Finite Group Databasebeta

← #14542766

Subgroups — #14542766

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#11763538C211yesmaximal, characteristic, complemented
2#11763528C213yesmaximal, complemented
3#11763516C213yesmaximal, complemented
4#11763511C213yesmaximal, complemented
5#11763501C213yesmaximal, complemented
6Q8×C22⋊D17C211yesmaximal, characteristic, complemented
7#11763466C211yesmaximal, characteristic, complemented
8#11763428C213yesmaximal, complemented
9#11763419C213yesmaximal, complemented
10#11763340C213yesmaximal, complemented
11#11763339C213yesmaximal, complemented
12#11763292C211yesmaximal, characteristic, complemented
13#11763285C211yesmaximal, characteristic, complemented
14C2×Q8×Dic17C211yesmaximal, characteristic, complemented
15C17×#548C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C2×Q8×D17C2211yescharacteristic
19C17×#124C2213yes—
20C17×C22⋊Q8C2213yes—
21C17×#121C2213yes—
22C4×Q8×C17C2211yescharacteristic
23C4×D4×C17C2213yes—
24#10587632C2213yes—
25Q8×Dic17C2211yescharacteristic
26Q8×Dic17C2211yescharacteristic
27Q8×Dic17C2213yes—
28Q8×Dic17C2211yescharacteristic
29Q8×Dic17C2213yes—
30#10587630C2213yes—
31#10587625C2213yes—
32#10587624C2213yes—
33#10587622C2213yes—
34#10587621C2213yes—
35D4×Dic17C2216yes—
36D4×Dic17C2213yes—
37D4×Dic17C2213yes—
38C4×C22⋊D17C2213yes—
39C4×C22⋊D17C2213yes—
40#10587612C2213yes—
41C2×C4⋊Dic17C2213yes—
42C2×C4⋊Dic17C2213yes—
43#10587610C2213yes—
44#10587610C2213yes—
45C2×C4×Dic17C2213yes—
46#10587584C2213yes—
47#10587578C2213yes—
48#10587576C2216yes—
49#10587576C2213yes—
50#10587576C2213yes—
51#10587575C2213yes—
52#10587573C2213yes—
53#10587572C2216yes—
54#10587572C2211yescharacteristic
55#10587571C2213yes—
56#10587565C2216yes—
57#10587565C2213yes—
58#10587565C2216yes—
59#10587564C2213yes—
60#10587564C2216yes—
61#10587563C2216yes—
62#10587558C2213yes—
63#10587555C2213yes—
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⋊D17C2311yescharacteristic
75C2×C22⋊D17C2313yes—
76C22×Dic17C2311yescharacteristic
77C22×Dic17C2316yes—
78C22×Dic17C2313yes—
79Q8×D17—24no—
80C2×C4×D17C2313yes—
81C2×Dic34C2313yes—
82C2×Dic34C2313yes—
83C4⋊C4×C17C2313yes—
84C4⋊C4×C17C2313yes—
85C22⋊C4×C17C2316yes—
86C22⋊C4×C17C2313yes—
87C22⋊C4×C17C2311yescharacteristic
88C42×C17C2313yes—
89C22⋊Dic17C2313yes—
90C22⋊Dic17C2316yes—
91C22⋊Dic17C2313yes—
92C22⋊Dic17C2313yes—
93C22⋊Dic17C2313yes—
94#63227C2311yescharacteristic
95#63227C2313yes—
96#63227C2313yes—
97#63227C2311yescharacteristic
98C4⋊Dic17C2311yescharacteristic
99C4⋊Dic17C2313yes—
100C4⋊Dic17C2316yes—
101C4⋊Dic17C2313yes—
102C4⋊Dic17C2313yes—
103C4⋊Dic17C2316yes—
104C4⋊Dic17C2313yes—
105C4⋊Dic17C2311yescharacteristic
106#63225C2313yes—
107#63225C2313yes—
