Finite Group Databasebeta

← #14542893

Subgroups — #14542893

Order 2176.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763542C213yesmaximal, complemented
2#11763535C211yesmaximal, characteristic, complemented
3#11763522C213yesmaximal, complemented
4#11763509C213yesmaximal, complemented
5#11763498C213yesmaximal, complemented
6#11763478C211yesmaximal, characteristic, complemented
7#11763471C211yesmaximal, characteristic, complemented
8#11763466C211yesmaximal, characteristic, complemented
9#11763424C213yesmaximal, complemented
10#11763410C213yesmaximal, complemented
11#11763343C213yesmaximal, complemented
12#11763343C213yesmaximal, complemented
13#11763291C211yesmaximal, characteristic, complemented
14#11763285C211yesmaximal, characteristic, complemented
15C4×#41×C17C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#63256C2211yescharacteristic
18C2×Q8×D17C2211yescharacteristic
19C4×Q8×C17C2211yescharacteristic
20C4×Q8×C17C2211yescharacteristic
21C4×D4×C17C2213yes—
22C4×D4×C17C2213yes—
23C17×#117C2213yes—
24C2×C42×C17C2213yes—
25Q8×Dic17C2211yescharacteristic
26Q8×Dic17C2211yescharacteristic
27#10587626C2213yes—
28#10587624C2213yes—
29#10587624C2213yes—
30#10587622C2213yes—
31#10587616C2213yes—
32#10587616C2213yes—
33#10587615C2213yes—
34#10587615C2213yes—
35#10587615C2216yes—
36C4×C22⋊D17C2213yes—
37C4×C22⋊D17C2213yes—
38C4×C22⋊D17C2216yes—
39#10587610C2213yes—
40#10587610C2213yes—
41#10587583C2213yes—
42#10587581C2213yes—
43#10587579C2213yes—
44#10587578C2213yes—
45C4⋊C4×D17C2213yes—
46#10587576C2213yes—
47#10587574C2213yes—
48#10587573C2213yes—
49#10587571C2216yes—
50#10587570C2216yes—
51#10587569C2216yes—
52#10587565C2216yes—
53#10587562C2211yescharacteristic
54#10587562C2213yes—
55#10587561C2213yes—
56#10587561C2213yes—
57C4×D68C2213yes—
58C4×D68C2213yes—
59#10587558C2216yes—
60#10587558C2213yes—
61#10587558C2213yes—
62#10587556C2213yes—
63#10587556C2213yes—
64C4×Dic34C2213yes—
65C4×Dic34C2213yes—
66#41×C17—24no—
67C2×Q8×C17C2311yescharacteristic
68C2×D4×C17C2313yes—
69C22×C4×C17C2313yes—
70C22×C4×C17C2313yes—
71C22×C4×C17C2313yes—
72C2×C22⋊D17C2313yes—
73C2×C22⋊D17C2313yes—
74C2×C22⋊D17C2316yes—
75#63256—24no—
76Q8×D17—24no—
77C2×D68C2313yes—
78C2×D68C2313yes—
79C2×C4×D17C2313yes—
80C2×C4×D17C2313yes—
81C2×C4×D17C2316yes—
82C2×Dic34C2313yes—
83C2×Dic34C2313yes—
84C4⋊C4×C17C2313yes—
85C4⋊C4×C17C2313yes—
86C22⋊C4×C17C2316yes—
87C42×C17C2313yes—
88C42×C17C2313yes—
89C42×C17C2313yes—
90C42×C17C2311yescharacteristic
91C22⋊Dic17C2313yes—
92C22⋊Dic17C2316yes—
93C22⋊Dic17C2313yes—
94#63227C2311yescharacteristic
95#63227C2313yes—
96#63227C2313yes—
97#63227C2316yes—
98#63227C2313yes—
99#63227C2313yes—
100#63227C2311yescharacteristic
101#63227C2313yes—
102#63227C2311yescharacteristic
