Finite Group Databasebeta

← #14540955

Subgroups — #14540955

Order 2176.

388 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#11763505C211yesmaximal, characteristic
2#11763177C211yesmaximal, characteristic, complemented
3#11763147C211yesmaximal, characteristic, complemented
4#11763142C211yesmaximal, characteristic, complemented
5#11763116C211yesmaximal, characteristic, complemented
6#11763057C211yesmaximal, characteristic, complemented
7#11763040C211yesmaximal, characteristic, complemented
8#11763011C211yesmaximal, characteristic, complemented
9#11762948C211yesmaximal, characteristic, complemented
10#11762948C211yesmaximal, characteristic, complemented
11#11762907C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762818C211yesmaximal, characteristic, complemented
14C2×#10587506C211yesmaximal, characteristic, complemented
15D4×Q8×C17C211yesmaximal, characteristic, complemented
16C22×Q8×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C17×C4⋊Q8—21no—
19C17×C4⋊Q8C2211yescharacteristic
20C17×C22⋊Q8—21no—
21C17×C22⋊Q8—21no—
22C17×C22⋊Q8C2211yescharacteristic
23C17×C22⋊Q8C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×D4×C17—21no—
26C4×D4×C17C2211yescharacteristic
27C2×#63230C2211yescharacteristic
28C2×#63230C2211yescharacteristic
29C2×#63230C2211yescharacteristic
30#10587625—21no—
31#10587616C2211yescharacteristic
32#10587616C2211yescharacteristic
33#10587612C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×#63223C2211yescharacteristic
36C22×#3622C2211yescharacteristic
37#10587584—21no—
38#10587569—21no—
39#10587567—21no—
40C4×D68C2211yescharacteristic
41#10587556C2211yescharacteristic
42#10587506C2211yescharacteristic
43#10587506C2211yescharacteristic
44#10587506C2211yescharacteristic
45#10587506C2211yescharacteristic
46#10587506C2211yescharacteristic
47#10587506C2211yescharacteristic
48#10587506C2211yescharacteristic
49#10587506C2211yescharacteristic
50#10587506C2211yescharacteristic
51C22⋊#3622C2211yescharacteristic
52C22⋊#3622C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587479C2211yescharacteristic
55#10587478C2211yescharacteristic
56#10587478C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58#10587474C2211yescharacteristic
59C2×Q8×C17—21no—
60C2×Q8×C17—21no—
61C2×Q8×C17—22no—
62C2×Q8×C17—22no—
63C2×Q8×C17—41no—
64C2×Q8×C17—41no—
65C2×Q8×C17C2311yescharacteristic
66C2×Q8×C17C2311yescharacteristic
67C2×Q8×C17C2311yescharacteristic
68C2×D4×C17D411yescharacteristic
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17C2311yescharacteristic
72C22×C4×C17C2311yescharacteristic
73C2×C22⋊D17—21no—
74C2×C22⋊D17—21no—
75C22×Dic17—21no—
76C2×D68C2311yescharacteristic
77C2×C4×D17—21no—
78C4⋊C4×C17—21no—
79C4⋊C4×C17—21no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17D411yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C22⋊C4×C17D411yescharacteristic
88C22⋊C4×C17D411yescharacteristic
89C42×C17—21no—
90C42×C17C2311yescharacteristic
91C22⋊Dic17—21no—
92#63230—22no—
93#63230—41no—
94#63230—41no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98C4⋊Dic17—21no—
99C4⋊Dic17C2311yescharacteristic
100C4⋊Dic17C2311yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102#63225—21no—
103#63225—21no—
104C4×Dic17—21no—
105#63223—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17C2×D411yescharacteristic
112C23×C17C2×D411yescharacteristic
113C22×D17—21no—
114Q8×C17—21no—
115Q8×C17—21no—
116Q8×C17—21no—
117Q8×C17—21no—
118Q8×C17—21no—
119Q8×C17—42no—
120Q8×C17—42no—
121Q8×C17#4112yes—
122D4×C17—22no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17C2×D411yescharacteristic
134C2×C4×C17C2×D411yescharacteristic
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2411yescharacteristic
138C22⋊D17—42no—
139C22⋊D17—81no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—81no—
144D68—21no—
145D68—21no—
146C4×D17—81no—
147#3622—21no—
148#3622—21no—
149#3622—21no—
150#3622—21no—
151#3320—171nomaximal, Sylow
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17#13712yes—
158C22×C17C22×D411yescharacteristic
159C2×D17—41no—
160C2×D17—41no—
161C2×D17—41no—
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—22no—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—41no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17C2×#4111yescharacteristic