108#63225C2313yes—
109#63225C2313yes—
110#63225C2316yes—
111#63225C2313yes—
112#63225C2313yes—
113C4×Dic17C2316yes—
114C4×Dic17C2313yes—
115C4×Dic17C2313yes—
116C4×Dic17C2313yes—
117C4×Dic17C2313yes—
118C4×Dic17C2313yes—
119C23×C17C2413yes—
120C23×C17C2411yescharacteristic
121C22×D17C2411yescharacteristic
122Q8×C17—26no—
123Q8×C17#4114yes—
124D4×C17—26no—
125D4×C17—26no—
126C2×C4×C17—26no—
127C2×C4×C17—26no—
128C2×C4×C17C2411yescharacteristic
129C2×C4×C17C2413yes—
130C2×C4×C17C2413yes—
131C2×C4×C17C2411yescharacteristic
132C2×C4×C17C2413yes—
133C22⋊D17—22no—
134C22⋊D17—43no—
135C2×Dic17—22no—
136C2×Dic17—46no—
137C2×Dic17—43no—
138C2×Dic17C2413yes—
139C2×Dic17C2411yescharacteristic
140C2×Dic17C2411yescharacteristic
141C2×Dic17C2411yescharacteristic
142C2×Dic17C2413yes—
143C2×Dic17C2413yes—
144C2×Dic17C2413yes—
145C4×D17—26no—
146Dic34—26no—
147Dic34—43no—
148#3516—171nomaximal, Sylow
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—23no—
152C22×C17—23no—
153C22×C17—23no—
154C22×C172−1+412yes—
155C22×C17C2511yescharacteristic
156C2×D17—21no—
157C2×D17—21no—
158C2×D17—21no—
159C4×C17—21no—
160C4×C17—23no—
161C4×C17—23no—
162C4×C17—21no—
163C4×C17C2×#4116yes—
164Dic17—23no—
165Dic17—21no—
166Dic17—21no—
167Dic17—21no—
168Dic17—23no—
169Dic17—23no—
170Dic17—23no—
171#557—171no—
172#556—173no—
173#554—173no—
174#553—173no—
175#552—173no—
176D4×Q8—171no—
177#548—171no—
178#548D1711yescharacteristic, complemented
179#542—173no—
180#541—173no—
181#529—173no—
182#529—173no—
183#520—171no—
184#519—171no—
185C2×C4×Q8—171no—
186C2×C17—22no—
187C2×C17—43no—
188C2×C17C22×#4111yescharacteristic
189C2×C17C2×2−1+411yescharacteristic
190C2×C17#58511yescharacteristic
191D17—41no—
192C2×#41C2×D1711yescharacteristic
193C22×Q8—171no—
194C22×Q8C2×D1711yescharacteristic
195C4⋊Q8—173no—
196C4⋊Q8—173no—
197#126—173no—
198#126—176no—
199#126—173no—
200#126—176no—
201#125—173no—
202#125—173no—
203#125—176no—
204#125—173no—
205#124—173no—
206#124—173no—
207#124C2×D1713yes—
208#123—173no—
209#123—171no—
210#123—176no—
211C22⋊Q8—173no—
212C22⋊Q8—173no—
213C22⋊Q8—173no—
214C22⋊Q8—173no—
215C22⋊Q8—176no—
216C22⋊Q8C2×D1713yes—
217#121—173no—
218#121—173no—
219#121C2×D1713yes—
220C4×Q8—171no—
221C4×Q8—173no—
222C4×Q8—171no—
223C4×Q8—173no—
224C4×Q8—171no—
225C4×Q8—173no—
226C4×Q8—173no—
227C4×Q8—173no—
228C4×Q8C2×D1711yescharacteristic
229C4×D4—173no—
230C4×D4—173no—
231C4×D4—176no—
232C4×D4—173no—
233C4×D4—173no—
234C4×D4C2×D1713yes—
235#117—173no—
236#117—173no—
237#117—176no—
238#117—173no—
239C2×C4⋊C4—173no—
240C2×C4⋊C4—173no—
241C2×C42—173no—
242C17#351611yescharacteristic, complemented, Sylow
243#41—24no—
244C2×Q8—24no—
245C2×Q8—173no—
246C2×Q8—173no—
