Group #29594688 — subgroups

Subgroups — #29594688

Order 15616.

258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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	#21448101	C₂	1	1	yes	maximal, characteristic
2	#21448072	C₂	1	1	yes	maximal, characteristic
3	#21448047	C₂	1	2	yes	maximal
4	#21445416	C₂	1	2	yes	maximal
5	C₆₁×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₈₈	—	2	2	no	—
7	#16840167	C₂²	1	2	yes	—
8	#16840160	—	2	2	no	—
9	#16840142	C₂²	1	2	yes	—
10	#16840113	—	2	2	no	—
11	#16839590	—	2	2	no	—
12	#16839567	—	2	4	no	—
13	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₈²×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₄₄	—	2	2	no	—
17	C₂×Dic₂₄₄	—	2	2	no	—
18	#14519777	—	2	2	no	—
19	#14519777	—	2	2	no	—
20	Dic₄₈₈	—	4	4	no	—
21	#14519711	—	2	2	no	—
22	#14519711	—	2	2	no	—
23	#14519709	—	4	4	no	—
24	#14519702	—	4	4	no	—
25	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
26	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
27	C₄×C₈×C₆₁	D₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₂₂	—	4	2	no	—
30	C₂×Dic₁₂₂	—	4	2	no	—
31	C₂×C₈×C₆₁	D₈	1	4	yes	—
32	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₆₁	—	4	2	no	—
36	C₄⋊Dic₆₁	—	4	2	no	—
37	C₄⋊Dic₆₁	—	4	2	no	—
38	C₄⋊Dic₆₁	—	4	2	no	—
39	Dic₂₄₄	—	4	4	no	—
40	Dic₂₄₄	—	4	4	no	—
41	C₁₆×C₆₁	—	2	2	no	—
42	C₁₆×C₆₁	SD₁₆	1	4	yes	—
43	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
45	C₂×Dic₆₁	—	8	2	no	—
46	C₂×Dic₆₁	—	8	2	no	—
47	Dic₁₂₂	—	8	2	no	—
48	Dic₁₂₂	—	8	2	no	—
49	Dic₁₂₂	—	8	4	no	—
50	C₈×C₆₁	—	2	4	no	—
51	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
52	C₈×C₆₁	#135	1	2	yes	—
53	#12507	—	61	1	no	maximal, Sylow
54	C₂²×C₆₁	#493	1	1	yes	characteristic
55	C₄×C₆₁	#492	1	1	yes	characteristic
56	C₄×C₆₁	#495	1	1	yes	characteristic
57	C₄×C₆₁	#510	1	4	yes	—
58	Dic₆₁	—	16	2	no	—
59	Dic₆₁	—	16	2	no	—
60	#2254	—	61	1	no	—
61	#2247	—	61	1	no	—
62	#2241	—	61	2	no	—
63	#1707	—	61	2	no	—
64	#1311	D₆₁	1	1	yes	characteristic
65	C₂×C₆₁	#1706	1	1	yes	characteristic
66	C₂×C₆₁	#2251	1	2	yes	—
67	C₂×Q₃₂	—	122	2	no	—
68	#500	—	61	2	no	—
69	#498	—	122	2	no	—
70	#494	—	61	2	no	—
71	#487	—	122	2	no	—
72	#367	—	122	2	no	—
73	#358	—	122	4	no	—
74	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
75	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
76	C₈²	C₂×D₆₁	1	1	yes	characteristic
77	C₆₁	#12507	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	122	2	no	—
79	C₂×Q₁₆	—	122	2	no	—
80	C₄⋊Q₈	—	122	2	no	—
81	C₄⋊Q₈	—	122	2	no	—
82	Q₃₂	—	244	4	no	—
83	C₂×C₁₆	D₂₄₄	1	2	yes	—
84	C₂×C₁₆	D₂₄₄	1	2	yes	—
85	#107	—	122	2	no	—
86	#107	—	122	2	no	—
87	#106	—	244	4	no	—
88	#103	—	244	4	no	—
89	C₄×C₈	D₂₄₄	1	2	yes	—
90	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
91	C₂×Q₈	—	244	2	no	—
92	C₂×Q₈	—	244	2	no	—
93	Q₁₆	—	244	4	no	—
94	Q₁₆	—	244	4	no	—
95	C₂×C₈	D₄₈₈	1	4	yes	—
96	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
97	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
98	C₄⋊C₄	—	244	2	no	—
99	C₄⋊C₄	—	244	2	no	—
100	C₄⋊C₄	—	244	2	no	—
101	C₄⋊C₄	—	244	2	no	—
102	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	#11758190	1	4	yes	—
105	Q₈	—	488	4	no	—
106	Q₈	—	488	2	no	—
107	Q₈	—	488	2	no	—
108	C₂×C₄	—	488	2	no	—
109	C₂×C₄	—	488	2	no	—
110	C₂×C₄	#14519774	1	1	yes	characteristic
111	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	C₂×#11758190	1	2	yes	—
114	C₈	#14519799	1	2	yes	—
115	C₂²	#16840139	1	1	yes	characteristic
116	C₄	—	976	2	no	—
117	C₄	—	976	2	no	—
118	C₄	#16840135	1	1	yes	characteristic
119	C₄	#16840146	1	1	yes	characteristic
120	C₄	#16840205	1	4	yes	—
121	C₂	#21445411	1	1	yes	characteristic
122	C₂	#21448090	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448101

