Finite Group Databasebeta

← #17949988

Subgroups — #17949988

Order 4352.

544 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544640C211yesmaximal, characteristic, complemented
2C4×C22⋊#3624C211yesmaximal, characteristic, complemented
3#14543603C212yesmaximal, complemented
4#14543600C212yesmaximal, complemented
5#14537110C211yesmaximal, characteristic
6C2×#10587544—21no—
7C2×C22⋊#3624—21no—
8C2×C22⋊#3624—21no—
9C2×C4×#3624—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587621—21no—
13C22×C4×Dic17C2211yescharacteristic
14#11762712C411yescharacteristic
15#11762712C2211yescharacteristic
16#11762705C411yescharacteristic
17#11762705C2211yescharacteristic
18#11762697—21no—
19#11762697—21no—
20#11762689—21no—
21#11762669C411yescharacteristic
22C2×C17×#121C411yescharacteristic
23C22×D4×C17—21no—
24C22×D4×C17C2×C411yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×Dic17D411yescharacteristic
27C23×Dic17D411yescharacteristic
28C22×#3624—22no—
29C22×#3624—22no—
30C17×#121—42no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17—21no—
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38#10587621—42no—
39#10587621—42no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×#63225—21no—
43C2×#63225—21no—
44C2×C4×Dic17—42no—
45C2×C4×Dic17C2×C411yescharacteristic
46C2×C4×Dic17D411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C2×C4×Dic17C2311yescharacteristic
49C22⋊#3624—24no—
50C22⋊#3624—42no—
51#10587544—42no—
52C4×#3624—42no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56C24×C17—21no—
57C24×C17C22⋊C411yescharacteristic
58C2×D4×C17—22no—
59C2×D4×C17—41no—
60C2×D4×C17—41no—
61C2×D4×C17—42no—
62C2×D4×C17—42no—
63C2×D4×C17—42no—
64C22×C4×C17—22no—
65C22×C4×C17—22no—
66C22×C4×C17—21no—
67C22×C4×C17—41no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×Dic17—22no—
71C22×Dic17—22no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—41no—
75C22×Dic17—41no—
76C22×Dic17—41no—
77C22×Dic17C22⋊C411yescharacteristic
78C22×Dic17C22⋊C411yescharacteristic
79C22×Dic17D812yes—
80C22×Dic17SD1612yes—
81C22×Dic17C2×D411yescharacteristic
82C22×Dic17C2×D411yescharacteristic
83C2×#3624—24no—
84C2×#3624—42no—
85C2×#3624—42no—
86C2×#3624—42no—
87C2×#3624—42no—
88C2×#3624—42no—
89C4⋊C4×C17—41no—
90C22⋊C4×C17—42no—
91C22⋊Dic17—42no—
92C22⋊Dic17—42no—
93C22⋊Dic17—42no—
94C22⋊Dic17—41no—
95C22⋊Dic17—41no—
96C22⋊Dic17—42no—
97C4⋊Dic17—22no—
98C4⋊Dic17—41no—
99#63225—42no—
100#63225—41no—
101#63225—41no—
102C4×Dic17—22no—
103C4×Dic17—22no—
104C4×Dic17—41no—
105C4×Dic17—41no—
106#7658—171nomaximal, Sylow
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—41no—
119C23×C17—41no—
120C23×C17#10212yes—
121C23×C17C2×C22⋊C411yescharacteristic
122D4×C17—82no—
123D4×C17—84no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—44no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17#9912yes—
133C2×C4×C17#13112yes—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—42no—
143C2×Dic17—42no—
144C2×Dic17—44no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17#10212yes—
152C2×Dic17#10412yes—
153C2×Dic17C2×D811yescharacteristic
154C2×Dic17C2×SD1611yescharacteristic
155C2×Dic17#13512yes—
156#3624—44no—
157#3624—82no—
158#2854—171no—
159#1797—171no—
160C4×C22⋊C8—171no—
161#1271—172no—
162#1270—172no—
163C22×C17—22no—
164C22×C17—22no—
165C22×C17—22no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—81no—
171C22×C17—81no—
172C22×C17—81no—
173C22×C17—81no—
174C22×C17—81no—
175C22×C17—81no—
176C22×C17—81no—
177C22×C17#40811yescharacteristic
178C22×C17C2×#9911yescharacteristic
179C22×C17#41311yescharacteristic
180C22×C17C2×#10211yescharacteristic
181C22×C17#41611yescharacteristic
182C22×C17C2×#10411yescharacteristic
183C22×C17#42111yescharacteristic
184C4×C17—22no—
185C4×C17—22no—
186C4×C17—81no—
187Dic17—22no—
188Dic17—22no—
189Dic17—21no—
190Dic17—22no—