247C2×Q8—344no—
248C2×Q8C22×D1713yes—
249C2×Q8C22×D1711yescharacteristic
250C2×D4—171no—
251C2×D4—173no—
252C2×D4C22×D1713yes—
253C2×D4C22×D1713yes—
254C22×C4—171no—
255C22×C4—173no—
256C22×C4—176no—
257C22×C4—173no—
258C22×C4C22×D1713yes—
259C22×C4C22×D1713yes—
260C4⋊C4—173no—
261C4⋊C4—173no—
262C4⋊C4—173no—
263C4⋊C4—173no—
264C4⋊C4—176no—
265C4⋊C4—171no—
266C4⋊C4—173no—
267C4⋊C4—173no—
268C4⋊C4—173no—
269C4⋊C4—176no—
270C4⋊C4—173no—
271C4⋊C4—176no—
272C4⋊C4—173no—
273C4⋊C4—171no—
274C4⋊C4—173no—
275C4⋊C4C22×D1713yes—
276C4⋊C4C22×D1713yes—
277C22⋊C4—173no—
278C22⋊C4—173no—
279C22⋊C4—173no—
280C22⋊C4—173no—
281C22⋊C4—176no—
282C22⋊C4—173no—
283C22⋊C4—171no—
284C22⋊C4—171no—
285C22⋊C4—173no—
286C22⋊C4C22×D1716yes—
287C22⋊C4C22×D1711yescharacteristic
288C22⋊C4C22×D1713yes—
289C42—173no—
290C42—173no—
291C42—173no—
292C42—173no—
293C42—176no—
294C42—173no—
295C42C22×D1713yes—
296C23—171no—
297C23C23×D1713yes—
298C23C23×D1711yescharacteristic
299Q8—26no—
300Q8—346no—
301Q8—683no—
302Q8#6325414yes—
303D4—26no—
304D4—26no—
305D4—342no—
306D4—683no—
307C2×C4—26no—
308C2×C4—26no—
309C2×C4—171no—
310C2×C4—171no—
311C2×C4—173no—
312C2×C4—171no—
313C2×C4—173no—
314C2×C4—173no—
315C2×C4—173no—
316C2×C4—342no—
317C2×C4—346no—
318C2×C4—686no—
319C2×C4—683no—
320C2×C4C23×D1713yes—
321C2×C4C23×D1713yes—
322C2×C4C23×D1711yescharacteristic
323C2×C4C23×D1711yescharacteristic
324C2×C4C23×D1713yes—
325C22—21no—
326C22—21no—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—341no—
331C22—341no—
332C22—341no—
333C22#1058769712yes—
334C22C24×D1711yescharacteristic
335C4—21no—
336C4—23no—
337C4—23no—
338C4—21no—
339C4—343no—
340C4—341no—
341C4—341no—
342C4—341no—
343C4—343no—
344C4—343no—
345C4—343no—
346C4C2×#6325416yes—
347C2—22no—
348C2—43no—
349C2—681no—
350C2C22×#6325411yescharacteristic
351C2C2×#1058769711yescharacteristic
352C2#1176364611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#548C2
hypercenter#548D17
hypocenterC17#3516
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #11763538C2semidirect
2. #11763528C2semidirect
3. #11763516C2semidirect
4. #11763511C2semidirect
5. #11763501C2semidirect
6. Q8×C22⋊D17C2semidirect
7. #11763466C2semidirect
8. #11763428C2semidirect
9. #11763419C2semidirect
10. #11763340C2semidirect
11. #11763339C2semidirect
12. #11763292C2semidirect
13. #11763285C2semidirect
14. C2×Q8×Dic17C2semidirect
15. C17×#548C2semidirect
178. #548D17semidirect
242. C17#3516semidirect

Maximal subgroups

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

subgroupindex
#117635382
#117635282
#117635162
#117635112
#117635012
Q8×C22⋊D172
#117634662
#117634282
#117634192
#117633402
#117633392
#117632922
#117632852
C2×Q8×Dic172
C17×#5482
#351617