Finite Group Databasebeta

← #16930285

Subgroups — #16930285

Order 4352.

532 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 99 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#14537813C211yesmaximal, characteristic
2#14534701C211yesmaximal, characteristic
3#14527303C212yesmaximal, complemented
4#14526637C212yesmaximal, complemented
5C4×C17×#99C211yesmaximal, characteristic
6C2×#10587610C2211yescharacteristic
7#11762881—24no—
8#11762866—21no—
9#11762773C2212yes—
10C2×C22⋊#3622—21no—
11#11762739C411yescharacteristic, complemented
12#11762737—21no—
13#11762734C411yescharacteristic, complemented
14#11762729C411yescharacteristic, complemented
15#11762723—21no—
16#11762721C411yescharacteristic, complemented
17#11762693—21no—
18#11762682—21no—
19#11762672C2211yescharacteristic
20#11762545—24no—
21C2×C4×D4×C17C2211yescharacteristic
22C2×C17×#99—21no—
23C4×C22⋊C4×C17C2212yes—
24C17×#342—21no—
25C22×D4×C17D411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C23×C4×C17C2311yescharacteristic
28C22×Dic34C2×C411yescharacteristic
29C4×D4×C17—22no—
30C4×D4×C17—22no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C42×C17—22no—
36C2×C42×C17D411yescharacteristic
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17—21no—
41C2×C4⋊Dic17C2×C411yescharacteristic
42#10587610—22no—
43#10587610—22no—
44C2×#63225—21no—
45C2×#63225—21no—
46C2×#63225—21no—
47C2×#63223—22no—
48C22×#3622—22no—
49C17×#99—24no—
50C17×#95—22no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502C2×C411yescharacteristic
57#10587502C2×C411yescharacteristic
58C22⋊#3622—22no—
59C22⋊#3622—22no—
60C22⋊#3622D414yes—
61C4⋊#3622—22no—
62#10587474—22no—
63C24×C17C22⋊C411yescharacteristic
64C24×C17C2×D411yescharacteristic
65C2×D4×C17—22no—
66C2×D4×C17—22no—
67C2×D4×C17—41no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—42no—
71C22×C4×C17—81no—
72C22×C4×C17—81no—
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D412yes—
79C22×C4×C17#4112yes—
80C22×Dic17—21no—
81C22×Dic17—21no—
82C22×Dic17—21no—
83C22×Dic17—21no—
84C2×Dic34—22no—
85C2×Dic34—81no—
86C4⋊C4×C17—22no—
87C22⋊C4×C17—22no—
88C22⋊C4×C17—22no—
89C22⋊C4×C17—24no—
90C22⋊C4×C17—42no—
91C42×C17—21no—
92C42×C17—21no—
93C42×C17—82no—
94C22⋊Dic17—42no—
95C22⋊Dic17—42no—
96C22⋊Dic17—81no—
97C22⋊Dic17—81no—
98C4⋊Dic17—22no—
99#63225—42no—
100#63225—41no—
101#63225—41no—
102#63225—81no—
103#63223—82no—
104C2×#3622—24no—
105C2×#3622—82no—
106#9183—171nomaximal, Sylow
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17#10412yes—
116C23×C17C2×C22⋊C411yescharacteristic
117C23×C17#12112yes—
118D4×C17—44no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—42no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—44no—
132C2×C4×C17—81no—
133C2×C4×C17—82no—
134C2×C4×C17—82no—
135C2×C4×C17C4×D412yes—
136C2×C4×C17#12011yescharacteristic
137C2×C4×C17#12311yescharacteristic
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150Dic34—82no—
151#3622—44no—
152#2973—171no—
153#2638—171no—
154C4×#99D1711yescharacteristic
155#1508—172no—
156#1284—172no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—22no—
160C22×C17—22no—
161C22×C17—22no—
162C22×C17—22no—
163C22×C17—42no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—81no—
167C22×C17—81no—
168C22×C17—81no—
169C22×C17#38611yescharacteristic
170C22×C17C2×#10411yescharacteristic
171C22×C17#42111yescharacteristic
172C4×C17—21no—
