Finite Group Databasebeta

← #26113455

Subgroups — #26113455

Order 12032.

500 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 107 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#20246273C211yesmaximal, characteristic
2#20244374C211yesmaximal, characteristic
3#20243487C211yesmaximal, characteristic
4#20241908C211yesmaximal, characteristic
5#20241293C211yesmaximal, characteristic
6#20241273C211yesmaximal, characteristic
7C47×#3095C211yesmaximal, characteristic
8#15689087C2211yescharacteristic
9#15689022—21no—
10#15689016—22no—
11#15688989—21no—
12#15688983C2211yescharacteristic
13C4⋊C4×Dic47—21no—
14C4×C22⋊Dic47C2211yescharacteristic
15C4×C22⋊Dic47C2211yescharacteristic
16#15688969—21no—
17#15688890—22no—
18#15688882—22no—
19#15688860—21no—
20#15688860—21no—
21C2×C22⋊Q8×C47C2211yescharacteristic
22C2×#117×C47C2211yescharacteristic
23C47×#485—22no—
24C47×#484—21no—
25C47×#483—21no—
26C47×#428C412yescomplemented
27C47×#419C412yescomplemented
28C47×#407C2211yescharacteristic
29C2×#10625329—21no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—22no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47D412yes—
35C2×C4×Dic47—22no—
36C2×C4×Dic47—21no—
37C2×C4×Dic47—21no—
38C22⋊(C47⋊C8)—21no—
39C22⋊(C47⋊C8)—21no—
40C47⋊#95—22no—
41C47⋊#95—22no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95D411yescharacteristic
45C47⋊#95D411yescharacteristic
46C22×Q8×C47C2×C411yescharacteristic
47C23×C4×C47C2311yescharacteristic
48C2×M16×C47C2×C412yes—
49C22⋊Q8×C47—21no—
50C22⋊Q8×C47—21no—
51C22⋊Q8×C47—21no—
52C22⋊Q8×C47—21no—
53#117×C47—21no—
54#117×C47—21no—
55#117×C47C2×C412yes—
56#117×C47D412yes—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47C2×C411yescharacteristic
59C2×C4⋊C4×C47D411yescharacteristic
60C2×C4⋊C4×C47D411yescharacteristic
61C2×C22⋊C4×C47—21no—
62C2×C22⋊C4×C47—21no—
63C2×C42×C47—21no—
64#107×C47—22no—
65#106×C47—22no—
66#103×C47—22no—
67#103×C47—22no—
68C22⋊C8×C47—21no—
69C22⋊C8×C47—21no—
70C24×C47C2×D411yescharacteristic
71C2×Q8×C47—21no—
72C2×Q8×C47—21no—
73C2×Q8×C47—42no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—42no—
79C22×C4×C47C22×C411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47#4111yescharacteristic
84C22×C4×C47#4111yescharacteristic
85C22×Dic47—21no—
86C22×Dic47—22no—
87C22×Dic47—21no—
88C22×Dic47—41no—
89M16×C47—42no—
90C2×C8×C47—22no—
91C4⋊C4×C47—21no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—21no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C4⋊C4×C47C22⋊C412yes—
98C4⋊C4×C47C22⋊C412yes—
99C22⋊C4×C47—22no—
100C22⋊C4×C47—22no—
101C22⋊C4×C47—41no—
102C22⋊C4×C47—41no—
103C42×C47—22no—
104C42×C47—22no—
105C22⋊Dic47—22no—
106C22⋊Dic47—42no—
107C22⋊Dic47—41no—
108C22⋊Dic47—42no—
109C4⋊Dic47—42no—
110C4⋊Dic47—82no—
111C4×Dic47—44no—
112C4×Dic47—42no—
113C4×Dic47—42no—
114#10625329—42no—
115C2×C47⋊C8—41no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47C4×D411yescharacteristic
121C23×C47#12011yescharacteristic
122C23×C47#12111yescharacteristic
123Q8×C47—42no—
124Q8×C47—81no—
125C2×C4×C47—22no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—81no—
139C2×C4×C47—81no—
140C2×C4×C47C2×C22⋊C411yescharacteristic
141C2×C4×C47C4×D411yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×C4×C47#12311yescharacteristic
144C2×Dic47—22no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—42no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C2×Dic47—41no—
152C2×Dic47—41no—
153C2×Dic47—42no—
154C2×Dic47—81no—
155C2×Dic47—81no—
156C8×C47—42no—
157C47⋊C8—81no—
158#9363—471nomaximal, Sylow
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—81no—
168C22×C47—81no—
169C22×C47#38611yescharacteristic
170C22×C47#45511yescharacteristic
171C22×C47#45611yescharacteristic
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—42no—
176C4×C47—42no—
177C4×C47—81no—
178C4×C47—81no—
179C4×C47#41012yes—
180Dic47—42no—
181Dic47—82no—
182Dic47—81no—
183Dic47—82no—
184#3095D4711yescharacteristic
185#2346—471no—
