Finite Group Databasebeta

← #17982448

Subgroups — #17982448

Order 4352.

474 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587670C211yesmaximal, characteristic
2C2×C22⋊#63241C211yesmaximal, characteristic
3C2×#11763903C211yesmaximal, characteristic, complemented
4C2×C4×#63241C211yesmaximal, characteristic, complemented
5#14545854C211yesmaximal, characteristic, complemented
6#14545854C211yesmaximal, characteristic, complemented
7C22⋊C4×#3624C211yesmaximal, characteristic
8C2×#10587670—22no—
9C2×#10587670—22no—
10C2×#10587670C414yescomplemented
11C22×#63241C2211yescharacteristic
12C22×#63241C2211yescharacteristic
13C22×#63241C2211yescharacteristic
14C22×#63241C2211yescharacteristic
15C22⋊#63241C414yescomplemented
16#11763903—22no—
17C4×#63241—22no—
18C23×#3624C2211yescharacteristic
19C2×C22⋊#3624C411yescharacteristic
20C2×C4×#3624C2211yescharacteristic
21C2×C4×#3624C2211yescharacteristic
22#11763675C411yescharacteristic
23#11763675C411yescharacteristic
24C22⋊C4×Dic17C411yescharacteristic
25C23×Dic17C2×C411yescharacteristic
26C22×#3624—21no—
27C22×#3624—21no—
28C22×#3624—21no—
29C22×#3624—21no—
30C22×#3624C2×C411yescharacteristic
31C22×#3624C2×C412yes—
32C22×#3624C2×C412yes—
33C22×#3624C2×C411yescharacteristic
34C22×#3624C2×C411yescharacteristic
35C22×#3624C2311yescharacteristic
36#10587670—28no—
37C2×#63241—21no—
38C2×#63241—21no—
39C2×#63241—21no—
40C2×#63241—21no—
41C2×#63241—21no—
42C2×#63241—21no—
43C2×#63241C2×C412yes—
44C2×#63241C2×C412yes—
45C2×#63241C2×C412yes—
46C2×#63241C2×C412yes—
47C2×#63241D412yes—
48C2×#63241D412yes—
49C2×C22⋊C4×C17C811yescharacteristic
50C2×C22⋊Dic17C811yescharacteristic
51C2×C4×Dic17C2×C411yescharacteristic
52C2×C4×Dic17C2×C411yescharacteristic
53C22⋊#3624C814yes—
54C4×#3624—21no—
55C4×#3624—21no—
56C4×#3624—21no—
57C4×#3624—21no—
58#10587502C811yescharacteristic
59#10587502C811yescharacteristic
60C24×C17C2×C811yescharacteristic
61C22×C4×C17C2×C811yescharacteristic
62C22×C4×C17C2×C811yescharacteristic
63C22×Dic17—21no—
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17C4212yes—
68C22×Dic17C2×C811yescharacteristic
69C22×Dic17C2×C811yescharacteristic
70C22×Dic17C2×C812yes—
71C22×Dic17C2×C811yescharacteristic
72C22×Dic17C22×C411yescharacteristic
73C2×#3624—21no—
74C2×#3624—21no—
75C2×#3624—21no—
76C2×#3624—21no—
77C2×#3624—21no—
78C2×#3624—21no—
79C2×#3624—22no—
80C2×#3624—22no—
81C2×#3624—22no—
82C2×#3624—22no—
83C2×#3624C4212yes—
84C2×#3624C2×C812yes—
85C2×#3624C2×C812yes—
86C2×#3624C22×C411yescharacteristic
87C2×#3624C22×C411yescharacteristic
88C2×#3624C2×D411yescharacteristic
89C2×#3624C2×D411yescharacteristic
90C2×#3624#4111yescharacteristic
91C2×#3624#4111yescharacteristic
92#63241—22no—
93#63241—22no—
94#63241—22no—
95#63241C22⋊C414yescomplemented
96C22⋊C4×C17—22no—
97C22⋊Dic17—22no—
98C4×Dic17—21no—
99C4×Dic17—21no—
100C4×Dic17—21no—
101C4×Dic17—21no—
102#12612—171nomaximal, Sylow
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17C4×C812yes—
110C23×C17C22×C811yescharacteristic
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17M3212yescomplemented
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—22no—
122C2×Dic17—22no—
123C2×Dic17—22no—
124C2×Dic17—22no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17C4×C812yes—
129C2×Dic17M3212yescomplemented
130C2×Dic17C2×C4211yescharacteristic
131C2×Dic17C4×D411yescharacteristic
132C2×Dic17C4×D411yescharacteristic
133C2×Dic17C22×C811yescharacteristic
134C2×Dic17#13111yescharacteristic
135C2×Dic17#13111yescharacteristic
136#3624—22no—
137#3624—22no—
138#3624—22no—
139#3624C2×C22⋊C411yescharacteristic
140#3624#11711yescharacteristic
141#3624C4×D411yescharacteristic
142#3624C4×D411yescharacteristic
143C22×M32—171no—
144C2×C22⋊C16—171no—
145C2×#346—171no—
146C2×C4×C16—171no—
147C8×C22⋊C4—171no—
148#1374—171no—
149#1374—171no—
150C22×C17—22no—
151C22×C17—22no—
152C22×C17—22no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17C2×C4×C811yescharacteristic
158C22×C17C8×D411yescharacteristic
159C22×C17C8×D411yescharacteristic
160C22×C17C2×M3211yescharacteristic
161C22×C17C2×M3211yescharacteristic
