Finite Group Databasebeta

← #14529323

Subgroups — #14529323

Order 2176.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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#11763368C211yesmaximal, characteristic
2#11763282C211yesmaximal, characteristic, complemented
3C2×#10587505C211yesmaximal, characteristic, complemented
4C2×#10587477C211yesmaximal, characteristic, complemented
5#11762576C211yesmaximal, characteristic, complemented
6#11762575C211yesmaximal, characteristic, complemented
7C2×C17×#99C211yesmaximal, characteristic, complemented
8C22×D4×C17C2211yescharacteristic
9C22×C22⋊D17C2211yescharacteristic
10C2×C22⋊C4×C17C2211yescharacteristic
11C2×C22⋊C4×C17C2211yescharacteristic
12#10587639C412yescomplemented
13C2×C22⋊Dic17C2211yescharacteristic
14C2×C22⋊Dic17C2211yescharacteristic
15C2×C22⋊Dic17C2211yescharacteristic
16#10587625—22no—
17#10587623—21no—
18#10587621—21no—
19#10587615C412yescomplemented
20C4×C22⋊D17—22no—
21#10587567—22no—
22C22⋊C4×D17—21no—
23#10587563—21no—
24C17×#99—22no—
25#10587505—22no—
26#10587477—22no—
27C24×C17C2×C411yescharacteristic, complemented
28C24×C17C2311yescharacteristic
29C23×D17D411yescharacteristic
30C2×D4×C17—21no—
31C2×D4×C17—21no—
32C2×D4×C17—21no—
33C2×D4×C17—21no—
34C2×D4×C17—21no—
35C2×D4×C17—21no—
36C22×C4×C17—21no—
37C22×C4×C17—21no—
38C22×C4×C17C2×C411yescharacteristic, complemented
39C2×C22⋊D17—21no—
40C2×C22⋊D17—21no—
41C2×C22⋊D17—21no—
42C2×C22⋊D17—21no—
43C2×C22⋊D17—22no—
44C2×C22⋊D17C2×C412yes—
45C2×C22⋊D17D412yes—
46C22×Dic17—21no—
47C22×Dic17—21no—
48C22×Dic17—21no—
49C22×Dic17D411yescharacteristic
50C2×C4×D17—22no—
51C22⋊C4×C17—21no—
52C22⋊C4×C17—21no—
53C22⋊C4×C17—21no—
54C22⋊C4×C17D412yescomplemented
55C22⋊Dic17—21no—
56C22⋊Dic17—21no—
57C22⋊Dic17—22no—
58C22⋊Dic17—21no—
59C22⋊Dic17—21no—
60C22⋊Dic17C2×C412yescomplemented
61C22⋊Dic17D412yescomplemented
62#63227—22no—
63#63227—22no—
64#63225—22no—
65#63225—22no—
66C4×Dic17—22no—
67C23×C17—21no—
68C23×C17—21no—
69C23×C17—21no—
70C23×C17—81no—
71C23×C17—81no—
72C23×C17C22×C411yescharacteristic
73C23×C17C2×D411yescharacteristic
74C23×C17C2×D411yescharacteristic
75C23×C17C2×D411yescharacteristic
76C23×C17C2×D411yescharacteristic
77C23×C17#4111yescharacteristic
78C23×C17#4111yescharacteristic
79C22×D17—21no—
80C22×D17—21no—
81C22×D17—81no—
82C22×D17C22⋊C412yes—
83D4×C17—41no—
84D4×C17—41no—
85D4×C17—41no—
86D4×C17—41no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—41no—
92C2×C4×C17—81no—
93C2×C4×C17—81no—
94C22⋊D17—22no—
95C22⋊D17—22no—
96C22⋊D17—22no—
97C22⋊D17—22no—
98C22⋊D17—82no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—41no—
104C2×Dic17—81no—
105C2×Dic17—81no—
106C2×Dic17—81no—
107C2×Dic17C22⋊C412yes—
108C4×D17—82no—
109#1913—171nomaximal, Sylow
110C22×C17—41no—
111C22×C17—41no—
112C22×C17—41no—
113C22×C17—41no—
114C22×C17—41no—
115C22×C17—41no—
116C22×C17—41no—
117C22×C17—42no—
118C22×C17—41no—
119C22×C17—81no—
120C22×C17—81no—
121C22×C17C2×C22⋊C411yescharacteristic
122C22×C17C4×D411yescharacteristic
123C22×C17C4×D411yescharacteristic
124C22×C17#12011yescharacteristic
125C22×C17#12111yescharacteristic
126C22×C17#12111yescharacteristic
127C22×C17#12311yescharacteristic
128C2×D17—22no—
129C2×D17—42no—
130C2×D17—42no—
131C2×D17—81no—
132C4×C17—42no—
133C4×C17—41no—
134C4×C17—81no—
135Dic17—22no—
136Dic17—42no—
137Dic17—42no—
138Dic17—81no—
139#534—171no—
140#518—171no—
141C2×#99—171no—
142C2×#99—171no—
143C2×#99D1711yescharacteristic, complemented
144#342—171no—
