Finite Group Databasebeta

← #16945228

Subgroups — #16945228

Order 4352.

572 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 83 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#14532119C211yesmaximal, characteristic
2#14528864C211yesmaximal, characteristic, complemented
3#14526815C212yesmaximal, complemented
4#14526788C212yesmaximal, complemented
5C2×C17×#324C211yesmaximal, characteristic, complemented
6C22×#63252C411yescharacteristic
7C2×#10587615—21no—
8#11762773—21no—
9#11762729C411yescharacteristic
10#11762729C2211yescharacteristic
11#11762726C411yescharacteristic
12#11762726C2211yescharacteristic
13#11762688—21no—
14#11762680—21no—
15#11762680—21no—
16#11762672C411yescharacteristic
17#11762578—21no—
18#11762578C2212yes—
19C22×C4⋊C4×C17C2211yescharacteristic
20C2×C17×C22⋊C8—21no—
21C17×#324—21no—
22C17×#324C2212yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×C22⋊D17—21no—
27C2×#63252—41no—
28C2×#63252—41no—
29C2×#63252—44no—
30C22×D68C2×C411yescharacteristic
31C22×C4×D17—21no—
32C22×Dic34C2×C411yescharacteristic
33C22×C8×C17—41no—
34C2×C4⋊C4×C17—41no—
35C2×C4⋊C4×C17—41no—
36C2×C4⋊C4×C17C2×C411yescharacteristic
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C4⋊C4×C17D411yescharacteristic
39C2×C4⋊C4×C17C2311yescharacteristic
40C2×C22⋊Dic17—21no—
41#10587615—44no—
42C2×#63227—21no—
43C2×#63227—21no—
44C2×#63227—21no—
45C2×#63227—21no—
46C2×#63225—21no—
47C2×#63225—21no—
48C2×#63223—22no—
49C17×C22⋊C8—22no—
50C17×C22⋊C8—41no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502C2×C411yescharacteristic
55#10587502C2×C411yescharacteristic
56C22⋊#3622—41no—
57C22⋊#3622—41no—
58C24×C17C22⋊C411yescharacteristic
59C23×D17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—22no—
67C22×C4×C17—41no—
68C22×C4×C17—42no—
69C22×C4×C17—42no—
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C22×C411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C2×C22⋊D17—24no—
77C2×C22⋊D17—42no—
78C2×C22⋊D17—41no—
79C2×C22⋊D17—41no—
80C22×Dic17—21no—
81C22×Dic17—21no—
82C22×Dic17—21no—
83#63252—44no—
84#63252—44no—
85#63252—44no—
86#63252—44no—
87C2×D68—41no—
88C2×D68—42no—
89C2×C4×D17—42no—
90C2×C4×D17—42no—
91C2×C4×D17—42no—
92C2×Dic34—41no—
93C2×Dic34—42no—
94C2×C8×C17—42no—
95C2×C8×C17—82no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17—41no—
98C4⋊C4×C17—41no—
99C4⋊C4×C17—41no—
100C22⋊Dic17—42no—
101#63227—42no—
102#63227—42no—
103#63227—42no—
104#63227—42no—
105#63225—42no—
106#63225—42no—
107#63223—42no—
108#63223—42no—
109C2×#3622—42no—
110#11848—171nomaximal, Sylow
111C23×C17—22no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—41no—
115C23×C17#9911yescharacteristic
116C23×C17#9911yescharacteristic
117C23×C17C2×C22⋊C411yescharacteristic
118C22×D17—22no—
119C22×D17—41no—
120C22×D17—41no—
121C22×D17—41no—
122C22×D17—41no—
123C22×D17—41no—
124C22×D17—41no—
125C2×C4×C17—21no—
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—21no—
134C2×C4×C17—42no—
135C2×C4×C17—42no—
136C2×C4×C17—42no—
137C2×C4×C17—42no—
138C2×C4×C17—42no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—81no—
144C2×C4×C17#13512yes—
145C22⋊D17—44no—
146C22⋊D17—44no—
147C2×Dic17—22no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156D68—42no—
157D68—42no—
158C4×D17—48no—
159Dic34—42no—
160Dic34—42no—
161C8×C17—82no—
162#3622—82no—
163#2406—171no—
164#1821—171no—
165C2×#324D1711yescharacteristic, complemented
166#1354—172no—
167#1345—172no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—21no—
172C22×C17—21no—
173C22×C17—21no—
174C22×C17—21no—
175C22×C17—42no—
176C22×C17—42no—
177C22×C17AΣL(1,𝔽16)12yes—
178C22×C17C2×#9911yescharacteristic
179C22×C17#41611yescharacteristic
180C22×C17#42111yescharacteristic
181C2×D17—42no—
182C2×D17—42no—
183C2×D17—42no—
184C2×D17—81no—
185C2×D17—81no—
186C2×D17—81no—
187C2×D17—81no—
188C4×C17—22no—
189C4×C17—41no—
190C4×C17—41no—
