Finite Group Databasebeta

← #16945255

Subgroups — #16945255

Order 4352.

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

normal subgroups only

This group has 412 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#14534164C211yesmaximal, characteristic
2C2×#11762535C211yesmaximal, characteristic
3#14526813C212yesmaximal, complemented
4#14526789C212yesmaximal, complemented
5C17×#1821C211yesmaximal, characteristic
6C2×#10587610C411yescharacteristic
7C2×C22⋊#3622—21no—
8#11762739C411yescharacteristic
9#11762739C2211yescharacteristic
10#11762729C411yescharacteristic
11#11762729C2211yescharacteristic
12#11762704—21no—
13#11762704—21no—
14#11762682—21no—
15#11762682—21no—
16C4×C22⋊Dic17C411yescharacteristic
17#11762535—21no—
18#11762535C2212yes—
19C22×C4⋊C4×C17C2211yescharacteristic
20C17×#407—21no—
21C17×#343—21no—
22C17×#343C2212yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C2×M16×C17—22no—
28C2×C4⋊C4×C17—41no—
29C2×C4⋊C4×C17—41no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17C2311yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C4⋊Dic17C2×C411yescharacteristic
37#10587610—41no—
38#10587610—41no—
39C2×#63225—21no—
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×C4×Dic17—21no—
45C22×#3622—41no—
46C17×C22⋊C8—41no—
47C17×C22⋊C8—41no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502C2×C411yescharacteristic
53#10587502C2×C411yescharacteristic
54C22⋊#3622—22no—
55C22⋊#3622—41no—
56C24×C17C22⋊C411yescharacteristic
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—22no—
64C22×C4×C17—41no—
65C22×C4×C17—81no—
66C22×C4×C17—81no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C2×Dic34—41no—
78C2×Dic34—81no—
79M16×C17—42no—
80M16×C17—42no—
81C2×C8×C17—42no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C22⋊Dic17—24no—
87C22⋊Dic17—41no—
88C22⋊Dic17—41no—
89C4⋊Dic17—41no—
90#63225—42no—
91#63225—42no—
92#63225—42no—
93#63225—81no—
94#63225—81no—
95C4×Dic17—81no—
96C2×#3622—42no—
97C2×#3622—82no—
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—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—81no—
123C2×C4×C17—81no—
124C2×C4×C17—81no—
125C2×C4×C17#13512yes—
126C2×Dic17—22no—
127C2×Dic17—22no—
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—41no—
137C2×Dic17—41no—
138Dic34—82no—
139C8×C17—82no—
140#3622—82no—
141#2583—171no—
142#1821D1711yescharacteristic
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—22no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17—41no—
165Dic17—42no—
166Dic17—42no—
167Dic17—81no—
168Dic17—81no—
169C2×C22⋊Q8—171no—
170C22×C4⋊C4C2×D1711yescharacteristic
171#407—21no—
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—171no—
182#343—21no—
183#343C2×D1712yes—
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×C4C4×D1711yescharacteristic
195C23×C4D6811yescharacteristic
196C23×C4C22⋊D1711yescharacteristic
197C2×M16—22no—
198C22×C8—681no—
199C22⋊Q8—681no—
200C22⋊Q8—681no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4C4×D1711yescharacteristic
210C2×C4⋊C4D6811yescharacteristic
211C2×C4⋊C4C22⋊D1711yescharacteristic
212C2×C4⋊C4C22×D1711yescharacteristic
213C2×C22⋊C4—341no—
214C2×C22⋊C4—341no—
215C2×C42—341no—
216C22⋊C8—41no—
217C22⋊C8—41no—
218C22⋊C8—342no—
219C22⋊C8—681no—
220#95—171no—
221#95—171no—
222#95—341no—
223#95—341no—
224#95—341no—
225#95—341no—
226C17#1185411yescharacteristic, complemented, Sylow
227C24#6322711yescharacteristic
228C2×Q8—681no—
229C2×Q8—1361no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—341no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—341no—
244C22×C4#6322711yescharacteristic
245C22×C4#6322711yescharacteristic
246C22×C4#6322711yescharacteristic
247C22×C4C2×C4×D1711yescharacteristic
248C22×C4C2×D6811yescharacteristic
249C22×C4C2×C22⋊D1711yescharacteristic
250M16—42no—
251M16—42no—
252C2×C8—42no—
253C2×C8—682no—
254C2×C8—1362no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—681no—
260C4⋊C4—682no—
261C4⋊C4—682no—
262C4⋊C4—682no—
263C4⋊C4—1361no—
264C4⋊C4—1361no—
265C22⋊C4—344no—
266C22⋊C4—681no—
267C22⋊C4—681no—
268C42—1361no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—41no—
273C23#1058747711yescharacteristic
274C23#1058747711yescharacteristic
275C23C2×#6322711yescharacteristic
276Q8—1362no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—342no—
297C2×C4—342no—
298C2×C4—681no—
299C2×C4—681no—
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#1058763612yes—
309C8—82no—
310C8—1362no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#1176260712yes—
321C22C2×#1058747711yescharacteristic
322C22#1176280311yescharacteristic
323C22#1176282711yescharacteristic
324C4—22no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—682no—
331C4—682no—
332C4—1361no—
333C4—1361no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2#1452733811yescharacteristic
339C2#1453043111yescharacteristic
340C2C2×#1176260711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1821C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1821D17
hypocenterC17#11854
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #14526813C2semidirect
4. #14526789C2semidirect
226. C17#11854semidirect

Maximal subgroups

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

subgroupindex
#145341642
C2×#117625352
#145268132
#145267892
C17×#18212
#1185417