Finite Group Databasebeta

← #16945249

Subgroups — #16945249

Order 4352.

368 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 69 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#14534477C211yesmaximal, characteristic
2#14528866C211yesmaximal, characteristic
3#14527097C211yesmaximal, characteristic
4#14526812C211yesmaximal, characteristic, complemented
5#14526810C211yesmaximal, characteristic, complemented
6#14526790C211yesmaximal, characteristic, complemented
7C17×#1346C211yesmaximal, characteristic, complemented
8C2×#10587576C411yescharacteristic
9C2×#10587578C2211yescharacteristic
10#11762859—21no—
11#11762772—21no—
12#11762748C2211yescharacteristic
13#11762746C411yescharacteristic
14#11762735C411yescharacteristic
15#11762708—21no—
16#11762701—21no—
17#11762692—21no—
18#11762680—21no—
19#11762654C411yescharacteristic
20#11762578C2211yescharacteristic
21#11762577—21no—
22#11762535C2211yescharacteristic
23#11762532—21no—
24C17×#398C2211yescharacteristic
25C17×#343C2211yescharacteristic
26C17×#324C2211yescharacteristic
27C22×C4×D17D411yescharacteristic
28C2×M16×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17C2311yescharacteristic
32C2×#63227—21no—
33C2×#63227—21no—
34C2×#63227C2×C411yescharacteristic
35C2×#63227D411yescharacteristic
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17D411yescharacteristic
40C2×#63225—21no—
41C2×C4×Dic17—21no—
42C2×C4×Dic17C2×C411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×#63223—21no—
45#10587578—41no—
46#10587578—41no—
47#10587576—41no—
48#10587576—41no—
49C17×C22⋊C8—21no—
50C17×#95—21no—
51C17×#95C2×C411yescharacteristic
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502C2×C411yescharacteristic
57C22⋊#3622—21no—
58#10587491—41no—
59#10587491—41no—
60#10587485—41no—
61C8×Dic17—41no—
62C23×D17C22⋊C411yescharacteristic
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C2×C4×D17—21no—
74C2×C4×D17—21no—
75C2×C4×D17—21no—
76C2×C4×D17—21no—
77C2×C4×D17—41no—
78M16×C17—41no—
79M16×C17—41no—
80C2×C8×C17—41no—
81C2×C8×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86#63227—41no—
87#63227—41no—
88#63227—42no—
89#63227—42no—
90C4⋊Dic17—41no—
91C4⋊Dic17—42no—
92C4⋊Dic17—42no—
93C4⋊Dic17—41no—
94#63225—41no—
95#63225—41no—
96C4×Dic17—21no—
97C4×Dic17—21no—
98C4×Dic17—21no—
99C4×Dic17—21no—
100C4×Dic17—42no—
101#63223—41no—
102#63223—41no—
103C2×#3622—41no—
104C2×#3622—41no—
105#11853—171nomaximal, Sylow
106C23×C17C2×C22⋊C411yescharacteristic
107C22×D17—21no—
108C22×D17—21no—
109C22×D17—21no—
110C22×D17—41no—
111C22×D17—41no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—22no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
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#13511yescharacteristic
125C2×C4×C17#13511yescharacteristic
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17#9911yescharacteristic
139C2×Dic17#9911yescharacteristic
140C4×D17—41no—
141C4×D17—41no—
142C4×D17—41no—
143C4×D17—41no—
144C8×C17—81no—
145C8×C17—81no—
146#3622—81no—
147#3622—81no—
148#2611—171no—
149#1822—171no—
150#1458—171no—
151#1353—171no—
152#1353—171no—
153#1346—171no—
154#1346D1711yescharacteristic, complemented
155C22×C17—21no—
156C22×C17—21no—
157C22×C17C2×#9911yescharacteristic
158C22×C17#41611yescharacteristic
159C22×C17#42111yescharacteristic
160C2×D17—42no—
161C2×D17—41no—
162C2×D17—41no—
163C2×D17—41no—
164C2×D17—81no—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—42no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17—81no—
