Finite Group Databasebeta

← #16932175

Subgroups — #16932175

Order 4352.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#14537864C211yesmaximal, characteristic, complemented
2#14533693C211yesmaximal, characteristic
3#14528777C211yesmaximal, characteristic, complemented
4#14527209C211yesmaximal, characteristic
5#14526644C211yesmaximal, characteristic, complemented
6#14526643C211yesmaximal, characteristic, complemented
7C17×#1486C211yesmaximal, characteristic
8C2×#10587610—21no—
9C2×C4×Dic34C2211yescharacteristic
10C8×Dic34—22no—
11#11762933—22no—
12#11762862C2211yescharacteristic
13#11762859—22no—
14#11762833C2211yescharacteristic
15C2×#10587495—21no—
16#11762740C412yes—
17#11762723C412yes—
18#11762697—22no—
19#11762675C2211yescharacteristic
20#11762669—21no—
21#11762577—21no—
22D4⋊#3622—22no—
23C2×C4×D4×C17C2211yescharacteristic
24C17×#431C2211yescharacteristic
25C17×#423C2211yescharacteristic
26C17×D4⋊C8—22no—
27C22×D4×C17C2×C411yescharacteristic
28C23×C4×C17—21no—
29C22×Dic34D411yescharacteristic
30C2×M16×C17—21no—
31C4×D4×C17—21no—
32C4×D4×C17—21no—
33C4×D4×C17—21no—
34C4×D4×C17—21no—
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C42×C17C2311yescharacteristic
38C2×C22⋊Dic17—22no—
39C2×C22⋊Dic17—22no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17D411yescharacteristic
42#10587610—42no—
43#10587610—42no—
44C2×#63225—22no—
45C2×#63225C2×C412yes—
46C2×C4×Dic17D412yes—
47C2×#63223—21no—
48C4×Dic34—21no—
49C4×Dic34—21no—
50C4×Dic34—21no—
51C4×Dic34—21no—
52C17×C4⋊C8—21no—
53C17×C4⋊C8—21no—
54C17×C4⋊C8—21no—
55C17×C22⋊C8—21no—
56C4×C8×C17D412yes—
57#10587502—21no—
58#10587502—22no—
59#10587502C2×C412yes—
60C22⋊#3622—21no—
61#10587495—42no—
62#10587485—24no—
63C8×Dic17—24no—
64C4⋊#3622—21no—
65C4⋊#3622—21no—
66C4⋊#3622—21no—
67C4×#3622D412yes—
68C24×C17—21no—
69C2×D4×C17—21no—
70C2×D4×C17—21no—
71C2×D4×C17—42no—
72C22×C4×C17—21no—
73C22×C4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17—41no—
76C22×C4×C17—41no—
77C22×C4×C17—41no—
78C22×C4×C17—41no—
79C22×C4×C17C22×C411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C22×Dic17—22no—
83C22×Dic17C22⋊C412yes—
84C22×Dic17C22⋊C412yes—
85C2×Dic34—21no—
86C2×Dic34—21no—
87C2×Dic34—21no—
88C2×Dic34—21no—
89C2×Dic34D812yes—
90C2×Dic34SD1612yes—
91M16×C17—41no—
92M16×C17—41no—
93C2×C8×C17—22no—
94C2×C8×C17—41no—
95C4⋊C4×C17—21no—
96C4⋊C4×C17—21no—
97C22⋊C4×C17—41no—
98C22⋊C4×C17—41no—
99C42×C17C2×D411yescharacteristic
100C42×C17C2×D411yescharacteristic
101C42×C17#4111yescharacteristic
102C42×C17#4111yescharacteristic
103C22⋊Dic17—44no—
104C22⋊Dic17—44no—
105C4⋊Dic17—21no—
106C4⋊Dic17—21no—
107C4⋊Dic17—41no—
108C4⋊Dic17—41no—
109#63225—22no—
110#63225—22no—
111#63225—44no—
112C4×Dic17—22no—
113C4×Dic17—22no—
114#63223—41no—
115#63223—41no—
116C2×#3622—22no—
117C2×#3622—41no—
118#9486—171nomaximal, Sylow
119C23×C17—22no—
120C23×C17—41no—
121C23×C17—41no—
122C23×C17—41no—
123C23×C17—41no—
124C23×C17—41no—
125C23×C17—41no—
126C23×C17C2×C22⋊C411yescharacteristic
127D4×C17—22no—
128D4×C17—22no—
129D4×C17—41no—
130D4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—81no—
137C2×C4×C17—81no—
138C2×C4×C17—81no—
139C2×C4×C17—81no—
140C2×C4×C17C4×D411yescharacteristic
141C2×C4×C17C4×D411yescharacteristic
142C2×C4×C17#12011yescharacteristic
143C2×C4×C17#12111yescharacteristic
144C2×C4×C17#12111yescharacteristic
145C2×C4×C17#12311yescharacteristic
146C2×C4×C17C2×D811yescharacteristic
147C2×C4×C17C2×SD1611yescharacteristic
148C2×C4×C17#13511yescharacteristic
149C2×C4×C17#13511yescharacteristic
150C2×Dic17—22no—
151C2×Dic17—22no—
152C2×Dic17—22no—
153C2×Dic17—42no—
154C2×Dic17—42no—
155C2×Dic17—42no—
156C2×Dic17—42no—
157C2×Dic17#10214yes—
158Dic34—22no—
159Dic34—22no—
160Dic34—22no—
161Dic34—22no—
162C8×C17—42no—
163C8×C17—81no—
164#3622—42no—
165#3622—81no—
166#2982—171no—
167#2536—171no—
168#1801—171no—
169#1486—171no—
170#1486D1711yescharacteristic
171#1286—171no—
172#1286—171no—
173C22×C17—21no—
174C22×C17—21no—
175C22×C17—42no—
176C22×C17—42no—
177C22×C17—42no—
178C22×C17—81no—
179C22×C17—81no—
180C22×C17—81no—
181C22×C17—81no—
182C22×C17#38611yescharacteristic
183C22×C17C2×#10211yescharacteristic
184C22×C17#41611yescharacteristic
