Finite Group Databasebeta

← #14542400

Subgroups — #14542400

Order 2176.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763535C211yesmaximal, characteristic, complemented
2#11763528C213yesmaximal, complemented
3#11763499C213yesmaximal, complemented
4#11763493C213yesmaximal, complemented
5Q8×C22⋊D17C211yesmaximal, characteristic, complemented
6#11763466C211yesmaximal, characteristic, complemented
7#11763459C213yesmaximal, complemented
8#11763361C213yesmaximal, complemented
9#11763340C213yesmaximal, complemented
10#11763338C213yesmaximal, complemented
11#41×Dic17C211yesmaximal, characteristic, complemented
12C4×#63254C213yesmaximal, complemented
13C2×Q8×Dic17C211yesmaximal, characteristic, complemented
14C4×Q8×D17C211yesmaximal, characteristic, complemented
15C17×#548C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C2×Q8×D17C2211yescharacteristic
19C2×#63254C2213yes—
20C17×#124C2213yes—
21C17×C22⋊Q8C2213yes—
22C17×#121C2213yes—
23C4×Q8×C17C2211yescharacteristic
24C4×D4×C17C2213yes—
25#10587632C2213yes—
26Q8×Dic17C2211yescharacteristic
27Q8×Dic17C2211yescharacteristic
28Q8×Dic17C2213yes—
29Q8×Dic17C2211yescharacteristic
30Q8×Dic17C2213yes—
31#10587630C2213yes—
32#10587624C2213yes—
33#10587622C2213yes—
34#10587621C2216yes—
35D4×Dic17C2213yes—
36D4×Dic17C2213yes—
37D4×Dic17C2213yes—
38C4×C22⋊D17C2213yes—
39C4×C22⋊D17C2213yes—
40#10587612C2213yes—
41C2×C4⋊Dic17C2213yes—
42#10587610C2213yes—
43#10587610C2213yes—
44C2×C4×Dic17C2213yes—
45C2×C4×Dic17C2213yes—
46#10587578C2213yes—
47C4⋊C4×D17C2213yes—
48#10587576C2213yes—
49#10587575C2216yes—
50#10587574C2213yes—
51#10587573C2213yes—
52#10587573C2213yes—
53#10587569C2216yes—
54#10587567C2211yescharacteristic
55#10587567C2213yes—
56#10587565C2216yes—
57#10587565C2216yes—
58#10587564C2216yes—
59#10587563C2213yes—
60#10587563C2213yes—
61#10587563C2216yes—
62C42×D17C2213yes—
63#10587556C2213yes—
64C4×Dic34C2213yes—
65C4×Dic34C2213yes—
66#41×C17—24no—
67C2×Q8×C17—24no—
68C2×Q8×C17C2313yes—
69C2×Q8×C17C2311yescharacteristic
70C2×D4×C17C2313yes—
71C2×D4×C17C2313yes—
72C22×C4×C17C2313yes—
73C22×C4×C17C2313yes—
74C2×C22⋊D17C2313yes—
75C2×C22⋊D17C2311yescharacteristic
76C22×Dic17C2313yes—
77C22×Dic17C2313yes—
78C22×Dic17C2313yes—
79C22×Dic17C2311yescharacteristic
80Q8×D17—24no—
81#63254—212no—
82C2×C4×D17C2313yes—
83C2×C4×D17C2313yes—
84C2×C4×D17C2311yescharacteristic
85C2×Dic34C2313yes—
86C2×Dic34C2313yes—
87C4⋊C4×C17C2313yes—
88C4⋊C4×C17C2313yes—
89C22⋊C4×C17C2316yes—
90C22⋊C4×C17C2313yes—
91C22⋊C4×C17C2311yescharacteristic
92C42×C17C2313yes—
93C22⋊Dic17C2316yes—
94C22⋊Dic17C2313yes—
95C22⋊Dic17C2313yes—
96C22⋊Dic17C2316yes—
97#63227C2313yes—
98#63227C2311yescharacteristic
99C4⋊Dic17C2313yes—
100C4⋊Dic17C2313yes—
101C4⋊Dic17C2313yes—
102C4⋊Dic17C2313yes—
103C4⋊Dic17C2316yes—
104#63225C2311yescharacteristic
105#63225C2313yes—
106#63225C2316yes—