C₂

yes

maximal, characteristic

#21448072

C₂

yes

maximal, characteristic

#21448047

C₂

yes

maximal

#21445416

C₂

yes

maximal

C₆₁×#1311

C₂

yes

maximal, characteristic

C₂×Dic₄₈₈

—

#16840167

C₂²

yes

—

#16840160

—

#16840142

C₂²

yes

—

#16840113

—

#16839590

—

#16839567

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

—

#14519777

—

Dic₄₈₈

—

—

—

—

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

—

C₁₆×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×SD₁₆

yes

—

C₈×C₆₁

#135

yes

—

—

C₂²×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×Q₃₂

—

122

—

#500

—

#498

—

122

—

#494

—

#487

—

122

—

#367

—

122

—

#358

—

122

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12507

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

Q₃₂

—

244

—

C₂×C₁₆

D₂₄₄

yes

—

C₂×C₁₆

D₂₄₄

yes

—

#107

—

122

—

#107

—

122

—

#106

—

244

—

#103

—

244

—

C₄×C₈

D₂₄₄

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

244

—

C₂×Q₈

—

244

—

Q₁₆

—

244

—

Q₁₆

—

244

—

C₂×C₈

D₄₈₈

yes

—

C₂×C₈

C₂×D₂₄₄

yes

characteristic

C₂×C₈

C₂×D₂₄₄

yes

characteristic

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

100

C₄⋊C₄

—

244

—

101

C₄⋊C₄

—

244

—

102

C₄²

C₂×D₂₄₄

yes

characteristic

103

C₁₆

—

104

C₁₆

#11758190

yes

—

105

Q₈

—

488

—

106

Q₈

—

488

—

107

Q₈

—

488

—

108

C₂×C₄

—

488

—

109

C₂×C₄

—

488

—

110

C₂×C₄

#14519774

yes

characteristic

111

C₂×C₄

C₂×D₄₈₈

yes

—

112

C₈

—

113

C₈

C₂×#11758190

yes

—

114

C₈

#14519799

yes

—

115

C₂²

#16840139

yes

characteristic

116

C₄

—

976

—

117

C₄

—

976

—

118

C₄

#16840135

yes

characteristic

119

C₄

#16840146

yes

characteristic

120

C₄

#16840205

yes

—

121

C₂

#21445411

yes

characteristic

122

C₂

#21448090

yes

—

	subgroup	G/H
center	C₂²	#16840139
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1311	C₂
socle, prime radical	C₂²×C₆₁	#493
hypercenter	#1311	D₆₁
hypocenter	C₆₁	#12507
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#16840139

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1311

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
77. C₆₁	#12507	semidirect

H ≅ G/N

type

77. C₆₁

#12507

semidirect

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

subgroup	index
#21448101	2
#21448072	2
#21448047	2
#21445416	2
C₆₁×#1311	2
#12507	61

subgroup

index

C₆₁×#1311

Subgroups — #29594688

Conjugacy classes of subgroups

Complements

Maximal subgroups