173C4×C17—22no—
174C4×C17—21no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17—84no—
179Dic17—81no—
180Dic17—81no—
181Dic17—81no—
182Dic17—81no—
183C2×C22⋊Q8—171no—
184C2×C4×D4C2×D1711yescharacteristic
185#435—344no—
186#432—341no—
187C2×#99—21no—
188#407—172no—
189C2×C22⋊C8—341no—
190#398—171no—
191#397—171no—
192#397—341no—
193#395—171no—
194#394—341no—
195#393—171no—
196#387—341no—
197#385—341no—
198#381—171no—
199C4×C22⋊C4C2×D1712yes—
200#342—21no—
201#328—344no—
202C2×C17—21no—
203C2×C17—21no—
204C2×C17—42no—
205C2×C17—42no—
206C2×C17#187111yescharacteristic
207C2×C17#190911yescharacteristic
208C2×C17#191311yescharacteristic
209C22×Q8—171no—
210C22×D4C22⋊D1711yescharacteristic
211C23×C4D6811yescharacteristic
212C23×C4C22×D1711yescharacteristic
213C2×M16—342no—
214C22×C8—342no—
215C22⋊Q8—342no—
216C22⋊Q8—342no—
217C4×D4—22no—
218C4×D4—22no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4C4×D1711yescharacteristic
224C2×C22⋊C4—22no—
225C2×C22⋊C4—341no—
226C2×C22⋊C4—341no—
227C2×C22⋊C4—341no—
228C2×C22⋊C4—341no—
229C2×C22⋊C4C4×D1711yescharacteristic
230C2×C22⋊C4C22⋊D1711yescharacteristic
231C2×C42—22no—
232C2×C42D6811yescharacteristic
233C4⋊C8—342no—
234#99—24no—
235C22⋊C8—174no—
236C22⋊C8—342no—
237C22⋊C8—342no—
238#97—342no—
239#95—22no—
240#95—171no—
241#95—171no—
242#95—341no—
243#95—341no—
244#95—341no—
245#95—341no—
246#95—341no—
247C17#918311yescharacteristic, complemented, Sylow
248C24#6322711yescharacteristic
249C24C2×C22⋊D1711yescharacteristic
250C2×Q8—342no—
251C2×Q8—1361no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4#6322711yescharacteristic
265C22×C4#6322711yescharacteristic
266C22×C4#6322711yescharacteristic
267C22×C4C2×C4×D1711yescharacteristic
268C22×C4C2×D6811yescharacteristic
269C22×C4D4×D1712yes—
270C22×C4#6325412yes—
271M16—1362no—
272C2×C8—344no—
273C2×C8—1362no—
274C4⋊C4—22no—
275C4⋊C4—342no—
276C4⋊C4—682no—
277C4⋊C4—681no—
278C4⋊C4—681no—
279C4⋊C4—1361no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—24no—
283C22⋊C4—42no—
284C22⋊C4—682no—
285C22⋊C4—682no—
286C22⋊C4—1361no—
287C22⋊C4—1361no—
288C42—21no—
289C42—21no—
290C42—82no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23#1058747612yes—
300C23C2×#6322711yescharacteristic
301C23#1058762512yes—
302Q8—1362no—
303D4—44no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4#1058756712yes—
333C2×C4#1058756811yescharacteristic
334C2×C4#1058757211yescharacteristic
335C8—684no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#1176268611yescharacteristic
349C22C2×#1058747611yescharacteristic
350C22#1176282711yescharacteristic
351C4—21no—
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—84no—
358C4—1361no—
359C4—1361no—
360C4—1361no—
361C4—1361no—
362C2—21no—
363C2—21no—
364C2—42no—
365C2—42no—
366C2#1452908411yescharacteristic
367C2#1452930411yescharacteristic
368C2#1452932411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC4×C17×#99C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenterC4×#99D17
hypocenterC17#9183
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #14527303C2semidirect
4. #14526637C2semidirect
11. #11762739C4semidirect
13. #11762734C4semidirect
14. #11762729C4semidirect
16. #11762721C4semidirect
247. C17#9183semidirect

Maximal subgroups

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

subgroupindex
#145378132
#145347012
#145273032
#145266372
C4×C17×#992
#918317