Finite Group Databasebeta

← #14542600

Subgroups — #14542600

Order 2176.

1060 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763539C212yesmaximal, complemented
2#11763534C212yesmaximal, complemented
3#11763520C212yesmaximal, complemented
4#11763514C214yesmaximal, complemented
5#11763474C212yesmaximal, complemented
6Q8×C22⋊D17C214yesmaximal, complemented
7#11763440C212yesmaximal, complemented
8#11763436C214yesmaximal, complemented
9#11763366C211yesmaximal, characteristic, complemented
10#11763364C212yesmaximal, complemented
11C4⋊Q8×D17C212yesmaximal, complemented
12#11763290C212yesmaximal
13C4×#63252C211yesmaximal, characteristic
14C2×C17×C4⋊Q8C211yesmaximal, characteristic, complemented
15C22×Q8×C17C2212yes—
16C2×Q8×D17C2214yes—
17C2×#63252C2211yescharacteristic
18C17×C4⋊Q8C2212yes—
19C17×C4⋊Q8C2212yes—
20C17×C4⋊Q8C2212yes—
21C17×C4⋊Q8C2212yes—
22C2×C4⋊C4×C17C2212yes—
23C2×C4⋊C4×C17C2212yes—
24C2×C42×C17C2211yescharacteristic
25#10587632C2214yes—
26#10587632C2218yes—
27Q8×Dic17C2214yes—
28#10587630C2214yes—
29#10587630C2218yes—
30C4×C22⋊D17C2212yes—
31C4×C22⋊D17C2218yes—
32C4×C22⋊D17C2212yes—
33#10587612C2211yescharacteristic
34#10587612C2212yes—
35#10587610C2212yes—
36#10587610C2218yes—
37#10587610C2212yes—
38#10587583C2214yes—
39#10587582C2218yes—
40#10587579C2214yes—
41#10587578C2214yes—
42C4⋊C4×D17C2214yes—
43C4⋊C4×D17C2214yes—
44#10587576C2214yes—
45#10587576C2214yes—
46#10587575C2218yes—
47#10587574C2214yes—
48#10587574C2214yes—
49#10587574C2214yes—
50#10587573C2214yes—
51#10587573C2214yes—
52C4×D68C2211yescharacteristic
53C4×D68C2211yescharacteristic
54#10587558C2212yes—
55C42×D17C2212yes—
56#10587556C2212yes—
57#10587555C2211yescharacteristic
58#10587555C2211yescharacteristic
59C4×Dic34C2212yes—
60C4×Dic34C2212yes—
61C4×Dic34C2211yescharacteristic
62C4×Dic34C2211yescharacteristic
63C2×Q8×C17—44no—
64C2×Q8×C17C2314yes—
65C2×Q8×C17C2314yes—
66C22×C4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C22×C4×C17C2314yes—
69C22×C4×C17C2311yescharacteristic
70C2×C22⋊D17C2312yes—
71Q8×D17—48no—
72#63252Q818yescomplemented
73C2×D68C2311yescharacteristic
74C2×C4×D17C2312yes—
75C2×C4×D17C2312yes—
76C2×C4×D17C2318yes—
77C2×C4×D17C2312yes—
78C2×Dic34C2312yes—
79C2×Dic34C2312yes—
80C2×Dic34C2318yes—
81C2×Dic34C2311yescharacteristic
82C4⋊C4×C17C2314yes—
83C4⋊C4×C17C2314yes—
84C4⋊C4×C17C2314yes—
85C4⋊C4×C17C2314yes—
86C42×C17C2311yescharacteristic
87C42×C17C2311yescharacteristic
88C42×C17C2311yescharacteristic
89C42×C17C2311yescharacteristic
90C22⋊Dic17C2312yes—
91C22⋊Dic17C2314yes—
92#63227C2312yes—
93#63227C2312yes—
94#63227C2318yes—
95C4⋊Dic17C2312yes—
96C4⋊Dic17C2312yes—
97C4⋊Dic17C2311yescharacteristic
98C4⋊Dic17C2311yescharacteristic
99C4⋊Dic17C2312yes—
100C4⋊Dic17C2312yes—
101C4⋊Dic17C2318yes—
102#63225C2314yes—
103#63225C2314yes—