145#342—171no—
146C2×C17—21no—
147C2×C17—21no—
148C2×C17—21no—
149C2×C17—21no—
150C2×C17—21no—
151C2×C17—21no—
152C2×C17—21no—
153C2×C17—81no—
154C2×C17#38611yescharacteristic
155D17—42no—
156C22×D4—171no—
157C22×D4C2×D1711yescharacteristic
158#123—172no—
159#123—341no—
160#121—342no—
161#120—172no—
162#120—341no—
163C4×D4—342no—
164C4×D4—342no—
165#117—341no—
166C2×C22⋊C4—171no—
167C2×C22⋊C4—171no—
168C2×C22⋊C4—171no—
169C2×C22⋊C4—341no—
170C2×C22⋊C4C2×D1711yescharacteristic
171C2×C22⋊C4C2×D1711yescharacteristic
172#99—22no—
173#99—342no—
174#99—342no—
175C17#191311yescharacteristic, complemented, Sylow
176C24—171no—
177C24C4×D1711yescharacteristic, complemented
178C24C22×D1711yescharacteristic
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—172no—
186C2×D4—172no—
187C2×D4—341no—
188C2×D4—341no—
189C2×D4—341no—
190C2×D4—341no—
191C2×D4—342no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—171no—
195C22×C4—342no—
196C22×C4—341no—
197C22×C4—341no—
198C22×C4—341no—
199C22×C4C4×D1711yescharacteristic, complemented
200C4⋊C4—342no—
201C4⋊C4—342no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—172no—
206C22⋊C4—172no—
207C22⋊C4—342no—
208C22⋊C4—342no—
209C22⋊C4—342no—
210C22⋊C4—341no—
211C22⋊C4—341no—
212C22⋊C4—341no—
213C22⋊C4—341no—
214C22⋊C4C22⋊D1712yescomplemented
215C42—342no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—81no—
220C23—81no—
221C23—172no—
222C23—341no—
223C23—341no—
224C23—1361no—
225C23C2×C4×D1711yescharacteristic
226C23#6325211yescharacteristic
227C23D4×D1711yescharacteristic
228C23D4×D1711yescharacteristic
229C23D4×D1711yescharacteristic
230C23#6325411yescharacteristic
231C23C2×C22⋊D1711yescharacteristic
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—342no—
237D4—342no—
238D4—342no—
239D4—342no—
240D4—1362no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—172no—
249C2×C4—341no—
250C2×C4—341no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—681no—
254C2×C4—1362no—
255C2×C4—1361no—
256C2×C4—1361no—
257C2×C4—1361no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—42no—
266C22—41no—
267C22—81no—
268C22—81no—
269C22—342no—
270C22—682no—
271C22—682no—
272C22—1361no—
273C22C22⋊C4×D1711yescharacteristic
274C22#1058756711yescharacteristic
275C22#1058756911yescharacteristic
276C22#1058757011yescharacteristic
277C22C4×C22⋊D1711yescharacteristic
278C22#1058762311yescharacteristic
279C22#1058762411yescharacteristic
280C4—42no—
281C4—41no—
282C4—81no—
283C4—342no—
284C4—682no—
285C4—682no—
286C4—1361no—
287C2—21no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—81no—
295C2—682no—
296C2#1176268511yescharacteristic
subgroupG/H
centerC2#11762685
commutatorC23×C17C22×C4
FrattiniC24C22×D17
FittingC2×C17×#99C2
socleC2×C17#386
prime radical#11763368C2
cosocleC24×C17C23
hypercenterC2×#99D17
hypocenterC17#1913
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #11763282C2semidirect
3. C2×#10587505C2semidirect
4. C2×#10587477C2semidirect
5. #11762576C2semidirect
6. #11762575C2semidirect
7. C2×C17×#99C2semidirect
12. #10587639C4semidirect
19. #10587615C4semidirect
27. C24×C17C2×C4semidirect
38. C22×C4×C17C2×C4semidirect
54. C22⋊C4×C17D4semidirect
60. C22⋊Dic17C2×C4semidirect
61. C22⋊Dic17D4semidirect
143. C2×#99D17semidirect
175. C17#1913semidirect
177. C24C4×D17semidirect
199. C22×C4C4×D17semidirect
214. C22⋊C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
#117633682
#117632822
C2×#105875052
C2×#105874772
#117625762
#117625752
C2×C17×#992
#191317