Finite Group Databasebeta

← #16931986

Subgroups — #16931986

Order 4352.

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

normal subgroups only

This group has 486 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#14537869C211yesmaximal, characteristic, complemented
2#14533760C211yesmaximal, characteristic
3#14527407C212yesmaximal
4#14526641C212yesmaximal, complemented
5C17×#1467C211yesmaximal, characteristic, complemented
6C2×#10587630C411yescharacteristic
7C2×#10587633C411yescharacteristic
8C2×#10587561C2211yescharacteristic
9#11762940—24no—
10#11762866—21no—
11#11762866C2212yes—
12#11762862—21no—
13#11762731—21no—
14#11762731—21no—
15#11762714C411yescharacteristic
16#11762710C411yescharacteristic
17#11762700—21no—
18#11762700—21no—
19C4×C4⋊Dic17C2211yescharacteristic
20#11762548—24no—
21C2×C4×Q8×C17C2211yescharacteristic
22C17×#432C2212yes—
23C17×#324—21no—
24C17×#324—21no—
25C22×Q8×C17C2×C411yescharacteristic
26C22×D68C2×C411yescharacteristic
27C22×Dic34C2×C411yescharacteristic
28C4×Q8×C17—22no—
29C4×Q8×C17—22no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C42×C17D411yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C42×C17C2311yescharacteristic
36#10587633—42no—
37#10587630—42no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×#63227—21no—
43C2×#63227—21no—
44C2×C4⋊Dic17—21no—
45C2×#63225—21no—
46C2×#63225—21no—
47C2×C4×Dic17C2×C411yescharacteristic
48C2×C4×Dic17C2×C411yescharacteristic
49#10587561—22no—
50#10587561—22no—
51C17×C4⋊C8—22no—
52C17×C22⋊C8—22no—
53C17×#97—22no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57C22⋊#3622—22no—
58C4⋊#3622—22no—
59C4⋊#3622—22no—
60C4⋊#3622—22no—
61#10587474—22no—
62#10587474D414yes—
63C4×#3622—22no—
64C23×D17—21no—
65C2×Q8×C17—22no—
66C2×Q8×C17—22no—
67C2×Q8×C17—41no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C2×D68—22no—
79C2×D68—41no—
80C2×D68—41no—
81C2×Dic34—22no—
82C2×Dic34—41no—
83C2×Dic34—41no—
84C2×C8×C17—42no—
85C4⋊C4×C17—22no—
86C4⋊C4×C17—22no—
87C4⋊C4×C17—22no—
88C42×C17—21no—
89C42×C17—21no—
90C42×C17—21no—
91C42×C17—21no—
92C42×C17C2×D412yes—
93C42×C17#4112yes—
94#63227—42no—
95#63227—42no—
96#63227—42no—
97#63227—42no—
98#63227—41no—
99#63227—42no—
100#63227—41no—
101C4⋊Dic17—42no—
102#63225—42no—
103#63225—42no—
104C4×Dic17—41no—
105C4×Dic17—41no—
106C2×#3622—24no—
107#9448—171nomaximal, Sylow
108C23×C17C2×C22⋊C411yescharacteristic
109C22×D17—41no—
110C22×D17—41no—
111C22×D17—41no—
112C22×D17—41no—
113C22×D17—41no—
114C22×D17—41no—
115C22×D17—41no—
116Q8×C17—44no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17#10012yes—
131C2×C4×C17C4×D412yes—
132C2×C4×C17#12011yescharacteristic
133C2×C4×C17#12112yes—
134C2×C4×C17#12311yescharacteristic
135C2×Dic17—22no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144D68—82no—
145Dic34—82no—
146C8×C17—82no—
147#3622—44no—
148#2983—171no—
149#2545—171no—
150#1533—172no—
151#1467D1711yescharacteristic, complemented
152#1285—172no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17#38611yescharacteristic
156C22×C17C2×#10011yescharacteristic
157C22×C17#41311yescharacteristic
158C2×D17—81no—
159C2×D17—81no—
160C2×D17—81no—
161C2×D17—81no—
162C2×D17—81no—
163C2×D17—81no—
164C2×D17—81no—
165C4×C17—22no—
166C4×C17—22no—
167C4×C17—22no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17—41no—
173Dic17—42no—
174Dic17—81no—
175Dic17—81no—
176C2×C4⋊Q8—171no—
177C2×#124—171no—
178C2×#124—171no—