191C4×C17—41no—
192C4×C17—41no—
193C4×C17—41no—
194Dic17—42no—
195Dic17—81no—
196Dic17—81no—
197C22×#41—171no—
198C2×#123—341no—
199C22×C4⋊C4C2×D1711yescharacteristic
200#407—341no—
201C2×C22⋊C8—21no—
202#395—171no—
203#395—171no—
204#394—171no—
205#394—171no—
206#386—341no—
207#385—341no—
208#385—341no—
209#381—171no—
210#343—172no—
211#343—341no—
212#324—21no—
213#324C2×D1712yes—
214C2×C17—21no—
215C2×C17—21no—
216C2×C17—22no—
217C2×C17—41no—
218C2×C17#151611yescharacteristic
219C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
220C2×C17#213311yescharacteristic
221D17—82no—
222C2×#41—681no—
223C2×#41—681no—
224C2×#41—684no—
225C22×Q8—171no—
226C22×D4—171no—
227C22×D4—341no—
228C23×C4—341no—
229C23×C4C4×D1711yescharacteristic
230C23×C4D6811yescharacteristic
231C23×C4C22⋊D1711yescharacteristic
232C2×M16—342no—
233C22×C8—41no—
234#123—684no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—341no—
238C2×C4⋊C4—341no—
239C2×C4⋊C4C4×D1711yescharacteristic
240C2×C4⋊C4D6811yescharacteristic
241C2×C4⋊C4C22⋊D1711yescharacteristic
242C2×C4⋊C4C22×D1711yescharacteristic
243C2×C22⋊C4—341no—
244C2×C22⋊C4—341no—
245C2×C22⋊C4—341no—
246C2×C22⋊C4—341no—
247C2×C22⋊C4—341no—
248C22⋊C8—22no—
249C22⋊C8—41no—
250C22⋊C8—681no—
251C22⋊C8—681no—
252#95—171no—
253#95—171no—
254#95—341no—
255#95—341no—
256#95—341no—
257C17#1184811yescharacteristic, complemented, Sylow
258C24—341no—
259C24#6322711yescharacteristic
260#41—684no—
261#41—684no—
262#41—684no—
263#41—684no—
264C2×Q8—681no—
265C2×Q8—682no—
266C2×D4—344no—
267C2×D4—681no—
268C2×D4—682no—
269C2×D4—682no—
270C2×D4—681no—
271C2×D4—681no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—341no—
283C22×C4—341no—
284C22×C4—341no—
285C22×C4—682no—
286C22×C4—682no—
287C22×C4—682no—
288C22×C4#6322711yescharacteristic
289C22×C4#6322711yescharacteristic
290C22×C4#6322711yescharacteristic
291C22×C4C2×C4×D1711yescharacteristic
292C22×C4C2×D6811yescharacteristic
293C22×C4C2×C22⋊D1711yescharacteristic
294M16—682no—
295M16—682no—
296C2×C8—42no—
297C2×C8—82no—
298C2×C8—682no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—682no—
304C4⋊C4—682no—
305C22⋊C4—682no—
306C22⋊C4—682no—
307C22⋊C4—682no—
308C22⋊C4—682no—
309C22⋊C4—682no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—41no—
314C23—342no—
315C23—681no—
316C23—681no—
317C23—681no—
318C23—681no—
319C23—681no—
320C23—681no—
321C23#1058747711yescharacteristic
322C23#1058747711yescharacteristic
323C23C2×#6322711yescharacteristic
324Q8—682no—
325Q8—682no—
326D4—682no—
327D4—682no—
328D4—684no—
329D4—684no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—342no—
350C2×C4—688no—
351C2×C4—681no—
352C2×C4—681no—
353C2×C4—681no—
354C2×C4—681no—
355C2×C4—681no—
356C2×C4—681no—
357C2×C4—681no—
358C2×C4—681no—
359C2×C4#1058759012yes—
360C8—82no—
361C8—1362no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—682no—
372C22—682no—
373C22—682no—
374C22—1361no—
375C22—1361no—
376C22—1361no—
377C22—1361no—
378C22#1176259512yes—
379C22C2×#1058747711yescharacteristic
380C22#1176280611yescharacteristic
381C22#1176282511yescharacteristic
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—682no—
389C4—1361no—
390C4—1361no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2—41no—
395C2—1362no—
396C2#1452734111yescharacteristic
397C2C2×#1176259511yescharacteristic
398C2#1453047111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC2×C17×#324C2
socleC22×C17C2×#99
prime radicalC22×#63252C4
cosocleC2×C4⋊C4×C17C23
hypercenterC2×#324D17
hypocenterC17#11848
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #14528864C2semidirect
3. #14526815C2semidirect
4. #14526788C2semidirect
5. C2×C17×#324C2semidirect
165. C2×#324D17semidirect
257. C17#11848semidirect

Maximal subgroups

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

subgroupindex
#145321192
#145288642
#145268152
#145267882
C2×C17×#3242
#1184817