Finite Group Databasebeta

← #14539952

Subgroups — #14539952

Order 2176.

334 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 121 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#11763343C211yesmaximal, characteristic
2#11763208C212yesmaximal, complemented
3#11763166C212yesmaximal, complemented
4#11763144C211yesmaximal, characteristic, complemented
5#11763139C211yesmaximal, characteristic, complemented
6#11763127C211yesmaximal, characteristic
7#11763122C211yesmaximal, characteristic
8#11763113C212yesmaximal, complemented
9#11763106C212yesmaximal, complemented
10#11762862C211yesmaximal, characteristic, complemented
11C2×C17×#125C211yesmaximal, characteristic, complemented
12C17×#125—21no—
13C17×#125—21no—
14C17×#125C2212yes—
15C17×#125C2212yes—
16C2×C4⋊C4×C17—21no—
17C2×C4⋊C4×C17—21no—
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C42×C17C2211yescharacteristic
21#10587616C2211yescharacteristic
22#10587615—21no—
23C4×C22⋊D17—21no—
24#10587610C2211yescharacteristic
25#10587562—21no—
26#10587561C2211yescharacteristic
27C4×D68C2211yescharacteristic
28#10587558—21no—
29#10587556C2211yescharacteristic
30C4×Dic34C2211yescharacteristic
31C22⋊#3622C2211yescharacteristic
32C22⋊#3622C2211yescharacteristic
33#10587481C2212yes—
34#10587481C2212yes—
35#10587480C2212yes—
36#10587480C2212yes—
37#10587479C2212yes—
38#10587479C2212yes—
39#10587478C2212yes—
40#10587478C2212yes—
41C4⋊#3622C2212yes—
42C4×#3622C2212yes—
43C22×C4×C17—21no—
44C22×C4×C17—21no—
45C22×C4×C17D411yescharacteristic
46C22×C4×C17D411yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—21no—
49C2×D68C2311yescharacteristic
50C2×C4×D17—21no—
51C2×Dic34C2311yescharacteristic
52C4⋊C4×C17—21no—
53C4⋊C4×C17—21no—
54C4⋊C4×C17—22no—
55C4⋊C4×C17—22no—
56C4⋊C4×C17—22no—
57C4⋊C4×C17—22no—
58C4⋊C4×C17C2312yes—
59C4⋊C4×C17C2312yes—
60C42×C17D411yescharacteristic
61C42×C17D411yescharacteristic
62C42×C17C2311yescharacteristic
63C42×C17C2311yescharacteristic
64C22⋊Dic17—21no—
65#63227—21no—
66#63227—21no—
67#63227—21no—
68#63227—21no—
69C4⋊Dic17C2311yescharacteristic
70C4⋊Dic17C2311yescharacteristic
71#63225—21no—
72#63225—21no—
73#63225—21no—
74#63225—21no—
75C4×Dic17—21no—
76C2×#3622C2312yes—
77C2×#3622C2312yes—
78C23×C17C2×D411yescharacteristic
79C22×D17—21no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—41no—
85C2×C4×C17—42no—
86C2×C4×C17—42no—
87C2×C4×C17C2×D411yescharacteristic
88C2×C4×C17C2×D411yescharacteristic
89C2×C4×C17C2×D411yescharacteristic
90C2×C4×C17C2×D411yescharacteristic
91C2×C4×C17C2×D411yescharacteristic
92C2×C4×C17C2411yescharacteristic
93C22⋊D17—42no—
94C2×Dic17—21no—
95C2×Dic17—21no—
96C2×Dic17—21no—
97D68—22no—
98C4×D17—42no—
99Dic34—22no—
100#3622—22no—
101#3622—22no—
102#3232—171nomaximal, Sylow
103C22×C17—21no—
104C22×C17—21no—
105C22×C17#13512yes—
106C22×C17C22×D411yescharacteristic
107C2×D17—41no—
108C2×D17—41no—
109C2×D17—41no—
110C4×C17—21no—
111C4×C17—21no—
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—21no—
115C4×C17—42no—
116C4×C17—42no—
117C4×C172−1+412yes—
118Dic17—41no—
119Dic17—41no—
