Finite Group Databasebeta

← #16936754

Subgroups — #16936754

Order 4352.

604 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 105 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#14539732C211yesmaximal, characteristic, complemented
2#14538581C211yesmaximal, characteristic
3#14527465C211yesmaximal, characteristic, complemented
4#14527282C211yesmaximal, characteristic, complemented
5#14527175C211yesmaximal, characteristic
6#14526687C211yesmaximal, characteristic, complemented
7C17×#2542C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9#11763188—21no—
10#11763159—21no—
11#11763109—22no—
12#11763099—21no—
13#11763089—21no—
14#11763016—22no—
15#11762965—22no—
16#11762862C2211yescharacteristic
17#11762862C2211yescharacteristic
18#11762849—21no—
19#11762833C2211yescharacteristic
20#11762814—21no—
21#11762558—22no—
22#11762545—22no—
23D4⋊#3622—22no—
24C2×C17×C4⋊Q8C2211yescharacteristic
25C2×C17×C22⋊Q8—22no—
26C2×C4×D4×C17C2211yescharacteristic
27C17×#399—22no—
28C17×#390—21no—
29C17×#384—21no—
30C4×C4⋊C4×C17C2211yescharacteristic
31C22×Q8×C17D412yes—
32C22×D4×C17D411yescharacteristic
33C23×C4×C17—21no—
34C17×C4⋊Q8—21no—
35C17×C4⋊Q8—21no—
36C17×C4⋊Q8—21no—
37C17×C4⋊Q8—21no—
38C17×C22⋊Q8—48no—
39C4×D4×C17—21no—
40C4×D4×C17—21no—
41C4×D4×C17—21no—
42C4×D4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—24no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17D411yescharacteristic
48C2×C22⋊C4×C17—24no—
49C2×C22⋊C4×C17—21no—
50C2×C42×C17—21no—
51C2×C42×C17C2311yescharacteristic
52C2×#63231—24no—
53#10587612—21no—
54#10587610—21no—
55#10587610—21no—
56C2×#63223—21no—
57#10587556—21no—
58#10587555D411yescharacteristic
59#10587555D411yescharacteristic
60C4×Dic34—21no—
61C4×Dic34—21no—
62C17×#95D412yes—
63#10587506—24no—
64#10587503—22no—
65#10587503—22no—
66C22⋊#3622—21no—
67C22⋊#3622—21no—
68#10587479—22no—
69#10587478—22no—
70C4⋊#3622—21no—
71C4⋊#3622—21no—
72C4⋊#3622D412yes—
73C4⋊#3622D412yes—
74C4×#3622—21no—
75C4×#3622—21no—
76C24×C17—21no—
77C2×Q8×C17—22no—
78C2×Q8×C17—22no—
79C2×Q8×C17—44no—
80C2×D4×C17—21no—
81C2×D4×C17—21no—
82C2×D4×C17—21no—
83C2×D4×C17—21no—
84C2×D4×C17—21no—
85C2×D4×C17Q1612yescomplemented
86C22×C4×C17—22no—
87C22×C4×C17—21no—
88C22×C4×C17—22no—
89C22×C4×C17—44no—
90C22×C4×C17—41no—
91C22×C4×C17—41no—
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17C2×D411yescharacteristic
95C2×Dic34—21no—
96C2×Dic34—21no—
97C4⋊C4×C17—24no—
98C4⋊C4×C17—21no—
99C4⋊C4×C17—21no—
100C4⋊C4×C17—48no—
101C4⋊C4×C17—41no—
102C4⋊C4×C17—41no—
103C4⋊C4×C17—41no—
104C4⋊C4×C17—41no—
105C22⋊C4×C17—48no—
106C22⋊C4×C17—41no—
107C22⋊C4×C17—41no—
108C42×C17—41no—
109C42×C17—41no—
110C42×C17C2×D411yescharacteristic
111C42×C17C2×D411yescharacteristic
112C42×C17C2×D411yescharacteristic
113C42×C17#4111yescharacteristic
114C22⋊Dic17—41no—
115#63231—84no—
116C4⋊Dic17—21no—
117C4⋊Dic17—21no—
118C4⋊Dic17—21no—
119C4⋊Dic17—21no—
120#63225—41no—
121#63225—41no—
122C4×Dic17—41no—