103C4⋊Dic17C2313yes—
104C4⋊Dic17C2313yes—
105C4⋊Dic17C2313yes—
106C4⋊Dic17C2313yes—
107#63225C2311yescharacteristic
108#63225C2313yes—
109#63225C2313yes—
110#63225C2311yescharacteristic
111#63225C2313yes—
112#63225C2313yes—
113#63225C2316yes—
114#63225C2313yes—
115#63225C2311yescharacteristic
116C4×Dic17C2316yes—
117C4×Dic17C2313yes—
118C4×Dic17C2313yes—
119C23×C17C2413yes—
120C22×D17C2411yescharacteristic
121C22×D17C2413yes—
122Q8×C17#4114yes—
123D4×C17—26no—
124C2×C4×C17—26no—
125C2×C4×C17—43no—
126C2×C4×C17—43no—
127C2×C4×C17C2411yescharacteristic
128C2×C4×C17C2411yescharacteristic
129C2×C4×C17C2411yescharacteristic
130C2×C4×C17C2413yes—
131C2×C4×C17C2413yes—
132C2×C4×C17C2413yes—
133C22⋊D17—46no—
134C22⋊D17—43no—
135C22⋊D17—43no—
136C2×Dic17C2411yescharacteristic
137C2×Dic17C2413yes—
138C2×Dic17C2411yescharacteristic
139C2×Dic17C2413yes—
140C2×Dic17C2413yes—
141C2×Dic17C2411yescharacteristic
142D68—26no—
143D68—43no—
144C4×D17—26no—
145C4×D17—26no—
146C4×D17—46no—
147Dic34—26no—
148Dic34—43no—
149#3524—171nomaximal, Sylow
150C22×C17—23no—
151C22×C17—23no—
152C22×C17—23no—
153C22×C17C2511yescharacteristic
154C2×D17—21no—
155C2×D17—23no—
156C2×D17—23no—
157C2×D17—21no—
158C2×D17—23no—
159C2×D17—21no—
160C4×C17—21no—
161C4×C17—23no—
162C4×C17—23no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17C2×#4116yes—
166Dic17—23no—
167Dic17—21no—
168Dic17—21no—
169Dic17—23no—
170Dic17—23no—
171Dic17—21no—
172#559—173no—
173#557—171no—
174#555—173no—
175#553—173no—
176#552—173no—
177#550—171no—
178#548—171no—
179#548—171no—
180#542—173no—
181#540—173no—
182#529—173no—
183#529—173no—
184#520—171no—
185#519—171no—
186C4×#41D1711yescharacteristic, complemented
187C2×C17—43no—
188C2×C17C22×#4111yescharacteristic
189C2×C17#58511yescharacteristic
190C2×C17#58511yescharacteristic
191D17—41no—
192D17—43no—
193C2×#41—171no—
194C2×#41C2×D1711yescharacteristic
195C22×Q8—171no—
196C4⋊Q8—173no—
197#127—173no—
198#126—176no—
199#126—171no—
200#126—173no—
201#125—173no—
202#125—173no—
203#125—173no—
204#124—173no—
205#124—176no—
206#124—173no—
207#124—173no—
208#123—176no—
209#123—173no—
210#123—173no—
211#123—176no—
212C22⋊Q8—173no—
213C22⋊Q8—173no—
214C22⋊Q8—173no—
215#121—173no—
216#121—173no—
217#121—173no—
218#121—173no—
219#121—173no—
220#121—176no—
221C4×Q8—171no—
222C4×Q8—171no—
223C4×Q8—173no—
224C4×Q8—173no—
225C4×Q8—173no—
226C4×Q8C2×D1711yescharacteristic
227C4×Q8C2×D1711yescharacteristic
228C4×D4—173no—
229C4×D4—173no—
230C4×D4—176no—
231C4×D4—173no—
232C4×D4—173no—
233C4×D4—173no—
234C4×D4C2×D1713yes—
235C4×D4C2×D1713yes—
236#117—173no—
237#117—173no—
238#117—173no—
239#117—176no—
240#117C2×D1713yes—
241C2×C4⋊C4—173no—
242C2×C42C2×D1713yes—
243C17#352411yescharacteristic, complemented, Sylow
244#41—24no—
