Finite Group Databasebeta

← #26383678

Subgroups — #26383678

Order 12032.

1570 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20249589C214yesmaximal
2#20249580C214yesmaximal, complemented
3#20249578C214yesmaximal, complemented
4#20249565C214yesmaximal
5#20246693C211yesmaximal, characteristic, complemented
6#20246682C212yesmaximal, complemented
7#20246664C214yesmaximal
8#20245864C214yesmaximal, complemented
9#20245784C212yesmaximal
10#20245727C211yesmaximal, characteristic
11C47×#2402C211yesmaximal, characteristic, complemented
12C2×C4×Dic94C2214yes—
13C2×#13093981C2212yes—
14C2×C42×D47C2211yescharacteristic
15#15689065C2214yes—
16#15689064C2214yes—
17#15689058C2214yes—
18#15689057C2214yes—
19#15689057C2214yes—
20#15689057C2214yes—
21#15689056C2214yes—
22#15689054C2214yes—
23#15689047C2214yes—
24#15689045C2214yes—
25#15689044C2214yes—
26#15689043C2214yes—
27#15689043C2214yes—
28#15689036C2214yes—
29#15689027C2214yes—
30#15689027C2214yes—
31#15689018C2214yes—
32#15689018C2214yes—
33#15689008C2214yes—
34#15689008C2214yes—
35#15689004C2214yes—
36#15689004C2214yes—
37#15688990C2212yes—
38#15688990C2212yes—
39#15688988C2214yes—
40#15688985C2212yes—
41#15688985C2212yes—
42#15688980C2214yes—
43C4×Dic47⋊C4C2214yes—
44C4×Dic47⋊C4C2214yes—
45C4×C4⋊Dic47C2212yes—
46#15688974C2214yes—
47C4×(C2×D47)⋊C4C2214yes—
48#15688970C2214yes—
49#15688969C2214yes—
50#15688969C2211yescharacteristic
51C42×Dic47C2212yes—
52C47×#400C2214yes—
53C47×#398C2214yes—
54C4×C4⋊C4×C47C2212yes—
55C4×C4⋊C4×C47C2214yes—
56C47×#376C2211yescharacteristic
57C22×Dic94C2314yes—
58C22×C4×D47C2311yescharacteristic
59C22×C4×D47C2312yes—
60C4×Dic94—216no—
61#13093981—28no—
62C2×C4⋊Dic47C2314yes—
63C2×C4⋊Dic47C2314yes—
64C2×Dic47⋊C4C2314yes—
65C2×Dic47⋊C4C2314yes—
66C2×Dic47⋊C4C2314yes—
67C2×Dic47⋊C4C2314yes—
68C2×Dic47⋊C4C2314yes—
69C2×Dic47⋊C4C2314yes—
70C2×Dic47⋊C4C2314yes—
71C2×(C2×D47)⋊C4C2314yes—
72C2×(C2×D47)⋊C4C2314yes—
73C2×(C2×D47)⋊C4C2314yes—
74C2×C4×Dic47C2314yes—
75C2×C4×Dic47C2314yes—
76C2×C4×Dic47C2314yes—
77C2×C4×Dic47C2311yescharacteristic
78C2×C4×Dic47C2312yes—
79C2×C4×Dic47C2312yes—
80C42×D47Q818yescomplemented
81C47⋊#95C2314yes—
82C47⋊#95C2314yes—
83C47⋊#95C2314yes—
84C47⋊#95C2314yes—
85C47⋊#95C2314yes—
86C47⋊#95C2314yes—
87C47⋊#95C2314yes—
88C47⋊#95C2312yes—
89C47⋊#95C2312yes—
90C47⋊#95C2314yes—
91C47⋊#95C2314yes—
92C47⋊#95C2314yes—
93C47⋊#95C2314yes—
94C2×C4⋊C4×C47C2314yes—
95C2×C4⋊C4×C47C2314yes—
96C2×C4⋊C4×C47C2314yes—
97C2×C42×C47C2312yes—
98C2×C42×C47C2314yes—
99C2×C42×C47C2311yescharacteristic
100#95×C47C2314yes—
101#95×C47C2312yes—
102#95×C47C2312yes—
103#95×C47C2314yes—
104#95×C47C2314yes—
105C23×D47C2411yescharacteristic
106C22×C4×C47C2414yes—
107C22×C4×C47C2414yes—
108C22×C4×C47C2414yes—
109C22×C4×C47C2412yes—
110C22×C4×C47C2411yescharacteristic
111C22×Dic47C2414yes—
112C22×Dic47C2414yes—
113C22×Dic47C2412yes—
114C22×Dic47C2411yescharacteristic
115C22×Dic47C2414yes—
