Finite Group Databasebeta

← #17950100

Subgroups — #17950100

Order 4352.

600 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 98 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#14544511C211yesmaximal, characteristic, complemented
2#14544445C211yesmaximal, characteristic, complemented
3#14543604C212yesmaximal, complemented
4#14543600C212yesmaximal, complemented
5#14537080C211yesmaximal, characteristic
6C22×#63242—21no—
7C2×#10587543—21no—
8C2×#10587542—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587632C411yescharacteristic
13C2×#10587624C411yescharacteristic
14C2×C42×D17C2211yescharacteristic
15#11762716C2211yescharacteristic
16#11762712C2211yescharacteristic
17#11762697—21no—
18#11762697—21no—
19#11762685—21no—
20#11762685—21no—
21#11762675C411yescharacteristic
22C2×C17×#124C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C22×D4×C17C2×C411yescharacteristic
25C22×C22⋊D17—21no—
26C22×C4×D17C2×C411yescharacteristic
27C22×C4×D17D411yescharacteristic
28C22×C4×D17D411yescharacteristic
29C22×#3624—22no—
30C22×#3624—22no—
31C2×#63242—24no—
32C2×#63242—42no—
33C2×#63242—42no—
34C17×#124—42no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C42×C17C2×C411yescharacteristic
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17—21no—
41#10587632—42no—
42#10587624—42no—
43C2×#63227—21no—
44C2×C4⋊Dic17C2×C411yescharacteristic
45C2×C4⋊Dic17C2×C411yescharacteristic
46C2×#63225—21no—
47C2×C4×Dic17D411yescharacteristic
48C2×C4×Dic17D411yescharacteristic
49C2×C4×Dic17C2311yescharacteristic
50C42×D17—42no—
51#10587543—24no—
52#10587542—42no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57C24×C17—21no—
58C23×D17C22⋊C411yescharacteristic
59C2×Q8×C17—22no—
60C2×Q8×C17—41no—
61C2×Q8×C17—41no—
62C2×D4×C17—22no—
63C2×D4×C17—41no—
64C2×D4×C17—41no—
65C22×C4×C17—21no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C2×C22⋊D17—42no—
70C2×C22⋊D17—42no—
71C2×C22⋊D17—42no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17C22⋊C411yescharacteristic
75C22×Dic17C2×D411yescharacteristic
76C22×Dic17C2×D411yescharacteristic
77C2×C4×D17—22no—
78C2×C4×D17—22no—
79C2×C4×D17—21no—
80C2×C4×D17—21no—
81C2×C4×D17—22no—
82C2×C4×D17—22no—
83C2×C4×D17—22no—
84C2×C4×D17—22no—
85C2×C4×D17—41no—
86C2×C4×D17—41no—
87C2×#3624—24no—
88C2×#3624—24no—
89C2×#3624—42no—
90C2×#3624—42no—
91C2×#3624—42no—
92C2×#3624—42no—
93#63242—48no—
94C22⋊C4×C17—42no—
95C22⋊C4×C17—42no—
96C42×C17—22no—
97C22⋊Dic17—42no—
98C22⋊Dic17—42no—
99C22⋊Dic17—41no—
100C22⋊Dic17—41no—
101#63227—42no—
102C4⋊Dic17—41no—
103C4⋊Dic17—41no—
104#63225—42no—
105C4×Dic17—22no—
106C4×Dic17—41no—
107C4×Dic17—41no—
108#7690—171nomaximal, Sylow
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17C2×C22⋊C411yescharacteristic
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—21no—
120C22×D17—21no—
121C22×D17#10012yes—
122C22×D17#10412yes—
123C22×D17#10412yes—
124Q8×C17—82no—
125D4×C17—82no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17#9912yes—
137C2×C4×C17#13112yes—
138C22⋊D17—84no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17#13512yes—
153C2×Dic17AGL(1,ℤ/8)11yescharacteristic
154C2×Dic17#13711yescharacteristic
155C4×D17—24no—
156C4×D17—24no—
157C4×D17—44no—
158C4×D17—44no—
159#3624—44no—
160#3624—44no—
161#2850—171no—
162#1749—171no—
163C2×#336—171no—
164#1272—172no—
165#1270—172no—
166C22×C17—81no—
167C22×C17—81no—
168C22×C17—81no—
169C22×C17—81no—
170C22×C17—81no—
171C22×C17—81no—
172C22×C17—81no—
173C22×C17#40811yescharacteristic
174C22×C17C2×#9911yescharacteristic
175C22×C17C2×#10011yescharacteristic
176C22×C17#41611yescharacteristic
177C22×C17#41711yescharacteristic
178C22×C17C2×#10411yescharacteristic
179C22×C17C2×#10411yescharacteristic
180C2×D17—22no—
181C2×D17—22no—
182C2×D17—22no—
183C2×D17—22no—
184C2×D17—22no—
185C2×D17—22no—
186C2×D17#32914yes—
187C4×C17—22no—
188C4×C17—22no—
189C4×C17—41no—
190C4×C17—81no—
191Dic17—22no—
192Dic17—21no—
193Dic17—21no—
194Dic17—41no—
195Dic17—81no—
196Dic17—81no—