123#63223—41no—
124#63223—41no—
125C2×#3622—22no—
126C2×#3622—22no—
127C2×#3622—41no—
128#10248—171nomaximal, Sylow
129C23×C17—22no—
130C23×C17—42no—
131C23×C17—42no—
132C23×C17—41no—
133C23×C17—41no—
134C23×C17#12011yescharacteristic
135Q8×C17—44no—
136Q8×C17—82no—
137D4×C17—22no—
138D4×C17—22no—
139D4×C17—41no—
140D4×C17—41no—
141C2×C4×C17—22no—
142C2×C4×C17—24no—
143C2×C4×C17—41no—
144C2×C4×C17—42no—
145C2×C4×C17—42no—
146C2×C4×C17—44no—
147C2×C4×C17—42no—
148C2×C4×C17—41no—
149C2×C4×C17—42no—
150C2×C4×C17—82no—
151C2×C4×C17—82no—
152C2×C4×C17#12011yescharacteristic
153C2×C4×C17#12011yescharacteristic
154C2×C4×C17#12111yescharacteristic
155C2×C4×C17#12111yescharacteristic
156C2×C4×C17#12111yescharacteristic
157C2×C4×C17#12711yescharacteristic
158C2×C4×C17C2×Q1611yescharacteristic
159C2×C4×C17#13511yescharacteristic
160C2×C4×C17#13711yescharacteristic
161C2×C4×C17#13711yescharacteristic
162C2×Dic17—41no—
163C2×Dic17—41no—
164Dic34—42no—
165Dic34—81no—
166#3622—42no—
167#3622—42no—
168#3622—81no—
169#3209—171no—
170#3077—171no—
171#2542D1711yescharacteristic
172#1548—171no—
173#1503—171no—
174#1478—171no—
175#1303—171no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—44no—
179C22×C17—42no—
180C22×C17—82no—
181C22×C17—81no—
182C22×C17—81no—
183C22×C17#39211yescharacteristic
184C22×C17#45111yescharacteristic
185C22×C17#45211yescharacteristic
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—21no—
189C4×C17—44no—
190C4×C17—81no—
191C4×C17—82no—
192C4×C17#45111yescharacteristic
193C4×C17#45211yescharacteristic
194C4×C17#45312yes—
195C4×C17#46711yescharacteristic
196C4×C17#47011yescharacteristic
197Dic17—81no—
198Dic17—81no—
199#533—171no—
200C2×C4⋊Q8C2×D1711yescharacteristic
201C2×C22⋊Q8—22no—
202C2×C4×D4C2×D1711yescharacteristic
203#494—341no—
204#487—341no—
205#478—342no—
206#476—341no—
207#474—341no—
208#462—342no—
209#451—342no—
210#431—171no—
211#431—171no—
212#428—341no—
213#423—171no—
214#418—341no—
215#399—22no—
216#390—21no—
217#384—21no—
218C4×C4⋊C4C2×D1711yescharacteristic
219#333—342no—
220#328—342no—
221D4⋊C8—342no—
222C2×C17—21no—
223C2×C17—21no—
224C2×C17—82no—
225C2×C17#200111yescharacteristic
226C2×C17#200211yescharacteristic
227C2×C17#200911yescharacteristic
228C22×Q8C22⋊D1712yes—
229C22×D4C22⋊D1711yescharacteristic
230C23×C4—21no—
231C2×Q16—344no—
232C2×M16—341no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—171no—
238C4⋊Q8—171no—
239#125—341no—
240C22⋊Q8—48no—
241C22⋊Q8—341no—
242C22⋊Q8—341no—
243C4×Q8—341no—
244C4×Q8—341no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249#117—341no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C22⋊D1711yescharacteristic
255C2×C22⋊C4—24no—
256C2×C22⋊C4—21no—
257C2×C42—21no—
258C2×C42C22×D1711yescharacteristic
259#107—342no—
260#106—342no—
261C4⋊C8—172no—
262C4⋊C8—172no—
263C4⋊C8—341no—
264C4⋊C8—341no—
265#103—344no—
266#102—342no—
267#102—342no—
268C22⋊C8—341no—
269C22⋊C8—341no—
270C4×C8—341no—