186#1996—471no—
187#1826—471no—
188#1508—471no—
189#1305—471no—
190#1297—471no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—41no—
194C2×C47—41no—
195C2×C47#188211yescharacteristic
196C2×C47#189911yescharacteristic
197C2×C47#190611yescharacteristic
198C2×C22⋊Q8C2×D4711yescharacteristic
199C2×#117C2×D4711yescharacteristic
200#485—22no—
201#484—21no—
202#483—21no—
203#428Dic4712yescomplemented
204#419Dic4712yescomplemented
205#407—471no—
206#407C2×D4711yescharacteristic
207#389—941no—
208#388—942no—
209#382—941no—
210#380—471no—
211C4×C4⋊C4—941no—
212C4×C22⋊C4—471no—
213C4×C22⋊C4—471no—
214#376—941no—
215#342—942no—
216#339—942no—
217#328—941no—
218#328—941no—
219C47#936311yescharacteristic, complemented, Sylow
220C22×Q8C2×Dic4711yescharacteristic
221C23×C4C22×D4711yescharacteristic
222C2×M16—941no—
223C2×M16C2×Dic4712yes—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228#117—21no—
229#117—21no—
230#117C2×Dic4712yes—
231#117C22⋊D4712yes—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—942no—
234C2×C4⋊C4—941no—
235C2×C4⋊C4C2×Dic4711yescharacteristic
236C2×C4⋊C4C22⋊D4711yescharacteristic
237C2×C4⋊C4C22⋊D4711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—472no—
241C2×C22⋊C4—941no—
242C2×C22⋊C4—941no—
243C2×C42—21no—
244C2×C42—942no—
245C2×C42—941no—
246C2×C42—941no—
247#107—22no—
248#106—22no—
249#103—22no—
250#103—22no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—941no—
254C22⋊C8—941no—
255#95—471no—
256#95—471no—
257#95—942no—
258#95—942no—
259#95—941no—
260#95—941no—
261C24D4×D4711yescharacteristic
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—42no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—941no—
271C22×C4—941no—
272C22×C4—942no—
273C22×C4—1881no—
274C22×C4D4×D4711yescharacteristic
275C22×C4#1062535211yescharacteristic
276C22×C4#1062535211yescharacteristic
277C22×C4C22×Dic4711yescharacteristic
278C22×C4C2×C22⋊D4711yescharacteristic
279C22×C4C2×C22⋊D4711yescharacteristic
280M16—42no—
281M16—1882no—
282C2×C8—22no—
283C2×C8—1881no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—3762no—
291C4⋊C4—1882no—
292C4⋊C4C22⋊Dic4712yes—
293C4⋊C4C22⋊Dic4712yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—942no—
299C22⋊C4—1882no—
300C22⋊C4—1881no—
301C22⋊C4—1882no—
302C42—22no—
303C42—22no—
304C42—1882no—
305C42—1882no—
306C42—1884no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23D4×Dic4711yescharacteristic
312C23#1309399211yescharacteristic
313C23#1309399611yescharacteristic
314Q8—42no—
315Q8—81no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—942no—
332C2×C4—3761no—
333C2×C4—3761no—
334C2×C4—1881no—
335C2×C4—1882no—
336C2×C4—1882no—
337C2×C4—1881no—
338C2×C4—1882no—
339C2×C4—1882no—
340C2×C4—1881no—
341C2×C4—1881no—
342C2×C4—1881no—
343C2×C4C2×C22⋊Dic4711yescharacteristic
344C2×C4D4×Dic4711yescharacteristic
345C2×C4#1309399611yescharacteristic
346C2×C4#1309400511yescharacteristic
347C8—42no—
348C8—3761no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22#1568900311yescharacteristic
360C22#1568929911yescharacteristic
361C22#1568930411yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—42no—
367C4—81no—
368C4—81no—
369C4—3761no—
370C4—3762no—
371C4—3762no—
372C4—1882no—
373C4#1568909712yes—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2024375911yescharacteristic
379C2#2024384711yescharacteristic
380C2#2024389111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#3095C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#3095D47
hypocenterC47#9363
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3095

Complements

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

NH ≅ G/Ntype
26. C47×#428C4semidirect
27. C47×#419C4semidirect
203. #428Dic47semidirect
204. #419Dic47semidirect
219. C47#9363semidirect

Maximal subgroups

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

subgroupindex
#202462732
#202443742
#202434872
#202419082
#202412932
#202412732
C47×#30952
#936347