Finite Group Databasebeta

← C4×C47×#557 #26073867

Subgroups — C4×C47×#557 #26073867

Order 12032.

2220 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C47×#557C211yesmaximal, characteristic
2C47×#2370C216yesmaximal
3C47×#2365C213yesmaximal
4C47×#2355C213yesmaximal
5C47×#2351C212yesmaximal
6Q8×C4⋊C4×C47C211yesmaximal, characteristic
7C4×C4⋊Q8×C47C213yesmaximal
8C4×#125×C47C216yesmaximal
9C42×Q8×C47C213yesmaximal
10C42×Q8×C47C212yesmaximal
11C42×Q8×C47C211yesmaximal, characteristic
12C47×#557C4116yesdirect product factor
13C2×C4⋊Q8×C47C2213yes—
14C2×#125×C47C2216yes—
15C2×C4×Q8×C47C2213yes—
16C2×C4×Q8×C47C2212yes—
17C2×C4×Q8×C47C2211yescharacteristic
18C2×C4×Q8×C47C2213yes—
19C2×C4×Q8×C47C2211yescharacteristic
20C2×C4×Q8×C47C2212yes—
21C2×C4×Q8×C47C2213yes—
22C2×C4×Q8×C47C2211yescharacteristic
23C47×#391C2216yes—
24C47×#389C2216yes—
25C47×#389C2216yes—
26C47×#389C2216yes—
27C47×#389C2216yes—
28C47×#387C22112yes—
29C47×#387C22112yes—
30C47×#384C2213yes—
31C47×#382C2216yes—
32C4×C4⋊C4×C47C2211yescharacteristic
33C4×C4⋊C4×C47C2216yes—
34C4×C4⋊C4×C47C2213yes—
35C4×C4⋊C4×C47C2216yes—
36C4×C4⋊C4×C47C2213yes—
37C4×C4⋊C4×C47C2216yes—
38C4×C4⋊C4×C47C2213yes—
39C4×C4⋊C4×C47C2213yes—
40C4×C4⋊C4×C47C2216yes—
41C4×C4⋊C4×C47C2216yes—
42C4×C4⋊C4×C47C2216yes—
43C4×C4⋊C4×C47C2213yes—
44C4×C4⋊C4×C47C2216yes—
45C43×C47C2216yes—
46C43×C47C2213yes—
47C22×Q8×C47C2313yes—
48C22×Q8×C47C2311yescharacteristic
49C4⋊Q8×C47C2×C4124yes—
50#125×C47C2×C4148yes—
51C4×Q8×C47—28no—
52C4×Q8×C47—24no—
53C4×Q8×C47—212no—
54C4×Q8×C47—212no—
55C4×Q8×C47C2×C4116yes—
56C4×Q8×C47C2×C4124yes—
57C4×Q8×C47C2×C418yes—
58C4×Q8×C47Q818yescomplemented
59C2×C4⋊C4×C47C2313yes—
60C2×C4⋊C4×C47C2316yes—
61C2×C4⋊C4×C47C2313yes—
62C2×C4⋊C4×C47C2316yes—
63C2×C4⋊C4×C47C2316yes—
64C2×C4⋊C4×C47C2316yes—
65C2×C4⋊C4×C47C2311yescharacteristic
66C2×C4⋊C4×C47C2316yes—
67C2×C4⋊C4×C47C2313yes—
68C2×C4⋊C4×C47C2316yes—
69C2×C4⋊C4×C47C2316yes—
70C2×C4⋊C4×C47C2313yes—
71C2×C4⋊C4×C47C2311yescharacteristic
72C2×C42×C47C2316yes—
73C2×C42×C47C2316yes—
74C2×C42×C47C2313yes—
75C2×C42×C47C2313yes—
76C2×C42×C47C2316yes—
77C2×C42×C47C2312yes—
78C2×C42×C47C2313yes—
79C2×C42×C47C2316yes—
80C2×C42×C47C2313yes—
81C2×C42×C47C2316yes—
82C2×C42×C47C2313yes—
83C2×C42×C47C2313yes—
84C2×C42×C47C2311yescharacteristic
85#95×C47C2312yes—
86#95×C47C2316yes—
87#95×C47C2316yes—
88#95×C47C2316yes—
89#95×C47C23112yes—
90#95×C47C23112yes—
91C2×Q8×C47—26no—
92C2×Q8×C47—26no—
93C2×Q8×C47C22×C4112yes—
94C2×Q8×C47C22×C414yes—
95C2×Q8×C47C2×Q814yes—
96C2×Q8×C47#4114yes—
97C22×C4×C47C2416yes—