107#63225C2316yes—
108#63225C2313yes—
109#63225C2316yes—
110#63225C2313yes—
111C4×Dic17C2313yes—
112C4×Dic17C2313yes—
113C4×Dic17C2313yes—
114C4×Dic17C2313yes—
115C4×Dic17C2313yes—
116C4×Dic17C2313yes—
117C4×Dic17C2313yes—
118C4×Dic17C2313yes—
119C4×Dic17C2311yescharacteristic
120C23×C17C2413yes—
121C23×C17C2411yescharacteristic
122C22×D17C2411yescharacteristic
123Q8×C17—26no—
124Q8×C17#4114yescomplemented
125D4×C17—26no—
126D4×C17—26no—
127C2×C4×C17—26no—
128C2×C4×C17—26no—
129C2×C4×C17C2411yescharacteristic
130C2×C4×C17C2413yes—
131C2×C4×C17C2413yes—
132C2×C4×C17C2411yescharacteristic
133C2×C4×C17C2413yes—
134C22⋊D17—26no—
135C22⋊D17#4114yescomplemented
136C2×Dic17—26no—
137C2×Dic17—26no—
138C2×Dic17—22no—
139C2×Dic17—26no—
140C2×Dic17C2413yes—
141C2×Dic17C2411yescharacteristic
142C2×Dic17C2413yes—
143C2×Dic17C2413yes—
144C2×Dic17C2413yes—
145C2×Dic17C2411yescharacteristic
146C2×Dic17C2411yescharacteristic
147C4×D17—26no—
148C4×D17—26no—
149C4×D17—22no—
150Dic34—26no—
151Dic34—26no—
152#3482—171nomaximal, Sylow
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—23no—
156C22×C17—23no—
157C22×C17—23no—
158C22×C17C2×#4112yes—
159C22×C17C2511yescharacteristic
160C2×D17—21no—
161C2×D17—21no—
162C2×D17C2×#4112yes—
163C4×C17—23no—
164C4×C17—21no—
165C4×C17—23no—
166C4×C17—21no—
167C4×C17C2×#4116yes—
168Dic17—23no—
169Dic17—23no—
170Dic17—23no—
171Dic17—21no—
172Dic17—23no—
173Dic17C2×#4112yes—
174Dic172−1+412yes—
175#557—171no—
176#556—173no—
177#552—173no—
178#552—173no—
179D4×Q8—171no—
180#548—171no—
181#548D1711yescharacteristic, complemented
182#547—173no—
183#533—173no—
184#529—173no—
185#529—173no—
186C4×#41—171no—
187C4×#41—173no—
188C2×C4×Q8—171no—
189C2×C4×Q8—171no—
190C2×C17—22no—
191C2×C17—43no—
192C2×C17C22×#4111yescharacteristic
193C2×C17C22×#4111yescharacteristic
194C2×C17C2×2−1+411yescharacteristic
195D17—22no—
196C2×#41—173no—
197C2×#41C2×D1711yescharacteristic
198C22×Q8—171no—
199C22×Q8C2×D1711yescharacteristic
200C4⋊Q8—173no—
201C4⋊Q8—173no—
202#126—176no—
203#126—176no—
204#125—173no—
205#125—176no—
206#125—173no—
207#124—173no—
208#124C2×D1713yes—
209#123—176no—
210#123—176no—
211C22⋊Q8—173no—
212C22⋊Q8—173no—
213C22⋊Q8—173no—
214C22⋊Q8—176no—
215C22⋊Q8C2×D1713yes—
216#121—173no—
217#121C2×D1713yes—
218C4×Q8—171no—
219C4×Q8—173no—
220C4×Q8—171no—
221C4×Q8—173no—
222C4×Q8—171no—
223C4×Q8—173no—
224C4×Q8—173no—
225C4×Q8—173no—
226C4×Q8—173no—
227C4×Q8C2×D1711yescharacteristic
228C4×D4—173no—
229C4×D4—173no—
230C4×D4—173no—
231C4×D4—173no—
232C4×D4—173no—
233C4×D4—171no—
234C4×D4—173no—
235C4×D4C2×D1713yes—
236#117—173no—
237#117—173no—
238#117—173no—
239#117—173no—
240#117—176no—
241C2×C4⋊C4—173no—
242C2×C4⋊C4—173no—
243C2×C42—173no—
244C2×C42—173no—
245C2×C42—173no—
246C17#348211yescharacteristic, complemented, Sylow
247#41—24no—
248#41—3412no—
249C2×Q8—24no—
