Group #28363877 — subgroups

Subgroups — #28363877

Order 13952.

292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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	#20295050	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812618	C₂	1	4	yes	maximal
3	C₄×#13182013	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	C₂²	1	2	yes	—
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₄⋊#11724784	—	2	2	no	—
8	#16812618	—	2	8	no	—
9	#16812610	—	2	1	no	—
10	C₄×#11724784	—	2	1	no	—
11	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
13	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
14	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
15	C₂×C₄⋊C₄×C₁₀₉	C₄	1	4	yes	—
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	#13182013	—	2	2	no	—
18	#13182013	—	2	4	no	—
19	#13182013	D₄	1	4	yes	—
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₂×Q₈×C₁₀₉	—	2	2	no	—
23	C₂×Q₈×C₁₀₉	—	2	2	no	—
24	C₂×Q₈×C₁₀₉	—	2	4	no	—
25	C₂×Q₈×C₁₀₉	—	2	4	no	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₉	—	2	4	no	—
30	C₄⋊C₄×C₁₀₉	—	2	4	no	—
31	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
32	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
33	C₄²×C₁₀₉	D₄	1	2	yes	—
34	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₀₉	—	4	4	no	—
36	Q₈×C₁₀₉	—	4	2	no	—
37	Q₈×C₁₀₉	—	4	2	no	—
38	C₂×C₄×C₁₀₉	—	2	4	no	—
39	C₂×C₄×C₁₀₉	—	2	4	no	—
40	C₂×C₄×C₁₀₉	—	2	8	no	—
41	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	#41	1	2	yes	—
47	#11724784	—	2	4	no	—
48	#11724784	—	4	2	no	—
49	C₂²×C₁₀₉	—	2	2	no	—
50	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
52	C₄×C₁₀₉	—	2	2	no	—
53	C₄×C₁₀₉	—	4	4	no	—
54	C₄×C₁₀₉	#101	1	4	yes	complemented
55	C₄×C₁₀₉	#121	1	2	yes	—
56	C₄×C₁₀₉	#124	1	1	yes	characteristic
57	C₄×C₁₀₉	#127	1	1	yes	characteristic
58	C₂×C₁₀₉	—	2	2	no	—
59	C₂×C₁₀₉	#390	1	1	yes	characteristic
60	C₂×C₁₀₉	C₂×#101	1	2	yes	—
61	#1966	—	109	1	no	maximal, Sylow
62	C₁₀₉	#1966	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic
64	#423	—	109	1	no	—
65	C₂×#101	—	109	4	no	—
66	C₄×M₁₆	—	109	1	no	—
67	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
68	C₂×M₁₆	—	109	2	no	—
69	C₂×M₁₆	—	109	2	no	—
70	C₄⋊Q₈	—	2	1	no	—
71	C₄⋊Q₈	—	2	1	no	—
72	C₄⋊Q₈	—	2	2	no	—
73	C₂×C₄⋊C₄	Dic₁₀₉	1	4	yes	—
74	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
75	C₄⋊C₈	—	218	2	no	—
76	#101	—	218	8	no	—
77	#97	—	218	1	no	—
78	C₄×C₈	—	218	1	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
84	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
86	M₁₆	—	109	4	no	—
87	M₁₆	—	218	4	no	—
88	M₁₆	—	218	2	no	—
89	C₂×C₈	—	218	2	no	—
90	C₂×C₈	—	218	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
94	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
95	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
96	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
97	Q₈	—	4	4	no	—
98	Q₈	—	4	2	no	—
99	Q₈	—	4	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
104	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
105	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
108	C₂×C₄	#13182028	1	2	yes	—
109	C₈	—	436	2	no	—
110	C₈	—	218	4	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
113	C₂²	D₄×Dic₁₀₉	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	#16812618	1	4	yes	complemented
117	C₄	#16812670	1	2	yes	—
118	C₄	#16812686	1	1	yes	characteristic
119	C₄	#16812695	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#20294929	1	1	yes	characteristic
122	C₂	C₂×#16812618	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295050

C₂

yes

maximal, characteristic

C₂×#16812618

C₂

yes

maximal

C₄×#13182013

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂×#13182013

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

—

D₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#101

yes

complemented

C₄×C₁₀₉

#121

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₂×C₁₀₉

—

C₂×C₁₀₉

#390

yes

characteristic

C₂×C₁₀₉

C₂×#101

yes

—

#1966

—

109

maximal, Sylow

C₁₀₉

#1966

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₀₉

yes

characteristic

#423

—

109

—

C₂×#101

—

109

—

C₄×M₁₆

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

218

—

#101

—

218

—

#97

—

218

—

C₄×C₈

—

218

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

—

C₂³

C₂²×Dic₁₀₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

104

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

105

C₂×C₄

D₄×D₁₀₉

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

108

C₂×C₄

#13182028

yes

—

109

C₈

—

436

—

110

C₈

—

218

—

111

C₂²

—

112

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

113

C₂²

D₄×Dic₁₀₉

yes

—

114

C₄

—

115

C₄

—

116

C₄

#16812618

yes

complemented

117

C₄

#16812670

yes

—

118

C₄

#16812686

yes

characteristic

119

C₄

#16812695

yes

characteristic

120

C₂

—

121

C₂

#20294929

yes

characteristic

122

C₂

C₂×#16812618

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₉
commutator, prime radical	C₂³×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	#1966
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₉

commutator, prime radical

C₂³×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×C₄⋊Q₈×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₀₉

hypocenter

C₁₀₉

#1966

LCS

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
54. C₄×C₁₀₉	#101	semidirect
62. C₁₀₉	#1966	semidirect
116. C₄	#16812618	semidirect

H ≅ G/N

type

54. C₄×C₁₀₉

#101

semidirect

62. C₁₀₉

#1966

semidirect

116. C₄

#16812618

semidirect

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

subgroup	index
#20295050	2
C₂×#16812618	2
C₄×#13182013	2
C₂×C₄⋊Q₈×C₁₀₉	2
#1966	109

subgroup

index

C₂×C₄⋊Q₈×C₁₀₉

109

Subgroups — #28363877

Conjugacy classes of subgroups

Complements

Maximal subgroups