Finite Group Databasebeta

← #16925938

Subgroups — #16925938

Order 4352.

782 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 115 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×#11763529C211yesmaximal, characteristic
2#14528881C212yesmaximal, complemented
3C4×#10587505C211yesmaximal, characteristic
4#14527305C212yesmaximal, complemented
5C17×#1465C211yesmaximal, characteristic
6#11763529—24no—
7#11763529—24no—
8C2×#10587555C411yescharacteristic, complemented
9C2×#10587565—21no—
10C2×#10587565—21no—
11C2×#10587622—21no—
12C2×#10587610C411yescharacteristic, complemented
13C2×#10587610C2211yescharacteristic
14C2×C4×Dic34C411yescharacteristic, complemented
15C2×#10587573—21no—
16C2×#10587612C411yescharacteristic, complemented
17C2×#10587612C2211yescharacteristic
18C2×#10587505—21no—
19C4×C22⋊Dic17C2212yes—
20#11762576—21no—
21#11762570—24no—
22#11762547—24no—
23C2×C4×D4×C17C2211yescharacteristic
24C17×#407C2212yes—
25C17×#324—21no—
26C17×#323—21no—
27C22×D4×C17D411yescharacteristic
28C23×C4×C17C2×C411yescharacteristic
29C23×C4×C17C2311yescharacteristic
30C22×Dic34—21no—
31C22×Dic34C2×C411yescharacteristic
32C2×M16×C17—22no—
33C4×D4×C17—22no—
34C4×D4×C17—22no—
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17D411yescharacteristic
37C2×C22⋊C4×C17D411yescharacteristic
38C2×C42×C17C2×C411yescharacteristic
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17—21no—
41C2×C22⋊Dic17—21no—
42C2×C22⋊Dic17—21no—
43C2×C22⋊Dic17—22no—
44#10587622—28no—
45#10587612—22no—
46#10587612—22no—
47#10587612—22no—
48#10587612D414yes—
49C2×C4⋊Dic17—21no—
50C2×C4⋊Dic17C2×C411yescharacteristic
51C2×C4⋊Dic17C2×C411yescharacteristic
52C2×C4⋊Dic17C2×C411yescharacteristic
53#10587610—22no—
54#10587610—22no—
55#10587610—22no—
56#10587610D414yes—
57C2×#63225—21no—
58C2×#63225—21no—
59C2×#63225—21no—
60C2×C4×Dic17—21no—
61C2×C4×Dic17—21no—
62C2×C4×Dic17—21no—
63C2×C4×Dic17—22no—
64#10587573—28no—
65#10587565—28no—
66#10587565—28no—
67#10587555—22no—
68#10587555—22no—
69C4×Dic34—22no—
70C4×Dic34—22no—
71C17×C22⋊C8—22no—
72C17×C22⋊C8—22no—
73#10587505—24no—
74#10587502—22no—
75C24×C17C22⋊C411yescharacteristic
76C24×C17C2×D411yescharacteristic
77C2×D4×C17—22no—
78C2×D4×C17—22no—
79C2×D4×C17—41no—
80C22×C4×C17—21no—
81C22×C4×C17—21no—
82C22×C4×C17—42no—
83C22×C4×C17—42no—
84C22×C4×C17C22⋊C411yescharacteristic
85C22×C4×C17C22⋊C411yescharacteristic
86C22×C4×C17C22⋊C411yescharacteristic
87C22×C4×C17C22×C411yescharacteristic
88C22×C4×C17C2×D412yes—
89C22×C4×C17C2×D412yes—
90C22×C4×C17C2×D411yescharacteristic
91C22×Dic17—21no—
92C22×Dic17—21no—
93C22×Dic17—21no—
94C22×Dic17—21no—
95C22×Dic17—42no—
96C2×Dic34—24no—
97C2×Dic34—21no—
98C2×Dic34—42no—
99C2×Dic34—42no—
100C2×Dic34—42no—
101C2×Dic34C22⋊C412yes—
102M16×C17—44no—
103C2×C8×C17—42no—
104C4⋊C4×C17—22no—
105C22⋊C4×C17—22no—
106C22⋊C4×C17—22no—
107C42×C17—21no—
108C42×C17—21no—
109C22⋊Dic17—24no—
110C22⋊Dic17—24no—
111C22⋊Dic17—24no—
112C22⋊Dic17—24no—
