Finite Group Databasebeta

← #17062461

Subgroups — #17062461

Order 4352.

1334 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 291 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531396C213yesmaximal
2#14530136C218yesmaximal, complemented
3C2×#11762576C213yesmaximal
4C2×C17×#543C211yesmaximal, characteristic
5C22×C22⋊Dic17C2213yes—
6C22×C22⋊Dic17C2211yescharacteristic
7#11762682—26no—
8#11762681C22112yes—
9#11762667—26no—
10#11762576C22112yes—
11C17×#543—26no—
12C17×#543C2214yes—
13C2×C17×C22⋊Q8—26no—
14C22×C22⋊C4×C17C2213yes—
15C25×C17C2311yescharacteristic
16C22×Q8×C17—22no—
17C23×C4×C17D416yes—
18C23×Dic17—23no—
19C23×Dic17C2312yes—
20C17×C22⋊Q8—224no—
21C17×C22⋊Q8—224no—
22C2×C4⋊C4×C17—26no—
23C2×C22⋊C4×C17—26no—
24C2×C22⋊C4×C17—26no—
25C2×C22⋊C4×C17D416yes—
26C2×C22⋊C4×C17C2316yes—
27C2×C22⋊Dic17—23no—
28C2×C22⋊Dic17—26no—
29C2×C22⋊Dic17—23no—
30C2×C22⋊Dic17—23no—
31C2×C22⋊Dic17Q8112yescomplemented
32C2×C22⋊Dic17C2316yes—
33C2×#63225—212no—
34#10587502—212no—
35C24×C17—82no—
36C24×C17C2×D413yes—
37C24×C17C2×D413yes—
38C24×C17C2×Q816yes—
39C24×C17#4112yes—
40C24×C17C2411yescharacteristic
41C2×Q8×C17—28no—
42C2×Q8×C17—84no—
43C22×C4×C17—26no—
44C22×C4×C17—26no—
45C22×C4×C17—86no—
46C22×C4×C17C2×D4112yes—
47C22×Dic17—26no—
48C22×Dic17—212no—
49C22×Dic17—26no—
50C22×Dic17—82no—
51C22×Dic17—86no—
52C4⋊C4×C17—224no—
53C22⋊C4×C17—212no—
54C22⋊C4×C17—212no—
55C22⋊Dic17—212no—
56C22⋊Dic17—26no—
57C22⋊Dic17—43no—
58C22⋊Dic17—41no—
59C22⋊Dic17C2×Q8112yescomplemented
60#63225—812no—
61#23565—171nomaximal, Sylow
62C23×C17—46no—
63C23×C17—42no—
64C23×C17—46no—
65C23×C17—86no—
66C23×C17—82no—
67C23×C17#12013yes—
68C23×C17#12011yescharacteristic
69C23×C17C22⋊Q816yes—
70C23×C17C22⋊Q816yes—
71C23×C17C4⋊Q816yes—
72C23×C17C4⋊Q816yes—
73C23×C17C22×D413yes—
74C23×C17C22×Q813yes—
75C23×C17C2×#4111yescharacteristic
76Q8×C17—88no—
77C2×C4×C17—212no—
78C2×C4×C17—412no—
79C2×C4×C17—412no—
80C2×C4×C17—86no—
81C2×Dic17—212no—
82C2×Dic17—412no—
83C2×Dic17—412no—
84C2×Dic17—42no—
85C2×Dic17—812no—
86C2×#543D1711yescharacteristic
87#2315—173no—
88#2062—178no—
89C2×#342—173no—
90C22×C17—26no—
91C22×C17—22no—
92C22×C17—26no—
93C22×C17—46no—
94C22×C17—46no—
95C22×C17—46no—
96C22×C17—46no—
97C22×C17—43no—
98C22×C17—41no—
99C22×C17#39516yes—
100C22×C17#39512yes—
101C22×C17C2×#12011yescharacteristic
102C22×C17C2×C22⋊Q813yes—
103C22×C17C2×C4⋊Q813yes—
104C4×C17—412no—
105Dic17—412no—
106Dic17—82no—
107#543—26no—
108#543C2×D1714yes—
109C2×C22⋊Q8—26no—
110C22×C22⋊C4—173no—
111C22×C22⋊C4—171no—
112C22×C22⋊C4C2×D1713yes—