162C22×C17#50411yescharacteristic
163C22×C17#50411yescharacteristic
164C4×C17—22no—
165C4×C17—41no—
166Dic17—22no—
167Dic17—22no—
168Dic17—41no—
169Dic17C4×C22⋊C411yescharacteristic
170Dic17#40511yescharacteristic
171Dic17C8×D411yescharacteristic
172Dic17C8×D411yescharacteristic
173C23×C8—171no—
174C2×M32—174no—
175C2×M32—342no—
176C2×M32—342no—
177C22×C16—171no—
178C22×C16—171no—
179C22×C16—171no—
180C22×C16—171no—
181C2×C22⋊C8—171no—
182C2×C4×C8—171no—
183C2×C4×C8—171no—
184C4×C22⋊C4—171no—
185C22⋊C16—174no—
186#346—342no—
187C4×C16—342no—
188#336—171no—
189#336—171no—
190C2×C17—42no—
191C2×C17C8×C22⋊C411yescharacteristic
192C2×C17C4×M3211yescharacteristic
193C2×C17#210811yescharacteristic
194C2×C17#216811yescharacteristic
195C2×C17#216811yescharacteristic
196C2×C17#216911yescharacteristic
197C2×C17#216911yescharacteristic
198C23×C4—171no—
199C22×C8—171no—
200C22×C8—171no—
201C22×C8—172no—
202C22×C8—172no—
203C22×C8—171no—
204C22×C8—171no—
205C22×C8—341no—
206C22×C8—341no—
207C22×C8—341no—
208C22×C8—341no—
209C2×C22⋊C4—171no—
210C2×C22⋊C4#363311yescharacteristic
211C2×C42—171no—
212C2×C42—171no—
213M32—348no—
214C2×C16—172no—
215C2×C16—172no—
216C2×C16—172no—
217C2×C16—172no—
218C2×C16—172no—
219C2×C16—172no—
220C2×C16—341no—
221C2×C16—341no—
222C2×C16—341no—
223C2×C16—341no—
224C2×C16—341no—
225C2×C16—341no—
226C22⋊C8—174no—
227C4×C8—341no—
228C4×C8—341no—
229C4×C8—341no—
230C4×C8—341no—
231#95—171no—
232#95—171no—
233C17#1261211yescharacteristic, complemented, Sylow
234C24C2×#363311yescharacteristic
235C22×C4—171no—
236C22×C4—172no—
237C22×C4—172no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4C2×#363311yescharacteristic
246C22×C4C2×#363311yescharacteristic
247C2×C8—171no—
248C2×C8—171no—
249C2×C8—171no—
250C2×C8—171no—
251C2×C8—171no—
252C2×C8—171no—
253C2×C8—172no—
254C2×C8—172no—
255C2×C8—172no—
256C2×C8—342no—
257C2×C8—342no—
258C2×C8—342no—
259C2×C8—342no—
260C2×C8—341no—
261C2×C8—341no—
262C2×C8—341no—
263C2×C8—341no—
264C2×C8—341no—
265C2×C8—341no—
266C22⋊C4—22no—
267C22⋊C4—342no—
268C42—341no—
269C42—341no—
270C42—341no—
271C42—341no—
272C16—174no—
273C16—342no—
274C16—342no—
275C16—342no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23C4×#363312yes—
283C23C22×#363311yescharacteristic
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—171no—
290C2×C4—171no—
291C2×C4—171no—
292C2×C4—171no—
293C2×C4—171no—
294C2×C4—171no—
295C2×C4—172no—
296C2×C4—172no—
297C2×C4—342no—
298C2×C4—342no—
299C2×C4—342no—
300C2×C4—342no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—341no—
308C2×C4#1058767012yescomplemented
309C8—171no—
310C8—171no—
311C8—171no—
312C8—171no—
313C8—342no—
314C8—342no—
315C8—342no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22C2×C4×#363311yescharacteristic
324C22D4×#363311yescharacteristic
325C22D4×#363311yescharacteristic
326C22C2×#1058767011yescharacteristic
327C22#1176392811yescharacteristic
328C22#1176393011yescharacteristic
329C22#1176393011yescharacteristic
330C4—22no—
331C4—41no—
332C4—171no—
333C4—171no—
334C4—171no—
335C4—171no—
336C4—342no—
337C4—342no—
338C4—681no—
339C2—42no—
340C2C22⋊C4×#363311yescharacteristic
341C2C4×#1058767011yescharacteristic
342C2#1454594511yescharacteristic
343C2#1454599611yescharacteristic
344C2#1454599711yescharacteristic
345C2#1454599811yescharacteristic
346C2#1454600011yescharacteristic
subgroupG/H
center, FrattiniC23C22×#3633
commutatorC22×C17C2×C4×C8
FittingC2×C22⋊C4×C17C8
socleC23×C17C22×C8
prime radicalC24×C17C2×C8
cosocleC22×#3624C23
hypercenterC2×C22⋊C4#3633
hypocenterC17#12612
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#11763903C2semidirect
4. C2×C4×#63241C2semidirect
5. #14545854C2semidirect
6. #14545854C2semidirect
10. C2×#10587670C4semidirect
15. C22⋊#63241C4semidirect
95. #63241C22⋊C4semidirect
116. C2×C4×C17M32semidirect
129. C2×Dic17M32semidirect
233. C17#12612semidirect
308. C2×C4#10587670semidirect

Maximal subgroups

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

subgroupindex
C22×#105876702
C2×C22⋊#632412
C2×#117639032
C2×C4×#632412
#145458542
C22⋊C4×#36242
#1261217