250C2×Q8—173no—
251C2×Q8—173no—
252C2×Q8—344no—
253C2×Q8C22×D1713yes—
254C2×Q8C22×D1711yescharacteristic
255C2×D4—171no—
256C2×D4—173no—
257C2×D4C22×D1713yes—
258C2×D4C22×D1713yes—
259C22×C4—171no—
260C22×C4—173no—
261C22×C4—173no—
262C22×C4—173no—
263C22×C4—171no—
264C22×C4—173no—
265C22×C4—173no—
266C22×C4C22×D1713yes—
267C22×C4C22×D1713yes—
268C4⋊C4—173no—
269C4⋊C4—173no—
270C4⋊C4—176no—
271C4⋊C4—176no—
272C4⋊C4—171no—
273C4⋊C4—173no—
274C4⋊C4—173no—
275C4⋊C4—173no—
276C4⋊C4—173no—
277C4⋊C4—173no—
278C4⋊C4—176no—
279C4⋊C4—176no—
280C4⋊C4C22×D1713yes—
281C4⋊C4C22×D1713yes—
282C22⋊C4—173no—
283C22⋊C4—176no—
284C22⋊C4—173no—
285C22⋊C4—176no—
286C22⋊C4—171no—
287C22⋊C4—173no—
288C22⋊C4C22×D1711yescharacteristic
289C22⋊C4C22×D1713yes—
290C22⋊C4C22×D1716yes—
291C42—171no—
292C42—173no—
293C42—173no—
294C42—173no—
295C42—173no—
296C42—173no—
297C42—173no—
298C42—173no—
299C42—173no—
300C42C22×D1713yes—
301C23—171no—
302C23C23×D1713yes—
303C23C23×D1711yescharacteristic
304Q8—26no—
305Q8—346no—
306Q8—346no—
307Q8#6325414yescomplemented
308D4—26no—
309D4—26no—
310D4—174no—
311D4—346no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—171no—
315C2×C4—171no—
316C2×C4—171no—
317C2×C4—173no—
318C2×C4—173no—
319C2×C4—173no—
320C2×C4—173no—
321C2×C4—342no—
322C2×C4—346no—
323C2×C4—346no—
324C2×C4—342no—
325C2×C4—346no—
326C2×C4—346no—
327C2×C4—346no—
328C2×C4C23×D1711yescharacteristic
329C2×C4C23×D1711yescharacteristic
330C2×C4C23×D1713yes—
331C2×C4C23×D1713yes—
332C2×C4C23×D1713yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—172no—
339C22—341no—
340C22—341no—
341C22#41×D1712yes—
342C22C24×D1711yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—21no—
347C4—172no—
348C4—172no—
349C4—343no—
350C4—343no—
351C4—343no—
352C4—341no—
353C4—343no—
354C4C2×#6325416yes—
355C2—22no—
356C2—43no—
357C2—342no—
358C2C2×#41×D1711yescharacteristic
359C2C22×#6325411yescharacteristic
360C22−1+4×D1711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#548C2
hypercenter#548D17
hypocenterC17#3482
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #11763535C2semidirect
2. #11763528C2semidirect
3. #11763499C2semidirect
4. #11763493C2semidirect
5. Q8×C22⋊D17C2semidirect
6. #11763466C2semidirect
7. #11763459C2semidirect
8. #11763361C2semidirect
9. #11763340C2semidirect
10. #11763338C2semidirect
11. #41×Dic17C2semidirect
12. C4×#63254C2semidirect
13. C2×Q8×Dic17C2semidirect
14. C4×Q8×D17C2semidirect
15. C17×#548C2semidirect
124. Q8×C17#41semidirect
135. C22⋊D17#41semidirect
181. #548D17semidirect
246. C17#3482semidirect
307. Q8#63254semidirect

Maximal subgroups

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

subgroupindex
#117635352
#117635282
#117634992
#117634932
Q8×C22⋊D172
#117634662
#117634592
#117633612
#117633402
#117633382
#41×Dic172
C4×#632542
C2×Q8×Dic172
C4×Q8×D172
C17×#5482
#348217