Group #17960296 — subgroups

Subgroups — #17960296

Order 4352.

214 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 70 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	#14544379	C₂	1	2	yes	maximal
2	#14543809	C₂	1	1	yes	maximal, characteristic
3	#14543745	C₂	1	2	yes	maximal
4	#14543700	C₂	1	1	yes	maximal, characteristic
5	#14530654	C₂	1	1	yes	maximal, characteristic
6	#11763858	—	2	2	no	—
7	#11763698	C₂²	1	1	yes	characteristic
8	#11763670	C₂²	1	2	yes	—
9	#11763659	—	2	2	no	—
10	#11763650	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₃₆	C₄	1	1	yes	characteristic
12	#11762903	C₂²	1	2	yes	—
13	#11762612	C₄	1	2	yes	—
14	#11762584	C₂²	1	1	yes	characteristic
15	C₁₇×#367	C₄	1	1	yes	characteristic
16	C₂×Dic₆₈	C₂×C₄	1	2	yes	—
17	#10587573	—	2	2	no	—
18	#10587544	—	4	2	no	—
19	C₄⋊#3624	—	2	2	no	—
20	C₄×#3624	—	4	2	no	—
21	C₄×#3624	D₄	1	2	yes	—
22	C₂×#63216	—	2	1	no	—
23	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₁₇×#107	C₂×C₄	1	2	yes	—
25	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
26	C₂×#63214	C₂×C₄	1	1	yes	characteristic
27	#10587481	—	2	2	no	—
28	Dic₁₃₆	C₈	1	4	yes	complemented
29	C₂×Dic₃₄	—	2	2	no	—
30	C₂×#3624	—	2	2	no	—
31	C₂×#3624	—	4	2	no	—
32	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₁₇	—	2	2	no	—
34	#63225	—	4	2	no	—
35	C₄×Dic₁₇	—	4	2	no	—
36	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×#3622	#41	1	1	yes	characteristic
38	Dic₆₈	—	2	2	no	—
39	Dic₆₈	C₂×C₈	1	4	yes	—
40	#63216	—	4	1	no	—
41	C₁₆×C₁₇	C₂×C₈	1	2	yes	—
42	#63214	—	2	1	no	—
43	#12066	—	17	1	no	maximal, Sylow
44	C₂×C₄×C₁₇	—	4	2	no	—
45	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
46	C₂×Dic₁₇	—	4	2	no	—
47	C₂×Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
48	Dic₃₄	—	2	4	no	—
49	Dic₃₄	—	4	2	no	—
50	#3624	—	4	2	no	—
51	#3624	—	8	2	no	—
52	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
53	C₈×C₁₇	#131	1	1	yes	characteristic
54	#3622	—	2	1	no	—
55	#2176	—	17	1	no	—
56	#1579	—	17	2	no	—
57	#1368	—	17	1	no	—
58	#1334	—	17	2	no	—
59	#1313	—	17	1	no	—
60	C₂²×C₁₇	#442	1	1	yes	characteristic
61	C₄×C₁₇	—	8	2	no	—
62	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
63	C₄×C₁₇	#444	1	1	yes	characteristic
64	Dic₁₇	—	2	2	no	—
65	Dic₁₇	—	4	4	no	—
66	C₂×Q₃₂	—	17	1	no	—
67	C₈×Q₈	—	34	2	no	—
68	C₄×Q₁₆	—	17	2	no	—
69	#367	#595	1	1	yes	characteristic
70	#358	—	17	2	no	—
71	#346	—	17	1	no	—
72	#346	—	17	1	no	—
73	#334	—	17	2	no	—
74	Q₈⋊C₈	—	34	2	no	—
75	#322	—	17	1	no	—
76	C₂×C₁₇	#1580	1	1	yes	characteristic
77	C₂×C₁₇	#2180	1	2	yes	—
78	C₂×Q₁₆	—	17	2	no	—
79	C₄×Q₈	—	34	2	no	—
80	Q₃₂	—	17	4	no	—
81	C₂×C₁₆	—	17	1	no	—
82	C₂×C₁₆	—	34	1	no	—
83	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
84	#107	C₂×#595	1	2	yes	—
85	C₄⋊C₈	—	34	2	no	—
86	C₄⋊C₈	—	68	2	no	—
87	#103	—	34	2	no	—
88	C₄×C₈	—	17	1	no	—
89	C₄×C₈	—	17	2	no	—
90	C₄×C₈	—	68	2	no	—
91	C₁₇	#12066	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	34	2	no	—
93	Q₁₆	—	17	4	no	—
94	Q₁₆	—	34	2	no	—
95	C₂×C₈	—	17	1	no	—
96	C₂×C₈	—	34	2	no	—
97	C₂×C₈	—	68	2	no	—
98	C₂×C₈	C₂²×#595	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	68	2	no	—
101	C₄²	—	17	1	no	—
102	C₄²	—	68	2	no	—
103	C₁₆	—	34	1	no	—
104	C₁₆	—	68	1	no	—
105	C₁₆	#63242	1	2	yes	—
106	Q₈	—	34	4	no	—
107	Q₈	—	68	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	17	2	no	—
110	C₂×C₄	—	68	2	no	—
111	C₂×C₄	D₄×#595	1	1	yes	characteristic
112	C₈	—	34	1	no	—
113	C₈	—	68	2	no	—
114	C₈	—	136	2	no	—
115	C₈	C₂×#63242	1	1	yes	characteristic
116	C₈	#10587680	1	1	yes	characteristic
117	C₂²	#11763850	1	1	yes	characteristic
118	C₄	—	8	2	no	—
119	C₄	—	34	2	no	—
120	C₄	—	68	4	no	—
121	C₄	#11763837	1	1	yes	characteristic
122	C₄	#11763855	1	1	yes	characteristic
123	C₂	#14544382	1	1	yes	characteristic
124	C₂	#14545189	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544379

