Finite Group Databasebeta

← #16936751

Subgroups — #16936751

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#14539729C211yesmaximal, characteristic, complemented
2#14533737C211yesmaximal, characteristic
3#14527466C211yesmaximal, characteristic, complemented
4#14527283C211yesmaximal, characteristic, complemented
5#14527175C211yesmaximal, characteristic
6#14526685C211yesmaximal, characteristic, complemented
7C17×#3077C211yesmaximal, characteristic
8C2×#10587555C2211yescharacteristic
9C2×#10587610—22no—
10#11763188—21no—
11#11763161—21no—
12#11763019—22no—
13#11762964—22no—
14#11762862C2211yescharacteristic
15#11762862C2211yescharacteristic
16#11762745—22no—
17#11762712—21no—
18#11762678—21no—
19C4×C4⋊Dic17C2211yescharacteristic
20#11762557—22no—
21#11762546—22no—
22D4⋊#3622—22no—
23C17×#533C2211yescharacteristic
24C2×C4×D4×C17C2211yescharacteristic
25C17×#478—22no—
26C17×#476—21no—
27C17×#474—21no—
28C17×#428—21no—
29C17×#423C2211yescharacteristic
30C17×#418—21no—
31C22×D4×C17D411yescharacteristic
32C23×C4×C17—21no—
33C22×Dic34D412yes—
34C2×M16×C17—21no—
35C17×C4⋊Q8D411yescharacteristic
36C17×C4⋊Q8D411yescharacteristic
37C17×#125—21no—
38C17×C22⋊Q8—21no—
39C17×C22⋊Q8—21no—
40C4×Q8×C17—21no—
41C4×Q8×C17—21no—
42C4×D4×C17—21no—
43C4×D4×C17—21no—
44C4×D4×C17—21no—
45C4×D4×C17—21no—
46C17×#117—21no—
47C2×C4⋊C4×C17D411yescharacteristic
48C2×C22⋊C4×C17—21no—
49C2×C42×C17C2311yescharacteristic
50C2×C22⋊Dic17—24no—
51C2×#63231—24no—
52C2×C4⋊Dic17—21no—
53C2×C4⋊Dic17—21no—
54C2×C4⋊Dic17—21no—
55#10587610—48no—
56C2×#63225—24no—
57C2×C4×Dic17—21no—
58#10587555—21no—
59#10587555—21no—
60#10587555—21no—
61#10587555—21no—
62C17×#107—22no—
63C17×#106—22no—
64C17×C4⋊C8—21no—
65C17×C4⋊C8D412yes—
66C17×#102—22no—
67C17×#102—22no—
68#10587502D412yes—
69C22⋊#3622—21no—
70C22⋊#3622—21no—
71#10587481—24no—
72C4⋊#3622—21no—
73C4⋊#3622D412yes—
74C4×#3622—21no—
75C4×#3622—21no—
76C24×C17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×D4×C17—21no—
80C2×D4×C17—21no—
81C2×D4×C17—21no—
82C2×D4×C17—21no—
83C2×D4×C17—21no—
84C2×D4×C17Q1612yescomplemented
85C22×C4×C17—21no—
86C22×C4×C17—44no—
87C22×C4×C17—41no—
88C22×C4×C17—41no—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×Dic17—22no—
93C22×Dic17—22no—
94C2×Dic34—22no—
95C2×Dic34—22no—
96C2×Dic34—44no—
97M16×C17—41no—
98M16×C17—41no—
99C2×C8×C17—22no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—21no—
103C4⋊C4×C17—21no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—21no—
106C4⋊C4×C17—41no—
107C4⋊C4×C17—41no—
108C22⋊C4×C17—41no—
109C22⋊C4×C17—41no—
110C22⋊C4×C17—41no—
111C42×C17—41no—
112C42×C17C2×D411yescharacteristic
113C42×C17C2×D411yescharacteristic
114C42×C17C2×D411yescharacteristic
115C42×C17#4111yescharacteristic
116C22⋊Dic17—48no—
117#63231—84no—
118C4⋊Dic17—24no—
119C4⋊Dic17—41no—
120C4⋊Dic17—41no—
121C4⋊Dic17—41no—
122C4⋊Dic17—41no—
123#63225—48no—
124C4×Dic17—41no—
125C4×Dic17—41no—
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—42no—
136Q8×C17—81no—
137D4×C17—22no—
138D4×C17—22no—
139D4×C17—41no—
140D4×C17—41no—
141C2×C4×C17—22no—
142C2×C4×C17—41no—
143C2×C4×C17—42no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—82no—
148C2×C4×C17—82no—
149C2×C4×C17#12011yescharacteristic
150C2×C4×C17#12011yescharacteristic
151C2×C4×C17#12111yescharacteristic
152C2×C4×C17#12111yescharacteristic
153C2×C4×C17#12111yescharacteristic
154C2×C4×C17#12711yescharacteristic
155C2×C4×C17C2×Q1611yescharacteristic
156C2×C4×C17#13511yescharacteristic
157C2×C4×C17#13711yescharacteristic
158C2×C4×C17#13711yescharacteristic
159C2×Dic17—24no—
160C2×Dic17—42no—
161C2×Dic17—42no—
162C2×Dic17—44no—
163C2×Dic17—42no—
164Dic34—44no—
165Dic34—82no—
166C8×C17—42no—
167#3622—42no—
168#3622—81no—
169#3209—171no—
170#3077D1711yescharacteristic
171#2542—171no—
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—81no—
190C4×C17—81no—
191C4×C17—81no—
192C4×C17#45111yescharacteristic
193C4×C17#45211yescharacteristic
