Finite Group Databasebeta

← #14539561

Subgroups — #14539561

Order 2176.

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

normal subgroups only

This group has 420 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×#10587610C211yesmaximal, characteristic, complemented
2#11763150C212yesmaximal, complemented
3#11763137C211yesmaximal, characteristic, complemented
4#11763120C211yesmaximal, characteristic, complemented
5#11763048C212yesmaximal, complemented
6#11763036C212yesmaximal, complemented
7#11762971C212yesmaximal, complemented
8#11762964C211yesmaximal, characteristic, complemented
9#11762947C211yesmaximal, characteristic, complemented
10C2×C22⋊#3622C211yesmaximal, characteristic, complemented
11C17×#525C211yesmaximal, characteristic
12C2×#41×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C22×Dic34C2211yescharacteristic
15C17×#123—21no—
16C17×C22⋊Q8C2211yescharacteristic
17C17×C22⋊Q8C2211yescharacteristic
18C17×#121C2211yescharacteristic
19C17×#121C2211yescharacteristic
20C17×#120—21no—
21C4×D4×C17—21no—
22C4×D4×C17—21no—
23C17×#117C2211yescharacteristic
24C2×C22⋊Dic17—21no—
25C2×#63231C2212yescomplemented
26C2×#63229C2212yescomplemented
27C2×C4⋊Dic17C2211yescharacteristic
28#10587610—21no—
29#10587610—21no—
30#10587610C2212yes—
31#10587610C2212yes—
32C2×#63225—21no—
33C22×#3622C2212yes—
34#10587506C2212yescomplemented
35#10587503C2212yescomplemented
36C22⋊#3622C2211yescharacteristic
37C22⋊#3622C2211yescharacteristic
38C22⋊#3622C2212yes—
39#10587481C2212yescomplemented
40#10587481C2212yescomplemented
41#10587479C2212yescomplemented
42#10587478C2212yescomplemented
43C24×C17D411yescharacteristic
44#41×C17—42no—
45C2×Q8×C17C2311yescharacteristic
46C2×D4×C17—21no—
47C2×D4×C17—21no—
48C2×D4×C17—21no—
49C2×D4×C17C2311yescharacteristic
50C22×C4×C17—21no—
51C22×C4×C17—42no—
52C22×C4×C17D411yescharacteristic
53C22×C4×C17D411yescharacteristic
54C22×C4×C17D411yescharacteristic
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×Dic17—21no—
59C22×Dic17—21no—
60C2×Dic34—21no—
61C2×Dic34—42no—
62C2×Dic34C2312yes—
63C4⋊C4×C17—21no—
64C4⋊C4×C17—21no—
65C4⋊C4×C17C2311yescharacteristic
66C4⋊C4×C17C2311yescharacteristic
67C22⋊C4×C17—21no—
68C22⋊C4×C17—21no—
69C22⋊C4×C17—21no—
70C22⋊C4×C17—21no—
71C22⋊C4×C17—21no—
72C42×C17—21no—
73C22⋊Dic17—22no—
74C22⋊Dic17—22no—
75#63231—42no—
76#63229—42no—
77C4⋊Dic17—21no—
78C4⋊Dic17C2312yes—
79#63225—22no—
80#63225—22no—
81C2×#3622—42no—
82C2×#3622C2312yes—
83C2×#3622C2312yes—
84C23×C17—22no—
85C23×C17—22no—
86C23×C17—21no—
87C23×C17—41no—
88C23×C17C2×D411yescharacteristic
89C23×C17C2×D411yescharacteristic
90C23×C17C2×D411yescharacteristic
91Q8×C17—22no—
92D4×C17—22no—
93D4×C17—42no—
94D4×C17—42no—
95D4×C17—42no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—41no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—42no—
106C2×C4×C17C2×D411yescharacteristic
107C2×C4×C17C2×D411yescharacteristic
108C2×C4×C17C2×D411yescharacteristic
109C2×C4×C17C2411yescharacteristic
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—42no—
113C2×Dic17—42no—
114Dic34—22no—
115Dic34—22no—
116Dic34—42no—
117#3622—22no—
118#3622—22no—
119#3190—171nomaximal, Sylow
120C22×C17—21no—
121C22×C17—21no—
122C22×C17—21no—
123C22×C17—21no—
124C22×C17—21no—
125C22×C17—21no—
126C22×C17—21no—
127C22×C17—44no—
128C22×C17—41no—
129C22×C17—41no—
130C22×C17—41no—
131C22×C17#13512yes—
132C22×C17#13712yes—
133C22×C17C22×D411yescharacteristic
134C4×C17—21no—
135C4×C17—21no—
136C4×C17—21no—
137C4×C17—41no—
138C4×C17—41no—
139C4×C17—41no—
140C4×C17H(5,𝔽2)12yes—
141Dic17—42no—
142Dic17—42no—
143#525D1711yescharacteristic
144C2×C22⋊Q8—171no—
