Finite Group Databasebeta

← #26104833

Subgroups — #26104833

Order 12032.

394 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 95 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#20251747C211yesmaximal, characteristic
2#20243530C211yesmaximal, characteristic
3#20243135C211yesmaximal, characteristic
4#20241212C211yesmaximal, characteristic
5#20241211C211yesmaximal, characteristic
6#20241210C211yesmaximal, characteristic
7C47×#1271C211yesmaximal, characteristic
8C2×#13094014—21no—
9C2×#13093934—21no—
10#15689022C411yescharacteristic
11#15689021C2211yescharacteristic
12#15689020—21no—
13#15689019C411yescharacteristic
14#15689018—21no—
15#15688992C411yescharacteristic
16#15688992C411yescharacteristic
17#15688987—21no—
18C42×Dic47C2211yescharacteristic
19#15688878C2211yescharacteristic
20#15688878C2211yescharacteristic
21#15688877—21no—
22#15688877—21no—
23#15688877—21no—
24C47×#389C2211yescharacteristic
25C47×#336C2211yescharacteristic
26C47×#336C2211yescharacteristic
27C22×C47⋊C8—21no—
28C22×C47⋊C8—21no—
29#13094014—42no—
30#13094014—42no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47C2×C411yescharacteristic
33C2×C4⋊Dic47C2×C411yescharacteristic
34C2×Dic47⋊C4—21no—
35C2×Dic47⋊C4—21no—
36C2×Dic47⋊C4—21no—
37C2×Dic47⋊C4—21no—
38C2×Dic47⋊C4—21no—
39C2×Dic47⋊C4—21no—
40C2×C4×Dic47C2×C411yescharacteristic
41C2×C4×Dic47C2×C411yescharacteristic
42C2×C4×Dic47D411yescharacteristic
43C2×C4×Dic47D411yescharacteristic
44C2×C4×Dic47D411yescharacteristic
45C2×C4×Dic47D411yescharacteristic
46#13093934—24no—
47C47⋊#95—21no—
48C47⋊#95—21no—
49C47⋊#95—21no—
50C22×C8×C47—21no—
51C22×C8×C47—21no—
52C2×C4⋊C4×C47—21no—
53C2×C4⋊C4×C47C2×C411yescharacteristic
54C2×C4⋊C4×C47C2×C411yescharacteristic
55C2×C42×C47C2311yescharacteristic
56#95×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—21no—
63C22×Dic47—21no—
64C22×Dic47C22⋊C411yescharacteristic
65C22×Dic47C22⋊C411yescharacteristic
66C22×Dic47C22⋊C411yescharacteristic
67C22×Dic47C22⋊C411yescharacteristic
68C2×C8×C47—22no—
69C2×C8×C47—41no—
70C2×C8×C47—41no—
71C2×C8×C47—41no—
72C2×C8×C47—41no—
73C2×C8×C47—41no—
74C4⋊C4×C47—21no—
75C4⋊C4×C47—21no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C42×C47—41no—
80C4⋊Dic47—21no—
81C4⋊Dic47—21no—
82C4⋊Dic47—42no—
83C4⋊Dic47—41no—
84Dic47⋊C4—42no—
85Dic47⋊C4—42no—
86Dic47⋊C4—42no—
87Dic47⋊C4—42no—
88Dic47⋊C4—41no—
89Dic47⋊C4—41no—
90Dic47⋊C4—42no—
91C4×Dic47—21no—
92C4×Dic47—21no—
93C4×Dic47—21no—
94C4×Dic47—21no—
95C4×Dic47—21no—
96C4×Dic47—21no—
97C4×Dic47—22no—
98C4×Dic47—22no—
99C4×Dic47—41no—
100C2×C47⋊C8—22no—
101C2×C47⋊C8—41no—
102C2×C47⋊C8—41no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C23×C47C2×C22⋊C411yescharacteristic
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47#13111yescharacteristic
120C2×C4×C47#13111yescharacteristic
121C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C47#13711yescharacteristic
124C2×C4×C47#13711yescharacteristic
125C2×Dic47—21no—
126C2×Dic47—21no—
127C2×Dic47—21no—
128C2×Dic47—21no—
129C2×Dic47—21no—
130C2×Dic47—21no—
131C2×Dic47—21no—
132C2×Dic47—21no—
133C2×Dic47—21no—
134C2×Dic47—21no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47#10112yes—
142C2×Dic47#10412yes—
143C8×C47—42no—
144C8×C47—81no—
145C47⋊C8—42no—
146C47⋊C8—81no—
147#7703—471nomaximal, Sylow
148C22×C47#40811yescharacteristic
149C22×C47#41011yescharacteristic
150C22×C47C2×#10111yescharacteristic
151C22×C47#41811yescharacteristic
152C22×C47#41911yescharacteristic
153C22×C47C2×#10411yescharacteristic
154C22×C47#42111yescharacteristic
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—21no—
160C4×C47—21no—
161C4×C47—81no—
162C4×C47—81no—
163Dic47—22no—
164Dic47—22no—
165Dic47—41no—
166Dic47—41no—
167Dic47—81no—
168Dic47—81no—
169#2859—471no—
170#1839—471no—
