Finite Group Databasebeta

← #26116332

Subgroups — #26116332

Order 12032.

556 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243368C211yesmaximal, characteristic
2#20243150C211yesmaximal, characteristic
3#20241231C212yesmaximal
4#20241231C212yesmaximal
5C2×C47×#387C211yesmaximal, characteristic
6C2×C4⋊(C47⋊C8)—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8C2×C22⋊(C47⋊C8)C2212yes—
9C2×#13093935—21no—
10#15688878—21no—
11#15688878—21no—
12C22×C4⋊C4×C47C411yescharacteristic
13C22×C4⋊C4×C47C411yescharacteristic
14C22×C42×C47C2211yescharacteristic
15C47×#387—21no—
16C47×#387—21no—
17C47×#387—21no—
18C47×#387—21no—
19C2×#95×C47C411yescharacteristic
20C2×#95×C47C411yescharacteristic
21C2×#95×C47C2211yescharacteristic
22C2×#95×C47C2211yescharacteristic
23C22×C47⋊C8—22no—
24C22×C47⋊C8—22no—
25C4⋊(C47⋊C8)—42no—
26C22⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—24no—
28C22⋊(C47⋊C8)Q814yes—
29#13093935—42no—
30C23×C4×C47C2×C411yescharacteristic
31C23×C4×C47C2×C411yescharacteristic
32C23×C4×C47C2×C411yescharacteristic
33C23×C4×C47C2×C411yescharacteristic
34C23×C4×C47C2×C411yescharacteristic
35C23×C4×C47C2×C411yescharacteristic
36C23×C4×C47C2311yescharacteristic
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—42no—
44C2×C42×C47—42no—
45C2×C42×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C2×C42×C47D411yescharacteristic
49#95×C47—21no—
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56#95×C47—21no—
57C24×C47C22×C411yescharacteristic
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47—41no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47—41no—
79C22×C4×C47—41no—
80C22×C4×C47—41no—
81C22×C4×C47—41no—
82C22×C4×C47—41no—
83C22×C4×C47C22⋊C411yescharacteristic
84C22×C4×C47C22⋊C411yescharacteristic
85C22×C4×C47C22⋊C411yescharacteristic
86C22×C4×C47C22⋊C411yescharacteristic
87C22×C4×C47M1612yes—
88C22×C4×C47M1612yes—
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×Q812yes—
92C22×C4×C47#4112yes—
93C4⋊C4×C47—24no—
94C4⋊C4×C47—24no—
95C4⋊C4×C47—42no—
96C4⋊C4×C47—42no—
97C42×C47—22no—
98C42×C47—22no—
99C42×C47—41no—
100C42×C47—41no—
101C2×C47⋊C8—24no—
102C2×C47⋊C8—42no—
103C2×C47⋊C8—42no—
104C2×C47⋊C8—42no—
105C2×C47⋊C8—42no—
106C2×C47⋊C8—42no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47C2×C22⋊C411yescharacteristic
112C23×C47C4×Q812yes—
113C23×C47C2×M1611yescharacteristic
114C23×C47C2×M1611yescharacteristic
115C23×C47#13112yes—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—22no—
125C2×C4×C47—22no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—42no—
134C2×C4×C47—42no—
135C2×C4×C47—44no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47—42no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—82no—
153C2×C4×C47—82no—
154C2×C4×C47#9912yes—
155C2×C4×C47C22⋊Q812yes—
156C2×C4×C47#12411yescharacteristic
157C2×C4×C47C4⋊Q811yescharacteristic
158C47⋊C8—44no—
159C47⋊C8—82no—
160#9881—471nomaximal, Sylow
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—22no—
166C22×C47—22no—
167C22×C47—22no—
168C22×C47—22no—
169C22×C47#39111yescharacteristic
170C22×C47#40711yescharacteristic
171C22×C47#40811yescharacteristic
172C22×C47C2×#9911yescharacteristic
173C22×C47#41011yescharacteristic
174C22×C47#41311yescharacteristic
175C22×C47#41311yescharacteristic
176C22×C47#44612yes—
177C22×C47#44612yes—
178C4×C47—22no—
179C4×C47—22no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—42no—
183C4×C47—41no—
184C4×C47—41no—
185C4×C47—42no—
186C4×C47—41no—
187C4×C47—81no—
188C4×C47—81no—
189C2×#387D4711yescharacteristic
190#1797—471no—
191#1749—471no—