145#485—172no—
146#483—171no—
147#480—171no—
148#467—172no—
149#465—172no—
150#452—172no—
151#451—171no—
152#448—171no—
153C2×C22⋊C8—171no—
154C2×C17—22no—
155C2×C17—22no—
156C2×C17—41no—
157C2×C17—81no—
158C2×C17#53411yescharacteristic
159C2×C17C2×#13511yescharacteristic
160C2×C17C2×#13711yescharacteristic
161C2×#41C2×D1711yescharacteristic
162C22×Q8—171no—
163C23×C4C2×D1711yescharacteristic
164C2×Q16—172no—
165C2×SD16—172no—
166C22×C8—172no—
167#123—21no—
168C22⋊Q8—172no—
169C22⋊Q8—172no—
170C22⋊Q8—341no—
171C22⋊Q8—341no—
172C22⋊Q8C2×D1711yescharacteristic
173C22⋊Q8C2×D1711yescharacteristic
174#121C2×D1711yescharacteristic
175#121C2×D1711yescharacteristic
176#120—21no—
177C4×D4—21no—
178C4×D4—21no—
179#117C2×D1711yescharacteristic
180C2×C4⋊C4—171no—
181C2×C4⋊C4—341no—
182C2×C22⋊C4—341no—
183#107—172no—
184#106—172no—
185#103—172no—
186#103—172no—
187#103—172no—
188#102—172no—
189C22⋊C8—171no—
190C22⋊C8—171no—
191C22⋊C8—172no—
192C17#319011yescharacteristic, complemented, Sylow
193C24C22⋊D1711yescharacteristic
194#41—42no—
195C2×Q8—172no—
196C2×Q8—341no—
197C2×Q8—682no—
198C2×Q8C22×D1711yescharacteristic
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4C22×D1711yescharacteristic
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—341no—
206C22×C4—341no—
207C22×C4C22⋊D1711yescharacteristic
208C22×C4C22⋊D1711yescharacteristic
209C22×C4C22⋊D1711yescharacteristic
210C22×C4C22×D1711yescharacteristic
211C22×C4C22×D1711yescharacteristic
212C22×C4C22×D1711yescharacteristic
213Q16—682no—
214SD16—682no—
215C2×C8—172no—
216C2×C8—172no—
217C2×C8—682no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—172no—
221C4⋊C4—341no—
222C4⋊C4—342no—
223C4⋊C4—342no—
224C4⋊C4C22×D1711yescharacteristic
225C4⋊C4C22×D1711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—21no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—342no—
232C22⋊C4—342no—
233C42—21no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—41no—
238C23C2×C22⋊D1711yescharacteristic
239C23C2×C22⋊D1711yescharacteristic
240C23C2×C22⋊D1711yescharacteristic
241Q8—22no—
242Q8—342no—
243Q8—342no—
244Q8—682no—
245D4—22no—
246D4—42no—
247D4—42no—
248D4—42no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—682no—
262C2×C4—682no—
263C2×C4C2×C22⋊D1711yescharacteristic
264C2×C4C2×C22⋊D1711yescharacteristic
265C2×C4C2×C22⋊D1711yescharacteristic
266C2×C4C23×D1711yescharacteristic
267C8—342no—
268C8—342no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22#1058763612yes—
281C22#1058763712yes—
282C22C22×C22⋊D1711yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—682no—
290C4—682no—
291C4#1058769112yes—
292C2—22no—
293C2—22no—
294C2—41no—
295C2—81no—
296C2#1176336811yescharacteristic
297C2C2×#1058763611yescharacteristic
298C2C2×#1058763711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#525C2
socleC22×C17C22×D4
hypercenter#525D17
hypocenterC17#3190
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#10587610C2semidirect
2. #11763150C2semidirect
3. #11763137C2semidirect
4. #11763120C2semidirect
5. #11763048C2semidirect
6. #11763036C2semidirect
7. #11762971C2semidirect
8. #11762964C2semidirect
9. #11762947C2semidirect
10. C2×C22⋊#3622C2semidirect
25. C2×#63231C22semidirect
26. C2×#63229C22semidirect
34. #10587506C22semidirect
35. #10587503C22semidirect
39. #10587481C22semidirect
40. #10587481C22semidirect
41. #10587479C22semidirect
42. #10587478C22semidirect
192. C17#3190semidirect

Maximal subgroups

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

subgroupindex
C2×#105876102
#117631502
#117631372
#117631202
#117630482
#117630362
#117629712
#117629642
#117629472
C2×C22⋊#36222
C17×#5252
#319017