Group #28364297 — subgroups

Subgroups — #28364297

Order 13952.

256 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20295052	C₂	1	1	yes	maximal, characteristic
2	#20295035	C₂	1	2	yes	maximal
3	C₂×#16812618	C₂	1	2	yes	maximal
4	#20294791	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	1	yes	characteristic
7	C₂×#13182013	C₂²	1	2	yes	—
8	#16812658	C₂²	1	1	yes	characteristic
9	#16812628	—	2	2	no	—
10	#16812623	—	2	4	no	—
11	#16812618	—	2	4	no	—
12	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₁₀₉	—	2	4	no	—
14	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
15	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	#13182013	—	2	2	no	—
17	#13182013	—	2	2	no	—
18	#13182013	—	2	2	no	—
19	C₂×#11724784	—	2	2	no	—
20	C₂×#11724784	—	2	1	no	—
21	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
22	C₂²⋊Dic₁₀₉	—	2	1	no	—
23	C₄×Dic₁₀₉	—	2	1	no	—
24	C₂×Q₈×C₁₀₉	—	2	2	no	—
25	C₂×Q₈×C₁₀₉	—	2	2	no	—
26	C₂×Q₈×C₁₀₉	—	2	2	no	—
27	C₂×Q₈×C₁₀₉	—	2	2	no	—
28	C₂×Q₈×C₁₀₉	—	2	2	no	—
29	C₂×Q₈×C₁₀₉	D₄	1	4	yes	—
30	C₂²×C₄×C₁₀₉	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	Q₁₆×C₁₀₉	—	2	8	no	—
33	C₂×C₈×C₁₀₉	—	2	2	no	—
34	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
35	C₂³×C₁₀₉	#41	1	1	yes	characteristic
36	Q₈×C₁₀₉	—	2	4	no	—
37	Q₈×C₁₀₉	—	2	4	no	—
38	Q₈×C₁₀₉	—	4	2	no	—
39	Q₈×C₁₀₉	—	4	2	no	—
40	C₂×C₄×C₁₀₉	—	2	4	no	—
41	C₂×C₄×C₁₀₉	—	4	2	no	—
42	C₂×C₄×C₁₀₉	—	4	2	no	—
43	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	#41	1	2	yes	—
47	C₂×Dic₁₀₉	—	2	2	no	—
48	C₈×C₁₀₉	—	2	2	no	—
49	#11724784	—	4	1	no	—
50	#11724784	—	4	2	no	—
51	C₂²×C₁₀₉	—	2	2	no	—
52	C₂²×C₁₀₉	#121	1	1	yes	characteristic
53	C₂²×C₁₀₉	#123	1	2	yes	—
54	C₄×C₁₀₉	—	4	4	no	—
55	C₄×C₁₀₉	#120	1	1	yes	characteristic
56	C₄×C₁₀₉	#121	1	2	yes	—
57	C₄×C₁₀₉	#124	1	1	yes	characteristic
58	Dic₁₀₉	—	4	2	no	—
59	C₂×C₁₀₉	—	2	2	no	—
60	C₂×C₁₀₉	#394	1	1	yes	characteristic
61	C₂×C₁₀₉	#473	1	2	yes	—
62	#2047	—	109	1	no	maximal, Sylow
63	C₁₀₉	#2047	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₁₆	D₁₀₉	1	1	yes	characteristic
65	#424	—	109	1	no	—
66	#419	—	109	2	no	—
67	C₂×#101	—	109	2	no	—
68	#341	—	109	1	no	—
69	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
70	C₂×Q₁₆	—	2	4	no	—
71	C₂×Q₁₆	—	2	2	no	—
72	C₂×M₁₆	—	109	2	no	—
73	C₂×M₁₆	—	109	1	no	—
74	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
75	#117	—	109	1	no	—
76	C₄⋊C₈	—	218	2	no	—
77	#103	—	218	4	no	—
78	#101	—	218	4	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	C₂²⋊D₁₀₉	1	4	yes	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
87	Q₁₆	—	2	8	no	—
88	M₁₆	—	218	2	no	—
89	M₁₆	—	218	2	no	—
90	M₁₆	—	218	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	218	2	no	—
93	C₂×C₈	—	218	1	no	—
94	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
95	C₄⋊C₄	—	109	2	no	—
96	C₂²⋊C₄	—	218	1	no	—
97	C₄²	—	218	1	no	—
98	C₂³	#13182028	1	1	yes	characteristic
99	Q₈	—	2	4	no	—
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	Q₈	—	4	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	218	2	no	—
107	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
110	C₂×C₄	#13182028	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	436	1	no	—
113	C₈	—	436	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#16812670	1	1	yes	characteristic
116	C₂²	#16812680	1	2	yes	—
117	C₄	—	4	4	no	—
118	C₄	—	436	2	no	—
119	C₄	#16812668	1	1	yes	characteristic
120	C₄	#16812671	1	2	yes	—
121	C₄	#16812686	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#20294940	1	1	yes	characteristic
124	C₂	#20295298	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295052

C₂

yes

maximal, characteristic

#20295035

C₂

yes

maximal

C₂×#16812618

C₂

yes

maximal

#20294791

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

characteristic

C₂×#13182013

C₂²

yes

—

#16812658

C₂²

yes

—

—

—

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

—

—

—

—

C₄⋊Dic₁₀₉

D₄

yes

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

#41

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

#123

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

#121

yes

—

C₄×C₁₀₉

#124

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#394

yes

characteristic

C₂×C₁₀₉

#473

yes

—

#2047

—

109

maximal, Sylow

C₁₀₉

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

D₁₀₉

yes

characteristic

#424

—

109

—

#419

—

109

—

C₂×#101

—

109

—

#341

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

C₄⋊C₈

—

218

—

#103

—

218

—

#101

—

218

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

Q₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

218

—

C₄²

—

218

—

C₂³

#13182028

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

218

—

107

C₂×C₄

D₄×D₁₀₉

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

110

C₂×C₄

#13182028

yes

—

111

C₈

—

112

C₈

—

436

—

113

C₈

—

436

—

114

C₂²

—

115

C₂²

#16812670

yes

characteristic

116

C₂²

#16812680

yes

—

117

C₄

—

118

C₄

—

436

—

119

C₄

#16812668

yes

characteristic

120

C₄

#16812671

yes

—

121

C₄

#16812686

yes

characteristic

122

C₂

—

123

C₂

#20294940

yes

characteristic

124

C₂

#20295298

yes

—

	subgroup	G/H
center	C₂²	#16812670
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂²×Q₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#121
prime radical	C₂³×C₁₀₉	#41
hypercenter	C₂²×Q₁₆	D₁₀₉
hypocenter	C₁₀₉	#2047
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#16812670

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂²×Q₁₆×C₁₀₉

C₂

C₂²×C₁₀₉

C₂³×C₁₀₉

C₂²×Q₁₆

D₁₀₉

hypocenter

C₁₀₉

#2047

LCS

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
63. C₁₀₉	#2047	semidirect

H ≅ G/N

type

63. C₁₀₉

#2047

semidirect

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

subgroup	index
#20295052	2
#20295035	2
C₂×#16812618	2
#20294791	2
C₂²×Q₁₆×C₁₀₉	2
#2047	109

subgroup

index

C₂²×Q₁₆×C₁₀₉

109

Subgroups — #28364297

Conjugacy classes of subgroups

Complements

Maximal subgroups