Finite Group Databasebeta

← #26114004

Subgroups — #26114004

Order 12032.

528 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 93 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#20247867C211yesmaximal, characteristic
2#20245989C211yesmaximal, characteristic
3#20243466C211yesmaximal, characteristic
4#20241921C211yesmaximal, characteristic
5#20241242C211yesmaximal, characteristic
6#20241240C211yesmaximal, characteristic
7C47×#1511C211yesmaximal, characteristic
8C22×C4⋊Dic47C2211yescharacteristic
9#15689087C2211yescharacteristic
10#15689053—22no—
11#15689030C412yescomplemented
12#15689008—22no—
13#15689006—22no—
14#15688995—21no—
15#15688995—21no—
16#15688992C412yescomplemented
17#15688986C2211yescharacteristic
18#15688981—21no—
19C22⋊C4×Dic47—21no—
20C4×C22⋊Dic47C2211yescharacteristic
21#15688891—22no—
22#15688860—22no—
23C2×C22⋊Q8×C47C2211yescharacteristic
24C47×#407C2211yescharacteristic
25C47×#381C2211yescharacteristic
26C47×#328—22no—
27C23×Dic47D411yescharacteristic
28C23×Dic47D411yescharacteristic
29C2×#10625329—21no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47C2×C412yes—
35C2×C4⋊Dic47D412yes—
36C2×Dic47⋊C4—22no—
37C2×Dic47⋊C4—22no—
38C2×Dic47⋊C4—22no—
39C2×C22⋊Dic47—21no—
40C2×C22⋊Dic47—21no—
41C2×C4×Dic47—22no—
42C2×C4×Dic47C2×C412yes—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—21no—
45C47⋊#95—22no—
46C47⋊#95—22no—
47C47⋊#95—21no—
48C47⋊#95—21no—
49C47⋊#95—22no—
50C47⋊#95—22no—
51C47⋊#95D412yes—
52C22×Q8×C47C2×C411yescharacteristic
53C23×C4×C47C2311yescharacteristic
54C2×M16×C47—21no—
55C22⋊Q8×C47—21no—
56C22⋊Q8×C47—21no—
57C22⋊Q8×C47—21no—
58C22⋊Q8×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47C2×C411yescharacteristic
61C2×C22⋊C4×C47—21no—
62C2×C22⋊C4×C47—21no—
63C22⋊C8×C47—21no—
64C22⋊C8×C47—21no—
65#95×C47—21no—
66#95×C47D412yes—
67C24×C47C2×D411yescharacteristic
68C2×Q8×C47—21no—
69C2×Q8×C47—21no—
70C2×Q8×C47—42no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—22no—
74C22×C4×C47—42no—
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47#4111yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×Dic47—22no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C22×Dic47—21no—
85C22×Dic47—21no—
86C22×Dic47—22no—
87C22×Dic47—81no—
88C22×Dic47—81no—
89C22×Dic47C22⋊C412yes—
90C22×Dic47C22⋊C412yes—
91M16×C47—42no—
92C2×C8×C47—41no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—21no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C22⋊C4×C47—42no—
98C22⋊C4×C47—41no—
99C22⋊C4×C47—41no—
100C22⋊Dic47—24no—
101C22⋊Dic47—42no—
102C4⋊Dic47—22no—
103C4⋊Dic47—22no—
104C4⋊Dic47—22no—
105C4⋊Dic47—22no—
106Dic47⋊C4—82no—
107Dic47⋊C4—82no—
108Dic47⋊C4—82no—
109C4×Dic47—42no—
110C4×Dic47—82no—
111#10625329—42no—
112C2×C47⋊C8—41no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—41no—
116C23×C47C2×C22⋊C411yescharacteristic
117C23×C47#12011yescharacteristic
118C23×C47#12311yescharacteristic
119Q8×C47—81no—
120Q8×C47—81no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47C4×D411yescharacteristic
135C2×C4×C47C4×D411yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C2×C4×C47#12111yescharacteristic
138C2×Dic47—24no—
139C2×Dic47—22no—
140C2×Dic47—22no—
141C2×Dic47—42no—
142C2×Dic47—42no—
143C2×Dic47—42no—
144C2×Dic47—42no—
145C2×Dic47—42no—
146C2×Dic47—42no—
147C2×Dic47—42no—
148C2×Dic47—42no—
149C2×Dic47—42no—
150C2×Dic47—81no—
151C2×Dic47—81no—
152C8×C47—81no—
153C47⋊C8—81no—
154#9461—471nomaximal, Sylow
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—44no—
163C22×C47#38611yescharacteristic
164C22×C47#41012yes—
165C22×C47#45612yes—
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—82no—
171C4×C47—81no—
172C4×C47—81no—
173Dic47—44no—
174Dic47—42no—
175Dic47—42no—
176Dic47—82no—
