Finite Group Databasebeta

← #16945205

Subgroups — #16945205

Order 4352.

432 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 79 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#14534471C211yesmaximal, characteristic
2#14527098C211yesmaximal, characteristic, complemented
3#14526804C212yesmaximal, complemented
4#14526780C212yesmaximal
5C17×#1820C211yesmaximal, characteristic, complemented
6C2×#10587561C411yescharacteristic, complemented
7#11762862—21no—
8#11762729C411yescharacteristic, complemented
9#11762726C411yescharacteristic, complemented
10#11762721C2211yescharacteristic
11#11762718C2211yescharacteristic
12#11762709—21no—
13#11762700—21no—
14#11762699—21no—
15#11762688—21no—
16#11762655C411yescharacteristic, complemented
17#11762535—21no—
18#11762533C2212yes—
19C2×C17×#117C2211yescharacteristic
20C17×#381—21no—
21C17×#342—21no—
22C17×#342C2212yes—
23C23×C4×C17D411yescharacteristic
24C22×C22⋊D17—21no—
25C22×D68C2×C411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C17×#117—41no—
28C17×#117—41no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17C2311yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×C42×C17D411yescharacteristic
36C2×C22⋊Dic17—21no—
37C2×#63227—21no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×#63225—21no—
43C2×C4×Dic17—21no—
44#10587561—41no—
45#10587561—41no—
46C17×#95—41no—
47C17×#95—41no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502C2×C411yescharacteristic
52#10587502C2×C411yescharacteristic
53C22⋊#3622—22no—
54C4⋊#3622—41no—
55C4×#3622—41no—
56C24×C17C2×D411yescharacteristic
57C23×D17—21no—
58C22×C4×C17—22no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—41no—
62C22×C4×C17—42no—
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22×C411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C2×C22⋊D17—42no—
70C2×C22⋊D17—42no—
71C2×C22⋊D17—81no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C2×D68—41no—
76C2×D68—81no—
77C2×Dic34—41no—
78C2×Dic34—81no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—42no—
84C22⋊C4×C17—42no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—21no—
88C42×C17—21no—
89C22⋊Dic17—81no—
90#63227—42no—
91#63227—42no—
92#63227—42no—
93#63227—41no—
94#63227—41no—
95#63227—81no—
96#63225—81no—
97C4×Dic17—42no—
98C2×#3622—42no—
99#11844—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17C2×C22⋊C411yescharacteristic
105C23×C17#13512yes—
106C22×D17—41no—
107C22×D17—41no—
108C22×D17—41no—
109C22×D17—41no—
110C22×D17—41no—
111C22×D17—41no—
112C22×D17—41no—
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—21no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—82no—
129C2×C4×C17—82no—
130C2×C4×C17—82no—
131C2×C4×C17#9911yescharacteristic
132C2×C4×C17#9911yescharacteristic
133C22⋊D17—84no—
134C2×Dic17—22no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143D68—82no—
144Dic34—82no—
145#3622—82no—
146#2610—171no—
147#1820D1711yescharacteristic, complemented
148#1458—171no—
149#1351—172no—
150#1343—172no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—22no—
154C22×C17—42no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—81no—
158C22×C17C2×#9911yescharacteristic
159C22×C17#41611yescharacteristic
160C22×C17#42111yescharacteristic
161C2×D17—81no—
162C2×D17—81no—
163C2×D17—81no—
164C2×D17—81no—
165C2×D17—81no—
166C2×D17—81no—
167C2×D17—81no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—162no—
176Dic17—42no—
177Dic17—81no—
178Dic17—81no—
179C2×#124—171no—
180C2×#117C2×D1711yescharacteristic
181#431—341no—
182#395—171no—
183#394—171no—
184#393—171no—
185#392—171no—
186#390—341no—
187#388—341no—
188#388—341no—
189#386—341no—
190#381—21no—
191#376—171no—
192#342—21no—
193#342C2×D1712yes—
194#324—341no—
195#323—172no—
196C2×C17—21no—
197C2×C17—21no—
198C2×C17—42no—
199C2×C17#151611yescharacteristic
200C2×C17#212211yescharacteristic
201C2×C17#212611yescharacteristic
202D17—161no—
203C22×Q8—171no—
204C22×D4—171no—
205C22×D4—341no—
206C23×C4C22⋊D1711yescharacteristic
207#124—681no—
208#124—681no—
209#117—41no—
210#117—41no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4C4×D1711yescharacteristic
213C2×C4⋊C4D6811yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—341no—
216C2×C22⋊C4—341no—
217C2×C22⋊C4—341no—
218C2×C22⋊C4—341no—
219C2×C22⋊C4—341no—
220C2×C22⋊C4—341no—
221C2×C22⋊C4C22⋊D1711yescharacteristic
222C2×C22⋊C4C22×D1711yescharacteristic
223C2×C42—341no—
224C2×C42C4×D1711yescharacteristic
225C2×C42D6811yescharacteristic
226C4⋊C8—681no—
227C22⋊C8—342no—
228C4×C8—681no—
229#95—41no—
230#95—41no—
231#95—171no—
232#95—171no—
233#95—341no—
234#95—341no—
235#95—341no—
236C17#1184411yescharacteristic, complemented, Sylow
237C24—341no—
238C24C2×C22⋊D1711yescharacteristic
239C2×Q8—681no—
240C2×Q8—1361no—
241C2×D4—681no—
242C2×D4—682no—
243C2×D4—682no—
244C2×D4—1361no—
245C2×D4—1361no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—42no—
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
260C2×C8—682no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—1361no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—682no—
269C22⋊C4—682no—
270C22⋊C4—682no—
271C22⋊C4—681no—
272C22⋊C4—681no—
273C22⋊C4—1361no—
274C22⋊C4—1361no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—682no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—681no—
285C23—681no—
286C23—681no—
287C23—681no—
288C23—681no—
289C23—681no—
290C23—681no—
291C23C2×#6322711yescharacteristic
292C23#1058763612yes—
293Q8—1362no—
294D4—1362no—
295D4—1364no—
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—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—342no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4#1058747711yescharacteristic
324C2×C4#1058747711yescharacteristic
325C8—1362no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—1361no—
334C22—1361no—
335C22—1361no—
336C22—1361no—
337C22—1361no—
338C22—1361no—
339C22—1361no—
340C22C2×#1058747711yescharacteristic
341C22#1176280311yescharacteristic
342C22#1176282711yescharacteristic
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—162no—
351C4—682no—
352C4—1361no—
353C4—1361no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—2721no—
358C2#1452733811yescharacteristic
359C2#1453043311yescharacteristic
360C2#1453044711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1820C2
socleC22×C17C2×#99
prime radical#11762718C22
cosocleC2×C22⋊C4×C17C23
hypercenter#1820D17
hypocenterC17#11844
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #14527098C2semidirect
3. #14526804C2semidirect
5. C17×#1820C2semidirect
6. C2×#10587561C4semidirect
8. #11762729C4semidirect
9. #11762726C4semidirect
16. #11762655C4semidirect
147. #1820D17semidirect
236. C17#11844semidirect

Maximal subgroups

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

subgroupindex
#145344712
#145270982
#145268042
#145267802
C17×#18202
#1184417