Finite Group Databasebeta

← #16943952

Subgroups — #16943952

Order 4352.

412 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 83 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#14533760C211yesmaximal, characteristic
2#14527128C211yesmaximal, characteristic, complemented
3#14526717C212yesmaximal, complemented
4#14526715C212yesmaximal
5C17×#1461C211yesmaximal, characteristic, complemented
6C2×#10587630C411yescharacteristic
7C2×#10587633C411yescharacteristic
8C2×#10587561C411yescharacteristic
9#11762866—21no—
10#11762731—21no—
11#11762731—21no—
12#11762714C2211yescharacteristic
13#11762710C2211yescharacteristic
14#11762700—21no—
15#11762700—21no—
16C4×C4⋊Dic17C411yescharacteristic
17#11762535—21no—
18#11762535C2212yes—
19C2×C4×Q8×C17C2211yescharacteristic
20C17×#431—21no—
21C17×#324—21no—
22C17×#324C2212yes—
23C22×Q8×C17C2×C411yescharacteristic
24C22×D68C2×C411yescharacteristic
25C22×Dic34C2×C411yescharacteristic
26C4×Q8×C17—41no—
27C4×Q8×C17—41no—
28C2×C4⋊C4×C17D411yescharacteristic
29C2×C4⋊C4×C17D411yescharacteristic
30C2×C4⋊C4×C17C2311yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×C42×C17D411yescharacteristic
33C2×C42×C17D411yescharacteristic
34#10587633—42no—
35#10587630—42no—
36C2×#63227—21no—
37C2×#63227—21no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×C4⋊Dic17—21no—
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17C2×C411yescharacteristic
46C2×C4×Dic17C2×C411yescharacteristic
47#10587561—41no—
48#10587561—41no—
49C17×C4⋊C8—41no—
50C17×C22⋊C8—22no—
51C4×C8×C17—41no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55C22⋊#3622—22no—
56C4⋊#3622—41no—
57#10587474—41no—
58C23×D17—21no—
59C2×Q8×C17—22no—
60C2×Q8×C17—22no—
61C2×Q8×C17—41no—
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—21no—
72C2×D68—22no—
73C2×D68—41no—
74C2×D68—41no—
75C2×Dic34—22no—
76C2×Dic34—41no—
77C2×Dic34—41no—
78C2×C8×C17—42no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—21no—
88#63227—42no—
89#63227—42no—
90#63227—42no—
91#63227—42no—
92#63227—41no—
93#63227—41no—
94#63227—42no—
95C4⋊Dic17—42no—
96#63225—42no—
97#63225—42no—
98C4×Dic17—41no—
99C4×Dic17—41no—
100C2×#3622—42no—
101#11565—171nomaximal, Sylow
102C23×C17C2×C22⋊C411yescharacteristic
103C22×D17—41no—
104C22×D17—41no—
105C22×D17—41no—
106C22×D17—41no—
107C22×D17—41no—
108C22×D17—41no—
109C22×D17—41no—
110Q8×C17—44no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17#9911yescharacteristic
126C2×C4×C17#9911yescharacteristic
127C2×C4×C17#10012yes—
128C2×C4×C17#13112yes—
129C2×Dic17—22no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138D68—82no—
139Dic34—82no—
140C8×C17—82no—
141#3622—82no—
142#2545—171no—
143#1467—171no—
144#1461D1711yescharacteristic, complemented
145#1319—172no—
146#1318—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17#40811yescharacteristic
150C22×C17C2×#9911yescharacteristic
151C22×C17C2×#10011yescharacteristic
152C2×D17—81no—
153C2×D17—81no—
154C2×D17—81no—
155C2×D17—81no—
156C2×D17—81no—
157C2×D17—81no—
158C2×D17—81no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—22no—
162C4×C17—22no—
163C4×C17—41no—
164C4×C17—41no—
165C4×C17—41no—
166C4×C17—41no—
167Dic17—42no—
168Dic17—81no—
169Dic17—81no—
170C2×C4⋊Q8—171no—
171C2×#124—171no—
172C2×#124—171no—