171C4×C17H(5,𝔽2)11yescharacteristic
172Dic17—41no—
173Dic17—41no—
174Dic17—41no—
175#553—171no—
176D4×Q8D1711yescharacteristic, complemented
177#491—171no—
178#485—171no—
179#484—171no—
180#479—171no—
181#468—171no—
182#465—171no—
183#461—171no—
184#448—171no—
185#448—171no—
186#440—171no—
187#435—171no—
188#419—171no—
189C2×#103—171no—
190C2×C17—22no—
191C2×C17—41no—
192C2×C17#54611yescharacteristic
193C2×C17C2×#13711yescharacteristic
194C2×C17#57711yescharacteristic
195D17—81no—
196C22×Q8C2×D1711yescharacteristic
197C22×Q8C2×D1711yescharacteristic
198C2×SD16—171no—
199C2×SD16—171no—
200C2×SD16—171no—
201C2×M16—171no—
202C22×C8—171no—
203C4⋊Q8—21no—
204C4⋊Q8C2×D1711yescharacteristic
205#126—341no—
206#125—171no—
207#123—341no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8C2×D1711yescharacteristic
211C22⋊Q8C2×D1711yescharacteristic
212#121—171no—
213#121—171no—
214#121—341no—
215C4×Q8C2×D1711yescharacteristic
216C4×D4—21no—
217C4×D4—171no—
218C4×D4—341no—
219C4×D4C2×D1711yescharacteristic
220#117—171no—
221C2×C4⋊C4—171no—
222#107—171no—
223#107—171no—
224#106—171no—
225C4⋊C8—171no—
226#103—171no—
227#103—171no—
228#103—171no—
229#103—171no—
230#103—171no—
231#103—171no—
232#103—171no—
233#103—171no—
234#103—171no—
235#102—171no—
236C22⋊C8—171no—
237C22⋊C8—171no—
238#97—171no—
239C17#332011yescharacteristic, complemented, Sylow
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C22×D1711yescharacteristic
247C2×Q8C22×D1711yescharacteristic
248C2×Q8C22×D1711yescharacteristic
249C2×D4—171no—
250C2×D4—341no—
251C2×D4—341no—
252C2×D4C22⋊D1711yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—341no—
256C22×C4—341no—
257C22×C4C22×D1711yescharacteristic
258C22×C4C22×D1711yescharacteristic
259SD16—342no—
260SD16—681no—
261SD16—681no—
262M16—681no—
263C2×C8—171no—
264C2×C8—171no—
265C2×C8—171no—
266C2×C8—171no—
267C2×C8—681no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—171no—
274C4⋊C4—171no—
275C4⋊C4—171no—
276C4⋊C4—341no—
277C4⋊C4—341no—
278C4⋊C4—341no—
279C4⋊C4C22⋊D1711yescharacteristic
280C4⋊C4C22×D1711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—341no—
284C22⋊C4—341no—
285C22⋊C4—341no—
286C22⋊C4—341no—
287C22⋊C4C22⋊D1711yescharacteristic
288C22⋊C4C22⋊D1711yescharacteristic
289C42—21no—
290C42—341no—
291C42C22×D1711yescharacteristic
292C23—341no—
293C23C2×C22⋊D1711yescharacteristic
294C23C2×C22⋊D1711yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—42no—
301Q8—42no—
302Q8#6325412yes—
303D4—22no—
304D4—341no—
305D4—341no—
306D4—682no—
307D4—1361no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—341no—
319C2×C4—341no—
320C2×C4—341no—
321C2×C4—1361no—
322C2×C4—1361no—
323C2×C4C2×C22⋊D1711yescharacteristic
324C2×C4C2×C22⋊D1711yescharacteristic
325C2×C4C2×C22⋊D1711yescharacteristic
326C2×C4C2×C22⋊D1711yescharacteristic
327C2×C4C23×D1711yescharacteristic
328C8—341no—
329C8—341no—
330C8—341no—
331C8—341no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—681no—
338C22—681no—
339C22—681no—
340C22#1058762812yes—
341C22C22×C22⋊D1711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—681no—
351C4—681no—
352C4—681no—
353C4C2×#6325411yescharacteristic
354C4#1058769111yescharacteristic
355C2—22no—
356C2—41no—
357C2—1361no—
358C2#1176345311yescharacteristic
359C2C2×#1058762811yescharacteristic
360C2#1176361711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingD4×Q8×C17C2
socleC22×C17C22×D4
prime radical#11763505C2
hypercenterD4×Q8D17
hypocenterC17#3320
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. #11763177C2semidirect
3. #11763147C2semidirect
4. #11763142C2semidirect
5. #11763116C2semidirect
6. #11763057C2semidirect
7. #11763040C2semidirect
8. #11763011C2semidirect
9. #11762948C2semidirect
10. #11762948C2semidirect
11. #11762907C2semidirect
12. #11762881C2semidirect
13. #11762818C2semidirect
14. C2×#10587506C2semidirect
15. D4×Q8×C17C2semidirect
176. D4×Q8D17semidirect
239. C17#3320semidirect

Maximal subgroups

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

subgroupindex
#117635052
#117631772
#117631472
#117631422
#117631162
#117630572
#117630402
#117630112
#117629482
#117629072
#117628812
#117628182
C2×#105875062
D4×Q8×C172
#332017