Finite Group Databasebeta

← Q16×#41×C47 #26101127

Subgroups — Q16×#41×C47 #26101127

Order 12032.

1826 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#41×C47C212yesmaximal
2C47×#3395C211yesmaximal, characteristic, complemented
3Q8×Q16×C47C211yesmaximal, characteristic, complemented
4C47×#3363C212yesmaximal, complemented
5C47×#3338C213yesmaximal, complemented
6C47×#3299C216yesmaximal, complemented
7D4×Q16×C47C213yesmaximal, complemented
8C47×#3170C213yesmaximal, complemented
9C47×#3104C216yesmaximal, complemented
10C8×#41×C47C211yesmaximal, characteristic
11C2×C4×Q16×C47C213yesmaximal, complemented
12C22×Q16×C47C2213yes—
13Q82×C47C2212yes—
14C47×#557—22no—
15C47×#557C2212yes—
16C47×#554—26no—
17C47×#554C2216yes—
18D4×Q8×C47C2216yes—
19C47×#533—26no—
20C47×#533C2216yes—
21C4×#41×C47—22no—
22C4×#41×C47C2211yescharacteristic
23C2×C4×Q8×C47C2216yes—
24C47×#500C2213yes—
25C47×#494C2213yes—
26C47×#487C2216yes—
27C47×#484C22112yes—
28C47×#477C2216yes—
29C47×#477C2216yes—
30C47×#467C2216yes—
31C47×#467C2216yes—
32C47×#462C2216yes—
33C47×#462C2216yes—
34C47×#451C22112yes—
35C8×Q8×C47C2211yescharacteristic
36C8×Q8×C47C2211yescharacteristic
37C4×Q16×C47C2213yes—
38C4×Q16×C47C2216yes—
39C4×Q16×C47C2213yes—
40C4×Q16×C47C2216yes—
41C4×Q16×C47C2216yes—
42C4×Q16×C47C2213yes—
43C4×Q16×C47C2211yescharacteristic
44C8×D4×C47C2213yes—
45C8×D4×C47C2213yes—
46C47×#431C2213yes—
47C2×#107×C47C2213yes—
48C2×#103×C47C2216yes—
49C2×C4×C8×C47C2213yes—
50C2×#41×C47D411yescharacteristic
51C22×Q8×C47C2316yes—
52C2×Q16×C47—212no—
53C2×Q16×C47C2316yes—
54C2×Q16×C47C2313yes—
55C2×Q16×C47C2316yes—
56C2×Q16×C47C2311yescharacteristic
57C22×C8×C47C2313yes—
58C22×C8×C47C2313yes—
59C4⋊Q8×C47—26no—
60C4⋊Q8×C47—26no—
61C4⋊Q8×C47C2316yes—
62C4⋊Q8×C47C2316yes—
63#125×C47—26no—
64#125×C47—26no—
65#125×C47—26no—
66C22⋊Q8×C47—26no—
67C22⋊Q8×C47—212no—
68C22⋊Q8×C47—212no—
69C22⋊Q8×C47C23112yes—
70C4×Q8×C47—22no—
71C4×Q8×C47—22no—
72C4×Q8×C47—26no—
73C4×Q8×C47—26no—
74C4×Q8×C47—26no—
75C4×Q8×C47D411yescharacteristic
76C4×Q8×C47C2316yes—
77C4×Q8×C47C2316yes—
78C4×Q8×C47C2312yes—
79C4×Q8×C47C2316yes—
80C4×Q8×C47C2311yescharacteristic
81C4×D4×C47—26no—
82C4×D4×C47—26no—
83C4×D4×C47D413yes—
84C4×D4×C47C2313yes—
85#117×C47—26no—
86#117×C47D413yes—
87C2×C4⋊C4×C47—26no—
88C2×C4⋊C4×C47C2316yes—
89C2×C42×C47—26no—
90C2×C42×C47C2313yes—
91#107×C47C2313yes—
92#107×C47C2316yes—
93#107×C47C2316yes—
94#107×C47C2311yescharacteristic
95C4⋊C8×C47C2313yes—
96C4⋊C8×C47C2313yes—
97#103×C47C2316yes—
98#103×C47C23112yes—
99#103×C47C2316yes—
100#103×C47C2316yes—
101#103×C47C2312yes—
102C22⋊C8×C47C2316yes—
103C4×C8×C47C2313yes—
104C4×C8×C47C2313yes—
105C4×C8×C47C2311yescharacteristic