173C2×C4×Q8C2×D1711yescharacteristic
174#432—341no—
175#431—21no—
176#396—341no—
177#396—341no—
178#391—171no—
179#390—171no—
180#388—341no—
181#388—341no—
182C4×C4⋊C4—171no—
183#324—21no—
184#324—172no—
185#324—341no—
186#324C2×D1712yes—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17#147111yescharacteristic
190C2×C17#212611yescharacteristic
191C2×C17#213311yescharacteristic
192D17—161no—
193C22×Q8—171no—
194C22×Q8C4×D1711yescharacteristic
195C22×D4—171no—
196C4⋊Q8—682no—
197#124—681no—
198#124—681no—
199#124—682no—
200C4×Q8—41no—
201C4×Q8—41no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4D6811yescharacteristic
206C2×C4⋊C4C22⋊D1711yescharacteristic
207C2×C4⋊C4C22×D1711yescharacteristic
208C2×C22⋊C4—341no—
209C2×C22⋊C4—341no—
210C2×C22⋊C4—341no—
211C2×C22⋊C4—341no—
212C2×C22⋊C4—341no—
213C2×C22⋊C4—341no—
214C2×C42—171no—
215C2×C42—171no—
216C2×C42C4×D1711yescharacteristic
217C2×C42D6811yescharacteristic
218C2×C42C22⋊D1711yescharacteristic
219C4⋊C8—41no—
220C4⋊C8—681no—
221C22⋊C8—22no—
222C22⋊C8—342no—
223#97—681no—
224C4×C8—41no—
225#95—341no—
226#95—341no—
227#95—341no—
228C17#1156511yescharacteristic, complemented, Sylow
229C24—341no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—342no—
234C2×Q8—681no—
235C2×Q8—681no—
236C2×D4—342no—
237C2×D4—681no—
238C2×D4—681no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4—341no—
242C22×C4#6322711yescharacteristic
243C22×C4#6322711yescharacteristic
244C22×C4#6322711yescharacteristic
245C22×C4#6322711yescharacteristic
246C22×C4C2×C4×D1711yescharacteristic
247C22×C4C2×D6811yescharacteristic
248C22×C4C2×C22⋊D1711yescharacteristic
249C2×C8—42no—
250C2×C8—682no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—682no—
255C4⋊C4—682no—
256C4⋊C4—682no—
257C22⋊C4—682no—
258C22⋊C4—682no—
259C22⋊C4—682no—
260C22⋊C4—682no—
261C22⋊C4—681no—
262C22⋊C4—681no—
263C22⋊C4—682no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—681no—
271C42—681no—
272C23—681no—
273C23—681no—
274C23—681no—
275C23—681no—
276C23—681no—
277C23—681no—
278C23—681no—
279C23C2×#6322711yescharacteristic
280Q8—44no—
281Q8—1362no—
282D4—1362no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—342no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4#1058747711yescharacteristic
307C2×C4#1058747711yescharacteristic
308C2×C4#1058749412yes—
309C2×C4#1058758712yes—
310C8—82no—
311C8—1362no—
312C22—21no—
313C22—21no—
314C22—1361no—
315C22—1361no—
316C22—1361no—
317C22—1361no—
318C22—1361no—
319C22—1361no—
320C22—1361no—
321C22#1176277511yescharacteristic
322C22C2×#1058747711yescharacteristic
323C22C2×#1058749411yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—682no—
333C4—1361no—
334C4—1361no—
335C2—21no—
336C2—21no—
337C2—2721no—
338C2#1452714311yescharacteristic
339C2#1453044711yescharacteristic
340C2#1453047111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1461C2
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1461D17
hypocenterC17#11565
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #14527128C2semidirect
3. #14526717C2semidirect
5. C17×#1461C2semidirect
144. #1461D17semidirect
228. C17#11565semidirect

Maximal subgroups

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

subgroupindex
#145337602
#145271282
#145267172
#145267152
C17×#14612
#1156517