Finite Group Databasebeta

← #16945072

Subgroups — #16945072

Order 4352.

386 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 87 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
2#14527114C211yesmaximal, characteristic
3#14526814C212yesmaximal, complemented
4#14526811C212yesmaximal, complemented
5C17×#2627C211yesmaximal, characteristic
6#11762773—21no—
7#11762773—21no—
8#11762578—21no—
9#11762578C2212yes—
10#11762535—21no—
11#11762535C2212yes—
12C2×C17×C22⋊Q8C411yescharacteristic
13C22×C4⋊C4×C17C2211yescharacteristic
14C17×#400C411yescharacteristic
15C17×#400C2211yescharacteristic
16C17×#398—21no—
17C17×#398—21no—
18C17×#395C411yescharacteristic
19C17×#395C2211yescharacteristic
20C17×#385—21no—
21C17×#381—21no—
22C17×#380C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C17×C22⋊Q8—41no—
28C17×C22⋊Q8—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—41no—
34C2×C4⋊C4×C17—41no—
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C4⋊C4×C17C2×C411yescharacteristic
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C4⋊C4×C17D411yescharacteristic
39C2×C4⋊C4×C17C2311yescharacteristic
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×#63223—22no—
43C2×#63223—41no—
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
57C2×Q8×C17—41no—
58C2×Q8×C17—81no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—22no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—41no—
71C22×C4×C17—81no—
72C22×C4×C17—81no—
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—42no—
85C4⋊C4×C17—81no—
86C4⋊C4×C17—81no—
87C4⋊C4×C17—81no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—41no—
90C22⋊C4×C17—41no—
91#63223—42no—
92#63223—42no—
93#63223—82no—
94C2×#3622—42no—
95C2×#3622—42no—
96#11817—171nomaximal, Sylow
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17#9911yescharacteristic
102C23×C17#9911yescharacteristic
103C23×C17C2×C22⋊C411yescharacteristic
104Q8×C17—82no—
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—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
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—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—42no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—81no—
133C2×C4×C17—81no—
134C2×C4×C17—81no—
135C2×C4×C17#13712yes—
136#3622—82no—
137#3622—82no—
138#2627D1711yescharacteristic
139#1821—171no—
140#1463—171no—
141#1354—172no—
142#1353—172no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—42no—
152C22×C17—42no—
153C22×C17C2×#9911yescharacteristic
154C22×C17#41911yescharacteristic
155C22×C17#42111yescharacteristic
156C4×C17—22no—
157C4×C17—41no—
158C4×C17—41no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—81no—
163C4×C17—81no—
164C4×C17—81no—
165C4×C17—81no—
166C2×C22⋊Q8Dic1711yescharacteristic
167C22×C4⋊C4C2×D1711yescharacteristic
168#407—341no—
169#407—341no—
170#400Dic1711yescharacteristic
171#400C2×D1711yescharacteristic
172#398—21no—
173#398—21no—
174#395Dic1711yescharacteristic
175#395C2×D1711yescharacteristic
176#385—21no—
177#381—21no—
178#380Dic1711yescharacteristic
179#343—172no—
180#343—341no—
181#324—172no—
182#324—341no—
183C2×C17—21no—
184C2×C17—21no—
185C2×C17—41no—
186C2×C17—41no—
187C2×C17#151111yescharacteristic
188C2×C17#212111yescharacteristic
189C2×C17#213311yescharacteristic
190C22×Q8C2×Dic1711yescharacteristic
191C23×C4C2×Dic1711yescharacteristic
192C23×C4C22⋊D1711yescharacteristic
193C23×C4C22⋊D1711yescharacteristic
194C2×M16—342no—
195C2×M16—681no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C2×Dic1711yescharacteristic
205C2×C4⋊C4C2×Dic1711yescharacteristic
206C2×C4⋊C4C22⋊D1711yescharacteristic
207C2×C4⋊C4C22⋊D1711yescharacteristic
208C2×C4⋊C4C22×D1711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C22⋊C8—342no—
212C22⋊C8—681no—
213C22⋊C8—681no—
214C22⋊C8—681no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95C2×Dic1711yescharacteristic
222#95C2×Dic1711yescharacteristic
223C17#1181711yescharacteristic, complemented, Sylow
224C24C22⋊Dic1711yescharacteristic
225C2×Q8—41no—
226C2×Q8—81no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4C22⋊Dic1711yescharacteristic
242C22×C4C22⋊Dic1711yescharacteristic
243C22×C4C22⋊Dic1711yescharacteristic
244C22×C4C22×Dic1711yescharacteristic
245C22×C4C2×C22⋊D1711yescharacteristic
246C22×C4C2×C22⋊D1711yescharacteristic
247M16—682no—
248M16—682no—
249M16—1362no—
250C2×C8—682no—
251C2×C8—682no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23#1058750511yescharacteristic
269C23#1058750511yescharacteristic
270C23C2×C22⋊Dic1711yescharacteristic
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4#1058762812yes—
303C8—1362no—
304C8—1362no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#1058750511yescharacteristic
316C22#1176281811yescharacteristic
317C22#1176282611yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#1452731711yescharacteristic
333C2#1453043011yescharacteristic
334C2#1453047311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2627C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2627D17
hypocenterC17#11817
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
3. #14526814C2semidirect
4. #14526811C2semidirect
223. C17#11817semidirect

Maximal subgroups

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

subgroupindex
#145288642
#145271142
#145268142
#145268112
C17×#26272
#1181717