Finite Group Databasebeta

← #16927867

Subgroups — #16927867

Order 4352.

530 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 123 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#14531694C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526662C212yesmaximal, complemented
4#14526651C212yesmaximal, complemented
5C17×#1799C211yesmaximal, characteristic
6C2×D4×Dic17—21no—
7C2×C22⋊#3622C2212yes—
8#11762708—21no—
9#11762684—21no—
10#11762684—21no—
11#11762675C411yescharacteristic, complemented
12#11762669C411yescharacteristic, complemented
13#11762669C2211yescharacteristic
14C4×C4⋊Dic17C411yescharacteristic, complemented
15C2×#10587502C411yescharacteristic, complemented
16C2×#10587502C2211yescharacteristic
17#11762535—21no—
18#11762533—21no—
19C2×C4×D4×C17C2211yescharacteristic
20C2×C17×#99—21no—
21C17×#380C2212yes—
22C17×#342—21no—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C23×Dic17—21no—
27C23×Dic17—21no—
28C4×D4×C17—22no—
29C4×D4×C17—22no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C42×C17C2×C411yescharacteristic
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38D4×Dic17—42no—
39D4×Dic17—42no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17—21no—
44C2×C4×Dic17—21no—
45C22×#3622—22no—
46C17×#99—42no—
47C17×#95—22no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502C2×C411yescharacteristic
52#10587502C2×C411yescharacteristic
53#10587502C2×C411yescharacteristic
54#10587502C2×C411yescharacteristic
55C22⋊#3622—22no—
56C22⋊#3622—22no—
57C24×C17C22⋊C411yescharacteristic
58C24×C17C2×D411yescharacteristic
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17D812yes—
62C2×D4×C17SD1612yes—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—42no—
66C22×C4×C17—81no—
67C22×C4×C17—81no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17#4112yes—
74C22×C4×C17#4112yes—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79C22×Dic17—42no—
80C22×Dic17—42no—
81C22×Dic17—42no—
82C22×Dic17—41no—
83C22×Dic17—41no—
84C22×Dic17—81no—
85C22×Dic17—81no—
86C4⋊C4×C17—22no—
87C4⋊C4×C17—82no—
88C22⋊C4×C17—22no—
89C22⋊C4×C17—22no—
90C22⋊C4×C17—42no—
91C22⋊C4×C17—42no—
92C42×C17—21no—
93C42×C17—21no—
94C22⋊Dic17—42no—
95C22⋊Dic17—41no—
96C22⋊Dic17—41no—
97C4⋊Dic17—42no—
98C4⋊Dic17—42no—
99C4⋊Dic17—81no—
100C4⋊Dic17—81no—
101C4×Dic17—42no—
102C2×#3622—42no—
103C2×#3622—82no—
104#8760—171nomaximal, Sylow
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17#10212yes—
112C23×C17C2×C22⋊C411yescharacteristic
113C23×C17#12312yes—
114C23×C17C2×D811yescharacteristic
115C23×C17C2×SD1611yescharacteristic
116C23×C17#13712yes—
117D4×C17—24no—
118D4×C17—24no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—42no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—81no—
130C2×C4×C17—81no—
131C2×C4×C17—82no—
132C2×C4×C17—82no—
133C2×C4×C17—82no—
134C2×C4×C17#10212yes—
135C2×C4×C17#11711yescharacteristic
136C2×C4×C17#11711yescharacteristic
137C2×C4×C17#12312yes—
138C2×Dic17—22no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—82no—
151C2×Dic17—82no—
152C2×Dic17—84no—
153#3622—82no—
154#2349—171no—
155#1799D1711yescharacteristic
156#1459—171no—
157#1294—172no—
158#1289—172no—
159C22×C17—21no—
160C22×C17—22no—
161C22×C17—21no—
162C22×C17—22no—
163C22×C17—22no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—42no—
167C22×C17—42no—
168C22×C17—42no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17#38111yescharacteristic
172C22×C17C2×#10211yescharacteristic
173C22×C17#41911yescharacteristic
174C22×C17#48012yes—
175C22×C17#48312yes—
176C4×C17—22no—
177C4×C17—22no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—162no—
183Dic17—42no—
184Dic17—81no—
185Dic17—81no—
186Dic17—81no—
187C2×C4×D4—341no—
188C2×C4×D4C2×D1711yescharacteristic
