Finite Group Databasebeta

← #17949905

Subgroups — #17949905

Order 4352.

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

normal subgroups only

This group has 556 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#14544510C211yesmaximal, characteristic
2#14544447C211yesmaximal, characteristic
3#14543604C212yesmaximal, complemented
4#14543602C212yesmaximal, complemented
5C22⋊Q8×Dic17C211yesmaximal, characteristic
6C22×#63247—21no—
7C2×C22⋊#3624—21no—
8C2×#10587542—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×Q8×Dic17C411yescharacteristic
13C22×C4×Dic17C2211yescharacteristic
14#11762716C2211yescharacteristic
15#11762708C2211yescharacteristic
16#11762695—21no—
17#11762684—21no—
18#11762669C411yescharacteristic
19C4⋊C4×Dic17—21no—
20C4⋊C4×Dic17C411yescharacteristic
21C22⋊C4×Dic17—21no—
22C2×C17×C22⋊Q8C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×Dic17D411yescharacteristic
26C23×Dic17D411yescharacteristic
27C2×#63247—24no—
28C2×#63247—42no—
29C2×#63247—42no—
30C22×#3624—22no—
31C22×#3624—22no—
32C17×C22⋊Q8—22no—
33C17×C22⋊Q8—22no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C22⋊Dic17—21no—
38Q8×Dic17—42no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×C4⋊Dic17C2×C411yescharacteristic
43C2×C4×Dic17—21no—
44C2×C4×Dic17—21no—
45C2×C4×Dic17—42no—
46C2×C4×Dic17C2×C411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C2×C4×Dic17D411yescharacteristic
49C2×C4×Dic17C2311yescharacteristic
50C22⋊#3624—24no—
51#10587542—42no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57C24×C17C22⋊C411yescharacteristic
58C2×Q8×C17—21no—
59C2×Q8×C17—41no—
60C2×Q8×C17—41no—
61C2×Q8×C17M1612yescomplemented
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—41no—
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—22no—
75C22×Dic17—22no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—41no—
79C22×Dic17C22⋊C411yescharacteristic
80C22×Dic17C22⋊C411yescharacteristic
81C22×Dic17C2×D411yescharacteristic
82C22×Dic17C2×D411yescharacteristic
83#63247—48no—
84C2×#3624—24no—
85C2×#3624—24no—
86C2×#3624—42no—
87C2×#3624—42no—
88C2×#3624—42no—
89C2×#3624—42no—
90C4⋊C4×C17—24no—
91C4⋊C4×C17—21no—
92C4⋊C4×C17M1612yescomplemented
93C22⋊C4×C17—24no—
94C22⋊Dic17—42no—
95C4⋊Dic17—42no—
96C4⋊Dic17—41no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100C4×Dic17—22no—
101C4×Dic17—22no—
102C4×Dic17—41no—
103C4×Dic17—41no—
104C4×Dic17—42no—
105C4×Dic17—42no—
106#7634—171nomaximal, Sylow
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17#10412yes—
114C23×C17C2×C22⋊C411yescharacteristic
115Q8×C17—42no—
116Q8×C17—42no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—22no—
120C2×C4×C17—22no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17C2×M1611yescharacteristic
131C2×C4×C17C2×M1611yescharacteristic
132C2×Dic17—24no—
133C2×Dic17—24no—
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—44no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17#9912yes—
150C2×Dic17#10412yes—
151C2×Dic17AGL(1,ℤ/8)11yescharacteristic
152C2×Dic17#13711yescharacteristic
153C2×Dic17#13712yes—
154#3624—44no—
155#3624—44no—
156C4×C22⋊Q8—171no—
157#1749—171no—
158#1728—171no—
159#1272—172no—
160#1271—172no—
161C22×C17—22no—
162C22×C17—22no—
163C22×C17—22no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17#40711yescharacteristic
169C22×C17C2×#9911yescharacteristic
170C22×C17#41311yescharacteristic
171C22×C17#41711yescharacteristic
172C22×C17#41911yescharacteristic
173C22×C17C2×#10411yescharacteristic
174C22×C17C2×#10411yescharacteristic
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—41no—
178C4×C17—42no—
179C4×C17—42no—
180Dic17—21no—
181Dic17—22no—
182Dic17—21no—
183Dic17—22no—
184Dic17—22no—
185Dic17—81no—
186Dic17—81no—
187C22×M16—341no—
188C2×C22⋊Q8#59511yescharacteristic
189C2×C4×Q8—171no—