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—21no—
189C4×C17—81no—
190C4×C17#46511yescharacteristic
191C4×C17#46511yescharacteristic
192C4×C17#46611yescharacteristic
193C4×C17#46611yescharacteristic
194Dic17—24no—
195Dic17—42no—
196Dic17—82no—
197C2×C22⋊Q8—341no—
198C2×C4×Q8—171no—
199C2×C4×D4C2×D1711yescharacteristic
200C8×Q8—342no—
201C8×Q8—342no—
202#431—171no—
203#431—342no—
204#431C2×D1711yescharacteristic
205#423—171no—
206#423C2×D1711yescharacteristic
207C2×#101—341no—
208#398—172no—
209#394—172no—
210#388—342no—
211#382—171no—
212#380—341no—
213#343—341no—
214D4⋊C8—22no—
215D4⋊C8—342no—
216C2×C17—21no—
217C2×C17—21no—
218C2×C17—82no—
219C2×C17#189111yescharacteristic
220C2×C17#189311yescharacteristic
221C2×C17#191211yescharacteristic
222C22×Q8—171no—
223C22×D4C4×D1711yescharacteristic
224C23×C4—21no—
225C2×M16—21no—
226C2×M16—341no—
227C22⋊Q8—682no—
228C22⋊Q8—682no—
229C4×Q8—341no—
230C4×Q8—341no—
231C4×Q8—341no—
232C4×Q8—341no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C2×C4⋊C4—172no—
238C2×C4⋊C4—171no—
239C2×C4⋊C4—342no—
240C2×C4⋊C4—341no—
241C2×C4⋊C4C4×D1711yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—342no—
244C2×C22⋊C4—342no—
245C2×C42—172no—
246C2×C42C22×D1711yescharacteristic
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—344no—
251C4⋊C8—341no—
252C4⋊C8—341no—
253C4⋊C8—341no—
254#101—682no—
255C22⋊C8—21no—
256C22⋊C8—341no—
257C4×C8—172no—
258C4×C8—344no—
259C4×C8C22⋊D1712yes—
260#95—172no—
261#95—342no—
262#95—341no—
263C17#948611yescharacteristic, complemented, Sylow
264C24—21no—
265C2×Q8—172no—
266C2×Q8—172no—
267C2×Q8—341no—
268C2×Q8—341no—
269C2×Q8—341no—
270C2×Q8—341no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—42no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—172no—
282C22×C4—172no—
283C22×C4—342no—
284C22×C4C2×C4×D1711yescharacteristic
285C22×C4D4×D1711yescharacteristic
286C22×C4D4×D1711yescharacteristic
287M16—41no—
288M16—41no—
289M16—681no—
290M16—681no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—342no—
294C2×C8—681no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—342no—
298C4⋊C4—341no—
299C4⋊C4—342no—
300C4⋊C4—341no—
301C4⋊C4—684no—
302C4⋊C4—681no—
303C4⋊C4—681no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—684no—
307C22⋊C4—684no—
308C42—342no—
309C42—342no—
310C42#6325211yescharacteristic
311C42D4×D1711yescharacteristic
312C42#6325411yescharacteristic
313C42C2×C22⋊D1711yescharacteristic
314C23—22no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C22⋊C4×D1711yescharacteristic
322Q8—342no—
323Q8—342no—
324Q8—342no—
325Q8—342no—
326D4—22no—
327D4—22no—
328D4—41no—
329D4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—174no—
340C2×C4—342no—
341C2×C4—342no—
342C2×C4—342no—
343C2×C4—682no—
344C2×C4—682no—
345C2×C4—682no—
346C2×C4—682no—
347C2×C4#1058756711yescharacteristic
348C2×C4#1058756911yescharacteristic
349C2×C4#1058757011yescharacteristic
350C2×C4D8×D1711yescharacteristic
351C2×C4#1058759811yescharacteristic
352C2×C4SD16×D1711yescharacteristic
353C2×C4#1058760211yescharacteristic
354C2×C4C4×C22⋊D1711yescharacteristic
355C2×C4#1058762311yescharacteristic
356C2×C4#1058762411yescharacteristic
357C8—42no—
358C8—81no—
359C8—682no—
360C8—1361no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1176268511yescharacteristic
371C22#102×D1711yescharacteristic
372C22#1176280211yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—344no—
379C4—682no—
380C4—1362no—
381C4#1176303511yescharacteristic
382C4#1176303711yescharacteristic
383C4#1176304211yescharacteristic
384C4#1176304411yescharacteristic
385C2—21no—
386C2—21no—
387C2—82no—
388C2#1452919911yescharacteristic
389C2#1452920911yescharacteristic
390C2#1452931811yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1486C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC2×C42×C17C23
hypercenter#1486D17
hypocenterC17#9486
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
1. #14537864C2semidirect
3. #14528777C2semidirect
5. #14526644C2semidirect
6. #14526643C2semidirect
263. C17#9486semidirect

Maximal subgroups

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

subgroupindex
#145378642
#145336932
#145287772
#145272092
#145266442
#145266432
C17×#14862
#948617