Finite Group Databasebeta

← #14536815

Subgroups — #14536815

Order 2176.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#10587582C211yesmaximal, characteristic, complemented
2C2×#10587624C211yesmaximal, characteristic
3#11762745C212yesmaximal, complemented
4#11762741C211yesmaximal, characteristic, complemented
5#11762739C211yesmaximal, characteristic, complemented
6#11762725C212yesmaximal, complemented
7#11762724C212yesmaximal, complemented
8#11762723C211yesmaximal, characteristic, complemented
9#11762721C212yesmaximal, complemented
10#11762671C211yesmaximal, characteristic, complemented
11C17×#394C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×C22⋊D17C2212yes—
14C22×C4×D17C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22#10587624—24no—
23C2×#63227C2211yescharacteristic
24C2×#63227C2211yescharacteristic
25C2×#63227C2212yes—
26C2×C4⋊Dic17C2211yescharacteristic
27C2×C4⋊Dic17C2212yes—
28C2×#63225C2211yescharacteristic
29C2×#63225C2211yescharacteristic
30C2×#63225C2212yes—
31#10587582—24no—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2312yes—
38C23×D17C2311yescharacteristic
39C2×D4×C17—22no—
40C2×D4×C17—42no—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—24no—
46C2×C22⋊D17—42no—
47C2×C22⋊D17—42no—
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2312yes—
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2311yescharacteristic
53C2×C4×D17—42no—
54C2×C4×D17D414yes—
55C2×Dic34—22no—
56C2×Dic34—42no—
57C4⋊C4×C17—22no—
58C22⋊C4×C17—42no—
59C22⋊C4×C17—42no—
60C22⋊Dic17—24no—
61C22⋊Dic17—42no—
62C22⋊Dic17—42no—
63#63227—22no—
64#63227—22no—
65#63227—42no—
66C4⋊Dic17—22no—
67C4⋊Dic17—42no—
68#63225—22no—
69#63225—22no—
70#63225—42no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17C2411yescharacteristic
79C22×D17—22no—
80C22×D17—21no—
81C22×D17—21no—
82C22×D17—22no—
83C22×D17C2×D412yes—
84D4×C17—44no—
85C2×C4×C17—22no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17C2×D412yes—
95C22⋊D17—48no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—21no—
107C2×Dic17—22no—
108C2×Dic17C2×D412yes—
109C4×D17—44no—
110Dic34—44no—
111#2816—171nomaximal, Sylow
112C22×C17—42no—
113C22×C17—42no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17—42no—
119C22×C17C22×D411yescharacteristic
120C22×C17H(5,𝔽2)12yes—
121C22×C17H(5,𝔽2)11yescharacteristic
122C22×C17H(5,𝔽2)11yescharacteristic
123C22×C172−1+412yes—
124C2×D17—22no—
125C2×D17—22no—
126C2×D17—22no—
127C2×D17—42no—
128C2×D17—42no—
129C4×C17—22no—
130C4×C17—41no—
131C4×C17—41no—
132C4×C17—42no—
133Dic17—22no—
134Dic17—41no—
135Dic17—41no—
136Dic17—42no—
137Dic17—42no—
138C2×C22⋊Q8—171no—
139C2×#121—171no—
140#399—172no—
141#398—171no—
142#398—171no—
143#394—171no—
144#394—172no—
145#394—172no—
146#394D1711yescharacteristic, complemented
147#393—172no—
148#381—171no—
149C2×C17—82no—
150C2×C17#53411yescharacteristic
151C2×C17#53511yescharacteristic
152C2×C17#53611yescharacteristic
153C2×C17#56212yes—
154C2×C17#56212yes—
155D17—42no—
156C22×Q8—171no—
157C22×D4—172no—
158C22×D4C2×D1711yescharacteristic
159C23×C4—171no—
160C22⋊Q8—344no—
161#121—344no—
162C2×C4⋊C4—171no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—171no—
165C2×C4⋊C4—172no—
166C2×C4⋊C4—172no—
167C2×C4⋊C4C2×D1711yescharacteristic
168C2×C22⋊C4—172no—
169C2×C22⋊C4—171no—
170C2×C22⋊C4—171no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4C2×D1712yes—
175C2×C22⋊C4C2×D1712yes—
176#95—172no—
177#95—171no—
178#95—172no—
179#95—172no—
180#95C2×D1711yescharacteristic
181C17#281611yescharacteristic, complemented, Sylow
182C24—171no—
183C24C22×D1712yes—
184C2×Q8—342no—
185C2×Q8—682no—
186C2×D4—22no—
187C2×D4—42no—
188C2×D4—344no—
189C2×D4—682no—
190C2×D4—682no—
191C22×C4—171no—
192C22×C4—174no—
193C22×C4—172no—
194C22×C4—171no—
195C22×C4—172no—
196C22×C4—171no—
197C22×C4—682no—
198C22×C4C22×D1712yes—
199C22×C4C22×D1711yescharacteristic
200C22×C4C22×D1711yescharacteristic
201C22×C4C22×D1711yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—342no—
204C4⋊C4—342no—
205C4⋊C4—342no—
206C4⋊C4—682no—
207C4⋊C4—682no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—342no—
211C22⋊C4—342no—
212C22⋊C4—344no—
213C22⋊C4—682no—
214C22⋊C4—682no—
215C22⋊C4—682no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—172no—
224C23—342no—
225C23—341no—
226C23—341no—
227C23—342no—
228C23C23×D1711yescharacteristic
229Q8—684no—
230D4—44no—
231D4—688no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—172no—
242C2×C4—342no—
243C2×C4—341no—
244C2×C4—342no—
245C2×C4—341no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—341no—
249C2×C4—342no—
250C2×C4—341no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—684no—
255C2×C4D4×D1712yes—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—342no—
264C22—342no—
265C22—342no—
266C22—682no—
267C22—682no—
268C22C2×D4×D1711yescharacteristic
269C22#1058769111yescharacteristic
270C22#1058769112yes—
271C22#1058769611yescharacteristic
272C22#1058769712yes—
273C4—22no—
274C4—42no—
275C4—41no—
276C4—41no—
277C4—342no—
278C4—682no—
279C4—681no—
280C4—682no—
281C4—681no—
282C2—82no—
283C2—682no—
284C2#1176336911yescharacteristic
285C2#1176337511yescharacteristic
286C2#1176337911yescharacteristic
287C2#1176355412yes—
288C2#1176355612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC2×#10587624C2
hypercenter#394D17
hypocenterC17#2816
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#10587582C2semidirect
3. #11762745C2semidirect
4. #11762741C2semidirect
5. #11762739C2semidirect
6. #11762725C2semidirect
7. #11762724C2semidirect
8. #11762723C2semidirect
9. #11762721C2semidirect
10. #11762671C2semidirect
11. C17×#394C2semidirect
146. #394D17semidirect
181. C17#2816semidirect

Maximal subgroups

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

subgroupindex
C2×#105875822
C2×#105876242
#117627452
#117627412
#117627392
#117627252
#117627242
#117627232
#117627212
#117626712
C17×#3942
#281617