Finite Group Databasebeta

← #16949951

Subgroups — #16949951

Order 4352.

810 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 127 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#14530453C212yesmaximal, complemented
2C2×#11762602C212yesmaximal, complemented
3#14528892C212yesmaximal, complemented
4C17×#3508C211yesmaximal, characteristic
5C2×#10587505C2214yes—
6#11762604—22no—
7#11762602—22no—
8#11762602—22no—
9#11762602—22no—
10#11762575—24no—
11C17×#559—22no—
12C17×#550—22no—
13C17×#546—24no—
14C17×#540—22no—
15C17×#535C411yescharacteristic, complemented
16C17×#535C2211yescharacteristic
17C17×#533C411yescharacteristic, complemented
18C17×#532—22no—
19C2×C17×#127C2211yescharacteristic
20C17×#518C411yescharacteristic, complemented
21C17×#518C2211yescharacteristic
22C4×#41×C17C411yescharacteristic, complemented
23C2×#41×C17—24no—
24C2×#41×C17C2×C411yescharacteristic
25C22×D4×C17—22no—
26C22×D4×C17C2×C411yescharacteristic
27C22×D4×C17C2311yescharacteristic
28C17×C4⋊Q8—21no—
29C17×#127—21no—
30C17×#127—22no—
31C17×#127D412yes—
32C17×#127D412yes—
33C17×#127D412yes—
34C17×#125—21no—
35C17×#124—24no—
36C17×#124—21no—
37C17×#124D412yes—
38C17×#123—24no—
39C17×#123—24no—
40C17×C22⋊Q8—22no—
41C17×#121—24no—
42C17×#121—24no—
43C17×#121—24no—
44C17×#121—22no—
45C17×#121—22no—
46C17×#120—22no—
47C17×#120—22no—
48C4×Q8×C17—21no—
49C4×Q8×C17—22no—
50C4×D4×C17—24no—
51C4×D4×C17—21no—
52C4×D4×C17—22no—
53C4×D4×C17—22no—
54C4×D4×C17—22no—
55C4×D4×C17—22no—
56C4×D4×C17—22no—
57C17×#117—21no—
58C17×#117C2×C411yescharacteristic
59C17×#117C2×C411yescharacteristic
60C17×#117C2×C411yescharacteristic
61C2×C22⋊C4×C17—22no—
62C2×C22⋊C4×C17D414yes—
63C2×C42×C17—21no—
64C2×C42×C17C2×C411yescharacteristic
65C2×C22⋊Dic17—24no—
66#10587505—24no—
67#10587505—24no—
68C24×C17—21no—
69C24×C17C2×D412yes—
70#41×C17—42no—
71#41×C17—88no—
72C2×Q8×C17—22no—
73C2×Q8×C17C22⋊C411yescharacteristic
74C2×D4×C17—21no—
75C2×D4×C17—24no—
76C2×D4×C17—24no—
77C2×D4×C17—24no—
78C2×D4×C17—24no—
79C2×D4×C17—21no—
80C2×D4×C17—42no—
81C2×D4×C17—42no—
82C2×D4×C17—84no—
83C2×D4×C17C22⋊C411yescharacteristic
84C2×D4×C17C22⋊C412yes—
85C2×D4×C17C2×D412yes—
86C2×D4×C17C2×D412yes—
87C22×C4×C17—22no—
88C22×C4×C17—21no—
89C22×C4×C17—24no—
90C22×C4×C17—24no—
91C22×C4×C17—21no—
92C22×C4×C17C22×C411yescharacteristic
93C22×Dic17—44no—
94C4⋊C4×C17—22no—
95C4⋊C4×C17—22no—
96C4⋊C4×C17—22no—
97C4⋊C4×C17—22no—
98C4⋊C4×C17—21no—
99C4⋊C4×C17—21no—
100C4⋊C4×C17—21no—
101C22⋊C4×C17—24no—
102C22⋊C4×C17—24no—
103C22⋊C4×C17—22no—
104C22⋊C4×C17—21no—
105C22⋊C4×C17—24no—
106C22⋊C4×C17—24no—
107C22⋊C4×C17C22⋊C412yes—
108C22⋊C4×C17C22⋊C412yes—
109C42×C17—22no—
110C42×C17—21no—
111C42×C17—21no—
112C42×C17—21no—
113C42×C17C22⋊C412yes—
114C42×C17C22⋊C411yescharacteristic
115C42×C17C22⋊C411yescharacteristic
116C22⋊Dic17—44no—
117C22⋊Dic17—44no—
118#12727—171nomaximal, Sylow
119C23×C17—24no—
120C23×C17—22no—
121C23×C17—21no—
122C23×C17—42no—
123C23×C17—82no—
124C23×C17—82no—
125C23×C17C2×C22⋊C411yescharacteristic
126C23×C17#12014yes—
127Q8×C17—41no—
128Q8×C17—44no—
129D4×C17—42no—
130D4×C17—42no—
131D4×C17—42no—
132D4×C17—41no—
133D4×C17—42no—
134D4×C17—48no—
135D4×C17—42no—
136D4×C17—84no—
137D4×C17—84no—
138D4×C17—84no—
139C2×C4×C17—21no—
140C2×C4×C17—22no—
141C2×C4×C17—21no—
142C2×C4×C17—22no—
143C2×C4×C17—21no—
144C2×C4×C17—22no—
145C2×C4×C17—22no—
146C2×C4×C17—42no—
147C2×C4×C17—42no—
148C2×C4×C17—44no—
149C2×C4×C17—84no—
150C2×C4×C17—84no—
151C2×C4×C17C2×C22⋊C411yescharacteristic
152C2×C4×C17C2×C22⋊C411yescharacteristic
153C2×Dic17—84no—
154C2×Dic17—164no—
155#3508D1711yescharacteristic
156#2127—172no—
157C2×#353—172no—
158#1828—172no—
159C22×C17—24no—
160C22×C17—41no—
161C22×C17—44no—
162C22×C17—42no—
163C22×C17—44no—
164C22×C17—41no—
165C22×C17—42no—
166C22×C17—84no—
167C22×C17—162no—
168C22×C17#37911yescharacteristic
169C22×C17#41012yes—
170C4×C17—22no—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—42no—
