Finite Group Databasebeta

← #16945237

Subgroups — #16945237

Order 4352.

396 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 91 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#14528864C211yesmaximal, characteristic
2C2×#11762535C211yesmaximal, characteristic
3#14526811C212yesmaximal, complemented
4#14526790C212yesmaximal, complemented
5C17×#2484C211yesmaximal, characteristic
6#11762773—21no—
7C2×C22⋊#3622—21no—
8#11762578—21no—
9#11762578C2212yes—
10#11762535—21no—
11#11762535C2212yes—
12C2×C17×#117C411yescharacteristic
13C22×C4⋊C4×C17C2211yescharacteristic
14C17×#400C411yescharacteristic
15C17×#400C2211yescharacteristic
16C17×#398C411yescharacteristic
17C17×#398C2211yescharacteristic
18C17×#387—21no—
19C17×#387—21no—
20C17×#385—21no—
21C17×#381—21no—
22C17×#381C411yescharacteristic
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C17×#117—41no—
27C17×#117—41no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—41no—
31C2×C4⋊C4×C17—41no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C4⋊C4×C17C2311yescharacteristic
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C42×C17—21no—
41C2×#63223—22no—
42C22×#3622—41no—
43C17×#95—21no—
44C17×#95—21no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95C2×C411yescharacteristic
51C17×#95C2×C411yescharacteristic
52C22⋊#3622—22no—
53C22⋊#3622—41no—
54C22⋊#3622—41no—
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—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—41no—
69C22×C4×C17—42no—
70C22×C4×C17—42no—
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22×C411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—42no—
84C4⋊C4×C17—42no—
85C22⋊C4×C17—42no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C42×C17—42no—
89#63223—42no—
90#63223—42no—
91C2×#3622—42no—
92C2×#3622—42no—
93C2×#3622—82no—
94#11850—171nomaximal, Sylow
95C23×C17—22no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—41no—
99C23×C17#9911yescharacteristic
100C23×C17#9911yescharacteristic
101C23×C17C2×C22⋊C411yescharacteristic
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—42no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—81no—
132C2×C4×C17#13512yes—
133#3622—82no—
134#3622—82no—
135#2484D1711yescharacteristic
136#1821—171no—
137C2×#324—171no—
138#1353—172no—
139#1346—172no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17#35212yes—
150C22×C17C2×#9911yescharacteristic
151C22×C17#41611yescharacteristic
152C22×C17#42111yescharacteristic
153C4×C17—22no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—41no—
159C4×C17—81no—
160C4×C17—81no—
161C4×C17—81no—
162C4×C17—81no—
163C2×#117Dic1711yescharacteristic
164C22×C4⋊C4C2×D1711yescharacteristic
165#407—341no—
166C2×C22⋊C8—341no—
167#400Dic1711yescharacteristic
168#400C2×D1711yescharacteristic
169#398Dic1711yescharacteristic
170#398C2×D1711yescharacteristic
171#387—21no—
172#387—21no—
173#385—21no—
174#381—21no—
175#381Dic1711yescharacteristic
176#343—172no—
177#343—341no—
178#324—172no—
179#324—341no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17—22no—
183C2×C17—41no—
184C2×C17#151611yescharacteristic
185C2×C17C2×#35211yescharacteristic
186C2×C17#213311yescharacteristic
187C23×C4C2×Dic1711yescharacteristic
188C23×C4C22⋊D1711yescharacteristic
189C23×C4C22⋊D1711yescharacteristic
190C2×M16—342no—
191C22×C8—681no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4C2×Dic1711yescharacteristic
199C2×C4⋊C4C2×Dic1711yescharacteristic
200C2×C4⋊C4C2×Dic1711yescharacteristic
201C2×C4⋊C4C22⋊D1711yescharacteristic
202C2×C4⋊C4C22⋊D1711yescharacteristic
203C2×C4⋊C4C22×D1711yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42—21no—
207C22⋊C8—342no—
208C22⋊C8—681no—
209C22⋊C8—681no—
210C22⋊C8—681no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95C2×Dic1711yescharacteristic
219#95C2×Dic1711yescharacteristic
220C17#1185011yescharacteristic, complemented, Sylow
221C24C22⋊Dic1711yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C22⋊Dic1711yescharacteristic
237C22×C4C22⋊Dic1711yescharacteristic
238C22×C4C22⋊Dic1711yescharacteristic
239C22×C4C22×Dic1711yescharacteristic
240C22×C4C2×C22⋊D1711yescharacteristic
241C22×C4C2×C22⋊D1711yescharacteristic
242M16—682no—
243M16—682no—
244C2×C8—682no—
245C2×C8—682no—
246C2×C8—1362no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42—42no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—41no—
263C23#1058750511yescharacteristic
264C23#1058750511yescharacteristic
265C23C2×C22⋊Dic1711yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4#1058763612yes—
297C8—1362no—
298C8—1362no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22#1176259912yes—
309C22C2×#1058750511yescharacteristic
310C22#1176280511yescharacteristic
311C22#1176282611yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—41no—
326C2#1452734011yescharacteristic
327C2C2×#1176259911yescharacteristic
328C2#1453047311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2484C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2484D17
hypocenterC17#11850
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
3. #14526811C2semidirect
4. #14526790C2semidirect
220. C17#11850semidirect

Maximal subgroups

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

subgroupindex
#145288642
C2×#117625352
#145268112
#145267902
C17×#24842
#1185017