98C22×C4×C47C2413yes—
99C22×C4×C47C2412yes—
100C22×C4×C47C2413yes—
101C22×C4×C47C2416yes—
102C22×C4×C47C2413yes—
103C22×C4×C47C2411yescharacteristic
104C22×C4×C47C2412yes—
105C22×C4×C47C2413yes—
106C22×C4×C47C2411yescharacteristic
107C22×C4×C47C2411yescharacteristic
108C4⋊C4×C47—212no—
109C4⋊C4×C47—212no—
110C4⋊C4×C47—26no—
111C4⋊C4×C47—212no—
112C4⋊C4×C47—26no—
113C4⋊C4×C47—46no—
114C4⋊C4×C47C22×C414yes—
115C4⋊C4×C47C22×C4124yes—
116C4⋊C4×C47C22×C4112yes—
117C4⋊C4×C47C22×C4124yes—
118C4⋊C4×C47C22×C4124yes—
119C4⋊C4×C47C2×Q8112yes—
120C4⋊C4×C47#4114yes—
121C42×C47—212no—
122C42×C47—26no—
123C42×C47—212no—
124C42×C47—26no—
125C42×C47—212no—
126C42×C47—24no—
127C42×C47—26no—
128C42×C47—43no—
129C42×C47—46no—
130C42×C47C22×C4124yes—
131C42×C47C22×C4112yes—
132C42×C47C2×Q8112yes—
133C42×C47#4114yes—
134C23×C47C2511yescharacteristic
135Q8×C47—224no—
136Q8×C47C4×Q8116yescomplemented
137C2×C4×C47—23no—
138C2×C4×C47—21no—
139C2×C4×C47—22no—
140C2×C4×C47—26no—
141C2×C4×C47—26no—
142C2×C4×C47—22no—
143C2×C4×C47—23no—
144C2×C4×C47—22no—
145C2×C4×C47—22no—
146C2×C4×C47—23no—
147C2×C4×C47—21no—
148C2×C4×C47—23no—
149C2×C4×C47—26no—
150C2×C4×C47—26no—
151C2×C4×C47—23no—
152C2×C4×C47—23no—
153C2×C4×C47—23no—
154C2×C4×C47—26no—
155C2×C4×C47C23×C414yes—
156C2×C4×C47C23×C416yes—
157C2×C4×C47C23×C4112yes—
158C2×C4×C47C23×C416yes—
159C2×C4×C47C23×C412yes—
160C2×C4×C47C22×Q816yes—
161C2×C4×C47C22×Q816yes—
162C2×C4×C47C22×Q812yes—
163C2×C4×C47C2×#4114yes—
164C2×C4×C47C2×#4112yes—
165C2×C4×C47C2×#4116yes—
166C2×C4×C47C2×#4112yes—
167C2×C4×C47C2×#4112yes—
168C2×C4×C47C2×#4112yes—
169C2×C4×C472−1+412yes—
170C4×#557C4711yesmaximal, characteristic, direct product factor, Sylow
171C22×C47C24×C411yescharacteristic
172C22×C47C23×Q811yescharacteristic
173C22×C47C22×#4111yescharacteristic
174C22×C47C22×#4111yescharacteristic
175C22×C47C22×#4111yescharacteristic
176C22×C47C2×2−1+411yescharacteristic
177C22×C47#58511yescharacteristic
178C4×C47—212no—
179C4×C47—24no—
180C4×C47—26no—
181C4×C47—26no—
182C4×C47—22no—
183C4×C47—24no—
184C4×C47—46no—
185C4×C47—43no—
186C4×C47C2×C4×Q8112yes—
187C4×C47C4×#4114yes—
188C4×C47#55714yesdirect product factor
189C2×#557C2×C4711yescharacteristic
190#2370C2×C4716yes—
191#2365C2×C4713yes—
192#2355C2×C4713yes—
193#2351C2×C4712yes—
194Q8×C4⋊C4C2×C4711yescharacteristic
195C4×C4⋊Q8C2×C4713yes—
196C4×#125C2×C4716yes—
197C42×Q8C2×C4713yes—
198C42×Q8C2×C4712yes—
199C42×Q8C2×C4711yescharacteristic
200C2×C47C22×C4×Q811yescharacteristic
201C2×C47C2×C4×#4111yescharacteristic
202C2×C47C4×2−1+411yescharacteristic
