Group #29573814 — subgroups

Subgroups — #29573814

Order 15616.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	#21445991	C₂	1	2	yes	maximal
2	#21444575	C₂	1	4	yes	maximal
3	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519713	—	2	4	no	—
5	#16839786	C₂²	1	4	yes	—
6	#16839513	—	2	8	no	—
7	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₈²×C₆₁	C₄	1	1	yes	characteristic
10	C₈²×C₆₁	C₄	1	2	yes	—
11	C₈²×C₆₁	C₄	1	1	yes	characteristic
12	C₂×#11758194	—	2	4	no	—
13	#14519713	—	4	8	no	—
14	C₄⋊#92418	—	2	8	no	—
15	C₂²×C₈×C₆₁	D₄	1	2	yes	—
16	C₂²×C₈×C₆₁	Q₈	1	4	yes	—
17	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₆₁	D₄	1	4	yes	—
23	C₄×C₈×C₆₁	Q₈	1	2	yes	—
24	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
26	C₂×C₈×C₆₁	—	2	8	no	—
27	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
28	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₆₁	SD₁₆	1	8	yes	—
32	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₆₁	C₂×D₄	1	2	yes	—
34	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
35	#11758194	—	4	8	no	—
36	C₂×#92418	—	4	4	no	—
37	C₂³×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₆₁	—	2	4	no	—
39	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
43	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
44	C₈×C₆₁	—	2	4	no	—
45	C₈×C₆₁	#106	1	4	yes	—
46	C₈×C₆₁	#106	1	4	yes	—
47	C₈×C₆₁	#108	1	8	yes	complemented
48	#92418	—	8	4	no	—
49	#8822	—	61	1	no	maximal, Sylow
50	C₂²×C₆₁	—	2	2	no	—
51	C₂²×C₆₁	#384	1	1	yes	characteristic
52	C₂²×C₆₁	#498	1	2	yes	—
53	C₄×C₆₁	—	2	2	no	—
54	C₄×C₆₁	C₂×#106	1	2	yes	—
55	C₄×C₆₁	C₂×#108	1	4	yes	—
56	C₄×C₆₁	#492	1	2	yes	—
57	#1848	—	61	2	no	—
58	#1566	—	61	4	no	—
59	C₂×C₈²	D₆₁	1	1	yes	characteristic
60	C₂×C₆₁	—	2	2	no	—
61	C₂×C₆₁	#1844	1	1	yes	characteristic
62	C₂×C₆₁	#1848	1	2	yes	—
63	C₂×#108	—	122	4	no	—
64	#423	—	61	4	no	—
65	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
67	#334	—	122	8	no	—
68	C₈²	Dic₆₁	1	1	yes	characteristic
69	C₈²	Dic₆₁	1	2	yes	—
70	C₈²	Dic₆₁	1	1	yes	characteristic
71	C₆₁	#8822	1	1	yes	characteristic, complemented, Sylow
72	C₂×M₁₆	—	122	4	no	—
73	C₂²×C₈	Dic₁₂₂	1	4	yes	—
74	C₂²×C₈	D₂₄₄	1	2	yes	—
75	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
76	#108	—	244	8	no	—
77	C₄⋊C₈	—	122	8	no	—
78	C₄×C₈	Dic₁₂₂	1	2	yes	—
79	C₄×C₈	D₂₄₄	1	4	yes	—
80	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
81	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
82	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
83	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
84	C₂²×C₄	C₂×Dic₁₂₂	1	2	yes	—
85	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
86	M₁₆	—	244	8	no	—
87	C₂×C₈	—	2	8	no	—
88	C₂×C₈	—	244	4	no	—
89	C₂×C₈	#11758190	1	8	yes	—
90	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
91	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
92	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
93	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
94	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
95	C₄²	C₂×D₂₄₄	1	2	yes	—
96	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
97	C₂³	#14519777	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
101	C₂×C₄	#14519774	1	1	yes	characteristic
102	C₂×C₄	#14519777	1	2	yes	—
103	C₂×C₄	C₂×#11758190	1	4	yes	—
104	C₈	—	2	4	no	—
105	C₈	—	488	4	no	—
106	C₈	#14519709	1	4	yes	—
107	C₈	#14519709	1	4	yes	—
108	C₈	#14519713	1	8	yes	complemented
109	C₂²	—	2	2	no	—
110	C₂²	#16839632	1	1	yes	characteristic
111	C₂²	#16840160	1	2	yes	—
112	C₄	—	2	2	no	—
113	C₄	C₂×#14519709	1	2	yes	—
114	C₄	C₂×#14519713	1	4	yes	—
115	C₄	#16840135	1	2	yes	—
116	C₂	—	2	2	no	—
117	C₂	#21445976	1	1	yes	characteristic
118	C₂	#21445991	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445991

