Finite Group Databasebeta

← #26114129

Subgroups — #26114129

Order 12032.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252468C211yesmaximal, characteristic, complemented
2#20248297C211yesmaximal, characteristic
3#20243381C211yesmaximal, characteristic, complemented
4#20241813C211yesmaximal, characteristic
5#20241248C211yesmaximal, characteristic, complemented
6#20241247C211yesmaximal, characteristic, complemented
7C47×#1486C211yesmaximal, characteristic
8C2×#13094003—21no—
9C2×C4×Dic94C2211yescharacteristic
10C8×Dic94—22no—
11#15689247—22no—
12#15689176C2211yescharacteristic
13#15689173—22no—
14#15689147C2211yescharacteristic
15C2×#13093942—21no—
16#15689054C412yes—
17#15689037C412yes—
18#15689011—22no—
19#15688989C2211yescharacteristic
20#15688983—21no—
21#15688891—21no—
22D4⋊(C47⋊C8)—22no—
23C2×C4×D4×C47C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C47×#423C2211yescharacteristic
26C47×D4⋊C8—22no—
27C22×Dic94D411yescharacteristic
28C2×#10625329—21no—
29#13094003—42no—
30#13094003—42no—
31C4×Dic94—21no—
32C4×Dic94—21no—
33C4×Dic94—21no—
34C4×Dic94—21no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47D411yescharacteristic
37C2×Dic47⋊C4—22no—
38C2×Dic47⋊C4C2×C412yes—
39C2×C22⋊Dic47—22no—
40C2×C22⋊Dic47—22no—
41C2×C4×Dic47D412yes—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45Dic47⋊C8—24no—
46#13093942—42no—
47C22⋊(C47⋊C8)—21no—
48C4×C47⋊C8D412yes—
49C8×Dic47—24no—
50C47⋊#95—21no—
51C47⋊#95—22no—
52C47⋊#95C2×C412yes—
53C22×D4×C47C2×C411yescharacteristic
54C23×C4×C47—21no—
55C2×M16×C47—21no—
56C4×D4×C47—21no—
57C4×D4×C47—21no—
58C4×D4×C47—21no—
59C4×D4×C47—21no—
60C2×C4⋊C4×C47C2×C411yescharacteristic
61C2×C22⋊C4×C47—21no—
62C2×C42×C47C2311yescharacteristic
63C4⋊C8×C47—21no—
64C4⋊C8×C47—21no—
65C4⋊C8×C47—21no—
66C22⋊C8×C47—21no—
67C4×C8×C47D412yes—
68C24×C47—21no—
69C2×D4×C47—21no—
70C2×D4×C47—21no—
71C2×D4×C47—42no—
72C22×C4×C47—21no—
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×C47C22×C411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×Dic47—22no—
83C22×Dic47C22⋊C412yes—
84C22×Dic47C22⋊C412yes—
85C2×Dic94—21no—
86C2×Dic94—21no—
87C2×Dic94—21no—
88C2×Dic94—21no—
89C2×Dic94D812yes—
90C2×Dic94SD1612yes—
91M16×C47—41no—
92M16×C47—41no—
93C2×C8×C47—22no—
94C2×C8×C47—41no—
95C4⋊C4×C47—21no—
96C4⋊C4×C47—21no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—41no—
99C42×C47C2×D411yescharacteristic
100C42×C47C2×D411yescharacteristic
101C42×C47#4111yescharacteristic
102C42×C47#4111yescharacteristic
103C22⋊Dic47—44no—
104C22⋊Dic47—44no—
105C4⋊Dic47—21no—
106C4⋊Dic47—21no—
107C4⋊Dic47—41no—
108C4⋊Dic47—41no—
109Dic47⋊C4—22no—
110Dic47⋊C4—22no—
111Dic47⋊C4—44no—
112C4×Dic47—22no—
113C4×Dic47—22no—
114#10625329—41no—
115#10625329—41no—
116C2×C47⋊C8—22no—
117C2×C47⋊C8—41no—
118C23×C47—22no—
119C23×C47—41no—
120C23×C47—41no—
121C23×C47—41no—
122C23×C47—41no—
123C23×C47—41no—
124C23×C47—41no—
125C23×C47C2×C22⋊C411yescharacteristic
126D4×C47—22no—
127D4×C47—22no—
128D4×C47—41no—
129D4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—81no—
136C2×C4×C47—81no—
137C2×C4×C47—81no—
138C2×C4×C47—81no—
139C2×C4×C47C4×D411yescharacteristic
140C2×C4×C47C4×D411yescharacteristic
141C2×C4×C47#12011yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×C4×C47#12111yescharacteristic
144C2×C4×C47#12311yescharacteristic
145C2×C4×C47C2×D811yescharacteristic
146C2×C4×C47C2×SD1611yescharacteristic
147C2×C4×C47#13511yescharacteristic
148C2×C4×C47#13511yescharacteristic
149C2×Dic47—22no—
150C2×Dic47—22no—
151C2×Dic47—22no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—42no—
156C2×Dic47#10214yes—
157Dic94—22no—
158Dic94—22no—
159Dic94—22no—
160Dic94—22no—
161C8×C47—42no—
162C8×C47—81no—
163C47⋊C8—42no—
164C47⋊C8—81no—
165#9486—471nomaximal, Sylow
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—42no—
169C22×C47—42no—
170C22×C47—42no—
171C22×C47—81no—
172C22×C47—81no—
173C22×C47—81no—
174C22×C47—81no—
175C22×C47#38611yescharacteristic
176C22×C47C2×#10211yescharacteristic
177C22×C47#41611yescharacteristic
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—81no—
183C4×C47#46511yescharacteristic
184C4×C47#46511yescharacteristic
185C4×C47#46611yescharacteristic