194C4×C17#45312yes—
195C4×C17#46711yescharacteristic
196C4×C17#47011yescharacteristic
197Dic17—44no—
198Dic17—82no—
199#533C2×D1711yescharacteristic
200C2×C4⋊Q8—171no—
201C2×C22⋊Q8—342no—
202C2×C4×D4C2×D1711yescharacteristic
203#494—341no—
204#487—341no—
205#478—22no—
206#476—21no—
207#474—21no—
208#462—342no—
209#451—342no—
210#431—171no—
211#431—171no—
212#428—21no—
213#423C2×D1711yescharacteristic
214#418—21no—
215#399—342no—
216#390—341no—
217#384—341no—
218C4×C4⋊C4—171no—
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×Q8—172no—
229C22×D4C22⋊D1711yescharacteristic
230C23×C4—21no—
231C2×Q16—344no—
232C2×M16—21no—
233C4⋊Q8—341no—
234C4⋊Q8—341no—
235C4⋊Q8—341no—
236C4⋊Q8—341no—
237C4⋊Q8C22⋊D1711yescharacteristic
238C4⋊Q8C22⋊D1711yescharacteristic
239#125—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—688no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249#117—21no—
250C2×C4⋊C4—341no—
251C2×C4⋊C4—344no—
252C2×C4⋊C4—341no—
253C2×C4⋊C4—341no—
254C2×C4⋊C4C22⋊D1711yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—344no—
257C2×C42—341no—
258C2×C42C22×D1711yescharacteristic
259#107—22no—
260#106—22no—
261C4⋊C8—21no—
262C4⋊C8—172no—
263C4⋊C8—341no—
264C4⋊C8C22⋊D1712yes—
265#103—344no—
266#102—22no—
267#102—22no—
268C22⋊C8—341no—
269C22⋊C8—341no—
270C4×C8—341no—
271C4×C8—341no—
272#95—172no—
273C17#1024811yescharacteristic, complemented, Sylow
274C24—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—342no—
278C2×Q8—342no—
279C2×Q8—684no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4#6323112yescomplemented
286C22×C4—21no—
287C22×C4—44no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—342no—
291C22×C4—342no—
292C22×C4D4×D1711yescharacteristic
293C22×C4D4×D1711yescharacteristic
294C22×C4C2×C22⋊D1711yescharacteristic
295Q16—1364no—
296M16—41no—
297M16—41no—
298C2×C8—22no—
299C2×C8—342no—
300C2×C8—681no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—344no—
310C4⋊C4—688no—
311C4⋊C4—681no—
312C4⋊C4—681no—
313C4⋊C4—681no—
314C4⋊C4—681no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—688no—
319C42—41no—
320C42—681no—
321C42—681no—
322C42D4×D1711yescharacteristic
323C42#6325411yescharacteristic
324C42C2×C22⋊D1711yescharacteristic
325C42C2×C22⋊D1711yescharacteristic
326C23—22no—
327C23—42no—
328C23—42no—
329C23—41no—
330C23—41no—
331C23#1058762311yescharacteristic
332Q8—42no—
333Q8—81no—
334Q8—684no—
335Q8—1362no—
336D4—22no—
337D4—22no—
338D4—41no—
339D4—41no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—344no—
349C2×C4—682no—
350C2×C4—682no—
351C2×C4—684no—
352C2×C4—682no—
353C2×C4#1058762311yescharacteristic
354C2×C4#1058762411yescharacteristic
355C2×C4#1058762511yescharacteristic
356C2×C4#1058762511yescharacteristic
357C2×C4#1058762611yescharacteristic
358C2×C4#1058762811yescharacteristic
359C2×C4C2×#6323111yescharacteristic
360C2×C4#1058763611yescharacteristic
361C2×C4#1058763711yescharacteristic
362C2×C4#1058763911yescharacteristic
363C8—42no—
364C8—682no—
365C8—1361no—
366C22—21no—
367C22—21no—
368C22—44no—
369C22—42no—
370C22—82no—
371C22—81no—
372C22—81no—
373C22#1176271711yescharacteristic
374C22#1176296411yescharacteristic
375C22#1176296911yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—684no—
383C4—1362no—
384C4#1176296511yescharacteristic
385C4#1176297111yescharacteristic
386C4#1176297512yes—
387C4#1176304811yescharacteristic
388C4#1176306511yescharacteristic
389C2—21no—
390C2—21no—
391C2—82no—
392C2#1452979211yescharacteristic
393C2#1452979611yescharacteristic
394C2#1452984111yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#3077C2
socleC22×C17#392
prime radicalC22×D4×C17D4
hypercenter#3077D17
hypocenterC17#10248
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #14539729C2semidirect
3. #14527466C2semidirect
4. #14527283C2semidirect
6. #14526685C2semidirect
84. 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
#145397292
#145337372
#145274662
#145272832
#145271752
#145266852
C17×#30772
#1024817