106C4×C8×C47C2313yes—
107#41×C47Q1618yesdirect product factor
108C2×Q8×C47—26no—
109C2×Q8×C47—26no—
110C2×Q8×C47—212no—
111C2×Q8×C47—46no—
112C2×Q8×C47C2×D411yescharacteristic
113C2×Q8×C47C2416yes—
114C2×Q8×C47C2412yes—
115C2×D4×C47C2×D413yes—
116C22×C4×C47—26no—
117C22×C4×C47—26no—
118C22×C4×C47C2×D413yes—
119C22×C4×C47C2×D413yes—
120C22×C4×C47C2413yes—
121Q16×C47—212no—
122Q16×C47—26no—
123Q16×C47—46no—
124Q16×C47#4114yesdirect product factor
125C2×C8×C47—26no—
126C2×C8×C47—43no—
127C2×C8×C47C2411yescharacteristic
128C2×C8×C47C2413yes—
129C2×C8×C47C2413yes—
130C2×C8×C47C2411yescharacteristic
131C4⋊C4×C47—22no—
132C4⋊C4×C47—26no—
133C4⋊C4×C47—26no—
134C4⋊C4×C47—26no—
135C4⋊C4×C47—26no—
136C4⋊C4×C47—26no—
137C4⋊C4×C47—26no—
138C4⋊C4×C47—26no—
139C4⋊C4×C47—212no—
140C4⋊C4×C47C2×D413yes—
141C4⋊C4×C47C2×D413yes—
142C4⋊C4×C47C2412yes—
143C4⋊C4×C47C2416yes—
144C22⋊C4×C47—212no—
145C22⋊C4×C47C2×D416yes—
146C42×C47—26no—
147C42×C47—26no—
148C42×C47—22no—
149C42×C47—26no—
150C42×C47C2×D413yes—
151C42×C47C2×D413yes—
152C42×C47C2413yes—
153C42×C47C2411yescharacteristic
154C23×C47C22×D413yes—
155Q8×C47—22no—
156Q8×C47—26no—
157Q8×C47—26no—
158Q8×C47—412no—
159Q8×C47—86no—
160Q8×C47C2×Q1614yescomplemented
161Q8×C47C2×#4114yes—
162D4×C47C2×Q16112yescomplemented
163C2×C4×C47—22no—
164C2×C4×C47—23no—
165C2×C4×C47—23no—
166C2×C4×C47—22no—
167C2×C4×C47—26no—
168C2×C4×C47—26no—
169C2×C4×C47—43no—
170C2×C4×C47—412no—
171C2×C4×C47—86no—
172C2×C4×C47C2×Q16112yescomplemented
173C2×C4×C47C22×D411yescharacteristic
174C2×C4×C47C22×D413yes—
175C2×C4×C47C22×D413yes—
176C2×C4×C47C22×D413yes—
177C2×C4×C47C22×D411yescharacteristic
178C2×C4×C47C2511yescharacteristic
179C8×C47—23no—
180C8×C47—21no—
181C8×C47—23no—
182C8×C47C2×#4112yes—
183Q16×#41C4711yesmaximal, characteristic, direct product factor, Sylow
184C22×C47—23no—
185C22×C47—23no—
186C22×C47C22×Q1616yes—
187C22×C47C23×D411yescharacteristic
188C4×C47—21no—
189C4×C47—23no—
190C4×C47—23no—
191C4×C47—24no—
192C4×C47—42no—
193C4×C47—46no—
194C4×C47—46no—
195C4×C47C22×Q1612yes—
196C4×C47C22×Q1616yes—
197C4×C47C22×#4111yescharacteristic
198C4×C47#58511yescharacteristic
199Q8×#41C2×C4712yes—
200#3395C2×C4711yescharacteristic, complemented
201Q8×Q16C2×C4711yescharacteristic, complemented
202#3363C2×C4712yescomplemented
203#3338C2×C4713yescomplemented
204#3299C2×C4716yescomplemented
205D4×Q16C2×C4713yescomplemented
206#3170C2×C4713yescomplemented
207#3104C2×C4716yescomplemented
208C8×#41C2×C4711yescharacteristic
209C2×C4×Q16C2×C4713yescomplemented
210C2×C47—26no—
211C2×C47D4×#4111yescharacteristic
212C2×C47C23×Q1611yescharacteristic
213C2×C47#358411yescharacteristic
214C22×Q16C22×C4713yes—
215Q82C22×C4712yes—
