Finite Group Databasebeta

← #26105511

Subgroups — #26105511

Order 12032.

350 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 83 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#20251341C211yesmaximal, characteristic
2#20244945C211yesmaximal, characteristic
3#20244797C211yesmaximal, characteristic
4#20241213C211yesmaximal, characteristic
5#20241211C211yesmaximal, characteristic
6#20241210C211yesmaximal, characteristic
7C47×#2084C211yesmaximal, characteristic
8C2×#13093947—21no—
9#15689062—21no—
10#15689062—21no—
11#15689044—21no—
12#15689029C2211yescharacteristic
13#15689022—21no—
14#15689021C2211yescharacteristic
15#15689010—21no—
16#15689009—21no—
17#15688885—21no—
18#15688885—21no—
19#15688885—21no—
20#15688878C2211yescharacteristic
21#15688878C2211yescharacteristic
22C2×#106×C47—21no—
23C47×#389C2211yescharacteristic
24C47×#384C2211yescharacteristic
25C47×#340—21no—
26C47×#336C2211yescharacteristic
27C22×Dic94D411yescharacteristic
28C22×C47⋊C8—21no—
29C22×C47⋊C8—21no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47D411yescharacteristic
33C2×C4⋊Dic47D411yescharacteristic
34C2×C4⋊Dic47D411yescharacteristic
35C2×Dic47⋊C4—21no—
36C2×C4×Dic47—21no—
37#13093947—42no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C22×C8×C47—21no—
47C2×C4⋊C4×C47—21no—
48C2×C4⋊C4×C47—21no—
49C2×C4⋊C4×C47—21no—
50C2×C4⋊C4×C47—21no—
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C42×C47C2311yescharacteristic
54#106×C47—42no—
55#95×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—41no—
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—21no—
63C22×Dic47—21no—
64C22×Dic47—21no—
65C22×Dic47—21no—
66C2×Dic94—21no—
67C2×Dic94—21no—
68C2×Dic94—41no—
69C2×Dic94—41no—
70C2×C8×C47—22no—
71C2×C8×C47—41no—
72C2×C8×C47—41no—
73C4⋊C4×C47—21no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47SD1612yes—
83C42×C47—41no—
84C4⋊Dic47—21no—
85C4⋊Dic47—21no—
86C4⋊Dic47—41no—
87C4⋊Dic47—41no—
88C4⋊Dic47—81no—
89C4⋊Dic47—81no—
90Dic47⋊C4—41no—
91Dic47⋊C4—41no—
92C4×Dic47—81no—
93C2×C47⋊C8—41no—
94C2×C47⋊C8—41no—
95C2×C47⋊C8—41no—
96C2×C47⋊C8—41no—
97C2×C47⋊C8—41no—
98C2×C47⋊C8—41no—
99C23×C47#12011yescharacteristic
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—81no—
113C2×C4×C47—81no—
114C2×C4×C47—81no—
115C2×C4×C47C2×SD1611yescharacteristic
116C2×C4×C47#13511yescharacteristic
117C2×C4×C47#13511yescharacteristic
118C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C47#13711yescharacteristic
120C2×C4×C47#13711yescharacteristic
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133Dic94—42no—
134Dic94—81no—
135C8×C47—42no—
136C47⋊C8—81no—
137C47⋊C8—81no—
138#7818—471nomaximal, Sylow
139C22×C47#44811yescharacteristic
140C22×C47#44911yescharacteristic
141C22×C47#45211yescharacteristic
142C22×C47#45411yescharacteristic
143C22×C47#45511yescharacteristic
144C22×C47#45611yescharacteristic
145C22×C47#45811yescharacteristic
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—21no—
151C4×C47—21no—
152C4×C47—42no—
153C4×C47—81no—
154C4×C47—161no—
155Dic47—81no—
156Dic47—81no—
157Dic47—81no—
158Dic47—81no—
159#2805—471no—
160#2099—471no—
161#2084D4711yescharacteristic
162#2072—471no—
163#1272—471no—
164#1271—471no—
165#1271—471no—
166C2×C47#161111yescharacteristic
