Finite Group Databasebeta

← #16945067

Subgroups — #16945067

Order 4352.

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

normal subgroups only

This group has 388 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#14528867C211yesmaximal, characteristic
2#14527118C211yesmaximal, characteristic
3#14526814C212yesmaximal, complemented
4#14526811C212yesmaximal, complemented
5C17×#2396C211yesmaximal, characteristic
6#11762866—21no—
7#11762773—21no—
8#11762578—21no—
9#11762578C2212yes—
10#11762535C2212yes—
11#11762533—21no—
12C2×C4×Q8×C17C411yescharacteristic
13C2×C17×#117C2211yescharacteristic
14C17×#400C2211yescharacteristic
15C17×#399C411yescharacteristic
16C17×#398—21no—
17C17×#396—21no—
18C17×#395C2211yescharacteristic
19C17×#393C411yescharacteristic
20C4×C4⋊C4×C17—21no—
21C4×C4⋊C4×C17C411yescharacteristic
22C4×C22⋊C4×C17—21no—
23C22×Q8×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C4×Q8×C17—41no—
26C4×Q8×C17—41no—
27C17×#117—41no—
28C17×#117—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C4⋊C4×C17C2311yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17C2×C411yescharacteristic
39C2×C22⋊C4×C17D411yescharacteristic
40C2×C42×C17—21no—
41C2×C42×C17—21no—
42C2×C42×C17C2×C411yescharacteristic
43C2×C42×C17D411yescharacteristic
44C2×#63223—22no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95C2×C411yescharacteristic
50C17×#95C2×C411yescharacteristic
51C22⋊#3622—22no—
52C22⋊#3622—41no—
53C22⋊#3622—41no—
54C4⋊#3622—41no—
55#10587474—41no—
56C24×C17C22⋊C411yescharacteristic
57C2×Q8×C17—42no—
58C2×Q8×C17—41no—
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—41no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—42no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—42no—
80C22⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—42no—
83C22⋊C4×C17—42no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—21no—
88C42×C17—42no—
89C42×C17—41no—
90C42×C17—41no—
91#63223—42no—
92#63223—42no—
93C2×#3622—42no—
94C2×#3622—42no—
95#11816—171nomaximal, Sylow
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17C2×C22⋊C411yescharacteristic
102Q8×C17—42no—
103Q8×C17—42no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
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—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—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17#9911yescharacteristic
131C2×C4×C17#9911yescharacteristic
132C2×C4×C17#13712yes—
133#3622—82no—
134#3622—82no—
135#2396D1711yescharacteristic
136#1822—171no—
137#1464—171no—
138#1354—172no—
139#1353—172no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—42no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—81no—
147C22×C17C2×#9911yescharacteristic
148C22×C17#41911yescharacteristic
149C22×C17#42111yescharacteristic
150C4×C17—22no—
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—42no—
154C4×C17—41no—
155C4×C17—42no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—41no—
159C4×C17—81no—
160C4×C17—81no—
161C2×C4×Q8Dic1711yescharacteristic
162C2×#117C2×D1711yescharacteristic
163#432—341no—
164#407—341no—
165#400C2×D1711yescharacteristic
166#399Dic1711yescharacteristic
167#398—21no—
168#396—21no—
169#395C2×D1711yescharacteristic
170#393Dic1711yescharacteristic
171C4×C4⋊C4—21no—
172C4×C4⋊C4Dic1711yescharacteristic
173C4×C22⋊C4—21no—
174#343—172no—
175#343—341no—
176#324—172no—
177#323—341no—
178C2×C17—21no—
179C2×C17—21no—
180C2×C17—81no—
181C2×C17#151111yescharacteristic
182C2×C17#212011yescharacteristic
183C2×C17#213211yescharacteristic
184C22×Q8C2×Dic1711yescharacteristic
185C23×C4C22⋊D1711yescharacteristic
186C2×M16—342no—
187C4×Q8—41no—
188C4×Q8—41no—
189#117—41no—
190#117—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×Dic1711yescharacteristic
196C2×C4⋊C4C22⋊D1711yescharacteristic
197C2×C4⋊C4C22×D1711yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4C2×Dic1711yescharacteristic
201C2×C22⋊C4C22⋊D1711yescharacteristic
202C2×C42—21no—
203C2×C42—21no—
204C2×C42C2×Dic1711yescharacteristic
205C2×C42C22⋊D1711yescharacteristic
206C4⋊C8—681no—
207C22⋊C8—342no—
208C22⋊C8—681no—
209C22⋊C8—681no—
210#97—681no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95C2×Dic1711yescharacteristic
216#95C2×Dic1711yescharacteristic
217C17#1181611yescharacteristic, complemented, Sylow
218C24C22⋊Dic1711yescharacteristic
219C2×Q8—42no—
220C2×Q8—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4C22⋊Dic1711yescharacteristic
230C22×C4C22⋊Dic1711yescharacteristic
231C22×C4C22⋊Dic1711yescharacteristic
232C22×C4C22×Dic1711yescharacteristic
233C22×C4C2×C22⋊D1711yescharacteristic
234C22×C4C2×C22⋊D1711yescharacteristic
235M16—682no—
236M16—682no—
237C2×C8—682no—
238C2×C8—682no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—42no—
255C42—41no—
256C42—41no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23C2×C22⋊Dic1711yescharacteristic
263Q8—42no—
264Q8—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
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—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4#1058750511yescharacteristic
292C2×C4#1058750511yescharacteristic
293C2×C4#1058762812yes—
294C8—1362no—
295C8—1362no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1058750511yescharacteristic
304C22#1176281811yescharacteristic
305C22#1176282611yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—42no—
310C4—41no—
311C4—42no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#1452731711yescharacteristic
321C2#1453042511yescharacteristic
322C2#1453046911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2396C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2396D17
hypocenterC17#11816
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2396

Complements

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

NH ≅ G/Ntype
3. #14526814C2semidirect
4. #14526811C2semidirect
217. C17#11816semidirect

Maximal subgroups

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

subgroupindex
#145288672
#145271182
#145268142
#145268112
C17×#23962
#1181617