C₂

yes

maximal

#14543809

C₂

yes

maximal, characteristic

#14543745

C₂

yes

maximal

#14543700

C₂

yes

maximal, characteristic

#14530654

C₂

yes

maximal, characteristic

#11763858

—

#11763698

C₂²

yes

characteristic

#11763670

C₂²

yes

—

#11763659

—

#11763650

C₂²

yes

characteristic

C₂×Dic₁₃₆

C₄

yes

characteristic

#11762903

C₂²

yes

—

#11762612

C₄

yes

—

#11762584

C₂²

yes

characteristic

C₁₇×#367

C₄

yes

characteristic

C₂×Dic₆₈

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₂×#63216

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#107

C₂×C₄

yes

—

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

#10587481

—

Dic₁₃₆

C₈

yes

complemented

C₂×Dic₃₄

—

C₂×#3624

—

C₂×#3624

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₁₇

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂×D₄

yes

yes

Dic₆₈

—

Dic₆₈

C₂×C₈

yes

—

#63216

—

C₁₆×C₁₇

C₂×C₈

yes

—

#63214

—

#12066

—

maximal, Sylow

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

AGL(1,ℤ/8)

yes

—

Dic₃₄

—

Dic₃₄

—

#3624

—

#3624

—

C₈×C₁₇

C₂²×C₈

yes

C₈×C₁₇

yes

—

—

—

—

—

—

C₂²×C₁₇

#442

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

#444

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂×Q₃₂

—

C₈×Q₈

—

C₄×Q₁₆

—

yes

—

—

—

—

Q₈⋊C₈

—

#322

—

C₂×C₁₇

#1580

yes

characteristic

C₂×C₁₇

#2180

yes

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#12066

yes

characteristic, complemented, Sylow

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×#595

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₁₆

—

104

C₁₆

—

105

C₁₆

#63242

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

D₄×#595

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

136

—

115

C₈

C₂×#63242

yes

characteristic

116

C₈

#10587680

yes

characteristic

117

C₂²

#11763850

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#11763837

yes

characteristic

122

C₄

#11763855

yes

characteristic

123

C₂

#14544382

yes

characteristic

124

C₂

#14545189

yes

—

	subgroup	G/H
center	C₂²	#11763850
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#367	C₄
socle, prime radical	C₂²×C₁₇	#442
cosocle	C₈×Dic₁₇	C₂³
hypercenter	#367	#595
hypocenter	C₁₇	#12066
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

#11763850

commutator

C₈×C₁₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#367

C₄

socle, prime radical

C₂²×C₁₇

#442

cosocle

C₈×Dic₁₇

C₂³

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
28. Dic₁₃₆	C₈	semidirect
91. C₁₇	#12066	semidirect

H ≅ G/N

type

28. Dic₁₃₆

C₈

semidirect

91. C₁₇

#12066

semidirect

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

subgroup	index
#14544379	2
#14543809	2
#14543745	2
#14543700	2
#14530654	2
#12066	17

subgroup

index

Subgroups — #17960296

Conjugacy classes of subgroups

Complements

Maximal subgroups