174C4×C17—41no—
175C4×C17—42no—
176C4×C17—42no—
177C4×C17—41no—
178Dic17—164no—
179#559—22no—
180#550—22no—
181#546—24no—
182#540—22no—
183#535Dic1711yescharacteristic, complemented
184#535C2×D1711yescharacteristic
185#533Dic1711yescharacteristic, complemented
186#532—22no—
187C2×#127C2×D1711yescharacteristic
188#518Dic1711yescharacteristic, complemented
189#518C2×D1711yescharacteristic
190C4×#41Dic1711yescharacteristic, complemented
191C2×#99—174no—
192#354—342no—
193#353—342no—
194#353—342no—
195#353—342no—
196#342—344no—
197C2×C17—21no—
198C2×C17—22no—
199C2×C17—44no—
200C2×C17—81no—
201C2×C17—82no—
202C2×C17#178211yescharacteristic
203C2×#41—24no—
204C2×#41C2×Dic1711yescharacteristic
205C22×D4—22no—
206C22×D4C2×Dic1711yescharacteristic
207C22×D4C22×D1711yescharacteristic
208C4⋊Q8—21no—
209#127—21no—
210#127—22no—
211#127C22⋊D1712yes—
212#127C22⋊D1712yes—
213#127C22⋊D1712yes—
214#125—21no—
215#124—24no—
216#124—21no—
217#124C22⋊D1712yes—
218#123—24no—
219#123—24no—
220C22⋊Q8—22no—
221#121—24no—
222#121—24no—
223#121—24no—
224#121—22no—
225#121—22no—
226#120—22no—
227#120—22no—
228C4×Q8—21no—
229C4×Q8—22no—
230C4×D4—24no—
231C4×D4—21no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237#117—21no—
238#117C2×Dic1711yescharacteristic
239#117C2×Dic1711yescharacteristic
240#117C2×Dic1711yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—344no—
243C2×C22⋊C4C22⋊D1714yes—
244C2×C42—21no—
245C2×C42C2×Dic1711yescharacteristic
246#99—344no—
247#99—344no—
248C17#1272711yescharacteristic, complemented, Sylow
249C24—21no—
250C24C2×C22⋊D1712yes—
251#41—42no—
252#41—88no—
253C2×Q8—22no—
254C2×Q8C22⋊Dic1711yescharacteristic
255C2×D4—21no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—21no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—84no—
264C2×D4C22⋊Dic1712yes—
265C2×D4C22⋊Dic1711yescharacteristic
266C2×D4C2×C22⋊D1712yes—
267C2×D4C2×C22⋊D1712yes—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—684no—
274C22×C4C22×Dic1711yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—24no—
285C22⋊C4—21no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—684no—
289C22⋊C4—684no—
290C22⋊C4C22⋊Dic1712yes—
291C22⋊C4C22⋊Dic1712yes—
292C42—22no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22⋊Dic1711yescharacteristic
297C42C22⋊Dic1711yescharacteristic
298C42C22⋊Dic1712yes—
299C23—24no—
300C23—22no—
301C23—21no—
302C23—42no—
303C23—82no—
304C23—82no—
305C23C2×C22⋊Dic1711yescharacteristic
306C23#1058763914yes—
307Q8—41no—
308Q8—44no—
309D4—42no—
310D4—42no—
311D4—42no—
312D4—41no—
313D4—42no—
314D4—48no—
315D4—42no—
316D4—84no—
317D4—84no—
318D4—84no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—2724no—
332C2×C4—1364no—
333C2×C4C2×C22⋊Dic1711yescharacteristic
334C2×C4C2×C22⋊Dic1711yescharacteristic
335C22—24no—
336C22—41no—
337C22—44no—
338C22—42no—
339C22—44no—
340C22—41no—
341C22—42no—
342C22—84no—
343C22—162no—
344C22#1176266511yescharacteristic
345C22#1176278312yes—
346C4—22no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—41no—
351C4—42no—
352C4—42no—
353C4—41no—
354C4—2724no—
355C2—21no—
356C2—22no—
357C2—44no—
358C2—82no—
359C2—81no—
360C2#1452870211yescharacteristic
subgroupG/H
centerC2#14528702
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
Fitting, prime radicalC17×#3508C2
socleC2×C17#1782
cosocleC22×D4×C17C23
hypercenter#3508D17
hypocenterC17#12727
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C23 ◁ #518 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #14530453C2semidirect
2. C2×#11762602C2semidirect
3. #14528892C2semidirect
15. C17×#535C4semidirect
17. C17×#533C4semidirect
20. C17×#518C4semidirect
22. C4×#41×C17C4semidirect
183. #535Dic17semidirect
185. #533Dic17semidirect
188. #518Dic17semidirect
190. C4×#41Dic17semidirect
248. C17#12727semidirect

Maximal subgroups

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

subgroupindex
#145304532
C2×#117626022
#145288922
C17×#35082
#1272717