271C4×C8—341no—
272#95C22⋊D1712yes—
273C17#1024811yescharacteristic, complemented, Sylow
274C24—21no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—44no—
278C2×Q8—341no—
279C2×Q8—341no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4#6323112yescomplemented
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—44no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4C2×C22⋊D1711yescharacteristic
293C22×C4C2×C22⋊D1711yescharacteristic
294C22×C4C2×C22⋊D1711yescharacteristic
295Q16—1364no—
296M16—681no—
297M16—681no—
298C2×C8—342no—
299C2×C8—342no—
300C2×C8—681no—
301C4⋊C4—24no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—48no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—341no—
310C4⋊C4—341no—
311C4⋊C4—341no—
312C4⋊C4—341no—
313C4⋊C4—681no—
314C4⋊C4—681no—
315C22⋊C4—48no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—681no—
319C42—41no—
320C42—41no—
321C42—681no—
322C42D4×D1711yescharacteristic
323C42D4×D1711yescharacteristic
324C42D4×D1711yescharacteristic
325C42#6325411yescharacteristic
326C23—22no—
327C23—42no—
328C23—42no—
329C23—41no—
330C23—41no—
331C23#1058763911yescharacteristic
332Q8—44no—
333Q8—82no—
334Q8—682no—
335Q8—1361no—
336D4—22no—
337D4—22no—
338D4—41no—
339D4—41no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—681no—
352C2×C4—681no—
353C2×C4#1058762311yescharacteristic
354C2×C4#1058762311yescharacteristic
355C2×C4#1058762411yescharacteristic
356C2×C4#1058762511yescharacteristic
357C2×C4#1058762511yescharacteristic
358C2×C4#1058762611yescharacteristic
359C2×C4#1058762811yescharacteristic
360C2×C4C2×#6323111yescharacteristic
361C2×C4#1058763611yescharacteristic
362C2×C4#1058763711yescharacteristic
363C8—682no—
364C8—682no—
365C8—1361no—
366C22—21no—
367C22—21no—
368C22—44no—
369C22—42no—
370C22—82no—
371C22—81no—
372C22—81no—
373C22#1176271711yescharacteristic
374C22#1176296511yescharacteristic
375C22#1176297111yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—44no—
380C4—81no—
381C4—82no—
382C4—1361no—
383C4—1361no—
384C4#1176296411yescharacteristic
385C4#1176296911yescharacteristic
386C4#1176297612yes—
387C4#1176304811yescharacteristic
388C4#1176306511yescharacteristic
389C2—21no—
390C2—21no—
391C2—82no—
392C2#1452979211yescharacteristic
393C2#1452979911yescharacteristic
394C2#1452984411yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2542C2
socleC22×C17#392
prime radicalC22×D4×C17D4
hypercenter#2542D17
hypocenterC17#10248
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2542

Complements

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

NH ≅ G/Ntype
1. #14539732C2semidirect
3. #14527465C2semidirect
4. #14527282C2semidirect
6. #14526687C2semidirect
85. C2×D4×C17Q16semidirect
273. C17#10248semidirect
285. C2×D4#63231semidirect

Maximal subgroups

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

subgroupindex
#145397322
#145385812
#145274652
#145272822
#145271752
#145266872
C17×#25422
#1024817