216#557—22no—
217#557C22×C4712yes—
218#554—26no—
219#554C22×C4716yes—
220D4×Q8C22×C4716yes—
221#533—26no—
222#533C22×C4716yes—
223C4×#41—22no—
224C4×#41C22×C4711yescharacteristic
225C2×C4×Q8C22×C4716yes—
226#500C22×C4713yes—
227#494C22×C4713yes—
228#487C22×C4716yes—
229#484C22×C47112yes—
230#477C22×C4716yes—
231#477C22×C4716yes—
232#467C22×C4716yes—
233#467C22×C4716yes—
234#462C22×C4716yes—
235#462C22×C4716yes—
236#451C22×C47112yes—
237C8×Q8C22×C4711yescharacteristic
238C8×Q8C22×C4711yescharacteristic
239C4×Q16C22×C4713yes—
240C4×Q16C22×C4716yes—
241C4×Q16C22×C4716yes—
242C4×Q16C22×C4716yes—
243C4×Q16C22×C4713yes—
244C4×Q16C22×C4713yes—
245C4×Q16C22×C4711yescharacteristic
246C8×D4C22×C4713yes—
247C8×D4C22×C4713yes—
248#431C22×C4713yes—
249C2×#107C22×C4713yes—
250C2×#103C22×C4716yes—
251C2×C4×C8C22×C4713yes—
252C47Q16×#4111yescharacteristic, direct product factor, Sylow
253C2×#41D4×C4711yescharacteristic
254C22×Q8C23×C4716yes—
255C2×Q16—212no—
256C2×Q16C23×C4716yes—
257C2×Q16C23×C4713yes—
258C2×Q16C23×C4716yes—
259C2×Q16C23×C4711yescharacteristic
260C22×C8C23×C4713yes—
261C22×C8C23×C4713yes—
262C4⋊Q8—26no—
263C4⋊Q8—26no—
264C4⋊Q8C23×C4716yes—
265C4⋊Q8C23×C4716yes—
266#125—26no—
267#125—26no—
268#125—26no—
269C22⋊Q8—26no—
270C22⋊Q8—212no—
271C22⋊Q8—212no—
272C22⋊Q8C23×C47112yes—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—26no—
276C4×Q8—26no—
277C4×Q8—26no—
278C4×Q8D4×C4711yescharacteristic
279C4×Q8C23×C4712yes—
280C4×Q8C23×C4716yes—
281C4×Q8C23×C4716yes—
282C4×Q8C23×C4716yes—
283C4×Q8C23×C4711yescharacteristic
284C4×D4—26no—
285C4×D4—26no—
286C4×D4D4×C4713yes—
287C4×D4C23×C4713yes—
288#117—26no—
289#117D4×C4713yes—
290C2×C4⋊C4—26no—
291C2×C4⋊C4C23×C4716yes—
292C2×C42—26no—
293C2×C42C23×C4713yes—
294#107C23×C4711yescharacteristic
295#107C23×C4713yes—
296#107C23×C4716yes—
297#107C23×C4716yes—
298C4⋊C8C23×C4713yes—
299C4⋊C8C23×C4713yes—
300#103C23×C4712yes—
301#103C23×C4716yes—
302#103C23×C4716yes—
303#103C23×C4716yes—
304#103C23×C47112yes—
305C22⋊C8C23×C4716yes—
306C4×C8C23×C4711yescharacteristic
307C4×C8C23×C4713yes—
308C4×C8C23×C4713yes—
309C4×C8C23×C4713yes—
310#41Q16×C4718yesdirect product factor
311C2×Q8—26no—
312C2×Q8—26no—
313C2×Q8—212no—
314C2×Q8—46no—
315C2×Q8C2×D4×C4711yescharacteristic
316C2×Q8C24×C4716yes—
317C2×Q8C24×C4712yes—
318C2×D4C2×D4×C4713yes—
319C22×C4—26no—
320C22×C4—26no—
321C22×C4C2×D4×C4713yes—
322C22×C4C2×D4×C4713yes—
323C22×C4C24×C4713yes—
324Q16—212no—
325Q16—26no—
326Q16—46no—
327Q16#41×C4714yesdirect product factor
328C2×C8—26no—
329C2×C8—43no—
330C2×C8C24×C4711yescharacteristic
331C2×C8C24×C4713yes—
332C2×C8C24×C4713yes—
333C2×C8C24×C4711yescharacteristic
334C4⋊C4—22no—
335C4⋊C4—26no—