191Dic17—21no—
192Dic17—81no—
193Dic17—81no—
194C2×#123—341no—
195C2×#121#59511yescharacteristic
196C22×C42—171no—
197C2×C4⋊C8—341no—
198C2×C22⋊C8—341no—
199C2×C22⋊C8—341no—
200C2×C4×C8—341no—
201#390—171no—
202#390—171no—
203#389—171no—
204#389—171no—
205#388—341no—
206#388—341no—
207#386—341no—
208#380—171no—
209#336—172no—
210#336—172no—
211C2×C17—42no—
212C2×C17—161no—
213C2×C17#146811yescharacteristic
214C2×C17#148211yescharacteristic
215C2×C17#149211yescharacteristic
216C2×C17#150111yescharacteristic
217C2×C17#151911yescharacteristic
218C2×C17#152911yescharacteristic
219C2×C17#154011yescharacteristic
220C22×D4—21no—
221C22×D4C2×#59511yescharacteristic
222C23×C4—171no—
223C23×C4—171no—
224C23×C4C2×#59511yescharacteristic
225C22×C8—342no—
226C22×C8—342no—
227#123—682no—
228#123—682no—
229#121—42no—
230C2×C4⋊C4—171no—
231C2×C4⋊C4—171no—
232C2×C4⋊C4—341no—
233C2×C4⋊C4—341no—
234C2×C4⋊C4C2×#59511yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—341no—
237C2×C22⋊C4—341no—
238C2×C22⋊C4—341no—
239C2×C22⋊C4—341no—
240C2×C22⋊C4—341no—
241C2×C42—171no—
242C2×C42—171no—
243C2×C42—171no—
244C2×C42—171no—
245C2×C42—682no—
246C4⋊C8—682no—
247C22⋊C8—344no—
248C22⋊C8—682no—
249C4×C8—682no—
250#95—341no—
251#95—341no—
252#95—341no—
253C17#765811yescharacteristic, complemented, Sylow
254C24—21no—
255C24C22⋊#59511yescharacteristic
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—171no—
267C22×C4—171no—
268C22×C4—171no—
269C22×C4—171no—
270C22×C4—172no—
271C22×C4—172no—
272C22×C4—342no—
273C22×C4—342no—
274C22×C4—341no—
275C22×C4—341no—
276C22×C4—681no—
277C22×C4—681no—
278C22×C4—681no—
279C22×C4C22⋊#59511yescharacteristic
280C22×C4C22×#59511yescharacteristic
281C2×C8—344no—
282C2×C8—682no—
283C2×C8—682no—
284C2×C8—682no—
285C2×C8—682no—
286C2×C8—682no—
287C4⋊C4—41no—
288C4⋊C4—342no—
289C4⋊C4—681no—
290C4⋊C4—681no—
291C4⋊C4—682no—
292C4⋊C4—681no—
293C22⋊C4—42no—
294C22⋊C4—682no—
295C22⋊C4—682no—
296C22⋊C4—681no—
297C22⋊C4—682no—
298C22⋊C4—681no—
299C22⋊C4—682no—
300C42—342no—
301C42—342no—
302C42—681no—
303C42—681no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#1058754712yes—
318C23C2×C22⋊#59511yescharacteristic
319D4—82no—
320D4—84no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—44no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—171no—
330C2×C4—172no—
331C2×C4—171no—
332C2×C4—172no—
333C2×C4—172no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—682no—
343C2×C4—682no—
344C2×C4—684no—
345C2×C4—681no—
346C2×C4—681no—
347C2×C4—681no—
348C2×C4—681no—
349C2×C4—681no—
350C2×C4—681no—
351C2×C4#1058755112yes—
352C2×C4#1058767812yes—
353C8—684no—
354C8—1362no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22#1176378211yescharacteristic
370C22C2×#1058755111yescharacteristic
371C22#1176379611yescharacteristic
372C22C2×#1058754711yescharacteristic
373C22#1176379911yescharacteristic
374C22D17⋊#10411yescharacteristic
375C22#1176380711yescharacteristic
376C4—22no—
377C4—22no—
378C4—81no—
379C4—342no—
380C4—342no—
381C4—341no—
382C4—342no—
383C4—341no—
384C4—1361no—
385C4—1361no—
386C2—42no—
387C2—161no—
388C2#1454404911yescharacteristic
389C2#1454409511yescharacteristic
390C2#1454412911yescharacteristic
391C2#1454415911yescharacteristic
392C2#1454422011yescharacteristic
393C2#1454425011yescharacteristic
394C2#1454428211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
Fitting, prime radicalC2×C17×#121C4
socleC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×#121#595
hypocenterC17#7658
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #14544640C2semidirect
2. C4×C22⋊#3624C2semidirect
3. #14543603C2semidirect
4. #14543600C2semidirect
253. C17#7658semidirect

Maximal subgroups

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

subgroupindex
#145446402
C4×C22⋊#36242
#145436032
#145436002
#145371102
#765817