Finite Group Databasebeta

← #14542777

Subgroups — #14542777

Order 2176.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763535C211yesmaximal, characteristic, complemented
2#11763531C213yesmaximal, complemented
3#11763514C213yesmaximal, complemented
4#11763507C213yesmaximal, complemented
5#11763500C213yesmaximal, complemented
6Q8×C22⋊D17C211yesmaximal, characteristic, complemented
7#11763471C211yesmaximal, characteristic, complemented
8#11763470C211yesmaximal, characteristic, complemented
9#11763431C213yesmaximal, complemented
10#11763417C213yesmaximal, complemented
11#11763343C213yesmaximal, complemented
12#11763343C213yesmaximal, complemented
13#11763291C211yesmaximal, characteristic, complemented
14#11763285C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C17C211yesmaximal, characteristic, complemented
16C22×Q8×C17C2211yescharacteristic
17C2×#63256C2211yescharacteristic
18C2×Q8×D17C2211yescharacteristic
19C4×Q8×C17C2211yescharacteristic
20C4×Q8×C17C2213yes—
21C4×Q8×C17C2211yescharacteristic
22C4×Q8×C17C2213yes—
23C2×C4⋊C4×C17C2213yes—
24C2×C42×C17C2213yes—
25#10587633C2213yes—
26#10587632C2213yes—
27#10587632C2213yes—
28Q8×Dic17C2211yescharacteristic
29Q8×Dic17C2211yescharacteristic
30#10587630C2213yes—
31#10587616C2213yes—
32#10587616C2213yes—
33#10587615C2211yescharacteristic
34#10587615C2213yes—
35C4×C22⋊D17C2213yes—
36C4×C22⋊D17C2213yes—
37C4×C22⋊D17C2216yes—
38#10587610C2213yes—
39#10587610C2213yes—
40#10587584C2216yes—
41#10587583C2213yes—
42#10587582C2216yes—
43#10587581C2213yes—
44#10587580C2216yes—
45#10587579C2213yes—
46#10587578C2213yes—
47C4⋊C4×D17C2213yes—
48#10587576C2213yes—
49#10587575C2216yes—
50#10587574C2213yes—
51#10587573C2213yes—
52#10587562C2213yes—
53#10587562C2213yes—
54#10587562C2216yes—
55#10587561C2213yes—
56#10587561C2213yes—
57C4×D68C2213yes—
58C4×D68C2213yes—
59#10587558C2216yes—
60#10587558C2213yes—
61#10587558C2213yes—
62#10587556C2213yes—
63#10587556C2213yes—
64C4×Dic34C2213yes—
65C4×Dic34C2213yes—
66C2×Q8×C17—24no—
67C2×Q8×C17C2313yes—
68C2×Q8×C17C2311yescharacteristic
69C22×C4×C17C2313yes—
70C22×C4×C17C2311yescharacteristic
71C22×C4×C17C2313yes—
72C2×C22⋊D17C2313yes—
73C2×C22⋊D17C2311yescharacteristic
74#63256—24no—
75Q8×D17—24no—
76C2×D68C2313yes—
77C2×D68C2313yes—
78C2×C4×D17C2313yes—
79C2×C4×D17C2313yes—
80C2×C4×D17C2316yes—
81C2×Dic34C2313yes—
82C2×Dic34C2313yes—
83C4⋊C4×C17C2313yes—
84C4⋊C4×C17C2313yes—
85C4⋊C4×C17C2316yes—
86C42×C17C2313yes—
87C42×C17C2313yes—
88C42×C17C2313yes—
89C42×C17C2313yes—
90C22⋊Dic17C2311yescharacteristic
91C22⋊Dic17C2313yes—
92#63227C2311yescharacteristic
93#63227C2313yes—
94#63227C2313yes—
95#63227C2313yes—
96#63227C2313yes—
97#63227C2316yes—
98#63227C2311yescharacteristic
99#63227C2313yes—
100#63227C2316yes—
101#63227C2313yes—
