Finite Group Databasebeta

← #17959632

Subgroups — #17959632

Order 4352.

412 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 70 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#14544683C211yesmaximal, characteristic
2C2×#11763655C211yesmaximal, characteristic
3#14543784C212yesmaximal, complemented
4#14543769C212yesmaximal, complemented
5#14534159C211yesmaximal, characteristic
6#11763779—21no—
7C2×C22⋊#3624—21no—
8#11763689—21no—
9#11763689C2212yes—
10#11763655—21no—
11#11763655C2212yes—
12C2×#10587610C411yescharacteristic
13C22×#63225C2211yescharacteristic
14#11762741C411yescharacteristic
15#11762741C2211yescharacteristic
16#11762730C411yescharacteristic
17#11762730C2211yescharacteristic
18#11762707—21no—
19#11762707—21no—
20#11762684—21no—
21#11762684—21no—
22C4×C22⋊C4×C17C411yescharacteristic
23C23×C4×C17C2×C411yescharacteristic
24C23×Dic17D411yescharacteristic
25C23×Dic17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C2×#63247—22no—
28C22×#3624—41no—
29C2×C22⋊C4×C17—21no—
30C2×C42×C17—21no—
31C2×C22⋊Dic17—21no—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17C2×C411yescharacteristic
37#10587610—41no—
38#10587610—41no—
39C2×#63225—21no—
40C2×#63225—41no—
41C2×#63225—41no—
42C2×#63225C2×C411yescharacteristic
43C2×#63225D411yescharacteristic
44C2×#63225D411yescharacteristic
45C2×#63225C2311yescharacteristic
46C22⋊#3624—22no—
47C22⋊#3624—41no—
48C22⋊#3624—41no—
49C22⋊#3624—41no—
50C17×#95C2×C411yescharacteristic
51C17×#95C2×C411yescharacteristic
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56C24×C17C22⋊C411yescharacteristic
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—81no—
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—22no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—41no—
72C22×Dic17—81no—
73C22×Dic17C22⋊C411yescharacteristic
74C22×Dic17C22⋊C411yescharacteristic
75C22×Dic17C2×D411yescharacteristic
76C22×Dic17C2×D411yescharacteristic
77C2×Dic34—41no—
78C2×Dic34—81no—
79#63247—42no—
80#63247—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—82no—
84C22⋊C4×C17—24no—
85C42×C17—81no—
86C22⋊Dic17—41no—
87C22⋊Dic17—41no—
88C4⋊Dic17—42no—
89C4⋊Dic17—42no—
90C4⋊Dic17—41no—
91C4⋊Dic17—81no—
92C4⋊Dic17—81no—
93#63225—41no—
94#63225—41no—
95#63225—41no—
96#63225—41no—
97#63225—42no—
98#11854—171nomaximal, Sylow
99C23×C17—22no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—41no—
103C23×C17#9911yescharacteristic
104C23×C17#9911yescharacteristic
105C23×C17C2×C22⋊C411yescharacteristic
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—81no—
117C2×C4×C17—81no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—81no—
137C2×Dic17#13512yes—
138Dic34—82no—
139#3624—82no—
140#3624—82no—
141#2583—171no—
142#1821—171no—
143C2×#324—171no—
144#1354—172no—
145#1346—172no—
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17#35612yes—
156C22×C17C2×#9911yescharacteristic
157C22×C17#41611yescharacteristic
158C22×C17#42111yescharacteristic
159C4×C17—42no—
160C4×C17—42no—
161C4×C17—41no—
162C4×C17—41no—
163Dic17—22no—
164Dic17—41no—
165Dic17—41no—
166Dic17—41no—
167Dic17—81no—
168Dic17—81no—
169C2×C22⋊Q8—171no—
170C22×C4⋊C4—171no—
171#407—341no—
172C2×C22⋊C8—341no—
173#398—171no—
174#398—171no—
175#395—171no—
176#395—171no—
177#389—341no—
178#389—341no—
179#385—341no—
180#385—341no—
181C4×C22⋊C4#59511yescharacteristic
182#343—172no—
183#343—341no—
184#324—172no—
185#324—341no—
186C2×C17—21no—
187C2×C17—21no—
188C2×C17—22no—
189C2×C17—41no—
190C2×C17#151611yescharacteristic
191C2×C17#212111yescharacteristic
192C2×C17C2×#35611yescharacteristic
193C22×Q8—171no—
194C23×C4—171no—
195C23×C4—171no—
196C23×C4C2×#59511yescharacteristic
197C2×M16—342no—
198C22×C8—681no—
199C22⋊Q8—681no—
200C22⋊Q8—681no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—171no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4—341no—
211C2×C4⋊C4—681no—
212C2×C4⋊C4—681no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—341no—
215C2×C42—21no—
216C22⋊C8—342no—
217C22⋊C8—681no—
218C22⋊C8—681no—
219C22⋊C8—681no—
220#95—341no—
221#95—341no—
222#95—341no—
223#95—341no—
224#95C2×#59511yescharacteristic
225#95C2×#59511yescharacteristic
226C17#1185411yescharacteristic, complemented, Sylow
227C24C22⋊#59511yescharacteristic
228C2×Q8—681no—
229C2×Q8—1361no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—81no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—342no—
240C22×C4—341no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4—341no—
246C22×C4—681no—
247C22×C4—1361no—
248C22×C4C22⋊#59511yescharacteristic
249C22×C4C22×#59511yescharacteristic
250M16—682no—
251M16—682no—
252C2×C8—682no—
253C2×C8—682no—
254C2×C8—1362no—
255C4⋊C4—681no—
256C4⋊C4—681no—
257C4⋊C4—681no—
258C4⋊C4—681no—
259C4⋊C4—681no—
260C4⋊C4—682no—
261C4⋊C4—682no—
262C4⋊C4—682no—
263C4⋊C4—1361no—
264C4⋊C4—1361no—
265C22⋊C4—24no—
266C22⋊C4—681no—
267C22⋊C4—681no—
268C42—81no—
269C23—22no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23#1058753911yescharacteristic
274C23#1058753911yescharacteristic
275C23C2×C22⋊#59511yescharacteristic
276Q8—1362no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—172no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—341no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—682no—
298C2×C4—682no—
299C2×C4—682no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—1361no—
309C8—1362no—
310C8—1362no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#1176372012yes—
321C22C2×#1058753911yescharacteristic
322C22#1176379911yescharacteristic
323C22#1176380611yescharacteristic
324C4—42no—
325C4—42no—
326C4—41no—
327C4—41no—
328C4—342no—
329C4—681no—
330C4—681no—
331C4—681no—
332C4—1361no—
333C4—1361no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2#1454421211yescharacteristic
339C2#1454504111yescharacteristic
340C2C2×#1176372011yescharacteristic
subgroupG/H
centerC22C2×#10587539
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC4×C22⋊C4×C17C4
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×#63225C23
hypercenterC4×C22⋊C4#595
hypocenterC17#11854
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #14543784C2semidirect
4. #14543769C2semidirect
226. C17#11854semidirect

Maximal subgroups

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

subgroupindex
#145446832
C2×#117636552
#145437842
#145437692
#145341592
#1185417