171Dic17—21no—
172Dic17—21no—
173Dic17—41no—
174Dic17—41no—
175Dic17—81no—
176Dic17—81no—
177C2×#125—171no—
178C2×#117—171no—
179#431—341no—
180#407—341no—
181#400—171no—
182#399—171no—
183#398C2×D1711yescharacteristic
184#397—171no—
185#389—341no—
186#388—341no—
187#387—341no—
188#385—341no—
189#376—171no—
190#343—171no—
191#343—341no—
192#343C2×D1711yescharacteristic
193#324—171no—
194#324C2×D1711yescharacteristic
195#323—341no—
196C2×C17—21no—
197C2×C17—21no—
198C2×C17#151611yescharacteristic
199C2×C17#212311yescharacteristic
200C2×C17#213211yescharacteristic
201D17—81no—
202C23×C4—171no—
203C2×M16—21no—
204C2×M16—341no—
205#125—681no—
206#125—681no—
207#117—681no—
208#117—681no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4C4×D1711yescharacteristic
216C2×C4⋊C4C22×D1711yescharacteristic
217C2×C22⋊C4—171no—
218C2×C22⋊C4—171no—
219C2×C22⋊C4—341no—
220C2×C22⋊C4—341no—
221C2×C42—171no—
222C2×C42—171no—
223C2×C42—341no—
224C4⋊C8—681no—
225C22⋊C8—21no—
226C22⋊C8—341no—
227C22⋊C8—681no—
228C22⋊C8—681no—
229C4×C8—681no—
230#95—21no—
231#95—171no—
232#95—341no—
233#95—341no—
234#95—341no—
235#95—341no—
236#95C4×D1711yescharacteristic
237C17#1185311yescharacteristic, complemented, Sylow
238C24—171no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—171no—
242C22×C4—171no—
243C22×C4—171no—
244C22×C4—341no—
245C22×C4—341no—
246C22×C4—341no—
247C22×C4—341no—
248C22×C4—341no—
249C22×C4—341no—
250C22×C4—681no—
251C22×C4C2×C4×D1711yescharacteristic
252C22×C4D4×D1711yescharacteristic
253C22×C4D4×D1711yescharacteristic
254M16—41no—
255M16—41no—
256M16—681no—
257M16—681no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—681no—
261C2×C8—681no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—681no—
267C4⋊C4—681no—
268C4⋊C4—682no—
269C4⋊C4—681no—
270C4⋊C4—681no—
271C4⋊C4—682no—
272C22⋊C4—681no—
273C22⋊C4—681no—
274C22⋊C4—682no—
275C22⋊C4—682no—
276C42—341no—
277C42—341no—
278C42—341no—
279C42—341no—
280C42—682no—
281C23—341no—
282C23—341no—
283C23—341no—
284C23—681no—
285C23—681no—
286C23C22⋊C4×D1711yescharacteristic
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—171no—
300C2×C4—171no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4#1058759811yescharacteristic
318C2×C4#1058760211yescharacteristic
319C8—81no—
320C8—81no—
321C8—1361no—
322C8—1361no—
323C22—21no—
324C22—21no—
325C22—682no—
326C22—681no—
327C22—681no—
328C22—681no—
329C22—1361no—
330C22#99×D1711yescharacteristic
331C22#1176280211yescharacteristic
332C22#1176282411yescharacteristic
333C4—21no—
334C4—21no—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—341no—
340C4—341no—
341C4—681no—
342C4—681no—
343C4—1361no—
344C4—1361no—
345C2—21no—
346C2—21no—
347C2—1361no—
348C2#1452733711yescharacteristic
349C2#1453043611yescharacteristic
350C2#1453046811yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1346C2
socleC22×C17C2×#99
prime radicalC23×D17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1346D17
hypocenterC17#11853
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1346

Complements

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

NH ≅ G/Ntype
4. #14526812C2semidirect
5. #14526810C2semidirect
6. #14526790C2semidirect
7. C17×#1346C2semidirect
154. #1346D17semidirect
237. C17#11853semidirect

Maximal subgroups

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

subgroupindex
#145344772
#145288662
#145270972
#145268122
#145268102
#145267902
C17×#13462
#1185317