113C22⋊Dic17—24no—
114C22⋊Dic17—42no—
115C4⋊Dic17—24no—
116C4⋊Dic17—21no—
117C4⋊Dic17—21no—
118C4⋊Dic17—21no—
119C4⋊Dic17C22⋊C412yes—
120C4⋊Dic17C22⋊C412yes—
121C4⋊Dic17C22⋊C412yes—
122#63225—24no—
123#63225—24no—
124#63225—24no—
125C4×Dic17—24no—
126C4×Dic17—24no—
127C4×Dic17—24no—
128C4×Dic17—44no—
129#8365—171nomaximal, Sylow
130C23×C17—21no—
131C23×C17—21no—
132C23×C17—21no—
133C23×C17—21no—
134C23×C17—21no—
135C23×C17—41no—
136C23×C17—41no—
137C23×C17—41no—
138C23×C17—41no—
139C23×C17C2×C22⋊C411yescharacteristic
140C23×C17#12012yes—
141D4×C17—44no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—24no—
145C2×C4×C17—22no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17—42no—
149C2×C4×C17—41no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17#10412yes—
155C2×C4×C17C2×C22⋊C411yescharacteristic
156C2×C4×C17C2×C22⋊C411yescharacteristic
157C2×C4×C17#12012yes—
158C2×Dic17—22no—
159C2×Dic17—22no—
160C2×Dic17—22no—
161C2×Dic17—22no—
162C2×Dic17—42no—
163C2×Dic17—42no—
164C2×Dic17—41no—
165C2×Dic17—41no—
166C2×Dic17—41no—
167C2×Dic17—41no—
168C2×Dic17—44no—
169C2×Dic17—82no—
170C2×Dic17—82no—
171Dic34—48no—
172Dic34—42no—
173Dic34—42no—
174C8×C17—82no—
175C2×#556—171no—
176#1826—172no—
177C4×#99—171no—
178#1508—172no—
179#1465D1711yescharacteristic
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—22no—
183C22×C17—22no—
184C22×C17—22no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17—81no—
189C22×C17—81no—
190C22×C17—81no—
191C22×C17—81no—
192C22×C17#37911yescharacteristic
193C22×C17C2×#10411yescharacteristic
194C22×C17#42111yescharacteristic
195C4×C17—22no—
196C4×C17—21no—
197C4×C17—22no—
198C4×C17—41no—
199C4×C17—41no—
200C4×C17#41012yes—
201Dic17—42no—
202Dic17—42no—
203Dic17—42no—
204Dic17—42no—
205Dic17—84no—
206#556—344no—
207#556—344no—
208C2×C4⋊Q8—171no—
209C2×#126—341no—
210C2×#126—341no—
211C2×#124—341no—
212C2×C22⋊Q8—171no—
213C2×C22⋊Q8—171no—
214C2×C4×Q8—171no—
215C2×C4×Q8—341no—
216C2×C4×D4C2×D1711yescharacteristic
217C2×#117—171no—
218C2×#117—171no—
219C2×#99—341no—
220#407C2×D1712yes—
221C4×C22⋊C4—172no—
222#342—341no—
223#339—344no—
224#328—344no—
225#324—21no—
226#323—21no—
227C2×C17—21no—
228C2×C17—21no—
229C2×C17—42no—
230C2×C17—81no—
231C2×C17#178211yescharacteristic
232C2×C17#178911yescharacteristic
233C2×C17#179611yescharacteristic
234C22×Q8—171no—
235C22×Q8—341no—
236C22×D4D6811yescharacteristic
237C23×C4C4×D1711yescharacteristic
238C23×C4C22×D1711yescharacteristic
239C2×M16—22no—
240C4⋊Q8—342no—
241C4⋊Q8—342no—
242#126—348no—
243#126—348no—
244#124—348no—
245C22⋊Q8—174no—
246C22⋊Q8—342no—
247C22⋊Q8—342no—
248C22⋊Q8—342no—
249C4×Q8—342no—
250C4×Q8—342no—
251C4×Q8—348no—
252C4×D4—22no—
253C4×D4—22no—
254#117—174no—
255#117—342no—
256#117—342no—
257#117—342no—
258C2×C4⋊C4—171no—
259C2×C4⋊C4—171no—
260C2×C4⋊C4—171no—
261C2×C4⋊C4—341no—
262C2×C4⋊C4—341no—
263C2×C4⋊C4—341no—
264C2×C4⋊C4—341no—
265C2×C4⋊C4C22⋊D1711yescharacteristic
266C2×C22⋊C4—341no—
267C2×C22⋊C4—341no—