203C2×C47C2×#55711yescharacteristic
204C2×C47Q8×#4111yescharacteristic
205C2×C47#348011yescharacteristic
206C2×C47#348211yescharacteristic
207#557C4×C47116yesdirect product factor
208C2×C4⋊Q8C22×C4713yes—
209C2×#125C22×C4716yes—
210C2×C4×Q8C22×C4713yes—
211C2×C4×Q8C22×C4712yes—
212C2×C4×Q8C22×C4711yescharacteristic
213C2×C4×Q8C22×C4713yes—
214C2×C4×Q8C22×C4711yescharacteristic
215C2×C4×Q8C22×C4712yes—
216C2×C4×Q8C22×C4713yes—
217C2×C4×Q8C22×C4711yescharacteristic
218#391C22×C4716yes—
219#389C22×C4716yes—
220#389C22×C4716yes—
221#389C22×C4716yes—
222#389C22×C4716yes—
223#387C22×C47112yes—
224#387C22×C47112yes—
225#384C22×C4713yes—
226#382C22×C4716yes—
227C4×C4⋊C4C22×C4716yes—
228C4×C4⋊C4C22×C4716yes—
229C4×C4⋊C4C22×C4713yes—
230C4×C4⋊C4C22×C4716yes—
231C4×C4⋊C4C22×C4713yes—
232C4×C4⋊C4C22×C4716yes—
233C4×C4⋊C4C22×C4713yes—
234C4×C4⋊C4C22×C4713yes—
235C4×C4⋊C4C22×C4716yes—
236C4×C4⋊C4C22×C4716yes—
237C4×C4⋊C4C22×C4716yes—
238C4×C4⋊C4C22×C4713yes—
239C4×C4⋊C4C22×C4711yescharacteristic
240C43C22×C4716yes—
241C43C22×C4713yes—
242C47C4×#55711yescharacteristic, direct product factor, Sylow
243C22×Q8C23×C4713yes—
244C22×Q8C23×C4711yescharacteristic
245C4⋊Q8C2×C4×C47124yes—
246#125C2×C4×C47148yes—
247C4×Q8—28no—
248C4×Q8—24no—
249C4×Q8—212no—
250C4×Q8—212no—
251C4×Q8C2×C4×C47116yes—
252C4×Q8C2×C4×C47124yes—
253C4×Q8C2×C4×C4718yes—
254C4×Q8Q8×C4718yescomplemented
255C2×C4⋊C4C23×C4713yes—
256C2×C4⋊C4C23×C4716yes—
257C2×C4⋊C4C23×C4713yes—
258C2×C4⋊C4C23×C4716yes—
259C2×C4⋊C4C23×C4716yes—
260C2×C4⋊C4C23×C4716yes—
261C2×C4⋊C4C23×C4711yescharacteristic
262C2×C4⋊C4C23×C4716yes—
263C2×C4⋊C4C23×C4713yes—
264C2×C4⋊C4C23×C4716yes—
265C2×C4⋊C4C23×C4716yes—
266C2×C4⋊C4C23×C4713yes—
267C2×C4⋊C4C23×C4711yescharacteristic
268C2×C42C23×C4716yes—
269C2×C42C23×C4716yes—
270C2×C42C23×C4713yes—
271C2×C42C23×C4716yes—
272C2×C42C23×C4713yes—
273C2×C42C23×C4712yes—
274C2×C42C23×C4713yes—
275C2×C42C23×C4716yes—
276C2×C42C23×C4713yes—
277C2×C42C23×C4716yes—
278C2×C42C23×C4713yes—
279C2×C42C23×C4713yes—
280C2×C42C23×C4711yescharacteristic
281#95C23×C4712yes—
282#95C23×C4716yes—
283#95C23×C4716yes—
284#95C23×C4716yes—
285#95C23×C47112yes—
286#95C23×C47112yes—
287C2×Q8—26no—
288C2×Q8—26no—
289C2×Q8C22×C4×C47112yes—
290C2×Q8C22×C4×C4714yes—
291C2×Q8C2×Q8×C4714yes—
292C2×Q8#41×C4714yes—
293C22×C4C24×C4713yes—
294C22×C4C24×C4716yes—
295C22×C4C24×C4712yes—
296C22×C4C24×C4713yes—
297C22×C4C24×C4716yes—
298C22×C4C24×C4713yes—
299C22×C4C24×C4712yes—
300C22×C4C24×C4711yescharacteristic
301C22×C4C24×C4713yes—
302C22×C4C24×C4711yescharacteristic