190C22×C42—171no—
191C2×C22⋊C8—341no—
192C2×#97—341no—
193#391—171no—
194#389—171no—
195#387—341no—
196#385—341no—
197#380—171no—
198C4×C4⋊C4—171no—
199C4×C4⋊C4—341no—
200C4×C22⋊C4—341no—
201#336—172no—
202#336—172no—
203#336—341no—
204C2×C17—42no—
205C2×C17#146511yescharacteristic
206C2×C17#148911yescharacteristic
207C2×C17#149111yescharacteristic
208C2×C17#150411yescharacteristic
209C2×C17#150811yescharacteristic
210C2×C17#152411yescharacteristic
211C2×C17#155911yescharacteristic
212C22×Q8C2×#59511yescharacteristic
213C23×C4—171no—
214C23×C4—171no—
215C23×C4C2×#59511yescharacteristic
216C2×M16—344no—
217C2×M16—682no—
218C2×M16—682no—
219C22×C8—342no—
220C22×C8—342no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C4×Q8—682no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—171no—
226C2×C4⋊C4—171no—
227C2×C4⋊C4—341no—
228C2×C4⋊C4—341no—
229C2×C4⋊C4C2×#59511yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—341no—
232C2×C42—171no—
233C2×C42—171no—
234C2×C42—171no—
235C2×C42—171no—
236C2×C42—341no—
237C2×C42—341no—
238C2×C42—682no—
239C22⋊C8—344no—
240#97—682no—
241#95—341no—
242#95—341no—
243#95—341no—
244#95—341no—
245#95—341no—
246C17#763411yescharacteristic, complemented, Sylow
247C24C22⋊#59511yescharacteristic
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8#6324712yescomplemented
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—171no—
259C22×C4—171no—
260C22×C4—171no—
261C22×C4—171no—
262C22×C4—342no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—342no—
266C22×C4—342no—
267C22×C4—342no—
268C22×C4—341no—
269C22×C4—341no—
270C22×C4—681no—
271C22×C4C22⋊#59511yescharacteristic
272C22×C4C22×#59511yescharacteristic
273M16—688no—
274C2×C8—344no—
275C2×C8—344no—
276C2×C8—682no—
277C2×C8—682no—
278C2×C8—682no—
279C2×C8—682no—
280C4⋊C4—24no—
281C4⋊C4—21no—
282C4⋊C4—681no—
283C4⋊C4—681no—
284C4⋊C4—682no—
285C4⋊C4—681no—
286C4⋊C4—681no—
287C4⋊C4#6324712yescomplemented
288C22⋊C4—24no—
289C22⋊C4—682no—
290C42—342no—
291C42—342no—
292C42—681no—
293C42—681no—
294C42—682no—
295C42—682no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23#1058755012yes—
303C23C2×C22⋊#59511yescharacteristic
304Q8—42no—
305Q8—42no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—172no—
320C2×C4—171no—
321C2×C4—171no—
322C2×C4—172no—
323C2×C4—172no—
324C2×C4—344no—
325C2×C4—344no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—684no—
335C2×C4—681no—
336C2×C4—681no—
337C2×C4—681no—
338C2×C4—681no—
339C2×C4—681no—
340C2×C4—681no—
341C2×C4C2×#6324711yescharacteristic
342C2×C4C2×#6324711yescharacteristic
343C8—684no—
344C8—684no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22#1176377911yescharacteristic
353C22D17⋊#9911yescharacteristic
354C22#1176379611yescharacteristic
355C22#1176380011yescharacteristic
356C22#1176380211yescharacteristic
357C22D17⋊#10411yescharacteristic
358C22C2×#1058755011yescharacteristic
359C4—21no—
360C4—21no—
361C4—41no—
362C4—42no—
363C4—42no—
364C4—341no—
365C4—342no—
366C4—341no—
367C4—342no—
368C4—342no—
369C4—1361no—
370C4—1361no—
371C2—42no—
372C2#1454404011yescharacteristic
373C2#1454411911yescharacteristic
374C2#1454412611yescharacteristic
375C2#1454417111yescharacteristic
376C2#1454418211yescharacteristic
377C2#1454423711yescharacteristic
378C2#1454432011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C22⋊Q8C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×C22⋊Q8#595
hypocenterC17#7634
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14543604C2semidirect
4. #14543602C2semidirect
61. C2×Q8×C17M16semidirect
92. C4⋊C4×C17M16semidirect
246. C17#7634semidirect
251. C2×Q8#63247semidirect
287. C4⋊C4#63247semidirect

Maximal subgroups

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

subgroupindex
#145445102
#145444472
#145436042
#145436022
C22⋊Q8×Dic172
#763417