Finite Group Databasebeta

← #17950214

Subgroups — #17950214

Order 4352.

444 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 88 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#14544675C211yesmaximal, characteristic
2#14544674C211yesmaximal, characteristic
3#14543604C212yesmaximal, complemented
4#14543603C212yesmaximal, complemented
5#14537118C211yesmaximal, characteristic
6C2×C22⋊#3624—21no—
7C2×C22⋊#3624—21no—
8#11763675—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C22×C4×Dic17C2211yescharacteristic
13#11762716C411yescharacteristic
14#11762716C2211yescharacteristic
15#11762705C411yescharacteristic
16#11762705C2211yescharacteristic
17#11762690—21no—
18#11762690—21no—
19#11762681—21no—
20#11762670—21no—
21#11762669C411yescharacteristic
22C2×C17×C22⋊Q8C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×Dic17D411yescharacteristic
26C23×Dic17D411yescharacteristic
27C22×#3624—22no—
28C22×#3624—22no—
29C17×C22⋊Q8—42no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17—21no—
33C2×C22⋊Dic17—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×#63225—21no—
37C2×#63225—21no—
38C2×C4×Dic17—42no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17C2311yescharacteristic
43C22⋊#3624—42no—
44C22⋊#3624—42no—
45#10587502—21no—
46#10587502—21no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52C24×C17C22⋊C411yescharacteristic
53C2×Q8×C17—22no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C22×C4×C17—22no—
57C22×C4×C17—22no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—41no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×Dic17—22no—
64C22×Dic17—22no—
65C22×Dic17—22no—
66C22×Dic17—22no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—41no—
70C22×Dic17—41no—
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C22×Dic17C2×D411yescharacteristic
75C2×#3624—42no—
76C2×#3624—42no—
77C2×#3624—42no—
78C2×#3624—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—42no—
83C22⋊C4×C17—42no—
84C22⋊Dic17—42no—
85C4⋊Dic17—22no—
86C4⋊Dic17—41no—
87#63225—42no—
88#63225—42no—
89C4×Dic17—41no—
90C4×Dic17—41no—
91C4×Dic17—41no—
92C4×Dic17—41no—
93#7721—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17#9912yes—
101C23×C17C2×C22⋊C411yescharacteristic
102Q8×C17—82no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—44no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17#13112yes—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—44no—
127C2×Dic17—44no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17#13512yes—
135C2×Dic17#13712yes—
136#3624—82no—
137#3624—82no—
138#2855—171no—
139#1817—171no—
140#1817—171no—
141#1272—172no—
142#1271—172no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17#40811yescharacteristic
151C22×C17C2×#9911yescharacteristic
152C22×C17#41311yescharacteristic
153C22×C17#41611yescharacteristic
154C22×C17#41911yescharacteristic
155C22×C17#42111yescharacteristic
156C22×C17#42111yescharacteristic
157C4×C17—22no—
158C4×C17—41no—
159C4×C17—81no—
160C4×C17—81no—
161Dic17—22no—
162Dic17—22no—
163Dic17—41no—
164Dic17—41no—
165Dic17—81no—
166Dic17—81no—
167C2×C22⋊Q8#59511yescharacteristic
168C22×C42—171no—
169C2×C22⋊C8—341no—
170C2×C22⋊C8—341no—
171#391—171no—
172#391—171no—
173#389—171no—
174#389—171no—
175#387—341no—
176#387—341no—
177#385—341no—
178#381—341no—
179#380—171no—
180#336—172no—
181#336—172no—
182#336—341no—
183#336—341no—
184C2×C17—42no—
185C2×C17#146811yescharacteristic
186C2×C17#148211yescharacteristic
187C2×C17#149711yescharacteristic
188C2×C17#151111yescharacteristic
189C2×C17#151611yescharacteristic
190C2×C17#153511yescharacteristic
191C2×C17#155411yescharacteristic
192C22×Q8C2×#59511yescharacteristic
193C23×C4—171no—
194C23×C4—171no—
195C23×C4C2×#59511yescharacteristic
196C22×C8—342no—
197C22×C8—342no—
198C22⋊Q8—42no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4C2×#59511yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—341no—
207C2×C42—171no—
208C2×C42—171no—
209C2×C42—171no—
210C2×C42—171no—
211C2×C42—682no—
212C22⋊C8—682no—
213C22⋊C8—682no—
214#95—341no—
215#95—341no—
216#95—341no—
217#95—341no—
218#95—341no—
219#95—341no—
220#95—341no—
221C17#772111yescharacteristic, complemented, Sylow
222C24C22⋊#59511yescharacteristic
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—41no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—171no—
232C22×C4—171no—
233C22×C4—171no—
234C22×C4—171no—
235C22×C4—342no—
236C22×C4—342no—
237C22×C4—342no—
238C22×C4—342no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4—681no—
242C22×C4—681no—
243C22×C4C22⋊#59511yescharacteristic
244C22×C4C22×#59511yescharacteristic
245C2×C8—682no—
246C2×C8—682no—
247C2×C8—682no—
248C2×C8—682no—
249C2×C8—682no—
250C2×C8—682no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—342no—
254C4⋊C4—681no—
255C4⋊C4—682no—
256C4⋊C4—682no—
257C22⋊C4—42no—
258C22⋊C4—682no—
259C42—681no—
260C42—681no—
261C42—681no—
262C42—681no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23#1058755112yes—
270C23C2×C22⋊#59511yescharacteristic
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—44no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—172no—
285C2×C4—172no—
286C2×C4—341no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—341no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—684no—
297C2×C4—684no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4#1058767812yes—
305C8—1362no—
306C8—1362no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22#1176378211yescharacteristic
315C22C2×#1058755111yescharacteristic
316C22#1176379611yescharacteristic
317C22#1176379911yescharacteristic
318C22#1176380211yescharacteristic
319C22#1176380611yescharacteristic
320C22#1176380811yescharacteristic
321C4—22no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4—342no—
326C4—342no—
327C4—681no—
328C4—681no—
329C4—1361no—
330C4—1361no—
331C2—42no—
332C2#1454404911yescharacteristic
333C2#1454409711yescharacteristic
334C2#1454414611yescharacteristic
335C2#1454419411yescharacteristic
336C2#1454421011yescharacteristic
337C2#1454427111yescharacteristic
338C2#1454431011yescharacteristic
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#7721
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. #14543603C2semidirect
221. C17#7721semidirect

Maximal subgroups

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

subgroupindex
#145446752
#145446742
#145436042
#145436032
#145371182
#772117