Finite Group Databasebeta

← #14539963

Subgroups — #14539963

Order 2176.

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

normal subgroups only

This group has 332 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#11763150C212yesmaximal, complemented
5#11763133C212yesmaximal
6#11763101C211yesmaximal, characteristic, complemented
7#11763096C211yesmaximal, characteristic, complemented
8#11763091C211yesmaximal, characteristic, complemented
9#11763086C211yesmaximal, characteristic, complemented
10#11762862C211yesmaximal, characteristic, complemented
11C17×#533C211yesmaximal, characteristic, complemented
12C17×C4⋊Q8C2211yescharacteristic
13C17×C4⋊Q8C2211yescharacteristic
14C17×#125C2212yes—
15C17×C22⋊Q8—21no—
16C17×C22⋊Q8—21no—
17C4×Q8×C17—21no—
18C4×Q8×C17—21no—
19C17×#117C2212yes—
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⋊#3622C2212yes—
32#10587481C2212yes—
33#10587481C2212yes—
34#10587480C2212yes—
35#10587480C2212yes—
36#10587479C2212yes—
37#10587479C2212yes—
38#10587478C2212yes—
39#10587478C2212yes—
40C4⋊#3622C2211yescharacteristic
41C4⋊#3622C2211yescharacteristic
42C4×#3622C2212yes—
43C2×Q8×C17—21no—
44C2×Q8×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—22no—
53C4⋊C4×C17—22no—
54C4⋊C4×C17—21no—
55C4⋊C4×C17—21no—
56C4⋊C4×C17C2312yes—
57C4⋊C4×C17C2312yes—
58C22⋊C4×C17—22no—
59C42×C17—22no—
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—
80Q8×C17—42no—
81Q8×C17—42no—
82C2×C4×C17—22no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—41no—
88C2×C4×C17C2×D411yescharacteristic
89C2×C4×C17C2×D411yescharacteristic
90C2×C4×C17C2×D411yescharacteristic
91C2×C4×C17C2×D411yescharacteristic
92C2×C4×C17C2×D411yescharacteristic
93C2×C4×C17C2411yescharacteristic
94C22⋊D17—42no—
95C2×Dic17—21no—
96C2×Dic17—21no—
97C2×Dic17—21no—
98D68—21no—
99D68—21no—
100C4×D17—42no—
101Dic34—21no—
102Dic34—21no—
103#3622—22no—
104#3622—22no—
105#3233—171nomaximal, Sylow
106C22×C17—21no—
107C22×C17—21no—
108C22×C17—21no—
109C22×C17C22×D411yescharacteristic
110C2×D17—41no—
111C2×D17—41no—
112C2×D17—41no—
113C4×C17—21no—
114C4×C17—21no—
115C4×C17—21no—
116C4×C17—21no—
117C4×C17—42no—
118C4×C17—42no—
119C4×C17#13512yes—
120C4×C172−1+411yescharacteristic
121C4×C172−1+411yescharacteristic
122Dic17—41no—
123Dic17—41no—
124Dic17—41no—
125#533D1711yescharacteristic, complemented
126#529—171no—
127#499—172no—
128#488—172no—
129#485—172no—
130#482—172no—
131#477—171no—
132#476—171no—
133#475—171no—
134#474—171no—
135#431—171no—
136C2×C17—41no—
137C2×C17#53611yescharacteristic
138C2×C17C2×#13511yescharacteristic
139C2×C17#57511yescharacteristic
140D17—81no—
141C4⋊Q8C2×D1711yescharacteristic
142C4⋊Q8C2×D1711yescharacteristic
143#126—341no—
144#125—171no—
145#125C2×D1712yes—
146#124—171no—
147#123—341no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150C22⋊Q8—171no—
151#121—171no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—171no—
155C4×D4—171no—
156C4×D4—341no—
157#117—341no—
158#117C2×D1712yes—
159C2×C42C2×D1711yescharacteristic
160#107—172no—
161#107—172no—
162#106—172no—
163#106—172no—
164C4⋊C8—171no—
165C4⋊C8—171no—
166#103—172no—
167#103—172no—
168#102—172no—
169#102—172no—
170C22⋊C8—172no—
171C4×C8—172no—
172C17#323311yescharacteristic, complemented, Sylow
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—171no—
176C2×D4—171no—
177C2×D4—341no—
178C22×C4—341no—
179C22×C4C22⋊D1711yescharacteristic
180C22×C4C22⋊D1711yescharacteristic
181C22×C4C22×D1711yescharacteristic
182C2×C8—172no—
183C2×C8—172no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—171no—
189C4⋊C4—171no—
190C4⋊C4—341no—
191C4⋊C4—341no—
192C4⋊C4—341no—
193C4⋊C4—341no—
194C4⋊C4C22×D1712yes—
195C4⋊C4C22×D1712yes—
196C22⋊C4—22no—
197C22⋊C4—341no—
198C22⋊C4—341no—
199C22⋊C4—341no—
200C22⋊C4—341no—
201C22⋊C4—341no—
202C42—22no—
203C42—341no—
204C42C22⋊D1711yescharacteristic
205C42C22⋊D1711yescharacteristic
206C42C22×D1711yescharacteristic
207C42C22×D1711yescharacteristic
208C23—341no—
209C23C2×C22⋊D1711yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—341no—
213Q8—341no—
214D4—341no—
215D4—341no—
216D4—682no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—341no—
224C2×C4—341no—
225C2×C4—341no—
226C2×C4—682no—
227C2×C4C2×C22⋊D1711yescharacteristic
228C2×C4C2×C22⋊D1711yescharacteristic
229C2×C4C2×C22⋊D1711yescharacteristic
230C2×C4C2×C22⋊D1711yescharacteristic
231C2×C4C2×C22⋊D1711yescharacteristic
232C2×C4C23×D1711yescharacteristic
233C8—342no—
234C8—342no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—681no—
239C22—681no—
240C22—681no—
241C22C22×C22⋊D1711yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—42no—
248C4—681no—
249C4—681no—
250C4—681no—
251C4#1058763612yes—
252C4#1058769411yescharacteristic
253C4#1058769411yescharacteristic
254C2—41no—
255C2—1361no—
256C2#1176338311yescharacteristic
257C2C2×#1058763611yescharacteristic
258C2#1176360711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#533C2
socleC22×C17C22×D4
prime radical#10587616C22
hypercenter#533D17
hypocenterC17#3233
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #11763208C2semidirect
3. #11763166C2semidirect
4. #11763150C2semidirect
6. #11763101C2semidirect
7. #11763096C2semidirect
8. #11763091C2semidirect
9. #11763086C2semidirect
10. #11762862C2semidirect
11. C17×#533C2semidirect
125. #533D17semidirect
172. C17#3233semidirect

Maximal subgroups

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

subgroupindex
#117633432
#117632082
#117631662
#117631502
#117631332
#117631012
#117630962
#117630912
#117630862
#117628622
C17×#5332
#323317