113#385—1712no—
114#385—346no—
115#380—346no—
116#342—1712no—
117C2×C17—23no—
118C2×C17—23no—
119C2×C17—22no—
120C2×C17—26no—
121C2×C17C2×#39511yescharacteristic
122C2×C17#302612yes—
123C25C22×D1711yescharacteristic
124C22×Q8—22no—
125C23×C4—172no—
126C23×C4—343no—
127C23×C4C22⋊D1716yes—
128C22⋊Q8—224no—
129C22⋊Q8—224no—
130C2×C4⋊C4—26no—
131C2×C4⋊C4—3412no—
132C2×C22⋊C4—26no—
133C2×C22⋊C4—26no—
134C2×C22⋊C4—1712no—
135C2×C22⋊C4—176no—
136C2×C22⋊C4—346no—
137C2×C22⋊C4—343no—
138C2×C22⋊C4—343no—
139C2×C22⋊C4—343no—
140C2×C22⋊C4C22⋊D1716yes—
141C2×C22⋊C4C22×D1716yes—
142#95—3412no—
143C17#2356511yescharacteristic, complemented, Sylow
144C24—82no—
145C24Q8×D1716yes—
146C24#6325612yes—
147C24C2×C22⋊D1713yes—
148C24C2×C22⋊D1713yes—
149C24C23×D1711yescharacteristic
150C2×Q8—28no—
151C2×Q8—84no—
152C22×C4—26no—
153C22×C4—26no—
154C22×C4—86no—
155C22×C4—346no—
156C22×C4—3412no—
157C22×C4—346no—
158C22×C4—1362no—
159C22×C4—1366no—
160C22×C4C2×C22⋊D17112yes—
161C4⋊C4—224no—
162C4⋊C4—13612no—
163C22⋊C4—212no—
164C22⋊C4—212no—
165C22⋊C4—1712no—
166C22⋊C4—346no—
167C22⋊C4—3412no—
168C22⋊C4—683no—
169C22⋊C4—681no—
170C23—46no—
171C23—46no—
172C23—42no—
173C23—86no—
174C23—82no—
175C23#1058763016yes—
176C23#1058763016yes—
177C23#1058763216yes—
178C23#1058763216yes—
179C23#1058763913yes—
180C23#1058763911yescharacteristic
181C23C2×Q8×D1713yes—
182C23C2×#6325611yescharacteristic
183C23C22×C22⋊D1713yes—
184Q8—88no—
185C2×C4—212no—
186C2×C4—412no—
187C2×C4—412no—
188C2×C4—86no—
189C2×C4—3412no—
190C2×C4—6812no—
191C2×C4—6812no—
192C2×C4—682no—
193C2×C4—13612no—
194C22—22no—
195C22—26no—
196C22—26no—
197C22—46no—
198C22—43no—
199C22—41no—
200C22—46no—
201C22—46no—
202C22—46no—
203C22#1176272812yes—
204C22#1176272816yes—
205C22C2×#1058763911yescharacteristic
206C22C2×#1058763213yes—
207C22C2×#1058763013yes—
208C4—412no—
209C4—6812no—
210C4—1362no—
211C2—23no—
212C2—22no—
213C2—26no—
214C2—23no—
215C2C2×#1176272811yescharacteristic
216C2#1453818112yes—
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC24×C17C24
FrattiniC24C23×D17
FittingC2×C17×#543C2
socleC22×C17C2×#120
prime radicalC25×C17C23
hypercenterC2×#543D17
hypocenterC17#23565
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C25 ◁ C2×#543

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14530136C2semidirect
31. C2×C22⋊Dic17Q8semidirect
59. C22⋊Dic17C2×Q8semidirect
143. C17#23565semidirect

Maximal subgroups

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

subgroupindex
#145313962
#145301362
C2×#117625762
C2×C17×#5432
#2356517