Finite Group Databasebeta

← #16929419

Subgroups — #16929419

Order 4352.

482 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 119 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#14537422C211yesmaximal, characteristic
2#14527213C212yesmaximal, complemented
3#14527098C211yesmaximal, characteristic
4#14526687C212yesmaximal, complemented
5C17×#2396C211yesmaximal, characteristic
6#11762870—24no—
7#11762862C2212yes—
8#11762833C2212yes—
9#11762773—21no—
10C4×#63223—21no—
11Q8⋊#3622—24no—
12#11762535—21no—
13#11762533—21no—
14C2×C4×Q8×C17C2211yescharacteristic
15C2×C17×#117C2211yescharacteristic
16C17×#400C411yescharacteristic
17C17×#399C411yescharacteristic
18C17×#398—21no—
19C17×#396—21no—
20C17×#395C411yescharacteristic
21C17×#393C411yescharacteristic
22C4×C4⋊C4×C17—21no—
23C4×C4⋊C4×C17C2211yescharacteristic
24C4×C22⋊C4×C17—21no—
25C22×Q8×C17C2×C411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C4×Q8×C17—22no—
28C4×Q8×C17—22no—
29C17×#117—22no—
30C17×#117—22no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C4⋊C4×C17C2×C411yescharacteristic
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17C2×C411yescharacteristic
41C2×C22⋊C4×C17D411yescharacteristic
42C2×C42×C17—21no—
43C2×C42×C17—21no—
44C2×C42×C17D411yescharacteristic
45C2×C42×C17C2311yescharacteristic
46C2×#63223—22no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95C2×C411yescharacteristic
52C17×#95C2×C411yescharacteristic
53C22⋊#3622—24no—
54C22⋊#3622—22no—
55C4⋊#3622—22no—
56C4⋊#3622—22no—
57C4⋊#3622D414yes—
58#10587474—22no—
59C4×#3622—22no—
60C4×#3622—22no—
61C24×C17C22⋊C411yescharacteristic
62C2×Q8×C17—22no—
63C2×Q8×C17—42no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22×C411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C4⋊C4×C17—22no—
79C4⋊C4×C17—22no—
80C4⋊C4×C17—22no—
81C4⋊C4×C17—42no—
82C4⋊C4×C17—42no—
83C4⋊C4×C17—42no—
84C4⋊C4×C17—42no—
85C22⋊C4×C17—22no—
86C22⋊C4×C17—22no—
87C22⋊C4×C17—42no—
88C22⋊C4×C17—42no—
89C42×C17—21no—
90C42×C17—21no—
91C42×C17—41no—
92C42×C17—41no—
93C42×C17—42no—
94C42×C17C2×D412yes—
95C42×C17#4112yes—
96#63223—42no—
97#63223—42no—
98C2×#3622—24no—
99C2×#3622—42no—
100#9034—171nomaximal, Sylow
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17C2×C22⋊C411yescharacteristic
107Q8×C17—24no—
108Q8×C17—42no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
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—42no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
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—41no—
132C2×C4×C17C4×D412yes—
133C2×C4×C17#12011yescharacteristic
134C2×C4×C17#12112yes—
135C2×C4×C17#12311yescharacteristic
136C2×C4×C17#13512yes—
137C2×C4×C17#13712yes—
138#3622—44no—
139#3622—82no—
140#2921—171no—
141#2396D1711yescharacteristic
142#1487—172no—
143#1458—171no—
144#1303—172no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—42no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—81no—
152C22×C17#38611yescharacteristic
153C22×C17#41611yescharacteristic
154C22×C17#41911yescharacteristic
155C4×C17—22no—
156C4×C17—22no—
157C4×C17—42no—
158C4×C17—41no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—42no—
162C4×C17—81no—
163C4×C17—81no—
164C4×C17#45212yes—
165C4×C17#48512yes—
166C2×C4×Q8C2×D1711yescharacteristic
167C2×#117C2×D1711yescharacteristic
168#433—344no—
169#431—172no—
170#423—172no—
171#407—341no—
172C4×M16—341no—
173#400Dic1711yescharacteristic
174#399Dic1711yescharacteristic
175#398—21no—
176#396—21no—
177#395Dic1711yescharacteristic
178#393Dic1711yescharacteristic
179C4×C4⋊C4—21no—
180C4×C4⋊C4C2×D1711yescharacteristic
181C4×C22⋊C4—21no—
182Q8⋊C8—344no—
183#324—341no—
184#323—341no—
185C2×C17—21no—
186C2×C17—21no—
187C2×C17—81no—
188C2×C17#187311yescharacteristic
189C2×C17#188411yescharacteristic
190C2×C17#189111yescharacteristic
191C22×Q8C2×Dic1711yescharacteristic
192C23×C4C22⋊D1711yescharacteristic
193C2×M16—342no—
194C4×Q8—22no—
195C4×Q8—22no—
196#117—22no—
197#117—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C2×Dic1711yescharacteristic
203C2×C4⋊C4C2×Dic1711yescharacteristic
204C2×C4⋊C4C22⋊D1711yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4C2×Dic1711yescharacteristic
208C2×C22⋊C4C22⋊D1711yescharacteristic
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C22⋊D1711yescharacteristic
212C2×C42C22×D1711yescharacteristic
213C4⋊C8—174no—
214C4⋊C8—342no—
215C4⋊C8—342no—
216C22⋊C8—344no—
217C22⋊C8—342no—
218#97—342no—
219C4×C8—342no—
220C4×C8—342no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95C2×Dic1711yescharacteristic
226#95C2×Dic1711yescharacteristic
227C17#903411yescharacteristic, complemented, Sylow
228C24C22⋊Dic1711yescharacteristic
229C2×Q8—22no—
230C2×Q8—42no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C22⋊Dic1711yescharacteristic
240C22×C4C22⋊Dic1711yescharacteristic
241C22×C4C22⋊Dic1711yescharacteristic
242C22×C4C22×Dic1711yescharacteristic
243C22×C4C2×C22⋊D1711yescharacteristic
244C22×C4C2×C22⋊D1711yescharacteristic
245M16—682no—
246M16—682no—
247C2×C8—344no—
248C2×C8—682no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C42—21no—
261C42—21no—
262C42—41no—
263C42—41no—
264C42—42no—
265C42D4×D1712yes—
266C42#6325412yes—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23C2×C22⋊Dic1711yescharacteristic
273Q8—24no—
274Q8—42no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4D4×Dic1712yes—
299C2×C4#1058762111yescharacteristic
300C2×C4#1058762311yescharacteristic
301C2×C4#1058762512yes—
302C2×C4#1058762812yes—
303C2×C4#1058763612yes—
304C8—684no—
305C8—1362no—
306C22—21no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—81no—
313C22#1176268911yescharacteristic
314C22#1176280511yescharacteristic
315C22#1176281811yescharacteristic
316C4—22no—
317C4—22no—
318C4—42no—
319C4—41no—
320C4—41no—
321C4—42no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4#1176296912yes—
326C4#1176314712yes—
327C2—21no—
328C2—21no—
329C2—81no—
330C2#1452909711yescharacteristic
331C2#1452916811yescharacteristic
332C2#1452920311yescharacteristic
subgroupG/H
centerC2×C4#10587623
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2396C2
socleC22×C17#386
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#2396D17
hypocenterC17#9034
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2396

Complements

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

NH ≅ G/Ntype
2. #14527213C2semidirect
4. #14526687C2semidirect
227. C17#9034semidirect

Maximal subgroups

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

subgroupindex
#145374222
#145272132
#145270982
#145266872
C17×#23962
#903417