Finite Group Databasebeta

← #14532604

Subgroups — #14532604

Order 2176.

584 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587567C211yesmaximal, characteristic
2#11762747C211yesmaximal, characteristic, complemented
3#11762742C211yesmaximal, characteristic, complemented
4#11762731C211yesmaximal, characteristic, complemented
5#11762720C211yesmaximal, characteristic, complemented
6#11762716C211yesmaximal, characteristic, complemented
7#11762690C211yesmaximal, characteristic, complemented
8#11762688C211yesmaximal, characteristic
9#11762686C211yesmaximal, characteristic
10#11762684C211yesmaximal, characteristic, complemented
11#11762681C211yesmaximal, characteristic, complemented
12#11762680C211yesmaximal, characteristic, complemented
13#11762666C211yesmaximal, characteristic, complemented
14C2×#10587502C211yesmaximal, characteristic, complemented
15C2×C17×C22⋊Q8C211yesmaximal, characteristic, complemented
16C22×Q8×C17C2211yescharacteristic
17C23×C4×C17C2211yescharacteristic
18C22×C22⋊D17C2211yescharacteristic
19C23×Dic17C2211yescharacteristic
20C23×Dic17C2211yescharacteristic
21C22×C4×D17C2211yescharacteristic
22C17×C22⋊Q8—24no—
23C2×C4⋊C4×C17C2211yescharacteristic
24C2×C4⋊C4×C17C2211yescharacteristic
25C2×C4⋊C4×C17C2211yescharacteristic
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C22⋊C4×C17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×C4⋊Dic17C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×#63225C2211yescharacteristic
39C2×C4×Dic17C2211yescharacteristic
40#10587567—24no—
41#10587502C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C24×C17C2311yescharacteristic
54C23×D17C2311yescharacteristic
55C2×Q8×C17—22no—
56C2×Q8×C17—22no—
57C2×Q8×C17—41no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—41no—
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C2×C22⋊D17—22no—
68C2×C22⋊D17—22no—
69C2×C22⋊D17Q814yescomplemented
70C22×Dic17—22no—
71C22×Dic17—22no—
72C22×Dic17—22no—
73C22×Dic17—41no—
74C22×Dic17—41no—
75C22×Dic17—41no—
76C22×Dic17C2311yescharacteristic
77C22×Dic17C2311yescharacteristic
78C22×Dic17C2311yescharacteristic
79C22×Dic17C2311yescharacteristic
80C22×Dic17C2311yescharacteristic
81C22×Dic17C2311yescharacteristic
82C22×Dic17C2311yescharacteristic
83C2×C4×D17—22no—
84C2×C4×D17—22no—
85C2×C4×D17—41no—
86C4⋊C4×C17—22no—
87C4⋊C4×C17—22no—
88C4⋊C4×C17—22no—
89C22⋊C4×C17—22no—
90C22⋊C4×C17D414yes—
91C22⋊Dic17—22no—
92#63227—22no—
93#63227—22no—
94#63227—41no—
95#63227—41no—
96#63227—41no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99#63225—22no—
100#63225—22no—
101#63225—41no—
102C4×Dic17—22no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17C2×D412yes—
108C23×C17C2×Q812yes—
109C23×C17#4112yes—
110C23×C17C2411yescharacteristic
111C22×D17—21no—
112C22×D17—21no—
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17C2×Q812yes—
118Q8×C17—44no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
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—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—44no—
135C2×C4×C17C2×D412yes—
136C2×C4×C17C2×D412yes—
137C2×C4×C17#4112yes—
138C22⋊D17—28no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—21no—
157C2×Dic17—21no—
158C2×Dic17—44no—
159C2×Dic17—44no—
160C2×Dic17C2×Q812yes—
161C2×Dic17#4112yes—
162C4×D17—44no—
163#2450—171nomaximal, Sylow
164C22×C17—22no—
165C22×C17—22no—
166C22×C17—22no—
167C22×C17—22no—
168C22×C17—22no—
169C22×C17—22no—
170C22×C17C22⋊Q814yes—
171C22×C17C22×D411yescharacteristic
172C22×C17C22×Q811yescharacteristic
173C22×C17C2×#4111yescharacteristic
174C22×C17C2×#4111yescharacteristic
175C22×C17C2×#4111yescharacteristic
176C22×C17H(5,𝔽2)11yescharacteristic
177C22×C17H(5,𝔽2)11yescharacteristic
178C2×D17—22no—
179C2×D17—22no—
180C2×D17—22no—
181C2×D17—41no—
182C2×D17—41no—
183C2×D17—41no—
184C2×D17—41no—
185C4×C17—22no—
186C4×C17—22no—
187C4×C17—22no—
188C4×C17—41no—
189C4×C17—41no—
190C4×C17—41no—
191Dic17—22no—
192Dic17—22no—
193Dic17—41no—
194Dic17—41no—
195Dic17—41no—
196Dic17—41no—
197Dic17—41no—
198C2×C22⋊Q8D1711yescharacteristic, complemented
199C2×C4×D4—171no—
200#400—171no—
201#398—171no—
202#396—171no—
203#393—171no—
204#391—171no—
205#387—171no—
206#386—171no—
207#386—171no—
208#385—171no—
209#385—171no—
210#385—171no—
211#380—171no—
212C2×#95—171no—
213C2×C17—24no—
