Finite Group Databasebeta

← #17950117

Subgroups — #17950117

Order 4352.

556 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 90 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, complemented
2#14544448C211yesmaximal, characteristic, complemented
3#14543602C212yesmaximal, complemented
4#14543600C212yesmaximal, complemented
5#14537114C211yesmaximal, characteristic
6C22×#63243—21no—
7C2×#10587543—21no—
8#11763675—21no—
9#11763675—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×#63243—24no—
31C2×#63243—42no—
32C2×#63243—42no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C42×C17C2×C411yescharacteristic
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17—21no—
40#10587624—42no—
41C2×#63227—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17C2×C411yescharacteristic
44C2×C4⋊Dic17C2×C411yescharacteristic
45C2×#63225—21no—
46C2×C4×Dic17D411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C2×C4×Dic17C2311yescharacteristic
49#10587569—44no—
50C42×D17—42no—
51#10587543—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—21no—
75C2×C4×D17—21no—
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—22no—
79C2×C4×D17—22no—
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—
88#63243—48no—
89C4⋊C4×C17—22no—
90C22⋊C4×C17—42no—
91C22⋊C4×C17—42no—
92C42×C17—41no—
93C22⋊Dic17—42no—
94C22⋊Dic17—42no—
95C22⋊Dic17—41no—
96C22⋊Dic17—41no—
97#63227—42no—
98C4⋊Dic17—42no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101#63225—42no—
102C4×Dic17—41no—
103C4×Dic17—41no—
104C4×Dic17—41no—
105#7694—171nomaximal, Sylow
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17C2×C22⋊C411yescharacteristic
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—21no—
120C22×D17#9912yes—
121D4×C17—82no—
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#10412yes—
133C2×C4×C17#13112yes—
134C22⋊D17—84no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17AGL(1,ℤ/8)11yescharacteristic
149C2×Dic17AGL(1,ℤ/8)12yes—
150C2×Dic17#13711yescharacteristic
151C4×D17—24no—
152C4×D17—24no—
153C4×D17—44no—
154C4×D17—44no—
155#3624—44no—
156#3624—82no—
157#2854—171no—
158#1817—171no—
159#1728—171no—
160#1271—172no—
161#1270—172no—
162C22×C17—81no—
163C22×C17—81no—
164C22×C17—81no—
165C22×C17—81no—
166C22×C17—81no—
167C22×C17—81no—
168C22×C17—81no—
169C22×C17#40811yescharacteristic
170C22×C17C2×#9911yescharacteristic
171C22×C17#41311yescharacteristic
172C22×C17#41711yescharacteristic
173C22×C17#41811yescharacteristic
174C22×C17C2×#10411yescharacteristic
175C22×C17#42111yescharacteristic
176C2×D17—22no—
177C2×D17—22no—
178C2×D17—22no—
179C2×D17—41no—
180C2×D17—41no—
181C2×D17—41no—
182C2×D17—41no—
183C4×C17—22no—
184C4×C17—22no—
185C4×C17—41no—
186C4×C17—81no—
187Dic17—21no—
188Dic17—21no—
189Dic17—22no—
190Dic17—41no—
191Dic17—81no—
192Dic17—81no—
193C22×M16—341no—
194C2×#123—341no—
195C2×#121—171no—
196C22×C42—171no—
197C2×C22⋊C8—341no—
198#390—171no—
199#390#59511yescharacteristic
200#389—171no—
201#389—171no—
202#388—341no—
203#388—341no—
204#386—341no—
205#380—171no—
206#336—172no—
207#336—172no—
208#336—341no—
209#336—341no—
210C2×C17—161no—
211C2×C17#146811yescharacteristic
212C2×C17#148411yescharacteristic
213C2×C17#149511yescharacteristic
214C2×C17#150411yescharacteristic
215C2×C17#151011yescharacteristic
216C2×C17#152911yescharacteristic