268C2×C22⋊C4—341no—
269C2×C22⋊C4—341no—
270C2×C22⋊C4—342no—
271C2×C22⋊C4D6811yescharacteristic
272C2×C22⋊C4C22⋊D1711yescharacteristic
273C2×C42—341no—
274C2×C42—341no—
275C2×C42—341no—
276C2×C42—342no—
277C2×C42C4×D1711yescharacteristic
278#99—344no—
279C22⋊C8—22no—
280C22⋊C8—22no—
281#95—342no—
282C17#836511yescharacteristic, complemented, Sylow
283C24#6322711yescharacteristic
284C24C2×D6811yescharacteristic
285C2×Q8—172no—
286C2×Q8—344no—
287C2×Q8—341no—
288C2×Q8—682no—
289C2×Q8—682no—
290C2×Q8—682no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—41no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—341no—
299C22×C4—341no—
300C22×C4—341no—
301C22×C4—341no—
302C22×C4—682no—
303C22×C4#6322711yescharacteristic
304C22×C4#6322711yescharacteristic
305C22×C4#6322711yescharacteristic
306C22×C4C2×C4×D1711yescharacteristic
307C22×C4D4×D1712yes—
308C22×C4D4×D1712yes—
309C22×C4C2×C22⋊D1711yescharacteristic
310M16—44no—
311C2×C8—42no—
312C4⋊C4—22no—
313C4⋊C4—172no—
314C4⋊C4—172no—
315C4⋊C4—172no—
316C4⋊C4—344no—
317C4⋊C4—344no—
318C4⋊C4—344no—
319C4⋊C4—341no—
320C4⋊C4—341no—
321C4⋊C4—341no—
322C4⋊C4—344no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—344no—
326C22⋊C4—344no—
327C22⋊C4—344no—
328C22⋊C4—344no—
329C22⋊C4—344no—
330C22⋊C4—682no—
331C42—21no—
332C42—21no—
333C42—344no—
334C42—344no—
335C42—344no—
336C42—684no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23#1058756812yes—
347C23C2×#6322711yescharacteristic
348Q8—682no—
349Q8—688no—
350Q8—682no—
351D4—44no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—342no—
365C2×C4—342no—
366C2×C4—342no—
367C2×C4—342no—
368C2×C4—682no—
369C2×C4—682no—
370C2×C4—681no—
371C2×C4—681no—
372C2×C4—681no—
373C2×C4—681no—
374C2×C4—684no—
375C2×C4—1362no—
376C2×C4—1362no—
377C2×C4#1058749612yes—
378C2×C4C22⋊C4×D1711yescharacteristic
379C2×C4C22⋊C4×D1711yescharacteristic
380C2×C4#1058762312yes—
381C8—82no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22#1176266411yescharacteristic
395C22C2×#1058749611yescharacteristic
396C22#1176282511yescharacteristic
397C4—22no—
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—682no—
403C4—682no—
404C4—682no—
405C4—682no—
406C4—1364no—
407C4#1176278112yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—81no—
412C2#1452870011yescharacteristic
413C2#1452872711yescharacteristic
414C2#1452876011yescharacteristic
subgroupG/H
centerC22#11762664
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1465C2
socleC22×C17#379
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenter#1465D17
hypocenterC17#8365
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #14528881C2semidirect
4. #14527305C2semidirect
8. C2×#10587555C4semidirect
12. C2×#10587610C4semidirect
14. C2×C4×Dic34C4semidirect
16. C2×#10587612C4semidirect
282. C17#8365semidirect

Maximal subgroups

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

subgroupindex
C2×#117635292
#145288812
C4×#105875052
#145273052
C17×#14652
#836517