214C2×C17C2×C22⋊Q811yescharacteristic
215C2×C17#52911yescharacteristic
216C2×C17#53811yescharacteristic
217C2×C17#54611yescharacteristic
218C2×C17#54611yescharacteristic
219C2×C17#55411yescharacteristic
220C2×C17#55411yescharacteristic
221D17—42no—
222C22×Q8C2×D1711yescharacteristic
223C22×D4—171no—
224C23×C4—171no—
225C23×C4—171no—
226C23×C4—171no—
227C23×C4C2×D1711yescharacteristic
228C22⋊Q8—24no—
229C4×D4—344no—
230C2×C4⋊C4—171no—
231C2×C4⋊C4—171no—
232C2×C4⋊C4—171no—
233C2×C4⋊C4—171no—
234C2×C4⋊C4—171no—
235C2×C4⋊C4C2×D1711yescharacteristic
236C2×C4⋊C4C2×D1711yescharacteristic
237C2×C4⋊C4C2×D1711yescharacteristic
238C2×C22⋊C4—171no—
239C2×C22⋊C4—171no—
240C2×C22⋊C4—171no—
241C2×C22⋊C4—171no—
242C2×C22⋊C4—171no—
243C2×C22⋊C4—171no—
244C2×C22⋊C4C2×D1711yescharacteristic
245C2×C22⋊C4C2×D1711yescharacteristic
246C2×C42—171no—
247#95—171no—
248#95—171no—
249#95—171no—
250#95—171no—
251#95—171no—
252#95—171no—
253#95—171no—
254#95—171no—
255#95—171no—
256#95—171no—
257#95—171no—
258#95—171no—
259C17#245011yescharacteristic, complemented, Sylow
260C24—171no—
261C24C22×D1711yescharacteristic
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×D4—174no—
266C2×D4—342no—
267C2×D4—342no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—171no—
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—171no—
276C22×C4—171no—
277C22×C4—171no—
278C22×C4—342no—
279C22×C4—342no—
280C22×C4—342no—
281C22×C4—342no—
282C22×C4—342no—
283C22×C4—681no—
284C22×C4—681no—
285C22×C4—681no—
286C22×C4—681no—
287C22×C4C22×D1711yescharacteristic
288C22×C4C22×D1711yescharacteristic
289C22×C4C22×D1711yescharacteristic
290C22×C4C22×D1711yescharacteristic
291C22×C4C22×D1711yescharacteristic
292C22×C4C22×D1711yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—342no—
297C4⋊C4—342no—
298C4⋊C4—681no—
299C4⋊C4—681no—
300C4⋊C4—681no—
301C22⋊C4—22no—
302C22⋊C4—342no—
303C22⋊C4—342no—
304C22⋊C4—342no—
305C22⋊C4—681no—
306C22⋊C4—681no—
307C22⋊C4—681no—
308C22⋊C4C22⋊D1714yes—
309C42—342no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—172no—
315C23—341no—
316C23—341no—
317C23—341no—
318C23—341no—
319C23—341no—
320C23—341no—
321C23Q8×D1712yes—
322C23#6325612yes—
323C23C2×C22⋊D1712yes—
324C23C23×D1711yescharacteristic
325Q8—44no—
326D4—348no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—172no—
344C2×C4—172no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—341no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—341no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—341no—
361C2×C4—341no—
362C2×C4—341no—
363C2×C4—341no—
364C2×C4—684no—
365C2×C4—684no—
366C2×C4—684no—
367C2×C4#6325412yes—
368C2×C4C2×C22⋊D1712yes—
369C2×C4C2×C22⋊D1712yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—342no—
377C22—342no—
378C22—342no—
379C22—681no—
380C22—681no—
381C22—681no—
382C22—681no—
383C22#1058763214yes—
384C22C2×#6325411yescharacteristic
385C22#1058769111yescharacteristic
386C22C2×Q8×D1711yescharacteristic
387C22C2×#6325611yescharacteristic
388C22#41×D1711yescharacteristic
389C22#1058769611yescharacteristic
390C22C22×C22⋊D1711yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—342no—
398C4—342no—
399C4—681no—
400C4—681no—
401C4—681no—
402C4—681no—
403C4—681no—
404C2—24no—
405C2—682no—
406C2C2×#1058763211yescharacteristic
407C2#1176334511yescharacteristic
408C2#1176339911yescharacteristic
409C2#1176344711yescharacteristic
410C2#1176345311yescharacteristic
411C2#1176351311yescharacteristic
412C2#1176351711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×C22⋊Q8C2
prime radicalC22×C22⋊D17C22
hypercenterC2×C22⋊Q8D17
hypocenterC17#2450
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11762747C2semidirect
3. #11762742C2semidirect
4. #11762731C2semidirect
5. #11762720C2semidirect
6. #11762716C2semidirect
7. #11762690C2semidirect
10. #11762684C2semidirect
11. #11762681C2semidirect
12. #11762680C2semidirect
13. #11762666C2semidirect
14. C2×#10587502C2semidirect
15. C2×C17×C22⋊Q8C2semidirect
69. C2×C22⋊D17Q8semidirect
198. C2×C22⋊Q8D17semidirect
259. C17#2450semidirect

Maximal subgroups

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

subgroupindex
C2×#105875672
#117627472
#117627422
#117627312
#117627202
#117627162
#117626902
#117626882
#117626862
#117626842
#117626812
#117626802
#117626662
C2×#105875022
C2×C17×C22⋊Q82
#245017