177#2502—471no—
178#2313—471no—
179#1821—471no—
180#1511—471no—
181#1511D4711yescharacteristic
182#1284—471no—
183#1284—471no—
184C2×C47—22no—
185C2×C47—22no—
186C2×C47—41no—
187C2×C47#188611yescharacteristic
188C2×C47#189911yescharacteristic
189C2×C47#191111yescharacteristic
190C2×C22⋊Q8C2×D4711yescharacteristic
191C22×C4⋊C4—471no—
192#407—471no—
193#407C2×D4711yescharacteristic
194#398—942no—
195#391—472no—
196#387—942no—
197#387—942no—
198#385—941no—
199#385—941no—
200#384—472no—
201#381—471no—
202#381C2×D4711yescharacteristic
203#380—941no—
204C4×C22⋊C4—471no—
205C4×C22⋊C4—941no—
206#343—942no—
207#328—22no—
208#328—942no—
209C47#946111yescharacteristic, complemented, Sylow
210C22×Q8C4×D4711yescharacteristic
211C23×C4—471no—
212C23×C4—471no—
213C23×C4C22×D4711yescharacteristic
214C2×M16—21no—
215C2×M16—941no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—472no—
222C2×C4⋊C4—472no—
223C2×C4⋊C4—941no—
224C2×C4⋊C4—941no—
225C2×C4⋊C4—941no—
226C2×C4⋊C4—941no—
227C2×C4⋊C4—942no—
228C2×C4⋊C4—942no—
229C2×C4⋊C4—942no—
230C2×C4⋊C4C4×D4711yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—941no—
234C2×C22⋊C4—941no—
235C2×C42—472no—
236C2×C42—942no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—941no—
240C22⋊C8—941no—
241#95—21no—
242#95—472no—
243#95—942no—
244#95—942no—
245#95—942no—
246#95—942no—
247#95—941no—
248#95—941no—
249#95C22⋊D4712yes—
250C24D4×D4711yescharacteristic
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—42no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—472no—
259C22×C4—472no—
260C22×C4—941no—
261C22×C4—941no—
262C22×C4—942no—
263C22×C4—941no—
264C22×C4—941no—
265C22×C4—942no—
266C22×C4—3761no—
267C22×C4—3761no—
268C22×C4C2×C4×D4711yescharacteristic
269C22×C4#1062535011yescharacteristic
270C22×C4D4×D4711yescharacteristic
271C22×C4D4×D4711yescharacteristic
272C22×C4#1062535211yescharacteristic
273C22×C4C2×C22⋊D4711yescharacteristic
274M16—42no—
275M16—1882no—
276C2×C8—41no—
277C2×C8—1881no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—942no—
283C4⋊C4—942no—
284C4⋊C4—942no—
285C4⋊C4—942no—
286C4⋊C4—3762no—
287C4⋊C4—3762no—
288C4⋊C4—3762no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—944no—
293C22⋊C4—1882no—
294C42—3762no—
295C42—1882no—
296C23—22no—
297C23—22no—
298C23—41no—
299C23C22⋊C4×D4711yescharacteristic
300C23#1309399211yescharacteristic
301C23#1309400611yescharacteristic
302Q8—81no—
303Q8—81no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—944no—
318C2×C4—942no—
319C2×C4—942no—
320C2×C4—3761no—
321C2×C4—3761no—
322C2×C4—1882no—
323C2×C4—1882no—
324C2×C4—1882no—
325C2×C4—1882no—
326C2×C4—1882no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1882no—
330C2×C4—1882no—
331C2×C4(C22⋊D47)⋊C411yescharacteristic
332C2×C4C4×C22⋊D4711yescharacteristic
333C2×C4#1309399511yescharacteristic
334C2×C4#1309399711yescharacteristic
335C8—81no—
336C8—3761no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—44no—
345C22#1568899911yescharacteristic
346C22#1568909512yes—
347C22#1568930412yes—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—82no—
353C4—81no—
354C4—81no—
355C4—3762no—
356C4—1884no—
357C4—1882no—
358C4—1882no—
359C2—22no—
360C2—22no—
361C2—41no—
362C2#2024377811yescharacteristic
363C2#2024384311yescharacteristic
364C2#2024391711yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1511C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1511D47
hypocenterC47#9461
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
11. #15689030C4semidirect
16. #15688992C4semidirect
209. C47#9461semidirect

Maximal subgroups

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

subgroupindex
#202478672
#202459892
#202434662
#202419212
#202412422
#202412402
C47×#15112
#946147