217C2×C17#155711yescharacteristic
218D17—42no—
219C22×D4—341no—
220C22×D4C2×#59511yescharacteristic
221C23×C4—171no—
222C23×C4—171no—
223C23×C4—171no—
224C2×M16—344no—
225C2×M16—682no—
226C2×M16—682no—
227C22×C8—342no—
228C22×C8—342no—
229#123—684no—
230#121—682no—
231C2×C4⋊C4—171no—
232C2×C4⋊C4—171no—
233C2×C4⋊C4—341no—
234C2×C4⋊C4—341no—
235C2×C4⋊C4C2×#59511yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—341no—
239C2×C22⋊C4—341no—
240C2×C22⋊C4—341no—
241C2×C22⋊C4—341no—
242C2×C42—171no—
243C2×C42—171no—
244C2×C42—171no—
245C2×C42—682no—
246C2×C42C2×#59511yescharacteristic
247C22⋊C8—682no—
248#95—341no—
249#95—341no—
250#95—341no—
251C17#769411yescharacteristic, complemented, Sylow
252C24—21no—
253C24—171no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—682no—
258C2×D4—682no—
259C2×D4—682no—
260C22×C4—21no—
261C22×C4—171no—
262C22×C4—171no—
263C22×C4—171no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4—342no—
267C22×C4—342no—
268C22×C4—342no—
269C22×C4—342no—
270C22×C4—342no—
271C22×C4—342no—
272C22×C4—341no—
273C22×C4—341no—
274C22×C4—681no—
275C22×C4—681no—
276C22×C4C22⋊#59511yescharacteristic
277C22×C4C22⋊#59511yescharacteristic
278C22×C4C22×#59511yescharacteristic
279M16—688no—
280C2×C8—344no—
281C2×C8—682no—
282C2×C8—682no—
283C2×C8—682no—
284C2×C8—682no—
285C2×C8—682no—
286C4⋊C4—22no—
287C4⋊C4—681no—
288C4⋊C4—681no—
289C4⋊C4—682no—
290C4⋊C4—682no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—682no—
294C22⋊C4—681no—
295C22⋊C4—681no—
296C22⋊C4—682no—
297C22⋊C4—682no—
298C42—41no—
299C42—681no—
300C42—681no—
301C42—681no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—172no—
310C23—341no—
311C23—341no—
312C23—341no—
313C23—341no—
314C23—341no—
315C23—341no—
316C23C2×C22⋊#59511yescharacteristic
317D4—82no—
318D4—1364no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—171no—
330C2×C4—171no—
331C2×C4—172no—
332C2×C4—344no—
333C2×C4—344no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—684no—
342C2×C4—684no—
343C2×C4—681no—
344C2×C4—681no—
345C2×C4—681no—
346C2×C4—681no—
347C2×C4—681no—
348C2×C4—681no—
349C2×C4#1058754812yes—
350C2×C4#1058767812yes—
351C8—684no—
352C8—1362no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—342no—
361C22—342no—
362C22—342no—
363C22—681no—
364C22—681no—
365C22—681no—
366C22—681no—
367C22#1176378311yescharacteristic
368C22D17⋊#9911yescharacteristic
369C22#1176379511yescharacteristic
370C22#1176380011yescharacteristic
371C22#1176380111yescharacteristic
372C22C2×#1058754811yescharacteristic
373C22#1176380611yescharacteristic
374C4—22no—
375C4—22no—
376C4—41no—
377C4—81no—
378C4—341no—
379C4—341no—
380C4—342no—
381C4—681no—
382C4—1361no—
383C4—1361no—
384C2—161no—
385C2—682no—
386C2#1454405011yescharacteristic
387C2#1454410411yescharacteristic
388C2#1454414011yescharacteristic
389C2#1454416911yescharacteristic
390C2#1454419011yescharacteristic
391C2#1454424911yescharacteristic
392C2#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#7694
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. #14544676C2semidirect
2. #14544448C2semidirect
3. #14543602C2semidirect
4. #14543600C2semidirect
251. C17#7694semidirect

Maximal subgroups

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

subgroupindex
#145446762
#145444482
#145436022
#145436002
#145371142
#769417