Finite Group Databasebeta

← #16944512

Subgroups — #16944512

Order 4352.

468 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 107 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#14534550C211yesmaximal, characteristic
2#14527098C211yesmaximal, characteristic
3#14526783C214yesmaximal
4C17×#1740C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic, complemented
6C2×#10587610—21no—
7#11762862—21no—
8#11762739—21no—
9#11762739C412yescomplemented
10#11762734—21no—
11#11762734C2212yes—
12#11762715—21no—
13#11762675C411yescharacteristic, complemented
14#11762535—21no—
15#11762533C2212yes—
16C2×C17×#117—21no—
17C2×C17×#117C2211yescharacteristic
18C17×#342—21no—
19C17×#342C2212yes—
20C23×C4×C17D411yescharacteristic
21C22×Dic34—21no—
22C17×#117—24no—
23C17×#117—41no—
24C17×#117—41no—
25C17×#117—42no—
26C2×C4⋊C4×C17—41no—
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C4⋊C4×C17D411yescharacteristic
29C2×C22⋊C4×C17—22no—
30C2×C22⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17C2311yescharacteristic
32C2×C42×C17—41no—
33C2×C42×C17C2×C411yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C22⋊Dic17—22no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C412yes—
39#10587610—42no—
40#10587610—42no—
41C2×#63225—22no—
42#10587555—41no—
43#10587555—41no—
44#10587502—22no—
45#10587502—21no—
46#10587502—22no—
47#10587502C2×C412yes—
48C22⋊#3622—22no—
49C4⋊#3622—41no—
50C4×#3622—41no—
51C24×C17C2×D411yescharacteristic
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—22no—
55C22×C4×C17—42no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17D812yes—
61C22×C4×C17SD1612yes—
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×Dic17—22no—
65C22×Dic17—22no—
66C2×Dic34—41no—
67C2×Dic34—41no—
68C2×Dic34—42no—
69C2×Dic34—42no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—41no—
72C4⋊C4×C17—44no—
73C22⋊C4×C17—24no—
74C22⋊C4×C17—41no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—42no—
77C42×C17—21no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17—21no—
81C42×C17—44no—
82C22⋊Dic17—42no—
83C22⋊Dic17—42no—
84C4⋊Dic17—42no—
85C4⋊Dic17—42no—
86C4⋊Dic17—42no—
87#63225—44no—
88C2×#3622—42no—
89#11694—171nomaximal, Sylow
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17C2×C22⋊C411yescharacteristic
95C23×C17C2×D811yescharacteristic
96C23×C17C2×SD1611yescharacteristic
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—44no—
112C2×C4×C17—82no—
113C2×C4×C17—82no—
114C2×C4×C17#9911yescharacteristic
115C2×C4×C17#9911yescharacteristic
116C2×C4×C17#10212yes—
117C2×C4×C17#10212yes—
118C2×C4×C17#10412yes—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125Dic34—82no—
126Dic34—82no—
127#3622—82no—
128#2621—171no—
129#1740D1711yescharacteristic
130#1458—171no—
131#1344—174no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—41no—
137C22×C17—41no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—81no—
141C22×C17C2×#9911yescharacteristic
142C22×C17C2×#10211yescharacteristic
143C22×C17C2×#10411yescharacteristic
144C4×C17—22no—
145C4×C17—41no—
146C4×C17—41no—
147C4×C17—41no—
148C4×C17—41no—
149C4×C17—84no—
150C4×C17#32712yes—
151C4×C17#32712yes—
152Dic17—82no—
153Dic17—82no—
154C2×C4⋊Q8—171no—
155C2×C22⋊Q8—341no—
156C2×#117—21no—
157C2×#117C2×D1711yescharacteristic
158#431—341no—
159#398—172no—
160#398—341no—
161#397—172no—
162#397—341no—
163#391—341no—
164#382—171no—
165#342—21no—
166#342C2×D1712yes—
167#324—341no—
168#323—172no—
169C2×C17—21no—
170C2×C17—21no—
171C2×C17—41no—
172C2×C17—41no—
173C2×C17C2×#32711yescharacteristic
174C2×C17#212311yescharacteristic
175C2×C17#212911yescharacteristic
176C22×Q8—341no—
177C23×C4C22⋊D1711yescharacteristic
178C4⋊Q8—681no—
179C4⋊Q8—681no—
180C22⋊Q8—682no—
181C22⋊Q8—682no—
182#117—24no—
183#117—41no—
184#117—41no—
185#117—42no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—172no—
188C2×C4⋊C4—342no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4C4×D1711yescharacteristic
192C2×C4⋊C4D6811yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—342no—
195C2×C22⋊C4C22⋊D1711yescharacteristic
196C2×C22⋊C4C22×D1711yescharacteristic
197C2×C42—41no—
198C2×C42C4×D1711yescharacteristic
199C2×C42D6811yescharacteristic
200C4⋊C8—681no—
201C22⋊C8—342no—
202C4×C8—681no—
203#95—172no—
204#95—342no—
205#95—341no—
206#95—342no—
207C17#1169411yescharacteristic, complemented, Sylow
208C24C2×C22⋊D1711yescharacteristic
209C2×Q8—682no—
210C2×Q8—681no—
211C2×Q8—681no—
212C2×Q8—682no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—342no—
218C22×C4—342no—
219C22×C4#6322711yescharacteristic
220C22×C4#6322711yescharacteristic
221C22×C4#6322711yescharacteristic
222C22×C4#6322711yescharacteristic
223C22×C4#6322812yes—
224C22×C4#6323012yes—
225C22×C4C2×C4×D1711yescharacteristic
226C22×C4C2×D6811yescharacteristic
227C2×C8—682no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—44no—
231C4⋊C4—682no—
232C4⋊C4—682no—
233C4⋊C4—684no—
234C4⋊C4—682no—
235C22⋊C4—24no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—682no—
240C22⋊C4—682no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—44no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23C2×#6322711yescharacteristic
251C23C2×#6322811yescharacteristic
252C23C2×#6323011yescharacteristic
253Q8—1362no—
254Q8—1362no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—682no—
273C2×C4—682no—
274C2×C4—682no—
275C2×C4—682no—
276C2×C4—682no—
277C2×C4—682no—
278C2×C4#1058747612yes—
279C2×C4#1058747711yescharacteristic
280C2×C4#1058747711yescharacteristic
281C2×C4#1058748012yes—
282C2×C4#1058748012yes—
283C8—1362no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22C2×#1058747711yescharacteristic
294C22C2×#1058748011yescharacteristic
295C22C2×#1058747611yescharacteristic
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—84no—
302C4—1362no—
303C4—1362no—
304C4#1176254212yes—
305C4#1176254212yes—
306C2—21no—
307C2—21no—
308C2—41no—
309C2—41no—
310C2C2×#1176254211yescharacteristic
311C2#1453043711yescharacteristic
312C2#1453045811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1740C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1740D17
hypocenterC17#11694
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
5. C2×#10587555C4semidirect
9. #11762739C4semidirect
13. #11762675C4semidirect
207. C17#11694semidirect

Maximal subgroups

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

subgroupindex
#145345502
#145270982
#145267832
C17×#17402
#1169417