197C23×C8—341no—
198C2×#124#59511yescharacteristic
199C2×C22⋊Q8—171no—
200C2×#121—171no—
201C22×C42—171no—
202C2×C22⋊C8—341no—
203C2×#97—341no—
204#391—171no—
205#390—171no—
206#388—341no—
207#388—341no—
208#386—341no—
209#386—341no—
210#382—171no—
211#336—172no—
212#336—172no—
213#336—341no—
214C2×C17—161no—
215C2×C17#147111yescharacteristic
216C2×C17#148311yescharacteristic
217C2×C17#149411yescharacteristic
218C2×C17#150111yescharacteristic
219C2×C17#150411yescharacteristic
220C2×C17C2×#32911yescharacteristic
221C2×C17#155111yescharacteristic
222D17—24no—
223C22×Q8C2×#59511yescharacteristic
224C22×D4—341no—
225C22×D4C2×#59511yescharacteristic
226C23×C4—171no—
227C23×C4—171no—
228C23×C4—171no—
229C22×C8—342no—
230C22×C8—342no—
231C22×C8—344no—
232C22×C8—682no—
233C22×C8—682no—
234#124—42no—
235C22⋊Q8—682no—
236#121—682no—
237C2×C4⋊C4—171no—
238C2×C4⋊C4—171no—
239C2×C4⋊C4—341no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—341no—
243C2×C22⋊C4—341no—
244C2×C22⋊C4—341no—
245C2×C22⋊C4—341no—
246C2×C42—171no—
247C2×C42—171no—
248C2×C42—171no—
249C2×C42—682no—
250C2×C42C2×#59511yescharacteristic
251C22⋊C8—344no—
252#97—682no—
253#95—341no—
254#95—341no—
255#95—341no—
256#95—341no—
257C17#769011yescharacteristic, complemented, Sylow
258C24—21no—
259C24—171no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—682no—
267C2×D4—682no—
268C2×D4—682no—
269C22×C4—21no—
270C22×C4—171no—
271C22×C4—171no—
272C22×C4—171no—
273C22×C4—341no—
274C22×C4—341no—
275C22×C4—342no—
276C22×C4—342no—
277C22×C4—342no—
278C22×C4—342no—
279C22×C4—342no—
280C22×C4—342no—
281C22×C4—341no—
282C22×C4—341no—
283C22×C4—681no—
284C22×C4—681no—
285C22×C4C22⋊#59511yescharacteristic
286C22×C4C22⋊#59511yescharacteristic
287C22×C4C22×#59511yescharacteristic
288C2×C8—344no—
289C2×C8—344no—
290C2×C8—682no—
291C2×C8—682no—
292C2×C8—688no—
293C2×C8—682no—
294C2×C8—682no—
295C4⋊C4—681no—
296C4⋊C4—681no—
297C4⋊C4—682no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—682no—
301C22⋊C4—682no—
302C22⋊C4—681no—
303C22⋊C4—681no—
304C22⋊C4—682no—
305C42—22no—
306C42—342no—
307C42—681no—
308C42—681no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—172no—
317C23—172no—
318C23—172no—
319C23—341no—
320C23—341no—
321C23—341no—
322C23—341no—
323C23C2×C22⋊#59511yescharacteristic
324Q8—82no—
325D4—82no—
326D4—1364no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—172no—
338C2×C4—171no—
339C2×C4—171no—
340C2×C4—344no—
341C2×C4—344no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—684no—
350C2×C4—684no—
351C2×C4—681no—
352C2×C4—681no—
353C2×C4—681no—
354C2×C4—681no—
355C2×C4—681no—
356C2×C4—681no—
357C2×C4#1058755112yes—
358C2×C4#1058767812yes—
359C8—684no—
360C8—684no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—174no—
369C22—342no—
370C22—342no—
371C22—342no—
372C22—342no—
373C22—342no—
374C22—342no—
375C22#1176378311yescharacteristic
376C22C2×#1058755111yescharacteristic
377C22D17⋊#10011yescharacteristic
378C22#1176379911yescharacteristic
379C22#1176380011yescharacteristic
380C22D17⋊#10411yescharacteristic
381C22D17⋊#10411yescharacteristic
382C4—22no—
383C4—22no—
384C4—41no—
385C4—81no—
386C4—342no—
387C4—341no—
388C4—341no—
389C4—681no—
390C4—1361no—
391C4—1361no—
392C2—161no—
393C2—344no—
394C2#1454405911yescharacteristic
395C2#1454410011yescharacteristic
396C2#1454413611yescharacteristic
397C2#1454415911yescharacteristic
398C2#1454417011yescharacteristic
399C2D17⋊#32911yescharacteristic
400C2#1454430411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×#124C4
socleC23×C17C2×C22⋊C4
prime radicalC2×#10587624C4
cosocleC2×C4×Dic17C23
hypercenterC2×#124#595
hypocenterC17#7690
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #14544511C2semidirect
2. #14544445C2semidirect
3. #14543604C2semidirect
4. #14543600C2semidirect
257. C17#7690semidirect

Maximal subgroups

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

subgroupindex
#145445112
#145444452
#145436042
#145436002
#145370802
#769017