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 86 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	2	yes	maximal
2	C₂×#16812769	C₂	1	2	yes	maximal
3	#20296031	C₂	1	1	yes	maximal, characteristic
4	#20296030	C₂	1	1	yes	maximal, characteristic
5	#97×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	C₂²	1	2	yes	—
7	C₂×#13182039	C₂²	1	1	yes	characteristic
8	C₂×#13182039	C₂²	1	1	yes	characteristic
9	#16812770	—	2	4	no	—
10	#16812769	—	2	4	no	—
11	#16812766	—	2	2	no	—
12	C₄⋊#11724786	—	2	2	no	—
13	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
14	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
15	#16812610	C₄	1	1	yes	characteristic
16	#16812609	C₄	1	2	yes	—
17	#97×C₁₀₉	C₄	1	1	yes	characteristic
18	#13182039	—	2	2	no	—
19	#13182039	—	2	2	no	—
20	#13182039	—	2	2	no	—
21	#13182039	—	2	2	no	—
22	C₂×#11724786	—	2	2	no	—
23	C₂×#11724786	—	2	1	no	—
24	C₂×#11724786	—	2	1	no	—
25	C₂×C₄×D₁₀₉	Q₈	1	2	yes	—
26	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×#11724784	C₂×C₄	1	2	yes	—
28	C₈×D₁₀₉	D₄	1	4	yes	—
29	C₈×D₁₀₉	Q₈	1	4	yes	—
30	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₀₉	D₄	1	2	yes	—
32	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
38	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₄×D₁₀₉	—	4	2	no	—
40	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
41	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
42	#11724786	—	4	2	no	—
43	#11724786	—	4	1	no	—
44	#11724786	—	4	1	no	—
45	C₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
46	#11724784	C₄⋊C₄	1	4	yes	—
47	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₁₀₉	—	2	2	no	—
49	C₂×D₁₀₉	C₄⋊Q₈	1	2	yes	—
50	C₄×C₁₀₉	—	2	2	no	—
51	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
52	Dic₁₀₉	—	2	2	no	—
53	Dic₁₀₉	#127	1	1	yes	characteristic
54	Dic₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₁₀₉	#384	1	1	yes	characteristic
56	C₂×C₁₀₉	#430	1	2	yes	—
57	D₁₀₉	—	2	2	no	—
58	#1849	—	109	1	no	maximal, Sylow
59	C₁₀₉	#1849	1	1	yes	characteristic, complemented, Sylow
60	C₂×#108	—	109	2	no	—
61	C₂×#108	—	109	2	no	—
62	#423	—	109	1	no	—
63	#423	—	109	1	no	—
64	C₂×#97	—	109	1	no	—
65	C₂×M₁₆	—	109	2	no	—
66	C₂×M₁₆	—	109	1	no	—
67	C₂×M₁₆	—	109	1	no	—
68	C₂²×C₈	—	109	2	no	—
69	C₂×C₄²	—	109	1	no	—
70	#108	—	218	4	no	—
71	#108	—	218	4	no	—
72	C₄⋊C₈	—	218	2	no	—
73	C₄⋊C₈	—	218	2	no	—
74	#97	—	109	2	no	—
75	#97	—	109	1	no	—
76	#97	#89137	1	1	yes	characteristic
77	C₂²×C₄	—	109	1	no	—
78	C₂²×C₄	—	109	2	no	—
79	M₁₆	—	218	2	no	—
80	M₁₆	—	218	2	no	—
81	M₁₆	—	218	2	no	—
82	M₁₆	—	218	2	no	—
83	C₂×C₈	—	109	2	no	—
84	C₂×C₈	—	109	4	no	—
85	C₂×C₈	—	109	4	no	—
86	C₂×C₈	—	218	2	no	—
87	C₂×C₈	—	218	1	no	—
88	C₂×C₈	—	218	1	no	—
89	C₂×C₈	C₂×#89137	1	2	yes	—
90	C₄²	—	109	2	no	—
91	C₄²	—	109	1	no	—
92	C₄²	C₂×#89137	1	1	yes	characteristic
93	C₂³	—	109	1	no	—
94	C₂×C₄	—	109	1	no	—
95	C₂×C₄	—	109	2	no	—
96	C₂×C₄	—	109	2	no	—
97	C₂×C₄	—	109	2	no	—
98	C₂×C₄	—	436	2	no	—
99	C₂×C₄	C₄⋊#89137	1	2	yes	—
100	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
101	C₈	—	109	4	no	—
102	C₈	—	436	2	no	—
103	C₈	—	436	1	no	—
104	C₈	—	436	1	no	—
105	C₈	C₄⋊#89137	1	4	yes	—
106	C₂²	—	109	2	no	—
107	C₂²	—	218	2	no	—
108	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	109	1	no	—
111	C₄	—	109	1	no	—
112	C₄	—	218	2	no	—
113	C₄	C₂×C₄⋊#89137	1	2	yes	—
114	C₂	—	218	2	no	—
115	C₂	#20295972	1	1	yes	characteristic
116	C₂	#20296041	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812770

C₂

yes

maximal

C₂×#16812769

C₂

yes

maximal

#20296031

C₂

yes

maximal, characteristic

#20296030

C₂

yes

maximal, characteristic

#97×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₂×#13182039

C₂²

yes

characteristic

C₂×#13182039

C₂²

yes

—

—

—

—

C₂×C₈×D₁₀₉

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812610

C₄

yes

characteristic

#16812609

C₄

yes

—

#97×C₁₀₉

C₄

yes

—

—

—

—

—

—

—

C₂×C₄×D₁₀₉

Q₈

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₈×D₁₀₉

D₄

yes

—

C₈×D₁₀₉

Q₈

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×D₄

yes

—

C₄×D₁₀₉

C₂×Q₈

yes

—

#11724786

—

#11724786

—

#11724786

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

#11724784

C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

#127

yes

characteristic

Dic₁₀₉

C₄⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

D₁₀₉

—

#1849

—

109

maximal, Sylow

C₁₀₉

#1849

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

C₂×#108

—

109

—

#423

—

109

—

#423

—

109

—

C₂×#97

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

#108

—

218

—

#108

—

218

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

#97

—

109

—

#97

—

109

—

#97

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

C₄⋊#89137

yes

—

100

C₂×C₄

C₂²×#89137

yes

characteristic

101

C₈

—

109

—

102

C₈

—

436

—

103

C₈

—

436

—

104

C₈

—

436

—

105

C₈

C₄⋊#89137

yes

—

106

C₂²

—

109

—

107

C₂²

—

218

—

108

C₂²

C₂×C₄⋊#89137

yes

characteristic

109

C₄

—

110

C₄

—

109

—

111

C₄

—

109

—

112

C₄

—

218

—

113

C₄

C₂×C₄⋊#89137

yes

—

114

C₂

—

218

—

115

C₂

#20295972

yes

characteristic

116

C₂

#20296041

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	#97×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₂×Q₈
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#97	#89137
hypocenter	C₁₀₉	#1849
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

#97×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₂×Q₈

cosocle

C₂×C₄×D₁₀₉

C₂³

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
59. C₁₀₉	#1849	semidirect

H ≅ G/N

type

59. C₁₀₉

#1849

semidirect

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

subgroup	index
C₂×#16812770	2
C₂×#16812769	2
#20296031	2
#20296030	2
#97×D₁₀₉	2
#1849	109

subgroup

index

#97×D₁₀₉

109

Subgroups — #28378974

Conjugacy classes of subgroups

Complements

Maximal subgroups