104#63225C2312yes—
105#63225C2312yes—
106#63225C2318yes—
107#63225C2318yes—
108#63225C2318yes—
109C4×Dic17C2314yes—
110C4×Dic17C2312yes—
111C4×Dic17C2312yes—
112C4×Dic17C2312yes—
113C4×Dic17C2318yes—
114C23×C17C2411yescharacteristic
115C22×D17C2412yes—
116Q8×C17—28no—
117Q8×C17—44no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—44no—
121C2×C4×C17C2×Q814yescomplemented
122C2×C4×C17C2411yescharacteristic
123C2×C4×C17C2411yescharacteristic
124C2×C4×C17C2418yes—
125C2×C4×C17C2411yescharacteristic
126C2×C4×C17C2411yescharacteristic
127C2×C4×C17C2411yescharacteristic
128C2×C4×C17C2411yescharacteristic
129C22⋊D17C2×Q818yescomplemented
130C2×Dic17C2412yes—
131C2×Dic17C2414yes—
132C2×Dic17C2414yes—
133C2×Dic17C2412yes—
134C2×Dic17C2412yes—
135D68C2×Q814yescomplemented
136C4×D17—24no—
137C4×D17—24no—
138C4×D17—48no—
139C4×D17C2×Q818yescomplemented
140Dic34—24no—
141Dic34—24no—
142Dic34—48no—
143Dic34C2×Q814yescomplemented
144#3503—171nomaximal, Sylow
145C22×C17—21no—
146C22×C17—21no—
147C22×C17C22×Q812yes—
148C22×C17C2511yescharacteristic
149C2×D17—22no—
150C2×D17—22no—
151C2×D17C22×Q814yes—
152C4×C17—28no—
153C4×C17—21no—
154C4×C17—21no—
155C4×C17C22×Q812yes—
156C4×C17C22×Q812yes—
157C4×C17H(5,𝔽2)12yes—
158C4×C172−1+412yes—
159Dic17—24no—
160Dic17—22no—
161Dic17—24no—
162Dic17—22no—
163Dic17C22×Q814yes—
164Q82—172no—
165#557—172no—
166#554—172no—
167#554—174no—
168D4×Q8—174no—
169D4×Q8—172no—
170#544—174no—
171#544—172no—
172#533—172no—
173#533—171no—
174C2×C4⋊Q8—172no—
175C2×C4⋊Q8D1711yescharacteristic, complemented
176#520—172no—
177C4×#41—171no—
178C2×C17—22no—
179C2×C17C23×Q811yescharacteristic
180C2×C17C2×H(5,𝔽2)11yescharacteristic
181C2×C17C2×2−1+411yescharacteristic
182D17—24no—
183C2×#41—171no—
184C22×Q8—174no—
185C22×Q8C2×D1712yes—
186C4⋊Q8—174no—
187C4⋊Q8—178no—
188C4⋊Q8—174no—
189C4⋊Q8—174no—
190C4⋊Q8—174no—
191C4⋊Q8—171no—
192C4⋊Q8—171no—
193C4⋊Q8C2×D1712yes—
194C4⋊Q8C2×D1712yes—
195C4⋊Q8C2×D1712yes—
196C4⋊Q8C2×D1712yes—
197#125—174no—
198#125—178no—
199#125—174no—
200#125—172no—
201C22⋊Q8—174no—
202C22⋊Q8—178no—
203C22⋊Q8—172no—
204C22⋊Q8—172no—
205C22⋊Q8—178no—
206C22⋊Q8—174no—
207C22⋊Q8—178no—
208C4×Q8—174no—
209C4×Q8—174no—
210C4×Q8—174no—
211C4×Q8—171no—
212C4×Q8—171no—
213C4×Q8—172no—
214C4×Q8—172no—
215C4×D4—172no—
216C4×D4—172no—
217C4×D4—178no—
218C4×D4—174no—
219C4×D4—171no—
220C4×D4—171no—
221#117—171no—
222#117—172no—
223#117—174no—
224#117—172no—
225C2×C4⋊C4—174no—
226C2×C4⋊C4—174no—
227C2×C4⋊C4C2×D1712yes—
228C2×C4⋊C4C2×D1712yes—
229C2×C42—172no—
230C2×C42C2×D1711yescharacteristic
231C17#350311yescharacteristic, complemented, Sylow
232#41—178no—
233C2×Q8—44no—
234C2×Q8—171no—
235C2×Q8—172no—
236C2×Q8—172no—
237C2×Q8—178no—
238C2×Q8—688no—
239C2×Q8C22×D1714yes—
240C2×Q8C22×D1714yes—
241C2×D4—172no—
242C2×D4—171no—
243C22×C4—172no—