186C4×C47#46611yescharacteristic
187Dic47—24no—
188Dic47—42no—
189Dic47—82no—
190#2982—471no—
191#2536—471no—
192#1801—471no—
193#1486—471no—
194#1486D4711yescharacteristic
195#1286—471no—
196#1286—471no—
197C2×C47—21no—
198C2×C47—21no—
199C2×C47—82no—
200C2×C47#189111yescharacteristic
201C2×C47#189311yescharacteristic
202C2×C47#191211yescharacteristic
203C2×C22⋊Q8—941no—
204C2×C4×Q8—471no—
205C2×C4×D4C2×D4711yescharacteristic
206C8×Q8—942no—
207C8×Q8—942no—
208#431—471no—
209#431—942no—
210#431C2×D4711yescharacteristic
211#423—471no—
212#423C2×D4711yescharacteristic
213C2×#101—941no—
214#398—472no—
215#394—472no—
216#388—942no—
217#382—471no—
218#380—941no—
219#343—941no—
220D4⋊C8—22no—
221D4⋊C8—942no—
222C47#948611yescharacteristic, complemented, Sylow
223C22×Q8—471no—
224C22×D4C4×D4711yescharacteristic
225C23×C4—21no—
226C2×M16—21no—
227C2×M16—941no—
228C22⋊Q8—1882no—
229C22⋊Q8—1882no—
230C4×Q8—941no—
231C4×Q8—941no—
232C4×Q8—941no—
233C4×Q8—941no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C2×C4⋊C4—472no—
239C2×C4⋊C4—471no—
240C2×C4⋊C4—942no—
241C2×C4⋊C4—941no—
242C2×C4⋊C4C4×D4711yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—942no—
245C2×C22⋊C4—942no—
246C2×C42—472no—
247C2×C42C22×D4711yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—944no—
252C4⋊C8—941no—
253C4⋊C8—941no—
254C4⋊C8—941no—
255#101—1882no—
256C22⋊C8—21no—
257C22⋊C8—941no—
258C4×C8—472no—
259C4×C8—944no—
260C4×C8C22⋊D4712yes—
261#95—472no—
262#95—942no—
263#95—941no—
264C24—21no—
265C2×Q8—472no—
266C2×Q8—472no—
267C2×Q8—941no—
268C2×Q8—941no—
269C2×Q8—941no—
270C2×Q8—941no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—42no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—472no—
282C22×C4—472no—
283C22×C4—942no—
284C22×C4C2×C4×D4711yescharacteristic
285C22×C4D4×D4711yescharacteristic
286C22×C4D4×D4711yescharacteristic
287M16—41no—
288M16—41no—
289M16—1881no—
290M16—1881no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—942no—
294C2×C8—1881no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—942no—
298C4⋊C4—942no—
299C4⋊C4—941no—
300C4⋊C4—941no—
301C4⋊C4—1884no—
302C4⋊C4—1881no—
303C4⋊C4—1881no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—1884no—
307C22⋊C4—1884no—
308C42—942no—
309C42—942no—
310C42#1062535011yescharacteristic
311C42D4×D4711yescharacteristic
312C42#1062535211yescharacteristic
313C42C2×C22⋊D4711yescharacteristic
314C23—22no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C22⋊C4×D4711yescharacteristic
322Q8—942no—
323Q8—942no—
324Q8—942no—
325Q8—942no—
326D4—22no—
327D4—22no—
328D4—41no—
329D4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—474no—
340C2×C4—942no—
341C2×C4—942no—
342C2×C4—942no—
343C2×C4—1882no—
344C2×C4—1882no—
345C2×C4—1882no—
346C2×C4—1882no—
347C2×C4(C22⋊D47)⋊C411yescharacteristic
348C2×C4C4×C22⋊D4711yescharacteristic
349C2×C4#1309399211yescharacteristic
350C2×C4#1309399511yescharacteristic
351C2×C4#1309399711yescharacteristic
352C2×C4#1309400611yescharacteristic
353C2×C4D8×D4711yescharacteristic
354C2×C4SD16×D4711yescharacteristic
355C2×C4#1309404311yescharacteristic
356C2×C4#1309404511yescharacteristic
357C8—42no—
358C8—81no—
359C8—3761no—
360C8—1882no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1568899911yescharacteristic
371C22#102×D4711yescharacteristic
372C22#1568911611yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—944no—
379C4—3762no—
380C4—1882no—
381C4#1568934911yescharacteristic
382C4#1568935111yescharacteristic
383C4#1568935611yescharacteristic
384C4#1568935811yescharacteristic
385C2—21no—
386C2—21no—
387C2—82no—
388C2#2024380311yescharacteristic
389C2#2024381311yescharacteristic
390C2#2024392211yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1486C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#1486D47
hypocenterC47#9486
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
1. #20252468C2semidirect
3. #20243381C2semidirect
5. #20241248C2semidirect
6. #20241247C2semidirect
222. C47#9486semidirect

Maximal subgroups

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

subgroupindex
#202524682
#202482972
#202433812
#202418132
#202412482
#202412472
C47×#14862
#948647