179C2×C4×Q8C2×D1711yescharacteristic
180#446—344no—
181#432—172no—
182#432—341no—
183#432C2×D1712yes—
184#431—341no—
185#396—341no—
186#396—341no—
187#391—171no—
188#390—171no—
189#388—341no—
190#388—341no—
191C4×C4⋊C4—171no—
192#329—344no—
193#324—21no—
194#324—21no—
195C2×C17—21no—
196C2×C17—21no—
197C2×C17#188411yescharacteristic
198C2×C17#190011yescharacteristic
199C2×C17#191211yescharacteristic
200D17—161no—
201C22×Q8—171no—
202C22×Q8C4×D1711yescharacteristic
203C22×D4—171no—
204C4⋊Q8—682no—
205#124—342no—
206#124—342no—
207#124—682no—
208C4×Q8—22no—
209C4×Q8—22no—
210C2×C4⋊C4—341no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4C4×D1711yescharacteristic
214C2×C4⋊C4C22⋊D1711yescharacteristic
215C2×C4⋊C4C22⋊D1711yescharacteristic
216C2×C22⋊C4—341no—
217C2×C22⋊C4—341no—
218C2×C22⋊C4—341no—
219C2×C22⋊C4—341no—
220C2×C22⋊C4—341no—
221C2×C22⋊C4—341no—
222C2×C42—171no—
223C2×C42—171no—
224C2×C42D6811yescharacteristic
225C2×C42D6811yescharacteristic
226C2×C42C22×D1711yescharacteristic
227C4⋊C8—22no—
228C4⋊C8—342no—
229C4⋊C8—342no—
230C4⋊C8—342no—
231C22⋊C8—22no—
232C22⋊C8—342no—
233#97—22no—
234#97—174no—
235#97—342no—
236C4×C8—342no—
237#95—341no—
238#95—341no—
239#95—341no—
240C17#944811yescharacteristic, complemented, Sylow
241C24—341no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—342no—
246C2×Q8—681no—
247C2×Q8—681no—
248C2×D4—342no—
249C2×D4—681no—
250C2×D4—681no—
251C22×C4—341no—
252C22×C4—341no—
253C22×C4—341no—
254C22×C4#6322711yescharacteristic
255C22×C4#6322711yescharacteristic
256C22×C4#6322711yescharacteristic
257C22×C4#6322711yescharacteristic
258C22×C4C2×C4×D1711yescharacteristic
259C22×C4C2×D6811yescharacteristic
260C22×C4C2×C22⋊D1711yescharacteristic
261C2×C8—42no—
262C2×C8—344no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—682no—
267C4⋊C4—682no—
268C4⋊C4—682no—
269C22⋊C4—682no—
270C22⋊C4—682no—
271C22⋊C4—682no—
272C22⋊C4—682no—
273C22⋊C4—681no—
274C22⋊C4—681no—
275C22⋊C4—682no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—681no—
281C42—681no—
282C42D4×D1712yes—
283C42#6325412yes—
284C23—681no—
285C23—681no—
286C23—681no—
287C23—681no—
288C23—681no—
289C23—681no—
290C23—681no—
291C23C2×#6322711yescharacteristic
292Q8—44no—
293Q8—1362no—
294D4—1362no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—342no—
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#1058749412yes—
318C2×C4#1058756712yes—
319C2×C4#1058756811yescharacteristic
320C2×C4#1058757211yescharacteristic
321C2×C4#1058762512yes—
322C8—82no—
323C8—684no—
324C22—21no—
325C22—21no—
326C22—1361no—
327C22—1361no—
328C22—1361no—
329C22—1361no—
330C22—1361no—
331C22—1361no—
332C22—1361no—
333C22#1176268611yescharacteristic
334C22C2×#1058749411yescharacteristic
335C22#1176279111yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—682no—
345C4—1361no—
346C4—1361no—
347C2—21no—
348C2—21no—
349C2—2721no—
350C2#1452916511yescharacteristic
351C2#1452924511yescharacteristic
352C2#1452931911yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1467C2
socleC22×C17#386
prime radicalC22×D68C2×C4
cosocleC2×C42×C17C23
hypercenter#1467D17
hypocenterC17#9448
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
1. #14537869C2semidirect
4. #14526641C2semidirect
5. C17×#1467C2semidirect
151. #1467D17semidirect
240. C17#9448semidirect

Maximal subgroups

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

subgroupindex
#145378692
#145337602
#145274072
#145266412
C17×#14672
#944817