Finite Group Databasebeta

← #17953575

Subgroups — #17953575

Order 4352.

588 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 108 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#14544676C211yesmaximal, characteristic
2#14543960C211yesmaximal, characteristic
3#14543657C212yesmaximal
4#14543657C212yesmaximal
5#14531188C211yesmaximal, characteristic
6C2×#10587543C2212yes—
7C2×(C2×#595)⋊C4—21no—
8C2×(C2×#595)⋊C4—21no—
9C2×(C2×#595)⋊C4C2212yes—
10#11763675—21no—
11#11763675—21no—
12C2×#10587558—21no—
13C2×C42×D17C2211yescharacteristic
14#11762678C411yescharacteristic, complemented
15#11762678C411yescharacteristic, complemented
16#11762673—21no—
17#11762668—21no—
18#11762666C2211yescharacteristic
19#11762666C2211yescharacteristic
20C4×#63225—21no—
21C4×C4⋊Dic17C411yescharacteristic, complemented
22C4×C4⋊C4×C17C411yescharacteristic, complemented
23C23×#595—22no—
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×C4×D17C2311yescharacteristic
27C22×#3624—22no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C42×C17—21no—
31C2×C42×C17C2×C411yescharacteristic
32C2×#63227—21no—
33C2×#63227—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×#63225—21no—
37C2×#63225—21no—
38C2×#63225—21no—
39C2×#63225—21no—
40C2×C4×Dic17—21no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44#10587558—44no—
45C42×D17—24no—
46(C2×#595)⋊C4—24no—
47(C2×#595)⋊C4—24no—
48(C2×#595)⋊C4—42no—
49#10587543—24no—
50C17×#95—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54C23×D17C2×D411yescharacteristic
55C22×#595—24no—
56C22×#595—42no—
57C22×#595—42no—
58C22×#595—42no—
59C22×#595—42no—
60C22×#595—42no—
61C22×#595—42no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C2×D411yescharacteristic
72C2×C4×D17—22no—
73C2×C4×D17—22no—
74C2×C4×D17—22no—
75C2×C4×D17—22no—
76C2×C4×D17—41no—
77C2×C4×D17—41no—
78C2×C4×D17—41no—
79C2×C4×D17SD1612yes—
80C2×C4×D17Q1612yes—
81C2×C4×D17#4114yes—
82C2×#3624—42no—
83C2×#3624—42no—
84C2×#3624—42no—
85C4⋊C4×C17—41no—
86C4⋊C4×C17—41no—
87C42×C17—22no—
88C42×C17—42no—
89#63227—42no—
90#63227—42no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93#63225—42no—
94#63225—42no—
95#63225—42no—
96#63225—42no—
97C4×Dic17—22no—
98C4×Dic17—22no—
99C4×Dic17—22no—
100C4×Dic17—42no—
101#8717—171nomaximal, Sylow
102C23×C17C2×C22⋊C411yescharacteristic
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17—21no—
107C22×D17#12312yes—
108C22×D17C2×SD1611yescharacteristic
109C22×D17C2×Q1611yescharacteristic
110C22×D17AGL(1,ℤ/8)12yes—
111C2×#595—44no—
112C2×#595—44no—
113C2×#595—44no—
114C2×#595—82no—
115C2×#595—82no—
116C2×#595—82no—
117C2×#595—82no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17#10312yes—
132C2×C4×C17#11712yes—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17#10312yes—
146C2×Dic17#10412yes—
147C2×Dic17#12312yes—
148C4×D17—24no—
149C4×D17—24no—
150C4×D17—44no—
151C4×D17—44no—
152#3624—82no—
153#2285—171no—
154#1817—171no—
155#1434—171no—
156#1294—172no—
157#1294—172no—
158C22×C17#38111yescharacteristic
159C22×C17#41011yescharacteristic
160C22×C17#41011yescharacteristic
161C22×C17C2×#10311yescharacteristic
162C22×C17#41811yescharacteristic
163C22×C17C2×#10411yescharacteristic
164C22×C17#42111yescharacteristic
165C2×D17—22no—
166C2×D17—22no—
167C2×D17—21no—
168C2×D17—21no—
169C2×D17—22no—
170C2×D17—22no—
171C2×D17—21no—
172C2×D17—21no—
173C2×D17#48112yes—
174C2×D17#48412yes—
175#595—84no—
176C4×C17—22no—
177C4×C17—22no—
178C4×C17—41no—
179C4×C17—81no—
180C4×C17—81no—
181Dic17—22no—
182Dic17—22no—
183Dic17—22no—
184Dic17—81no—
185Dic17—81no—
186C2×#117—341no—
187C22×C42—171no—
188C2×C22⋊C8—172no—
189#384—171no—
190#384—171no—
191#382—341no—
192#380—171no—
193#380—171no—
194#380—341no—
195C4×C4⋊C4—171no—
196C4×C4⋊C4—341no—