189C2×#99—21no—
190C2×C22⋊C8—172no—
191#389—341no—
192#385—341no—
193#385—341no—
194#382—171no—
195#380—171no—
196#380—171no—
197#380C2×D1712yes—
198C4×C4⋊C4—171no—
199C2×#95—171no—
200C2×#95—171no—
201#342—21no—
202#324—341no—
203#323—341no—
204C2×C17—21no—
205C2×C17—22no—
206C2×C17—21no—
207C2×C17—22no—
208C2×C17—42no—
209C2×C17#182311yescharacteristic
210C2×C17#182711yescharacteristic
211C2×C17#182811yescharacteristic
212C22×D4C22⋊D1711yescharacteristic
213C23×C4—341no—
214C23×C4—341no—
215C23×C4C4×D1711yescharacteristic
216C23×C4C22×D1711yescharacteristic
217C22×C8—342no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—682no—
221C4×D4—682no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4—341no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4—341no—
227C2×C4⋊C4D6811yescharacteristic
228C2×C22⋊C4—22no—
229C2×C22⋊C4—341no—
230C2×C22⋊C4—341no—
231C2×C22⋊C4D6811yescharacteristic
232C2×C22⋊C4C22⋊D1711yescharacteristic
233C2×C42—341no—
234C2×C42C4×D1711yescharacteristic
235#99—42no—
236C22⋊C8—342no—
237C22⋊C8—342no—
238#95—22no—
239#95—171no—
240#95—171no—
241#95—171no—
242#95—171no—
243#95—341no—
244#95—341no—
245#95—341no—
246C17#876011yescharacteristic, complemented, Sylow
247C24#6322711yescharacteristic
248C24C2×C22⋊D1711yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4#6322812yes—
252C2×D4#6323012yes—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—341no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—682no—
263C22×C4—682no—
264C22×C4—682no—
265C22×C4—681no—
266C22×C4—681no—
267C22×C4—1361no—
268C22×C4—1361no—
269C22×C4#6322711yescharacteristic
270C22×C4#6322711yescharacteristic
271C22×C4#6322711yescharacteristic
272C22×C4C2×C4×D1711yescharacteristic
273C22×C4C2×D6811yescharacteristic
274C22×C4#6325412yes—
275C22×C4#6325412yes—
276C2×C8—682no—
277C2×C8—1362no—
278C4⋊C4—22no—
279C4⋊C4—82no—
280C4⋊C4—682no—
281C4⋊C4—682no—
282C4⋊C4—1361no—
283C4⋊C4—1361no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—682no—
289C22⋊C4—681no—
290C22⋊C4—681no—
291C42—21no—
292C42—21no—
293C42—682no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23#1058748012yes—
301C23C2×#6322711yescharacteristic
302C23C2×#6322811yescharacteristic
303C23#1058762112yes—
304C23C2×#6323011yescharacteristic
305C23#1058763712yes—
306D4—24no—
307D4—24no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—342no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4—681no—
334C2×C4—681no—
335C2×C4—1362no—
336C2×C4—1362no—
337C2×C4—1364no—
338C2×C4#1058748012yes—
339C2×C4#1058756311yescharacteristic
340C2×C4#1058756311yescharacteristic
341C2×C4#1058757212yes—
342C8—1362no—
343C22—21no—
344C22—22no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22#1176267011yescharacteristic
356C22C2×#1058748011yescharacteristic
357C22#1176281711yescharacteristic
358C22#1176312012yes—
359C22#1176313712yes—
360C4—22no—
361C4—22no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—162no—
367C4—682no—
368C4—1361no—
369C4—1361no—
370C4—1361no—
371C2—22no—
372C2—21no—
373C2—21no—
374C2—22no—
375C2—42no—
376C2#1452886811yescharacteristic
377C2#1452888511yescharacteristic
378C2#1452889011yescharacteristic
subgroupG/H
centerC22#11762670
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1799C2
socleC22×C17#381
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenter#1799D17
hypocenterC17#8760
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #14526662C2semidirect
4. #14526651C2semidirect
11. #11762675C4semidirect
12. #11762669C4semidirect
14. C4×C4⋊Dic17C4semidirect
15. C2×#10587502C4semidirect
246. C17#8760semidirect

Maximal subgroups

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

subgroupindex
#145316942
#145271022
#145266622
#145266512
C17×#17992
#876017