Finite Group Databasebeta

← #17950133

Subgroups — #17950133

Order 4352.

532 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 88 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#14544641C211yesmaximal, characteristic, complemented
2#14544511C211yesmaximal, characteristic, complemented
3#14543602C212yesmaximal, complemented
4#14543600C212yesmaximal, complemented
5#14537114C211yesmaximal, characteristic
6C2×#10587544—21no—
7C2×#10587543—21no—
8C2×#10587543—21no—
9C2×#10587542—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587569—21no—
13C2×#10587624C411yescharacteristic
14C2×C42×D17C2211yescharacteristic
15#11762712C2211yescharacteristic
16#11762708C2211yescharacteristic
17#11762707C411yescharacteristic
18#11762703—21no—
19#11762702—21no—
20#11762685—21no—
21#11762666C411yescharacteristic
22C17×#390C411yescharacteristic
23C22×D4×C17C2×C411yescharacteristic
24C22×C22⋊D17—21no—
25C22×C4×D17C2×C411yescharacteristic
26C22×C4×D17D411yescharacteristic
27C22×C4×D17D411yescharacteristic
28C22×#3624—22no—
29C22×#3624—22no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17—21no—
32C2×C22⋊C4×C17—21no—
33C2×C42×C17C2×C411yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37#10587624—42no—
38C2×#63227—21no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×#63225—21no—
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17C2311yescharacteristic
46#10587569—44no—
47C42×D17—42no—
48#10587544—42no—
49#10587543—24no—
50#10587543—42no—
51#10587542—42no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55C24×C17—21no—
56C23×D17C22⋊C411yescharacteristic
57C2×D4×C17—22no—
58C2×D4×C17—41no—
59C2×D4×C17—41no—
60C22×C4×C17—21no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C2×C22⋊D17—42no—
65C2×C22⋊D17—42no—
66C2×C22⋊D17—42no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17C2×D411yescharacteristic
71C22×Dic17C2×D411yescharacteristic
72C2×C4×D17—22no—
73C2×C4×D17—22no—
74C2×C4×D17—22no—
75C2×C4×D17—21no—
76C2×C4×D17—22no—
77C2×C4×D17—21no—
78C2×C4×D17—41no—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81C2×C4×D17—41no—
82C2×#3624—24no—
83C2×#3624—42no—
84C2×#3624—42no—
85C2×#3624—42no—
86C2×#3624—42no—
87C2×#3624—42no—
88C4⋊C4×C17—22no—
89C22⋊C4×C17—42no—
90C22⋊C4×C17—42no—
91C42×C17—41no—
92C22⋊Dic17—42no—
93C22⋊Dic17—42no—
94C22⋊Dic17—41no—
95C22⋊Dic17—41no—
96#63227—42no—
97C4⋊Dic17—42no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100#63225—42no—
101C4×Dic17—22no—
102C4×Dic17—22no—
103C4×Dic17—41no—
104#7698—171nomaximal, Sylow
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17C2×C22⋊C411yescharacteristic
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17#10412yes—
120D4×C17—82no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17#13112yes—
133C22⋊D17—84no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17#9912yes—
147C2×Dic17AGL(1,ℤ/8)11yescharacteristic
148C2×Dic17AGL(1,ℤ/8)12yes—
149C2×Dic17#13711yescharacteristic
150C4×D17—44no—
151C4×D17—44no—
152C4×D17—42no—
153C4×D17—42no—
154#3624—44no—
155#3624—82no—
156#2854—171no—
157#1797—171no—
158#1749—171no—
159#1271—172no—
160#1270—172no—
161C22×C17—81no—
162C22×C17—81no—
163C22×C17—81no—
164C22×C17—81no—
165C22×C17—81no—
166C22×C17—81no—
167C22×C17—81no—
168C22×C17#40811yescharacteristic
169C22×C17C2×#9911yescharacteristic
170C22×C17#41311yescharacteristic
171C22×C17#41711yescharacteristic
172C22×C17#41811yescharacteristic
173C22×C17C2×#10411yescharacteristic
174C22×C17#42111yescharacteristic
175C2×D17—22no—
176C2×D17—22no—
177C2×D17—22no—
178C2×D17—41no—
179C2×D17—41no—
180C2×D17—41no—
181C2×D17—41no—
182C4×C17—22no—
183C4×C17—41no—
184C4×C17—41no—
185C4×C17—81no—
186Dic17—22no—
187Dic17—22no—
188Dic17—21no—
189Dic17—21no—
190Dic17—81no—
191Dic17—81no—
192C2×#123—341no—
193C2×#121—171no—
194C22×C42—171no—
195C2×C4⋊C8—341no—
196C2×C22⋊C8—341no—