192#1280—472no—
193#1280—472no—
194C2×C47—22no—
195C2×C47—22no—
196C2×C47#146411yescharacteristic
197C2×C47#146511yescharacteristic
198C2×C47#146811yescharacteristic
199C2×C47#147311yescharacteristic
200C2×C47#198511yescharacteristic
201C2×C47#199611yescharacteristic
202C2×C47#199811yescharacteristic
203C22×C4⋊C4Dic4711yescharacteristic
204C22×C4⋊C4Dic4711yescharacteristic
205C22×C42C2×D4711yescharacteristic
206C2×C4⋊C8—941no—
207C2×C22⋊C8—472no—
208C2×C22⋊C8—472no—
209C2×#97—941no—
210#387—21no—
211#387—21no—
212#387—21no—
213#387—21no—
214C2×#95Dic4711yescharacteristic
215C2×#95Dic4711yescharacteristic
216C2×#95C2×D4711yescharacteristic
217C2×#95C2×D4711yescharacteristic
218#336—941no—
219#336—941no—
220C47#988111yescharacteristic, complemented, Sylow
221C23×C4C2×Dic4711yescharacteristic
222C23×C4C2×Dic4711yescharacteristic
223C23×C4C2×Dic4711yescharacteristic
224C23×C4C2×Dic4711yescharacteristic
225C23×C4C2×Dic4711yescharacteristic
226C23×C4C2×Dic4711yescharacteristic
227C23×C4C22×D4711yescharacteristic
228C22×C8—942no—
229C22×C8—942no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—42no—
237C2×C42—42no—
238C2×C42C22⋊D4711yescharacteristic
239C2×C42C22⋊D4711yescharacteristic
240C2×C42C22⋊D4711yescharacteristic
241C2×C42C22⋊D4711yescharacteristic
242C4⋊C8—1882no—
243C22⋊C8—474no—
244C22⋊C8—942no—
245C22⋊C8—944no—
246#97—1882no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255C24C22×Dic4711yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4#1062532912yes—
282C22×C4#1062532912yes—
283C22×C4C22⋊Dic4711yescharacteristic
284C22×C4C22⋊Dic4711yescharacteristic
285C22×C4C22⋊Dic4711yescharacteristic
286C22×C4C22⋊Dic4711yescharacteristic
287C22×C4Q8×D4712yes—
288C22×C4#1062535412yes—
289C22×C4C2×C22⋊D4711yescharacteristic
290C22×C4C2×C22⋊D4711yescharacteristic
291C2×C8—944no—
292C2×C8—1882no—
293C2×C8—1882no—
294C2×C8—1882no—
295C2×C8—1882no—
296C2×C8—1882no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C42—22no—
302C42—22no—
303C42—41no—
304C42—41no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×C22⋊Dic4711yescharacteristic
310C23Q8×Dic4712yes—
311C23C2×#1062532911yescharacteristic
312C23C2×#1062532911yescharacteristic
313C23#1309403112yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4#1309393912yes—
353C2×C4#1309400012yes—
354C2×C4#1309401311yescharacteristic
355C2×C4#1309402211yescharacteristic
356C8—3762no—
357C8—1884no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22#1568903011yescharacteristic
367C22#1568908711yescharacteristic
368C22#1568909011yescharacteristic
369C22C2×#1309393911yescharacteristic
370C22#1568909711yescharacteristic
371C22#1568910711yescharacteristic
372C22#1568910711yescharacteristic
373C22#1568925412yes—
374C22#1568925412yes—
375C4—22no—
376C4—21no—
377C4—22no—
378C4—21no—
379C4—42no—
380C4—41no—
381C4—41no—
382C4—42no—
383C4—41no—
384C4—81no—
385C4—81no—
386C2—22no—
387C2—22no—
388C2#2024172211yescharacteristic
389C2#2024172611yescharacteristic
390C2#2024173611yescharacteristic
391C2#2024175711yescharacteristic
392C2#2024432911yescharacteristic
393C2#2024437411yescharacteristic
394C2#2024438211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutator, prime radicalC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#387C2
socleC23×C47C2×C22⋊C4
cosocleC23×C4×C47C23
hypercenterC2×#387D47
hypocenterC47#9881
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
220. C47#9881semidirect

Maximal subgroups

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

subgroupindex
#202433682
#202431502
#202412312
C2×C47×#3872
#988147