197C4×C4⋊C4#59511yescharacteristic, complemented
198C2×#95—172no—
199C2×#95—341no—
200C2×#95—341no—
201#336—341no—
202#336—341no—
203C2×C17#150311yescharacteristic
204C2×C17#150511yescharacteristic
205C2×C17#151411yescharacteristic
206C2×C17#151811yescharacteristic
207C2×C17#182011yescharacteristic
208C2×C17#182411yescharacteristic
209C2×C17#182611yescharacteristic
210D17—22no—
211D17—22no—
212C23×C4—171no—
213C23×C4—171no—
214C23×C4—171no—
215C23×C4—342no—
216C22×C8—342no—
217#117—684no—
218C2×C4⋊C4—171no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4C2×#59511yescharacteristic
225C2×C4⋊C4C2×#59511yescharacteristic
226C2×C22⋊C4—341no—
227C2×C22⋊C4—341no—
228C2×C42—21no—
229C2×C42—171no—
230C2×C42—171no—
231C2×C42—171no—
232C2×C42—344no—
233C2×C42—341no—
234C2×C42C2×#59511yescharacteristic
235C22⋊C8—344no—
236#95—21no—
237#95—341no—
238#95—341no—
239#95—344no—
240#95—344no—
241#95—341no—
242#95—682no—
243C17#871711yescharacteristic, complemented, Sylow
244C24—171no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—171no—
248C22×C4—171no—
249C22×C4—171no—
250C22×C4—172no—
251C22×C4—172no—
252C22×C4—174no—
253C22×C4—342no—
254C22×C4—342no—
255C22×C4—342no—
256C22×C4—342no—
257C22×C4—344no—
258C22×C4—341no—
259C22×C4—341no—
260C22×C4—681no—
261C22×C4—681no—
262C22×C4—681no—
263C22×C4—682no—
264C22×C4—682no—
265C22×C4—682no—
266C22×C4—682no—
267C22×C4—682no—
268C22×C4—682no—
269C22×C4C22⋊#59511yescharacteristic
270C22×C4C22⋊#59511yescharacteristic
271C22×C4C22×#59511yescharacteristic
272C2×C8—682no—
273C2×C8—682no—
274C2×C8—682no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—681no—
278C4⋊C4—681no—
279C4⋊C4—682no—
280C4⋊C4—682no—
281C4⋊C4—682no—
282C4⋊C4—682no—
283C22⋊C4—682no—
284C22⋊C4—682no—
285C42—22no—
286C42—42no—
287C42—342no—
288C42—342no—
289C42—342no—
290C42—682no—
291C23—172no—
292C23—171no—
293C23—171no—
294C23—172no—
295C23—341no—
296C23—341no—
297C23—341no—
298C23—341no—
299C23C2×C22⋊#59511yescharacteristic
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—172no—
314C2×C4—172no—
315C2×C4—172no—
316C2×C4—344no—
317C2×C4—344no—
318C2×C4—341no—
319C2×C4—341no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—341no—
324C2×C4—684no—
325C2×C4—684no—
326C2×C4—684no—
327C2×C4—684no—
328C2×C4—684no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4—681no—
334C2×C4—681no—
335C2×C4—1362no—
336C2×C4—1362no—
337C2×C4—1362no—
338C2×C4—1362no—
339C2×C4#1058754912yes—
340C2×C4#1058767712yes—
341C8—1362no—
342C22—172no—
343C22—172no—
344C22—341no—
345C22—341no—
346C22—342no—
347C22—342no—
348C22—342no—
349C22—342no—
350C22—341no—
351C22—341no—
352C22#1176375211yescharacteristic
353C22#1176378811yescharacteristic
354C22#1176378811yescharacteristic
355C22C2×#1058754911yescharacteristic
356C22#1176380111yescharacteristic
357C22D17⋊#10411yescharacteristic
358C22#1176380611yescharacteristic
359C4—22no—
360C4—22no—
361C4—41no—
362C4—81no—
363C4—81no—
364C4—342no—
365C4—342no—
366C4—342no—
367C4—1364no—
368C4—1361no—
369C4—1361no—
370C2—342no—
371C2—342no—
372C2#1454416611yescharacteristic
373C2#1454417411yescharacteristic
374C2#1454420311yescharacteristic
375C2#1454421711yescharacteristic
376C2#1454468111yescharacteristic
377C2#1454468811yescharacteristic
378C2#1454469211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC4×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenterC4×C4⋊C4#595
hypocenterC17#8717
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
14. #11762678C4semidirect
15. #11762678C4semidirect
21. C4×C4⋊Dic17C4semidirect
22. C4×C4⋊C4×C17C4semidirect
197. C4×C4⋊C4#595semidirect
243. C17#8717semidirect

Maximal subgroups

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

subgroupindex
#145446762
#145439602
#145436572
#145311882
#871717