102C4⋊Dic17C2313yes—
103C4⋊Dic17C2313yes—
104C4⋊Dic17C2313yes—
105C4⋊Dic17C2313yes—
106#63225C2311yescharacteristic
107#63225C2313yes—
108#63225C2313yes—
109#63225C2311yescharacteristic
110#63225C2316yes—
111#63225C2313yes—
112#63225C2313yes—
113#63225C2313yes—
114#63225C2313yes—
115#63225C2316yes—
116C4×Dic17C2313yes—
117C4×Dic17C2316yes—
118C4×Dic17C2313yes—
119C23×C17C2411yescharacteristic
120C22×D17C2411yescharacteristic
121C22×D17C2413yes—
122Q8×C17—26no—
123Q8×C17#4114yes—
124C2×C4×C17—22no—
125C2×C4×C17—26no—
126C2×C4×C17—43no—
127C2×C4×C17C2411yescharacteristic
128C2×C4×C17C2411yescharacteristic
129C2×C4×C17C2413yes—
130C2×C4×C17C2413yes—
131C2×C4×C17C2413yes—
132C2×C4×C17C2413yes—
133C22⋊D17—26no—
134C22⋊D17—22no—
135C2×Dic17C2411yescharacteristic
136C2×Dic17C2413yes—
137C2×Dic17C2413yes—
138C2×Dic17C2411yescharacteristic
139C2×Dic17C2413yes—
140C2×Dic17C2411yescharacteristic
141D68—26no—
142D68—43no—
143C4×D17—26no—
144C4×D17—26no—
145C4×D17—46no—
146Dic34—26no—
147Dic34—43no—
148#3516—171nomaximal, Sylow
149C22×C17—21no—
150C22×C17—21no—
151C22×C172−1+412yes—
152C22×C17C2511yescharacteristic
153C2×D17—21no—
154C2×D17—23no—
155C2×D17—21no—
156C2×D17—23no—
157C2×D17—23no—
158C2×D17—21no—
159C4×C17—23no—
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—23no—
163C4×C17—23no—
164C4×C17C2×#4116yes—
165Dic17—23no—
166Dic17—21no—
167Dic17—23no—
168Dic17—21no—
169Dic17—21no—
170Dic17—23no—
171#557—171no—
172#556—173no—
173#554—173no—
174#553—173no—
175#552—173no—
176D4×Q8—171no—
177#548—171no—
178#548—171no—
179#542—173no—
180#541—173no—
181#529—173no—
182#529—173no—
183#520—171no—
184#519—171no—
185C2×C4×Q8D1711yescharacteristic, complemented
186C2×C17—22no—
187C2×C17C22×#4111yescharacteristic
188C2×C17C2×2−1+411yescharacteristic
189C2×C17#58511yescharacteristic
190D17—41no—
191D17—43no—
192C2×#41—171no—
193C22×Q8—171no—
194C22×Q8C2×D1711yescharacteristic
195C4⋊Q8—173no—
196C4⋊Q8—173no—
197#126—176no—
198#126—176no—
199#126—173no—
200#126—173no—
201#125—176no—
202#125—173no—
203#125—173no—
204#125—173no—
205#124—173no—
206#124—173no—
207#124—173no—
208#123—171no—
209#123—173no—
210#123—176no—
211C22⋊Q8—173no—
212C22⋊Q8—173no—
213C22⋊Q8—173no—
214C22⋊Q8—173no—
215C22⋊Q8—173no—
216C22⋊Q8—176no—
217#121—173no—
218#121—173no—
219#121—173no—
220C4×Q8—171no—
221C4×Q8—171no—
222C4×Q8—173no—
223C4×Q8—173no—
224C4×Q8—173no—
225C4×Q8C2×D1711yescharacteristic
226C4×Q8C2×D1713yes—
227C4×Q8C2×D1711yescharacteristic
228C4×Q8C2×D1713yes—
229C4×D4—173no—
230C4×D4—173no—
231C4×D4—176no—
232C4×D4—173no—
233C4×D4—173no—
234C4×D4—173no—
235#117—173no—
236#117—173no—
237#117—173no—
238#117—176no—
239C2×C4⋊C4—173no—
240C2×C4⋊C4C2×D1713yes—
241C2×C42C2×D1713yes—
242C17#351611yescharacteristic, complemented, Sylow