336C4⋊C4—26no—
337C4⋊C4—26no—
338C4⋊C4—26no—
339C4⋊C4—26no—
340C4⋊C4—26no—
341C4⋊C4—26no—
342C4⋊C4—212no—
343C4⋊C4C2×D4×C4713yes—
344C4⋊C4C2×D4×C4713yes—
345C4⋊C4C24×C4712yes—
346C4⋊C4C24×C4716yes—
347C22⋊C4—212no—
348C22⋊C4C2×D4×C4716yes—
349C42—22no—
350C42—26no—
351C42—26no—
352C42—26no—
353C42C2×D4×C4713yes—
354C42C2×D4×C4713yes—
355C42C24×C4711yescharacteristic
356C42C24×C4713yes—
357C23C22×D4×C4713yes—
358Q8—22no—
359Q8—26no—
360Q8—26no—
361Q8—412no—
362Q8—86no—
363Q8C2×Q16×C4714yescomplemented
364Q8C2×#41×C4714yes—
365D4C2×Q16×C47112yescomplemented
366C2×C4—22no—
367C2×C4—23no—
368C2×C4—23no—
369C2×C4—22no—
370C2×C4—26no—
371C2×C4—26no—
372C2×C4—43no—
373C2×C4—412no—
374C2×C4—86no—
375C2×C4C2×Q16×C47112yescomplemented
376C2×C4C22×D4×C4711yescharacteristic
377C2×C4C22×D4×C4713yes—
378C2×C4C22×D4×C4713yes—
379C2×C4C22×D4×C4713yes—
380C2×C4C22×D4×C4711yescharacteristic
381C2×C4C25×C4711yescharacteristic
382C8—23no—
383C8—21no—
384C8—23no—
385C8C2×#41×C4712yes—
386C22—23no—
387C22—23no—
388C22C22×Q16×C4716yes—
389C22C23×D4×C4711yescharacteristic
390C4—21no—
391C4—23no—
392C4—23no—
393C4—24no—
394C4—42no—
395C4—46no—
396C4—46no—
397C4C22×Q16×C4712yes—
398C4C22×Q16×C4716yes—
399C4C22×#41×C4711yescharacteristic
400C4C47×#58511yescharacteristic
401C2—26no—
402C2D4×#41×C4711yescharacteristic
403C2C23×Q16×C4711yescharacteristic
404C2C47×#358411yescharacteristic
subgroupG/H
centerC2×C4×C47C22×D4
commutator, Frattini, cosocleC2×C4C25×C47
socleC22×C47C23×D4
prime radicalC2×#41×C47D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C47 ◁ C4×#41×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C47×#3395C2semidirect
3. Q8×Q16×C47C2semidirect
4. C47×#3363C2semidirect
5. C47×#3338C2semidirect
6. C47×#3299C2semidirect
7. D4×Q16×C47C2semidirect
8. C47×#3170C2semidirect
9. C47×#3104C2semidirect
11. C2×C4×Q16×C47C2semidirect
107. #41×C47Q16direct (N × H)
124. Q16×C47#41direct (N × H)
160. Q8×C47C2×Q16semidirect
162. D4×C47C2×Q16semidirect
172. C2×C4×C47C2×Q16semidirect
183. Q16×#41C47direct (N × H)
200. #3395C2×C47semidirect
201. Q8×Q16C2×C47semidirect
202. #3363C2×C47semidirect
203. #3338C2×C47semidirect
204. #3299C2×C47semidirect
205. D4×Q16C2×C47semidirect
206. #3170C2×C47semidirect
207. #3104C2×C47semidirect
209. C2×C4×Q16C2×C47semidirect
252. C47Q16×#41direct (N × H)
310. #41Q16×C47direct (N × H)
327. Q16#41×C47direct (N × H)
363. Q8C2×Q16×C47semidirect
365. D4C2×Q16×C47semidirect
375. C2×C4C2×Q16×C47semidirect

Maximal subgroups

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

subgroupindex
Q8×#41×C472
C47×#33952
Q8×Q16×C472
C47×#33632
C47×#33382
C47×#32992
D4×Q16×C472
C47×#31702
C47×#31042
C8×#41×C472
C2×C4×Q16×C472
Q16×#4147