116C2×C4×D47—24no—
117C2×C4×D47—44no—
118C2×C4×D47C2×Q814yes—
119C2×C4×D47C2×Q818yes—
120C2×Dic94—28no—
121C2×Dic94—44no—
122C2×Dic94—44no—
123C4⋊C4×C47—44no—
124C4⋊C4×C47—44no—
125C4⋊C4×C47—44no—
126C42×C47—24no—
127C42×C47—28no—
128C42×C47C2×Q814yes—
129(C2×D47)⋊C4—28no—
130(C2×D47)⋊C4—44no—
131(C2×D47)⋊C4—44no—
132C4⋊Dic47—28no—
133C4⋊Dic47—44no—
134Dic47⋊C4—28no—
135Dic47⋊C4—28no—
136Dic47⋊C4—28no—
137Dic47⋊C4—44no—
138Dic47⋊C4—44no—
139Dic47⋊C4—44no—
140Dic47⋊C4—44no—
141C4×Dic47—28no—
142C4×Dic47—44no—
143C4×Dic47—44no—
144C4×Dic47C2×Q818yes—
145C4×Dic47C2×Q814yes—
146C4×Dic47#4118yes—
147C23×C47C2511yescharacteristic
148C22×D47—22no—
149C22×D47—24no—
150C22×D47C22×Q812yes—
151C2×C4×C47—24no—
152C2×C4×C47—24no—
153C2×C4×C47—24no—
154C2×C4×C47—24no—
155C2×C4×C47—24no—
156C2×C4×C47—24no—
157C2×C4×C47—24no—
158C2×C4×C47—24no—
159C2×C4×C47—24no—
160C2×C4×C47—24no—
161C2×C4×C47C22×Q814yes—
162C2×C4×C47C22×Q812yes—
163C2×C4×C47C2×#4114yes—
164C2×Dic47—22no—
165C2×Dic47—24no—
166C2×Dic47—24no—
167C2×Dic47—24no—
168C2×Dic47—24no—
169C2×Dic47—24no—
170C2×Dic47—24no—
171C2×Dic47—24no—
172C2×Dic47—24no—
173C2×Dic47—24no—
174C2×Dic47C22×Q812yes—
175C2×Dic47C22×Q814yes—
176C2×Dic47C2×#4118yes—
177C4×D47—28no—
178C4×D47—48no—
179Dic94—416no—
180#34644—471nomaximal, Sylow
181C22×C47C23×Q811yescharacteristic
182C22×C47C22×#4112yes—
183C22×C47#58514yes—
184C2×D47—22no—
185C2×D47—24no—
186C2×D47—44no—
187C4×C47—24no—
188C4×C47—44no—
189C4×C47—44no—
190C4×C47—44no—
191C4×C47#53314yes—
192Dic47—22no—
193Dic47—28no—
194Dic47—24no—
195Dic47—44no—
196Dic47—44no—
197#2667—474no—
198#2666—474no—
199#2666—474no—
200#2665—474no—
201#2402—471no—
202#2402D4711yescharacteristic, complemented
203#2399—472no—
204#2396—474no—
205#2295—474no—
206#2283—472no—
207C4×#117—471no—
208C2×C47C2×#53311yescharacteristic
209C2×C47#345514yes—
210C2×C47#345812yes—
211D47—42no—
212C2×C4×Q8—474no—
213C2×#117—472no—
214C22×C42—471no—
215#400—474no—
216#400—474no—
217#400C2×D4714yes—
218#399—474no—
219#398—474no—
220#398—474no—
221#398—474no—
222#398—474no—
223#398—474no—
224#398C2×D4714yes—
225#396—474no—
226#396—474no—
227#395—474no—
228#395—474no—
229#395—474no—
230#393—474no—
231#391—474no—
232#391—474no—
233#389—474no—
234#389—474no—
235#387—474no—
236#387—474no—
237#387—474no—
238#387—474no—
239#383—472no—
240#383—472no—
241#382—474no—
242#381—472no—
243#381—472no—
244#380—474no—
245C4×C4⋊C4—472no—
246C4×C4⋊C4—474no—
247C4×C4⋊C4—474no—
248C4×C4⋊C4—474no—
249C4×C4⋊C4C2×D4714yes—
250C4×C4⋊C4C2×D4712yes—
251C4×C22⋊C4—474no—
252C4×C22⋊C4—474no—
253#376—471no—
254#376—474no—
255#376C2×D4711yescharacteristic
256C43—472no—
257C47#3464411yescharacteristic, complemented, Sylow
258C22×Q8—474no—
259C23×C4—471no—
260C23×C4—472no—
261C4×Q8—9416no—
262#117—948no—
263C2×C4⋊C4—474no—
264C2×C4⋊C4—474no—
265C2×C4⋊C4—474no—
266C2×C4⋊C4—474no—
267C2×C4⋊C4—474no—
268C2×C4⋊C4—474no—
269C2×C4⋊C4—474no—
270C2×C4⋊C4—474no—
