Finite Group Databasebeta

← #16945360

Subgroups — #16945360

Order 4352.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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#14533776C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526783C214yesmaximal, complemented
4C17×#1799C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic, complemented
6C2×#10587622—21no—
7C2×C22⋊#3622—21no—
8#11762734C412yescomplemented
9#11762734C2212yes—
10#11762714—21no—
11#11762669—21no—
12#11762669—21no—
13C4×C4⋊Dic17C411yescharacteristic, complemented
14#11762535—21no—
15#11762533C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C2×C17×#99—21no—
18C17×#380—21no—
19C17×#342C2212yes—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C22×Dic34—21no—
24C4×D4×C17—41no—
25C4×D4×C17—41no—
26C2×C4⋊C4×C17—41no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17C2311yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34#10587622—42no—
35#10587622—42no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C412yes—
39C2×C4×Dic17—21no—
40C22×#3622—41no—
41#10587555—41no—
42#10587555—41no—
43C17×#99—42no—
44C17×#95—41no—
45#10587502—22no—
46#10587502—22no—
47#10587502C2×C412yes—
48C22⋊#3622—22no—
49C22⋊#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C2×D411yescharacteristic
52C2×D4×C17—41no—
53C2×D4×C17—41no—
54C2×D4×C17D812yes—
55C2×D4×C17SD1612yes—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—42no—
61C22×C4×C17—42no—
62C22×C4×C17—42no—
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×Dic17—22no—
69C22×Dic17—22no—
70C2×Dic34—42no—
71C2×Dic34—42no—
72C2×Dic34—42no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—82no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—42no—
78C22⋊C4×C17—42no—
79C42×C17—21no—
80C42×C17—21no—
81C22⋊Dic17—44no—
82C22⋊Dic17—42no—
83C22⋊Dic17—42no—
84C4⋊Dic17—42no—
85C4⋊Dic17—41no—
86C4⋊Dic17—41no—
87C4⋊Dic17—42no—
88C4×Dic17—42no—
89C2×#3622—42no—
90C2×#3622—82no—
91#11878—171nomaximal, Sylow
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17#10212yes—
101C23×C17C2×C22⋊C411yescharacteristic
102C23×C17C2×D811yescharacteristic
103C23×C17C2×SD1611yescharacteristic
104D4×C17—42no—
105D4×C17—42no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—81no—
118C2×C4×C17—81no—
119C2×C4×C17—82no—
120C2×C4×C17—82no—
121C2×C4×C17—82no—
122C2×C4×C17#9911yescharacteristic
123C2×C4×C17#9911yescharacteristic
124C2×C4×C17#10212yes—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131Dic34—84no—
132#3622—82no—
133#2547—171no—
134#1799D1711yescharacteristic
135#1459—171no—
136#1344—174no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—22no—
141C22×C17—21no—
142C22×C17—42no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—81no—
150C22×C17—81no—
151C22×C17C2×#9911yescharacteristic
152C22×C17C2×#10211yescharacteristic
153C22×C17#42111yescharacteristic
154C4×C17—21no—
155C4×C17—22no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—41no—
159C4×C17—162no—
160C4×C17#35412yes—
161Dic17—82no—
162Dic17—82no—
163C2×C4⋊Q8—171no—
164C2×#124—341no—
165C2×C4×D4C2×D1711yescharacteristic
166C2×#99—21no—
167C2×C22⋊C8—341no—
168#397—172no—
169#397—172no—
170#391—341no—
171#380—21no—
172#380—341no—
173#380—341no—
174C4×C4⋊C4—171no—
175#342C2×D1712yes—
176#324—341no—
177#323—172no—
178C2×C17—21no—
179C2×C17—21no—
180C2×C17—41no—
181C2×C17—42no—
182C2×C17—41no—
183C2×C17#151911yescharacteristic
184C2×C17#212311yescharacteristic
185C2×C17C2×#35411yescharacteristic
186C22×Q8—341no—
187C22×D4C22⋊D1711yescharacteristic
188C23×C4D6811yescharacteristic
189C23×C4C22⋊D1711yescharacteristic
190C22×C8—681no—
191C4⋊Q8—681no—
192C4⋊Q8—681no—
193#124—682no—
194#124—682no—
195C4×D4—41no—
196C4×D4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—172no—
199C2×C4⋊C4—341no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4D6811yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—342no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4C4×D1711yescharacteristic
206C2×C22⋊C4C22×D1711yescharacteristic
207C2×C42—341no—
208C2×C42C4×D1711yescharacteristic
209#99—42no—
210C22⋊C8—342no—
211C22⋊C8—681no—
212#95—41no—
213#95—172no—
214#95—342no—
215#95—342no—
216C17#1187811yescharacteristic, complemented, Sylow
217C24#6322711yescharacteristic
218C24C2×C22⋊D1711yescharacteristic
219C2×Q8—682no—
220C2×Q8—682no—
221C2×Q8—682no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#6322812yes—
225C2×D4#6323012yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—342no—
234C22×C4—342no—
235C22×C4#6322711yescharacteristic
236C22×C4#6322711yescharacteristic
237C22×C4#6322711yescharacteristic
238C22×C4C2×C4×D1711yescharacteristic
239C22×C4C2×D6811yescharacteristic
240C2×C8—682no—
241C2×C8—1362no—
242C4⋊C4—41no—
243C4⋊C4—82no—
244C4⋊C4—682no—
245C4⋊C4—682no—
246C4⋊C4—681no—
247C4⋊C4—681no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—684no—
253C22⋊C4—682no—
254C22⋊C4—682no—
255C42—21no—
256C42—21no—
257C42—682no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#1058748012yes—
267C23C2×#6322711yescharacteristic
268C23C2×#6322811yescharacteristic
269C23C2×#6323011yescharacteristic
270Q8—1364no—
271D4—42no—
272D4—42no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—682no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4—682no—
294C2×C4—682no—
295C2×C4#1058747711yescharacteristic
296C2×C4#1058747711yescharacteristic
297C2×C4#1058748012yes—
298C8—1362no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—21no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22C2×#1058747711yescharacteristic
314C22C2×#1058748011yescharacteristic
315C22#1176282711yescharacteristic
316C4—22no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—162no—
322C4—1362no—
323C4—1362no—
324C4#1176260312yes—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—42no—
329C2—41no—
330C2#1452735111yescharacteristic
331C2#1453043711yescharacteristic
332C2C2×#1176260311yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1799C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1799D17
hypocenterC17#11878
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #14526783C2semidirect
5. C2×#10587555C4semidirect
8. #11762734C4semidirect
13. C4×C4⋊Dic17C4semidirect
216. C17#11878semidirect

Maximal subgroups

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

subgroupindex
#145337762
#145271022
#145267832
C17×#17992
#1187817