C₂

yes

maximal

#21444575

C₂

yes

maximal

C₂×C₈²×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519713

—

#16839786

C₂²

yes

—

#16839513

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₈²×C₆₁

C₄

yes

characteristic

C₈²×C₆₁

C₄

yes

—

C₈²×C₆₁

C₄

yes

—

—

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

Q₈

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×Q₈

yes

characteristic

#11758194

—

C₂×#92418

—

C₂³×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₈×C₆₁

—

C₈×C₆₁

#106

yes

—

C₈×C₆₁

#106

yes

—

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#384

yes

characteristic

C₂²×C₆₁

#498

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#106

yes

—

C₄×C₆₁

C₂×#108

yes

—

C₄×C₆₁

#492

yes

—

#1848

—

#1566

—

C₂×C₈²

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#1844

yes

characteristic

C₂×C₆₁

#1848

yes

—

C₂×#108

—

122

—

#423

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄×C₈

C₂×D₆₁

yes

—

#334

—

122

—

C₈²

Dic₆₁

yes

characteristic

C₈²

Dic₆₁

yes

—

C₈²

Dic₆₁

yes

characteristic

C₆₁

#8822

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

Dic₁₂₂

yes

—

C₂²×C₈

D₂₄₄

yes

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

#108

—

244

—

C₄⋊C₈

—

122

—

C₄×C₈

Dic₁₂₂

yes

—

C₄×C₈

D₂₄₄

yes

—

C₄×C₈

C₂×Dic₆₁

yes

—

C₄×C₈

C₂×Dic₆₁

yes

characteristic

C₄×C₈

C₂×Dic₆₁

yes

characteristic

C₄×C₈

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₁₂₂

yes

—

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

#11758190

yes

—

C₂×C₈

C₄⋊Dic₆₁

yes

—

C₂×C₈

C₄⋊Dic₆₁

yes

—

C₂×C₈

C₄⋊Dic₆₁

yes

—

C₂×C₈

C₄⋊Dic₆₁

yes

—

C₄²

C₂×Dic₁₂₂

yes

characteristic

C₄²

C₂×D₂₄₄

yes

—

C₄²

C₂²×Dic₆₁

yes

characteristic

C₂³

#14519777

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

100

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

—

101

C₂×C₄

#14519774

yes

characteristic

102

C₂×C₄

#14519777

yes

—

103

C₂×C₄

C₂×#11758190

yes

—

104

C₈

—

105

C₈

—

488

—

106

C₈

#14519709

yes

—

107

C₈

#14519709

yes

—

108

C₈

#14519713

yes

complemented

109

C₂²

—

110

C₂²

#16839632

yes

characteristic

111

C₂²

#16840160

yes

—

112

C₄

—

113

C₄

C₂×#14519709

yes

—

114

C₄

C₂×#14519713

yes

—

115

C₄

#16840135

yes

—

116

C₂

—

117

C₂

#21445976

yes

characteristic

118

C₂

#21445991

yes

—

	subgroup	G/H
center	C₂×C₄	#14519774
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈²×C₆₁	C₂
socle	C₂²×C₆₁	#384
prime radical	C₂³×C₆₁	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈²	D₆₁
hypocenter	C₆₁	#8822
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#14519774

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₈²×C₆₁

C₂

socle

C₂²×C₆₁

#384

prime radical

C₂³×C₆₁

C₄⋊Q₈

cosocle

C₂×C₄²×C₆₁

C₂³

hypercenter

C₂×C₈²

D₆₁

hypocenter

C₆₁

#8822

LCS

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
47. C₈×C₆₁	#108	semidirect
71. C₆₁	#8822	semidirect
108. C₈	#14519713	semidirect

H ≅ G/N

type

47. C₈×C₆₁

#108

semidirect

71. C₆₁

#8822

semidirect

108. C₈

#14519713

semidirect

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

subgroup	index
#21445991	2
#21444575	2
C₂×C₈²×C₆₁	2
#8822	61

subgroup

index

#21445991

#21444575

C₂×C₈²×C₆₁

#8822

Subgroups — #29573814

Conjugacy classes of subgroups

Complements

Maximal subgroups