171#1745—471no—
172#1271—471no—
173#1271—471no—
174#1271—471no—
175#1271D4711yescharacteristic
176C2×C47#147011yescharacteristic
177C2×C47#148411yescharacteristic
178C2×C47#149711yescharacteristic
179C2×C47#150611yescharacteristic
180C2×C47#151411yescharacteristic
181C2×C47#152811yescharacteristic
182C2×C47#155611yescharacteristic
183C2×#125—941no—
184C2×#97—941no—
185#389—471no—
186#389—471no—
187#389—471no—
188#389—941no—
189#389—941no—
190#389C2×D4711yescharacteristic
191#384—471no—
192#384—471no—
193#382—941no—
194C43—471no—
195#336—471no—
196#336—471no—
197#336—941no—
198#336—941no—
199#336—941no—
200#336C2×D4711yescharacteristic
201#336C2×D4711yescharacteristic
202C47#770311yescharacteristic, complemented, Sylow
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—941no—
206C22×C8—941no—
207#125—1882no—
208#125—1882no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—471no—
211C2×C4⋊C4—471no—
212C2×C4⋊C4—941no—
213C2×C4⋊C4—941no—
214C2×C4⋊C4—941no—
215C2×C4⋊C4—941no—
216C2×C4⋊C4—941no—
217C2×C4⋊C4—941no—
218C2×C4⋊C4—941no—
219C2×C4⋊C4C4×D4711yescharacteristic
220C2×C4⋊C4C4×D4711yescharacteristic
221C2×C42—471no—
222C2×C42—471no—
223C2×C42—471no—
224C2×C42—471no—
225C2×C42—471no—
226C2×C42—471no—
227C2×C42C22×D4711yescharacteristic
228#97—944no—
229#95—21no—
230#95—941no—
231#95—941no—
232#95—941no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—471no—
236C22×C4—471no—
237C22×C4—471no—
238C22×C4—471no—
239C22×C4—941no—
240C22×C4—941no—
241C22×C4C2×C4×D4711yescharacteristic
242C22×C4D4×D4711yescharacteristic
243C22×C4D4×D4711yescharacteristic
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—942no—
251C2×C8—1881no—
252C2×C8—1881no—
253C2×C8—1881no—
254C2×C8—1881no—
255C2×C8—1881no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—941no—
262C4⋊C4—941no—
263C4⋊C4—1881no—
264C4⋊C4—1882no—
265C4⋊C4—1882no—
266C4⋊C4—1882no—
267C4⋊C4—1882no—
268C4⋊C4—1882no—
269C4⋊C4—1882no—
270C4⋊C4—1881no—
271C4⋊C4—1881no—
272C42—41no—
273C42—942no—
274C42—942no—
275C42—941no—
276C42—941no—
277C42—941no—
278C42—941no—
279C42—941no—
280C42—941no—
281C42—1881no—
282C23C22⋊C4×D4711yescharacteristic
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—472no—
296C2×C4—472no—
297C2×C4—941no—
298C2×C4—941no—
299C2×C4—941no—
300C2×C4—941no—
301C2×C4—941no—
302C2×C4—941no—
303C2×C4—941no—
304C2×C4—941no—
305C2×C4—941no—
306C2×C4—941no—
307C2×C4—1881no—
308C2×C4—1881no—
309C2×C4—1881no—
310C2×C4—1881no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4#1309403011yescharacteristic
314C2×C4#1309403211yescharacteristic
315C2×C4#1309404811yescharacteristic
316C2×C4#1309405111yescharacteristic
317C2×C4#1309405311yescharacteristic
318C2×C4#1309405611yescharacteristic
319C8—42no—
320C8—81no—
321C8—3761no—
322C8—1882no—
323C22#1568908811yescharacteristic
324C22#1568909511yescharacteristic
325C22#101×D4711yescharacteristic
326C22#1568912611yescharacteristic
327C22#1568913011yescharacteristic
328C22#104×D4711yescharacteristic
329C22#1568913811yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—81no—
337C4—81no—
338C4—942no—
339C4—942no—
340C4—3761no—
341C4—3761no—
342C4—1881no—
343C4—1881no—
344C2#2024174211yescharacteristic
345C2#2024180311yescharacteristic
346C2#2024185911yescharacteristic
347C2#2024189811yescharacteristic
348C2#2024193311yescharacteristic
349C2#2024198811yescharacteristic
350C2#2024210311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1271C2
socle, prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1271D47
hypocenterC47#7703
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
202. C47#7703semidirect

Maximal subgroups

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

subgroupindex
#202517472
#202435302
#202431352
#202412122
#202412112
#202412102
C47×#12712
#770347