120Dic17—41no—
121#529—171no—
122C2×#125D1711yescharacteristic, complemented
123#499—172no—
124#488—172no—
125#484—171no—
126#483—171no—
127#481—171no—
128#480—171no—
129#479—172no—
130#478—172no—
131#431—171no—
132C2×C17—22no—
133C2×C17#53611yescharacteristic
134C2×C17C2×#13511yescharacteristic
135C2×C17#57511yescharacteristic
136D17—81no—
137#126—341no—
138#125—21no—
139#125—21no—
140#125—171no—
141#125C2×D1712yes—
142#125C2×D1712yes—
143#124—171no—
144#123—341no—
145C22⋊Q8—171no—
146#121—171no—
147C4×Q8—171no—
148C4×D4—171no—
149C4×D4—341no—
150#117—341no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4C2×D1711yescharacteristic
154C2×C4⋊C4C2×D1711yescharacteristic
155C2×C42C2×D1711yescharacteristic
156#107—172no—
157#107—172no—
158#106—172no—
159#106—172no—
160C4⋊C8—172no—
161#103—172no—
162#103—172no—
163#102—172no—
164#102—172no—
165C22⋊C8—171no—
166C22⋊C8—171no—
167C4×C8—172no—
168C17#323211yescharacteristic, complemented, Sylow
169C2×Q8—171no—
170C2×D4—171no—
171C2×D4—341no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—341no—
175C22×C4C22⋊D1711yescharacteristic
176C22×C4C22⋊D1711yescharacteristic
177C22×C4C22×D1711yescharacteristic
178C2×C8—172no—
179C2×C8—172no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—171no—
187C4⋊C4—171no—
188C4⋊C4—341no—
189C4⋊C4—341no—
190C4⋊C4—341no—
191C4⋊C4—341no—
192C4⋊C4C22×D1712yes—
193C4⋊C4C22×D1712yes—
194C22⋊C4—341no—
195C22⋊C4—341no—
196C22⋊C4—341no—
197C22⋊C4—341no—
198C22⋊C4—341no—
199C42—341no—
200C42C22⋊D1711yescharacteristic
201C42C22⋊D1711yescharacteristic
202C42C22×D1711yescharacteristic
203C42C22×D1711yescharacteristic
204C23—341no—
205C23C2×C22⋊D1711yescharacteristic
206Q8—342no—
207D4—342no—
208D4—682no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—341no—
217C2×C4—341no—
218C2×C4—341no—
219C2×C4—682no—
220C2×C4C2×C22⋊D1711yescharacteristic
221C2×C4C2×C22⋊D1711yescharacteristic
222C2×C4C2×C22⋊D1711yescharacteristic
223C2×C4C2×C22⋊D1711yescharacteristic
224C2×C4C2×C22⋊D1711yescharacteristic
225C2×C4C23×D1711yescharacteristic
226C8—342no—
227C8—342no—
228C22—21no—
229C22—21no—
230C22—681no—
231C22—681no—
232C22—681no—
233C22#1058763612yes—
234C22C22×C22⋊D1711yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—681no—
243C4—681no—
244C4—681no—
245C4#1058769412yes—
246C2—22no—
247C2—1361no—
248C2#1176338311yescharacteristic
249C2C2×#1058763611yescharacteristic
250C2#1176360711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×#125C2
socleC22×C17C22×D4
prime radical#10587616C22
hypercenterC2×#125D17
hypocenterC17#3232
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #11763208C2semidirect
3. #11763166C2semidirect
4. #11763144C2semidirect
5. #11763139C2semidirect
8. #11763113C2semidirect
9. #11763106C2semidirect
10. #11762862C2semidirect
11. C2×C17×#125C2semidirect
122. C2×#125D17semidirect
168. C17#3232semidirect

Maximal subgroups

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

subgroupindex
#117633432
#117632082
#117631662
#117631442
#117631392
#117631272
#117631222
#117631132
#117631062
#117628622
C2×C17×#1252
#323217