271C2×C4⋊C4—474no—
272C2×C4⋊C4C22×D4714yes—
273C2×C4⋊C4C22×D4714yes—
274C2×C4⋊C4C22×D4714yes—
275C2×C22⋊C4—474no—
276C2×C22⋊C4—474no—
277C2×C22⋊C4—474no—
278C2×C42—472no—
279C2×C42—472no—
280C2×C42—471no—
281C2×C42—478no—
282C2×C42—474no—
283C2×C42—474no—
284C2×C42—474no—
285C2×C42C22×D4712yes—
286C2×C42C22×D4714yes—
287C2×C42C22×D4711yescharacteristic
288#95—474no—
289#95—474no—
290#95—474no—
291#95—474no—
292#95—474no—
293#95—474no—
294#95—474no—
295#95—472no—
296#95—472no—
297#95—474no—
298#95—474no—
299#95—474no—
300#95—474no—
301#95C22×D4714yes—
302#95C22×D4712yes—
303#95C22×D4712yes—
304#95C22×D4714yes—
305#95C22×D4714yes—
306C24—471no—
307C2×Q8—948no—
308C2×Q8—1884no—
309C2×Q8—1884no—
310C22×C4—472no—
311C22×C4—471no—
312C22×C4—474no—
313C22×C4—474no—
314C22×C4—478no—
315C22×C4—474no—
316C22×C4—474no—
317C22×C4—944no—
318C22×C4—1884no—
319C22×C4C23×D4714yes—
320C22×C4C23×D4714yes—
321C22×C4C23×D4714yes—
322C22×C4C23×D4712yes—
323C22×C4C23×D4711yescharacteristic
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—948no—
328C4⋊C4—948no—
329C4⋊C4—948no—
330C4⋊C4—948no—
331C4⋊C4—1884no—
332C4⋊C4—1884no—
333C4⋊C4—1884no—
334C4⋊C4—1884no—
335C4⋊C4—1884no—
336C22⋊C4—948no—
337C22⋊C4—1884no—
338C22⋊C4—1884no—
339C42—24no—
340C42—28no—
341C42—478no—
342C42—474no—
343C42—478no—
344C42—948no—
345C42—1884no—
346C42—1884no—
347C42Q8×D4714yes—
348C23—472no—
349C23—942no—
350C23—944no—
351C23C24×D4711yescharacteristic
352Q8—18816no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—474no—
364C2×C4—472no—
365C2×C4—478no—
366C2×C4—942no—
367C2×C4—944no—
368C2×C4—944no—
369C2×C4—944no—
370C2×C4—948no—
371C2×C4—944no—
372C2×C4—944no—
373C2×C4—944no—
374C2×C4—944no—
375C2×C4—944no—
376C2×C4—944no—
377C2×C4—1888no—
378C2×C4C2×Q8×D4714yes—
379C2×C4C2×Q8×D4712yes—
380C2×C4#41×D4714yes—
381C22—944no—
382C22—942no—
383C22—1884no—
384C22C22×Q8×D4711yescharacteristic
385C22C2×#41×D4712yes—
386C22#1568995914yes—
387C4—24no—
388C4—44no—
389C4—44no—
390C4—44no—
391C4—948no—
392C4—944no—
393C4—942no—
394C4—1884no—
395C4—1884no—
396C4#1568967914yes—
397C2—1882no—
398C2C2×#1568967911yescharacteristic
399C2#2025677114yes—
400C2#2025679112yes—
subgroupG/H
centerC22×C4C23×D47
commutator, socle, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC47×#2402C2
prime radicalC23×D47C24
hypercenter#2402D47
hypocenterC47#34644
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #2402

Complements

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

NH ≅ G/Ntype
2. #20249580C2semidirect
3. #20249578C2semidirect
5. #20246693C2semidirect
6. #20246682C2semidirect
8. #20245864C2semidirect
11. C47×#2402C2semidirect
80. C42×D47Q8semidirect
202. #2402D47semidirect
257. C47#34644semidirect

Maximal subgroups

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

subgroupindex
#202495892
#202495802
#202495782
#202495652
#202466932
#202466822
#202466642
#202458642
#202457842
#202457272
C47×#24022
#3464447