244C22×C4—172no—
245C22×C4—172no—
246C22×C4—178no—
247C22×C4C22×D1711yescharacteristic
248C22×C4C22×D1711yescharacteristic
249C22×C4C22×D1714yes—
250C22×C4C22×D1711yescharacteristic
251C4⋊C4—172no—
252C4⋊C4—172no—
253C4⋊C4—178no—
254C4⋊C4—178no—
255C4⋊C4—172no—
256C4⋊C4—172no—
257C4⋊C4—171no—
258C4⋊C4—172no—
259C4⋊C4—171no—
260C4⋊C4—172no—
261C4⋊C4—174no—
262C4⋊C4—178no—
263C4⋊C4—178no—
264C4⋊C4—174no—
265C4⋊C4C22×D1714yes—
266C4⋊C4C22×D1714yes—
267C4⋊C4C22×D1714yes—
268C4⋊C4C22×D1714yes—
269C22⋊C4—174no—
270C22⋊C4—172no—
271C22⋊C4—178no—
272C22⋊C4—172no—
273C22⋊C4—172no—
274C42—172no—
275C42—172no—
276C42—172no—
277C42—178no—
278C42—174no—
279C42C22×D1711yescharacteristic
280C42C22×D1711yescharacteristic
281C42C22×D1711yescharacteristic
282C42C22×D1711yescharacteristic
283C23—172no—
284C23C23×D1711yescharacteristic
285Q8—28no—
286Q8—44no—
287Q8—174no—
288Q8—344no—
289Q8—344no—
290Q8—688no—
291D4—178no—
292D4—174no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—44no—
296C2×C4—172no—
297C2×C4—178no—
298C2×C4—174no—
299C2×C4—172no—
300C2×C4—172no—
301C2×C4—174no—
302C2×C4—344no—
303C2×C4—344no—
304C2×C4—688no—
305C2×C4Q8×D1714yescomplemented
306C2×C4C23×D1718yes—
307C2×C4C23×D1711yescharacteristic
308C2×C4C23×D1711yescharacteristic
309C2×C4C23×D1711yescharacteristic
310C2×C4C23×D1711yescharacteristic
311C2×C4C23×D1711yescharacteristic
312C2×C4C23×D1711yescharacteristic
313C22—21no—
314C22—21no—
315C22—174no—
316C22—342no—
317C22—342no—
318C22C2×Q8×D1712yes—
319C22C24×D1711yescharacteristic
320C4—21no—
321C4—28no—
322C4—21no—
323C4—174no—
324C4—342no—
325C4—344no—
326C4—342no—
327C4—344no—
328C4#1058769112yes—
329C4C2×Q8×D1712yes—
330C4C2×Q8×D1712yes—
331C4#1058769412yes—
332C2—22no—
333C2—344no—
334C2C22×Q8×D1711yescharacteristic
335C2C2×#1058769111yescharacteristic
336C2C2×#1058769411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC2×C17×C4⋊Q8C2
prime radicalC2×#63252C22
hypercenterC2×C4⋊Q8D17
hypocenterC17#3503
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #11763539C2semidirect
2. #11763534C2semidirect
3. #11763520C2semidirect
4. #11763514C2semidirect
5. #11763474C2semidirect
6. Q8×C22⋊D17C2semidirect
7. #11763440C2semidirect
8. #11763436C2semidirect
9. #11763366C2semidirect
10. #11763364C2semidirect
11. C4⋊Q8×D17C2semidirect
14. C2×C17×C4⋊Q8C2semidirect
72. #63252Q8semidirect
121. C2×C4×C17C2×Q8semidirect
129. C22⋊D17C2×Q8semidirect
135. D68C2×Q8semidirect
139. C4×D17C2×Q8semidirect
143. Dic34C2×Q8semidirect
175. C2×C4⋊Q8D17semidirect
231. C17#3503semidirect
305. C2×C4Q8×D17semidirect

Maximal subgroups

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

subgroupindex
#117635392
#117635342
#117635202
#117635142
#117634742
Q8×C22⋊D172
#117634402
#117634362
#117633662
#117633642
C4⋊Q8×D172
#117632902
C4×#632522
C2×C17×C4⋊Q82
#350317