Finite Group Databasebeta

← #14539957

Subgroups — #14539957

Order 2176.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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#11763345C211yesmaximal, characteristic
2C2×#10587576C211yesmaximal, characteristic, complemented
3#11763210C212yesmaximal, complemented
4#11763165C212yesmaximal, complemented
5#11763137C211yesmaximal, characteristic, complemented
6#11763127C211yesmaximal, characteristic
7#11763123C211yesmaximal, characteristic
8#11763118C212yesmaximal, complemented
9#11763112C212yesmaximal, complemented
10#11762859C211yesmaximal, characteristic, complemented
11C17×#484C211yesmaximal, characteristic, complemented
12C17×C22⋊Q8C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14#10587633C2211yescharacteristic
15Q8×Dic17C2211yescharacteristic
16#10587616C2211yescharacteristic
17C2×C4⋊Dic17—21no—
18C2×C4⋊Dic17C2211yescharacteristic
19C2×#63225—21no—
20C2×C4×Dic17C2211yescharacteristic
21#10587584—21no—
22#10587579C2211yescharacteristic
23#10587578—21no—
24#10587576—21no—
25#10587576—21no—
26#10587576C2211yescharacteristic
27#10587576C2212yes—
28#10587576C2212yes—
29#10587572—21no—
30#10587567—21no—
31C17×#107C2212yes—
32C17×#103C2212yes—
33C17×C22⋊C8C2211yescharacteristic
34#10587506C2212yes—
35C22⋊#3622C2211yescharacteristic
36#10587492C2212yes—
37#10587489C2212yes—
38#10587485C2212yes—
39C8×Dic17C2212yes—
40#10587480C2212yes—
41#10587479C2212yes—
42#10587479C2212yes—
43C2×Q8×C17C2311yescharacteristic
44C22×C4×C17—21no—
45C22×C4×C17C2311yescharacteristic
46C2×C22⋊D17—21no—
47C22×Dic17—21no—
48C22×Dic17D411yescharacteristic
49C22×Dic17D411yescharacteristic
50C2×D68C2311yescharacteristic
51C2×C4×D17—21no—
52C2×C8×C17C2312yes—
53C4⋊C4×C17—21no—
54C4⋊C4×C17—21no—
55C4⋊C4×C17C2311yescharacteristic
56C4⋊C4×C17C2312yes—
57C22⋊C4×C17—21no—
58#63227—21no—
59#63227—21no—
60#63227—21no—
61#63227—21no—
62C4⋊Dic17—21no—
63C4⋊Dic17—21no—
64C4⋊Dic17—21no—
65C4⋊Dic17—22no—
66C4⋊Dic17—22no—
67C4⋊Dic17C2311yescharacteristic
68C4⋊Dic17C2312yes—
69#63225—21no—
70#63225—22no—
71#63225—22no—
72C4×Dic17—21no—
73C4×Dic17D411yescharacteristic
74C4×Dic17D411yescharacteristic
75C4×Dic17C2311yescharacteristic
76C4×Dic17C2311yescharacteristic
77C2×#3622C2312yes—
78C23×C17C2×D411yescharacteristic
79C22×D17—21no—
80Q8×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—22no—
85C2×C4×C17—42no—
86C2×C4×C17C2×D411yescharacteristic
87C2×C4×C17C2411yescharacteristic
88C22⋊D17—42no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—21no—
92C2×Dic17—41no—
93C2×Dic17—42no—
94C2×Dic17C2×D411yescharacteristic
95C2×Dic17C2×D411yescharacteristic
96C2×Dic17C2×D411yescharacteristic
97C2×Dic17C2×D411yescharacteristic
98D68—22no—
99C4×D17—42no—
100C8×C17—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—41no—
112C4×C17—41no—
113C4×C17—42no—
114C4×C172−1+412yes—
115Dic17—21no—
116Dic17—21no—
117Dic17—21no—
118Dic17—21no—
119Dic17—42no—
120Dic17—41no—
121#529—171no—
122C2×#125—171no—
123#499—172no—
124#488—172no—
125#484D1711yescharacteristic, complemented
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—172no—
139#125—172no—
140#125—171no—
141#125—341no—
142#125—341no—
143#124—171no—
144#123—341no—
145C22⋊Q8C2×D1711yescharacteristic
146#121—171no—
147C4×Q8—171no—
148C4×D4—171no—
149C4×D4—341no—
150#117—341no—
151C2×C4⋊C4—171no—
152C2×C4⋊C4—341no—
153C2×C4⋊C4—341no—
154C2×C4⋊C4C2×D1711yescharacteristic
155C2×C42—171no—
156#107—172no—
157#107C2×D1712yes—
158#106—172no—
159#106—172no—
160C4⋊C8—172no—
161#103—172no—
162#103C2×D1712yes—
163#102—172no—
164#102—172no—
165C22⋊C8—171no—
166C22⋊C8C2×D1711yescharacteristic
167C4×C8—172no—
168C17#323211yescharacteristic, complemented, Sylow
169C2×Q8C22×D1711yescharacteristic
170C2×D4—171no—
171C2×D4—341no—
172C22×C4—21no—
173C22×C4—171no—
174C22×C4—171no—
175C22×C4—341no—
176C22×C4—341no—
177C22×C4C22×D1711yescharacteristic
178C2×C8—172no—
179C2×C8C22×D1712yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—172no—
183C4⋊C4—171no—
184C4⋊C4—342no—
185C4⋊C4—342no—
186C4⋊C4—342no—
187C4⋊C4—342no—
188C4⋊C4—341no—
189C4⋊C4—341no—
190C4⋊C4—341no—
191C4⋊C4—341no—
192C4⋊C4C22×D1712yes—
193C4⋊C4C22×D1711yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—341no—
196C22⋊C4—341no—
197C22⋊C4—341no—
198C22⋊C4—341no—
199C42—171no—
200C42—171no—
201C42—171no—
202C42—171no—
203C42—341no—
204C23—341no—
205C23D4×D1711yescharacteristic
206Q8—22no—
207D4—342no—
208D4—682no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—171no—
215C2×C4—171no—
216C2×C4—171no—
217C2×C4—171no—
218C2×C4—342no—
219C2×C4—341no—
220C2×C4—342no—
221C2×C4—682no—
222C2×C4—681no—
223C2×C4—682no—
224C2×C4D4×D1711yescharacteristic
225C2×C4C23×D1711yescharacteristic
226C8—22no—
227C8—342no—
228C22—21no—
229C22—21no—
230C22—681no—
231C22—681no—
232C22—681no—
233C22#1058760512yes—
234C22C2×D4×D1711yescharacteristic
235C4—21no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—341no—
240C4—341no—
241C4—341no—
242C4—341no—
243C4—682no—
244C4—681no—
245C4#1058769712yes—
246C2—22no—
247C2—1361no—
248C2#1176338111yescharacteristic
249C2C2×#1058760511yescharacteristic
250C2#1176360511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#484C2
socleC22×C17C22×D4
prime radical#10587616C22
hypercenter#484D17
hypocenterC17#3232
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#10587576C2semidirect
3. #11763210C2semidirect
4. #11763165C2semidirect
5. #11763137C2semidirect
8. #11763118C2semidirect
9. #11763112C2semidirect
10. #11762859C2semidirect
11. C17×#484C2semidirect
125. #484D17semidirect
168. C17#3232semidirect

Maximal subgroups

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

subgroupindex
#117633452
C2×#105875762
#117632102
#117631652
#117631372
#117631272
#117631232
#117631182
#117631122
#117628592
C17×#4842
#323217