245#41—344no—
246C2×Q8—173no—
247C2×Q8—173no—
248C2×Q8—344no—
249C2×Q8C22×D1711yescharacteristic
250C2×D4—173no—
251C2×D4—176no—
252C2×D4—173no—
253C2×D4—173no—
254C2×D4—173no—
255C2×D4C22×D1713yes—
256C22×C4—173no—
257C22×C4—173no—
258C22×C4—176no—
259C22×C4C22×D1713yes—
260C22×C4C22×D1713yes—
261C22×C4C22×D1713yes—
262C4⋊C4—173no—
263C4⋊C4—173no—
264C4⋊C4—171no—
265C4⋊C4—173no—
266C4⋊C4—173no—
267C4⋊C4—173no—
268C4⋊C4—173no—
269C4⋊C4—176no—
270C4⋊C4—171no—
271C4⋊C4—173no—
272C4⋊C4—173no—
273C4⋊C4—171no—
274C4⋊C4—173no—
275C4⋊C4C22×D1713yes—
276C4⋊C4C22×D1713yes—
277C22⋊C4—173no—
278C22⋊C4—176no—
279C22⋊C4—173no—
280C22⋊C4—171no—
281C22⋊C4—173no—
282C22⋊C4—173no—
283C22⋊C4—173no—
284C22⋊C4—173no—
285C22⋊C4—176no—
286C22⋊C4—171no—
287C22⋊C4—173no—
288C22⋊C4—171no—
289C22⋊C4C22×D1716yes—
290C42—176no—
291C42—173no—
292C42—173no—
293C42C22×D1711yescharacteristic
294C42C22×D1713yes—
295C42C22×D1713yes—
296C42C22×D1713yes—
297C23—171no—
298C23—173no—
299C23C23×D1713yes—
300Q8—346no—
301Q8—683no—
302Q8#6325214yes—
303D4—26no—
304D4—346no—
305D4—683no—
306D4—683no—
307D4—683no—
308D4—686no—
309C2×C4—26no—
310C2×C4—43no—
311C2×C4—43no—
312C2×C4—171no—
313C2×C4—173no—
314C2×C4—171no—
315C2×C4—173no—
316C2×C4—173no—
317C2×C4—171no—
318C2×C4—346no—
319C2×C4—346no—
320C2×C4—686no—
321C2×C4C23×D1713yes—
322C2×C4C23×D1711yescharacteristic
323C2×C4C23×D1711yescharacteristic
324C2×C4C23×D1711yescharacteristic
325C2×C4C23×D1713yes—
326C2×C4C23×D1713yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—341no—
331C22—343no—
332C22—343no—
333C22—341no—
334C22—343no—
335C22—341no—
336C22C24×D1711yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—23no—
341C4—23no—
342C4—341no—
343C4—343no—
344C4—343no—
345C4—341no—
346C4—343no—
347C4—341no—
348C4C2×#6325216yes—
349C2—43no—
350C2—681no—
351C2—683no—
352C2C22×#6325211yescharacteristic
353C2#1176364511yescharacteristic
354C2#1176364511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC4×#41×C17C2
hypercenterC4×#41D17
hypocenterC17#3524
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #11763542C2semidirect
2. #11763535C2semidirect
3. #11763522C2semidirect
4. #11763509C2semidirect
5. #11763498C2semidirect
6. #11763478C2semidirect
7. #11763471C2semidirect
8. #11763466C2semidirect
9. #11763424C2semidirect
10. #11763410C2semidirect
11. #11763343C2semidirect
12. #11763343C2semidirect
13. #11763291C2semidirect
14. #11763285C2semidirect
15. C4×#41×C17C2semidirect
186. C4×#41D17semidirect
243. C17#3524semidirect

Maximal subgroups

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

subgroupindex
#117635422
#117635352
#117635222
#117635092
#117634982
#117634782
#117634712
#117634662
#117634242
#117634102
#117633432
#117632912
#117632852
C4×#41×C172
#352417