303C22×C4C24×C4711yescharacteristic
304C4⋊C4—212no—
305C4⋊C4—212no—
306C4⋊C4—26no—
307C4⋊C4—212no—
308C4⋊C4—26no—
309C4⋊C4—46no—
310C4⋊C4C22×C4×C4714yes—
311C4⋊C4C22×C4×C47124yes—
312C4⋊C4C22×C4×C47112yes—
313C4⋊C4C22×C4×C47124yes—
314C4⋊C4C22×C4×C47124yes—
315C4⋊C4C2×Q8×C47112yes—
316C4⋊C4#41×C4714yes—
317C42—212no—
318C42—26no—
319C42—212no—
320C42—26no—
321C42—212no—
322C42—24no—
323C42—26no—
324C42—43no—
325C42—46no—
326C42C22×C4×C47124yes—
327C42C22×C4×C47112yes—
328C42C2×Q8×C47112yes—
329C42#41×C4714yes—
330C23C25×C4711yescharacteristic
331Q8—224no—
332Q8C4×Q8×C47116yescomplemented
333C2×C4—23no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—26no—
337C2×C4—26no—
338C2×C4—22no—
339C2×C4—23no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—23no—
343C2×C4—21no—
344C2×C4—23no—
345C2×C4—26no—
346C2×C4—26no—
347C2×C4—23no—
348C2×C4—23no—
349C2×C4—23no—
350C2×C4—26no—
351C2×C4C23×C4×C4714yes—
352C2×C4C23×C4×C4716yes—
353C2×C4C23×C4×C47112yes—
354C2×C4C23×C4×C4716yes—
355C2×C4C23×C4×C4712yes—
356C2×C4C22×Q8×C4716yes—
357C2×C4C22×Q8×C4716yes—
358C2×C4C22×Q8×C4712yes—
359C2×C4C2×#41×C4714yes—
360C2×C4C2×#41×C4712yes—
361C2×C4C2×#41×C4716yes—
362C2×C4C2×#41×C4712yes—
363C2×C4C2×#41×C4712yes—
364C2×C4C2×#41×C4712yes—
365C2×C42−1+4×C4712yes—
366C22C24×C4×C4711yescharacteristic
367C22C23×Q8×C4711yescharacteristic
368C22C22×#41×C4711yescharacteristic
369C22C22×#41×C4711yescharacteristic
370C22C22×#41×C4711yescharacteristic
371C22C2×2−1+4×C4711yescharacteristic
372C22C47×#58511yescharacteristic
373C4—212no—
374C4—24no—
375C4—26no—
376C4—26no—
377C4—22no—
378C4—24no—
379C4—46no—
380C4—43no—
381C4C2×C4×Q8×C47112yes—
382C4C4×#41×C4714yes—
383C4C47×#55714yesdirect product factor
384C2C22×C4×Q8×C4711yescharacteristic
385C2C2×C4×#41×C4711yescharacteristic
386C2C4×2−1+4×C4711yescharacteristic
387C2C2×C47×#55711yescharacteristic
388C2Q8×#41×C4711yescharacteristic
389C2C47×#348011yescharacteristic
390C2C47×#348211yescharacteristic
subgroupG/H
centerC22×C4×C47C24
commutatorC22C24×C4×C47
Frattini, cosocleC23C25×C47
socle, prime radicalC23×C47C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
12. C47×#557C4direct (N × H)
58. C4×Q8×C47Q8semidirect
136. Q8×C47C4×Q8semidirect
170. C4×#557C47direct (N × H)
188. C4×C47#557direct (N × H)
207. #557C4×C47direct (N × H)
242. C47C4×#557direct (N × H)
254. C4×Q8Q8×C47semidirect
332. Q8C4×Q8×C47semidirect
383. C4C47×#557direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C47×#5572
C47×#23702
C47×#23652
C47×#23552
C47×#23512
Q8×C4⋊C4×C472
C4×C4⋊Q8×C472
C4×#125×C472
C42×Q8×C472
C4×#55747