167C2×C47#161211yescharacteristic
168C2×C47#161511yescharacteristic
169C2×C47#162611yescharacteristic
170C2×C47#164311yescharacteristic
171C2×C47#165211yescharacteristic
172C2×C47#166411yescharacteristic
173C2×#106—21no—
174C2×#103—941no—
175#400—941no—
176#400—941no—
177#395—941no—
178#391—471no—
179#389—471no—
180#389—941no—
181#389C2×D4711yescharacteristic
182#387—941no—
183#387—941no—
184#384C2×D4711yescharacteristic
185#340—21no—
186#340—941no—
187#340—941no—
188#340—941no—
189#336—471no—
190#336—471no—
191#336C2×D4711yescharacteristic
192C47#781811yescharacteristic, complemented, Sylow
193C22×Q8—471no—
194C22×C8—21no—
195C22×C8—941no—
196C22×C8—941no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—471no—
202C2×C4⋊C4—471no—
203C2×C4⋊C4—471no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4—941no—
206C2×C4⋊C4—941no—
207C2×C4⋊C4C22⋊D4711yescharacteristic
208C2×C4⋊C4C22⋊D4711yescharacteristic
209C2×C42—941no—
210C2×C42C22×D4711yescharacteristic
211#106—42no—
212#103—1882no—
213#95—21no—
214#95—941no—
215#95—941no—
216#95—941no—
217#95—941no—
218#95—941no—
219#95—941no—
220#95—941no—
221#95—941no—
222C2×Q8—941no—
223C2×Q8—941no—
224C2×Q8—1881no—
225C2×Q8—1881no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—941no—
230C22×C4—941no—
231C22×C4—941no—
232C22×C4—941no—
233C22×C4D4×D4711yescharacteristic
234C22×C4D4×D4711yescharacteristic
235C22×C4C2×C22⋊D4711yescharacteristic
236C2×C8—22no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—1881no—
240C2×C8—1881no—
241C2×C8—1881no—
242C2×C8—1881no—
243C2×C8—1881no—
244C2×C8—1881no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—941no—
255C4⋊C4—941no—
256C4⋊C4—3761no—
257C4⋊C4—3761no—
258C4⋊C4—1881no—
259C4⋊C4—1881no—
260C4⋊C4—1881no—
261C4⋊C4—1881no—
262C4⋊C4#1062533512yes—
263C42—41no—
264C42—3761no—
265C23#1309399211yescharacteristic
266Q8—3761no—
267Q8—1882no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1881no—
290C2×C4—1881no—
291C2×C4—1881no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4—1881no—
295C2×C4C2×#1062533511yescharacteristic
296C2×C4#1309404511yescharacteristic
297C2×C4#1309404511yescharacteristic
298C2×C4#1309405111yescharacteristic
299C2×C4#1309405311yescharacteristic
300C2×C4#1309405411yescharacteristic
301C8—42no—
302C8—3761no—
303C8—3761no—
304C22#1568926111yescharacteristic
305C22#1568926711yescharacteristic
306C22#1568928211yescharacteristic
307C22#1568929411yescharacteristic
308C22#1568929911yescharacteristic
309C22#1568930511yescharacteristic
310C22#1568931211yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—42no—
318C4—81no—
319C4—161no—
320C4—3761no—
321C4—3761no—
322C4—3761no—
323C4—3761no—
324C2#2024239511yescharacteristic
325C2#2024240111yescharacteristic
326C2#2024241611yescharacteristic
327C2#2024248311yescharacteristic
328C2#2024259311yescharacteristic
329C2#2024265511yescharacteristic
330C2#2024272011yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2084C2
socle, prime radicalC23×C47#120
hypercenter#2084D47
hypocenterC47#7818
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
192. C47#7818semidirect

Maximal subgroups

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

subgroupindex
#202513412
#202449452
#202447972
#202412132
#202412112
#202412102
C47×#20842
#781847