197C2×C22⋊C8—341no—
198C2×#97—341no—
199#390—171no—
200#390#59511yescharacteristic
201#389—171no—
202#389—171no—
203#388—341no—
204#388—341no—
205#386—341no—
206#380—171no—
207#336—172no—
208#336—172no—
209C2×C17—161no—
210C2×C17#146811yescharacteristic
211C2×C17#148311yescharacteristic
212C2×C17#149611yescharacteristic
213C2×C17#150711yescharacteristic
214C2×C17#151311yescharacteristic
215C2×C17#153211yescharacteristic
216C2×C17#155711yescharacteristic
217D17—42no—
218C22×D4—341no—
219C22×D4C2×#59511yescharacteristic
220C23×C4—171no—
221C23×C4—171no—
222C23×C4—171no—
223C22×C8—342no—
224C22×C8—342no—
225#123—684no—
226#121—682no—
227C2×C4⋊C4—171no—
228C2×C4⋊C4—171no—
229C2×C4⋊C4—341no—
230C2×C4⋊C4—341no—
231C2×C4⋊C4C2×#59511yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—341no—
235C2×C22⋊C4—341no—
236C2×C22⋊C4—341no—
237C2×C22⋊C4—341no—
238C2×C42—171no—
239C2×C42—171no—
240C2×C42—171no—
241C2×C42—682no—
242C2×C42C2×#59511yescharacteristic
243C4⋊C8—682no—
244C22⋊C8—344no—
245C22⋊C8—682no—
246#97—682no—
247#95—341no—
248#95—341no—
249#95—341no—
250C17#769811yescharacteristic, complemented, Sylow
251C24—21no—
252C24—171no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—682no—
257C2×D4—682no—
258C2×D4—682no—
259C22×C4—21no—
260C22×C4—171no—
261C22×C4—171no—
262C22×C4—171no—
263C22×C4—341no—
264C22×C4—342no—
265C22×C4—342no—
266C22×C4—341no—
267C22×C4—342no—
268C22×C4—342no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4—681no—
272C22×C4—681no—
273C22×C4—681no—
274C22×C4—681no—
275C22×C4C22⋊#59511yescharacteristic
276C22×C4C22⋊#59511yescharacteristic
277C22×C4C22×#59511yescharacteristic
278C2×C8—344no—
279C2×C8—682no—
280C2×C8—682no—
281C2×C8—682no—
282C2×C8—682no—
283C2×C8—682no—
284C4⋊C4—22no—
285C4⋊C4—681no—
286C4⋊C4—681no—
287C4⋊C4—682no—
288C4⋊C4—682no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—682no—
292C22⋊C4—681no—
293C22⋊C4—681no—
294C22⋊C4—682no—
295C22⋊C4—682no—
296C42—41no—
297C42—342no—
298C42—342no—
299C42—681no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—172no—
308C23—341no—
309C23—341no—
310C23—341no—
311C23—341no—
312C23—341no—
313C23—341no—
314C23C2×C22⋊#59511yescharacteristic
315D4—82no—
316D4—1364no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—172no—
329C2×C4—171no—
330C2×C4—171no—
331C2×C4—172no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—684no—
339C2×C4—684no—
340C2×C4—682no—
341C2×C4—682no—
342C2×C4—681no—
343C2×C4—681no—
344C2×C4—681no—
345C2×C4—681no—
346C2×C4—681no—
347C2×C4—681no—
348C2×C4#1058767812yes—
349C8—684no—
350C8—1362no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—342no—
359C22—342no—
360C22—342no—
361C22—681no—
362C22—681no—
363C22—681no—
364C22—681no—
365C22#1176378311yescharacteristic
366C22D17⋊#9911yescharacteristic
367C22#1176379511yescharacteristic
368C22#1176380011yescharacteristic
369C22#1176380111yescharacteristic
370C22D17⋊#10411yescharacteristic
371C22#1176380711yescharacteristic
372C4—22no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—342no—
377C4—341no—
378C4—341no—
379C4—342no—
380C4—1361no—
381C4—1361no—
382C2—161no—
383C2—682no—
384C2#1454405011yescharacteristic
385C2#1454410011yescharacteristic
386C2#1454414411yescharacteristic
387C2#1454417911yescharacteristic
388C2#1454420011yescharacteristic
389C2#1454425911yescharacteristic
390C2#1454431511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#390C4
socleC23×C17C2×C22⋊C4
prime radicalC2×#10587624C4
cosocleC2×C4×Dic17C23
hypercenter#390#595
hypocenterC17#7698
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #14544641C2semidirect
2. #14544511C2semidirect
3. #14543602C2semidirect
4. #14543600C2semidirect
250. C17#7698semidirect

Maximal subgroups

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

subgroupindex
#145446412
#145445112
#145436022
#145436002
#145371142
#769817