243#41—344no—
244C2×Q8—24no—
245C2×Q8—173no—
246C2×Q8—173no—
247C2×Q8—344no—
248C2×Q8C22×D1713yes—
249C2×Q8C22×D1711yescharacteristic
250C2×D4—173no—
251C2×D4—171no—
252C2×D4—173no—
253C2×D4—173no—
254C22×C4—173no—
255C22×C4—173no—
256C22×C4—176no—
257C22×C4C22×D1711yescharacteristic
258C22×C4C22×D1713yes—
259C22×C4C22×D1713yes—
260C4⋊C4—173no—
261C4⋊C4—173no—
262C4⋊C4—171no—
263C4⋊C4—173no—
264C4⋊C4—173no—
265C4⋊C4—173no—
266C4⋊C4—173no—
267C4⋊C4—176no—
268C4⋊C4—173no—
269C4⋊C4—173no—
270C4⋊C4—176no—
271C4⋊C4—173no—
272C4⋊C4—171no—
273C4⋊C4—173no—
274C4⋊C4C22×D1713yes—
275C4⋊C4C22×D1713yes—
276C4⋊C4C22×D1716yes—
277C22⋊C4—171no—
278C22⋊C4—173no—
279C22⋊C4—173no—
280C22⋊C4—171no—
281C22⋊C4—173no—
282C22⋊C4—173no—
283C22⋊C4—173no—
284C22⋊C4—176no—
285C22⋊C4—171no—
286C22⋊C4—176no—
287C22⋊C4—173no—
288C22⋊C4—173no—
289C42—176no—
290C42—173no—
291C42—173no—
292C42C22×D1713yes—
293C42C22×D1713yes—
294C42C22×D1713yes—
295C42C22×D1713yes—
296C23—171no—
297C23—173no—
298C23C23×D1711yescharacteristic
299Q8—26no—
300Q8—346no—
301Q8—683no—
302Q8#6325214yes—
303D4—346no—
304D4—342no—
305D4—346no—
306D4—683no—
307C2×C4—22no—
308C2×C4—26no—
309C2×C4—43no—
310C2×C4—171no—
311C2×C4—173no—
312C2×C4—171no—
313C2×C4—173no—
314C2×C4—173no—
315C2×C4—171no—
316C2×C4—346no—
317C2×C4—346no—
318C2×C4—686no—
319C2×C4C23×D1713yes—
320C2×C4C23×D1711yescharacteristic
321C2×C4C23×D1711yescharacteristic
322C2×C4C23×D1713yes—
323C2×C4C23×D1713yes—
324C2×C4C23×D1713yes—
325C22—21no—
326C22—21no—
327C22—341no—
328C22—343no—
329C22—341no—
330C22—343no—
331C22—343no—
332C22—341no—
333C22#1058769412yes—
334C22C24×D1711yescharacteristic
335C4—23no—
336C4—21no—
337C4—21no—
338C4—23no—
339C4—23no—
340C4—341no—
341C4—343no—
342C4—343no—
343C4—341no—
344C4—343no—
345C4—341no—
346C4C2×#6325216yes—
347C2—22no—
348C2—681no—
349C2—683no—
350C2C22×#6325211yescharacteristic
351C2C2×#1058769411yescharacteristic
352C2#1176364511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC2×C4×Q8×C17C2
hypercenterC2×C4×Q8D17
hypocenterC17#3516
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #11763535C2semidirect
2. #11763531C2semidirect
3. #11763514C2semidirect
4. #11763507C2semidirect
5. #11763500C2semidirect
6. Q8×C22⋊D17C2semidirect
7. #11763471C2semidirect
8. #11763470C2semidirect
9. #11763431C2semidirect
10. #11763417C2semidirect
11. #11763343C2semidirect
12. #11763343C2semidirect
13. #11763291C2semidirect
14. #11763285C2semidirect
15. C2×C4×Q8×C17C2semidirect
185. C2×C4×Q8D17semidirect
242. C17#3516semidirect

Maximal subgroups

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

subgroupindex
#117635352
#117635312
#117635142
#117635072
#117635002
Q8×C22⋊D172
#117634712
#117634702
#117634312
#117634172
#117633432
#117632912
#117632852
C2×C4×Q8×C172
#351617