Finite Group Databasebeta

← #26126993

Subgroups — #26126993

Order 12032.

402 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 79 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#20246656C211yesmaximal, characteristic
2#20241722C211yesmaximal, characteristic
3#20241411C212yesmaximal
4#20241408C212yesmaximal
5C47×#1820C211yesmaximal, characteristic
6C2×C4×Dic94C411yescharacteristic, complemented
7#15689180—21no—
8#15689062C411yescharacteristic, complemented
9#15689059C2211yescharacteristic
10#15689056—21no—
11#15689046—21no—
12#15689043C411yescharacteristic, complemented
13#15689035C2211yescharacteristic
14C4×Dic47⋊C4—21no—
15C4×C4⋊Dic47C411yescharacteristic, complemented
16C4×C22⋊Dic47—21no—
17#15688849—21no—
18#15688847C2212yes—
19C2×#117×C47C2211yescharacteristic
20C47×#381—21no—
21C47×#342—21no—
22C47×#342C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C4×Dic94—41no—
25C4×Dic94—41no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—21no—
33C2×C4×Dic47—21no—
34C2×C4×Dic47—21no—
35C4⋊(C47⋊C8)—41no—
36C22⋊(C47⋊C8)—22no—
37#13093935—41no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45#117×C47—41no—
46#117×C47—41no—
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C22⋊C4×C47—22no—
50C2×C22⋊C4×C47D411yescharacteristic
51C2×C22⋊C4×C47C2311yescharacteristic
52C2×C42×C47C2×C411yescharacteristic
53C2×C42×C47D411yescharacteristic
54#95×C47—41no—
55#95×C47—41no—
56C24×C47C2×D411yescharacteristic
57C22×C4×C47—22no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—41no—
61C22×C4×C47—42no—
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C22×Dic47—21no—
72C2×Dic94—41no—
73C2×Dic94—42no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—41no—
78C22⋊C4×C47—42no—
79C22⋊C4×C47—42no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84C22⋊Dic47—42no—
85C22⋊Dic47—42no—
86C4⋊Dic47—42no—
87C4⋊Dic47—41no—
88Dic47⋊C4—42no—
89Dic47⋊C4—42no—
90Dic47⋊C4—42no—
91C4×Dic47—42no—
92C4×Dic47—41no—
93C4×Dic47—41no—
94C2×C47⋊C8—42no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47C2×C22⋊C411yescharacteristic
100C23×C47#13712yes—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—82no—
117C2×C4×C47—82no—
118C2×C4×C47—82no—
119C2×C4×C47#9911yescharacteristic
120C2×C4×C47#9911yescharacteristic
121C2×Dic47—22no—
122C2×Dic47—22no—
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—42no—
135C47⋊C8—82no—
136#11810—471nomaximal, Sylow
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—22no—
140C22×C47—42no—
141C22×C47—41no—
142C22×C47—41no—
143C22×C47—81no—
144C22×C47C2×#9911yescharacteristic
145C22×C47#41911yescharacteristic
146C22×C47#42111yescharacteristic
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—41no—
154C4×C47—162no—
155Dic47—42no—
156Dic47—42no—
157Dic47—81no—
158Dic47—81no—
159#2396—471no—
160#1820D4711yescharacteristic
161#1464—471no—
162#1352—472no—
163#1351—472no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—42no—
167C2×C47#151111yescharacteristic
168C2×C47#212011yescharacteristic
169C2×C47#212611yescharacteristic
170C2×C4×Q8—471no—
171C2×#117C2×D4711yescharacteristic
172#432—941no—
173#400—471no—
174#399—471no—
175#398—941no—
176#396—941no—
177#395—471no—
178#393—471no—
179#381—21no—
180C4×C4⋊C4—471no—
181C4×C4⋊C4—941no—
182C4×C22⋊C4—941no—
183#342—21no—
184#342C2×D4712yes—
185#324—941no—
186#323—472no—
187C47#1181011yescharacteristic, complemented, Sylow
188C22×Q8—471no—
189C23×C4C22⋊D4711yescharacteristic
190C4×Q8—1881no—
191C4×Q8—1881no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—471no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4—941no—
197C2×C4⋊C4—941no—
198C2×C4⋊C4—941no—
199C2×C4⋊C4C4×D4711yescharacteristic
200C2×C4⋊C4D18811yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—941no—
203C2×C22⋊C4—941no—
204C2×C22⋊C4C22⋊D4711yescharacteristic
205C2×C22⋊C4C22×D4711yescharacteristic
206C2×C42—941no—
207C2×C42—941no—
208C2×C42C4×D4711yescharacteristic
209C2×C42D18811yescharacteristic
210C4⋊C8—1881no—
211C22⋊C8—942no—
212#97—1881no—
213#95—41no—
214#95—41no—
215#95—471no—
216#95—471no—
217#95—941no—
218#95—941no—
219#95—941no—
220#95—941no—
221C24C2×C22⋊D4711yescharacteristic
222C2×Q8—1881no—
223C2×Q8—1882no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—941no—
230C22×C4—941no—
231C22×C4—941no—
232C22×C4—941no—
233C22×C4(C2×D47)⋊C411yescharacteristic
234C22×C4(C2×D47)⋊C411yescharacteristic
235C22×C4(C2×D47)⋊C411yescharacteristic
236C22×C4(C2×D47)⋊C411yescharacteristic
237C22×C4C2×C4×D4711yescharacteristic
238C22×C4C2×D18811yescharacteristic
239C2×C8—1882no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—1881no—
243C4⋊C4—1882no—
244C4⋊C4—1882no—
245C4⋊C4—1882no—
246C4⋊C4—1882no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—1882no—
252C22⋊C4—1882no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—1882no—
258C42—1881no—
259C42—1881no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×(C2×D47)⋊C411yescharacteristic
265C23#1309405412yes—
266Q8—1882no—
267Q8—1882no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—942no—
287C2×C4—942no—
288C2×C4—1881no—
289C2×C4—1881no—
290C2×C4—1881no—
291C2×C4—1881no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4—1881no—
295C2×C4—1881no—
296C2×C4—1881no—
297C2×C4—1881no—
298C2×C4#1309393811yescharacteristic
299C2×C4#1309393811yescharacteristic
300C8—3762no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×#1309393811yescharacteristic
309C22#1568913111yescharacteristic
310C22#1568914111yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—162no—
319C4—3761no—
320C4—3761no—
321C4—1882no—
322C4—1882no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2#2024192011yescharacteristic
327C2#2024502711yescharacteristic
328C2#2024505111yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1820C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1820D47
hypocenterC47#11810
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic94C4semidirect
8. #15689062C4semidirect
12. #15689043C4semidirect
15. C4×C4⋊Dic47C4semidirect
187. C47#11810semidirect

Maximal subgroups

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

subgroupindex
#202466562
#202417222
#202414112
#202414082
C47×#18202
#1181047