Finite Group Databasebeta

← #26127152

Subgroups — #26127152

Order 12032.

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

normal subgroups only

This group has 390 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#20248965C211yesmaximal, characteristic
2#20243465C211yesmaximal, characteristic
3#20241410C212yesmaximal
4#20241389C212yesmaximal
5C47×#1458C211yesmaximal, characteristic
6C2×C4×Dic94C411yescharacteristic, complemented
7#15689053C411yescharacteristic, complemented
8#15689048C2211yescharacteristic
9#15689043C411yescharacteristic, complemented
10#15689035C2211yescharacteristic
11#15689011—21no—
12#15689007—21no—
13#15689004—21no—
14#15688996—21no—
15#15688991C411yescharacteristic, complemented
16#15688986—21no—
17#15688890—21no—
18#15688890C2212yes—
19C2×#117×C47C2211yescharacteristic
20C47×#431—21no—
21C47×#324—21no—
22C47×#323C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C4×Dic94—41no—
25C4×Dic94—41no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—22no—
33C2×C4×Dic47—21no—
34C47⋊#95—21no—
35C47⋊#95—21no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—41no—
41C47⋊#95—41no—
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×C47D411yescharacteristic
50C2×C22⋊C4×C47C2311yescharacteristic
51C2×C42×C47C2×C411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C4⋊C8×C47—41no—
54C22⋊C8×C47—22no—
55C4×C8×C47—41no—
56C24×C47C2×D411yescharacteristic
57C22×C4×C47—22no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—41no—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×Dic47—21no—
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C22×Dic47—42no—
72C2×Dic94—41no—
73C2×Dic94—42no—
74C2×C8×C47—42no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C22⋊C4×C47—41no—
78C22⋊C4×C47—41no—
79C42×C47—21no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47—21no—
83C22⋊Dic47—42no—
84C22⋊Dic47—42no—
85C22⋊Dic47—42no—
86C22⋊Dic47—42no—
87C4⋊Dic47—41no—
88Dic47⋊C4—42no—
89Dic47⋊C4—42no—
90Dic47⋊C4—41no—
91Dic47⋊C4—41no—
92C4×Dic47—42no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47C2×C22⋊C411yescharacteristic
98C23×C47#13512yes—
99C2×C4×C47—21no—
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—21no—
107C2×C4×C47—21no—
108C2×C4×C47—42no—
109C2×C4×C47—42no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47#9911yescharacteristic
115C2×C4×C47#9911yescharacteristic
116C2×Dic47—22no—
117C2×Dic47—41no—
118C2×Dic47—41no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—82no—
129C2×Dic47—82no—
130C2×Dic47—82no—
131Dic94—42no—
132Dic94—42no—
133C8×C47—82no—
134#11842—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—22no—
138C22×C47—42no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—81no—
142C22×C47C2×#9911yescharacteristic
143C22×C47#41611yescharacteristic
144C22×C47#42111yescharacteristic
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152Dic47—42no—
153Dic47—81no—
154Dic47—81no—
155Dic47—81no—
156Dic47—162no—
157#2599—471no—
158#1820—471no—
159#1458D4711yescharacteristic
160#1351—472no—
161#1344—472no—
162C2×C47—21no—
163C2×C47—21no—
164C2×C47—42no—
165C2×C47#151611yescharacteristic
166C2×C47#212011yescharacteristic
167C2×C47#212911yescharacteristic
168C2×C4×Q8—471no—
169C2×#117C2×D4711yescharacteristic
170#431—21no—
171#398—471no—
172#397—471no—
173#395—471no—
174#393—471no—
175#388—941no—
176#387—941no—
177#387—941no—
178#385—941no—
179#383—471no—
180#381—941no—
181#342—472no—
182#342—941no—
183#324—21no—
184#323C2×D4712yes—
185C47#1184211yescharacteristic, complemented, Sylow
186C22×Q8—471no—
187C23×C4D18811yescharacteristic
188C4×Q8—1881no—
189C4×Q8—1881no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—471no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4C4×D4711yescharacteristic
197C2×C4⋊C4C22⋊D4711yescharacteristic
198C2×C22⋊C4—941no—
199C2×C22⋊C4—941no—
200C2×C22⋊C4—942no—
201C2×C22⋊C4D18811yescharacteristic
202C2×C22⋊C4C22×D4711yescharacteristic
203C2×C42—941no—
204C2×C42C4×D4711yescharacteristic
205C2×C42C22⋊D4711yescharacteristic
206C4⋊C8—41no—
207C22⋊C8—22no—
208C4×C8—41no—
209#95—471no—
210#95—471no—
211#95—941no—
212#95—941no—
213#95—941no—
214#95—941no—
215#95—941no—
216#95—941no—
217#95—1881no—
218#95—1881no—
219C24C2×D18811yescharacteristic
220C2×Q8—1881no—
221C2×Q8—1882no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—941no—
227C22×C4—941no—
228C22×C4—941no—
229C22×C4—941no—
230C22×C4—1882no—
231C22×C4(C2×D47)⋊C411yescharacteristic
232C22×C4(C2×D47)⋊C411yescharacteristic
233C22×C4(C2×D47)⋊C411yescharacteristic
234C22×C4(C2×D47)⋊C411yescharacteristic
235C22×C4C2×C4×D4711yescharacteristic
236C22×C4C2×C22⋊D4711yescharacteristic
237C2×C8—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—1881no—
241C4⋊C4—1882no—
242C4⋊C4—1882no—
243C4⋊C4—1881no—
244C4⋊C4—1881no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—1882no—
248C22⋊C4—1882no—
249C22⋊C4—1882no—
250C22⋊C4—1882no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—1882no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23C2×(C2×D47)⋊C411yescharacteristic
261C23#1309404612yes—
262Q8—1882no—
263Q8—1882no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—942no—
280C2×C4—3762no—
281C2×C4—3762no—
282C2×C4—3762no—
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#1309393811yescharacteristic
295C2×C4#1309393811yescharacteristic
296C8—82no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22C2×#1309393811yescharacteristic
305C22#1568912011yescharacteristic
306C22#1568913911yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—3761no—
315C4—3761no—
316C4—3761no—
317C4—1882no—
318C4—7522no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2#2024194511yescharacteristic
323C2#2024503011yescharacteristic
324C2#2024506211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1458C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1458D47
hypocenterC47#11842
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic94C4semidirect
7. #15689053C4semidirect
9. #15689043C4semidirect
15. #15688991C4semidirect
185. C47#11842semidirect

Maximal subgroups

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

subgroupindex
#202489652
#202434652
#202414102
#202413892
C47×#14582
#1184247