Finite Group Databasebeta

← #26127189

Subgroups — #26127189

Order 12032.

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

normal subgroups only

This group has 396 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#20247675C211yesmaximal, characteristic
2C2×#15688849C211yesmaximal, characteristic
3#20241414C212yesmaximal, complemented
4#20241393C212yesmaximal, complemented
5C47×#1821C211yesmaximal, characteristic
6C2×#13093983C411yescharacteristic
7C2×C22⋊(C47⋊C8)—21no—
8#15689062C411yescharacteristic
9#15689062C2211yescharacteristic
10#15689053C411yescharacteristic
11#15689053C2211yescharacteristic
12#15689004—21no—
13#15689004—21no—
14#15688996—21no—
15#15688986—21no—
16#15688986C411yescharacteristic
17#15688849—21no—
18#15688849C2212yes—
19C22×C4⋊C4×C47C2211yescharacteristic
20C47×#407—21no—
21C47×#343—21no—
22C47×#343C2212yes—
23C22×C47⋊C8—41no—
24#13093983—41no—
25#13093983—41no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—21no—
32C2×C4×Dic47—21no—
33C22⋊(C47⋊C8)—22no—
34C22⋊(C47⋊C8)—41no—
35C47⋊#95—21no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C2×M16×C47—22no—
48C2×C4⋊C4×C47—41no—
49C2×C4⋊C4×C47—41no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C4⋊C4×C47C2311yescharacteristic
54C22⋊C8×C47—41no—
55C22⋊C8×C47—41no—
56C24×C47C22⋊C411yescharacteristic
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—41no—
65C22×C4×C47—42no—
66C22×C4×C47—42no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76C22×Dic47—21no—
77M16×C47—42no—
78M16×C47—42no—
79C2×C8×C47—42no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84C22⋊Dic47—42no—
85C22⋊Dic47—41no—
86C22⋊Dic47—41no—
87C4⋊Dic47—41no—
88C4⋊Dic47—41no—
89Dic47⋊C4—42no—
90Dic47⋊C4—42no—
91C4×Dic47—42no—
92C2×C47⋊C8—42no—
93C2×C47⋊C8—82no—
94C23×C47—22no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—41no—
98C23×C47#9911yescharacteristic
99C23×C47#9911yescharacteristic
100C23×C47C2×C22⋊C411yescharacteristic
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—42no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—81no—
120C2×C4×C47#13512yes—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—42no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C8×C47—82no—
133C47⋊C8—82no—
134#11850—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—42no—
143C22×C47—42no—
144C22×C47#35212yes—
145C22×C47C2×#9911yescharacteristic
146C22×C47#41611yescharacteristic
147C22×C47#42111yescharacteristic
148C4×C47—22no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—41no—
154Dic47—81no—
155Dic47—81no—
156Dic47—81no—
157Dic47—81no—
158#2484—471no—
159#1821D4711yescharacteristic
160C2×#324—471no—
161#1353—472no—
162#1346—472no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47—22no—
166C2×C47—41no—
167C2×C47#151611yescharacteristic
168C2×C47C2×#35211yescharacteristic
169C2×C47#213311yescharacteristic
170C2×#117—471no—
171C22×C4⋊C4C2×D4711yescharacteristic
172#407—21no—
173C2×C22⋊C8—941no—
174#400—471no—
175#400—471no—
176#398—471no—
177#398—471no—
178#387—941no—
179#387—941no—
180#385—941no—
181#381—471no—
182#381—941no—
183#343—21no—
184#343C2×D4712yes—
185#324—472no—
186#324—941no—
187C47#1185011yescharacteristic, complemented, Sylow
188C23×C4C4×D4711yescharacteristic
189C23×C4D18811yescharacteristic
190C23×C4C22⋊D4711yescharacteristic
191C2×M16—22no—
192C22×C8—1881no—
193#117—1881no—
194#117—1881no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—471no—
198C2×C4⋊C4—471no—
199C2×C4⋊C4—941no—
200C2×C4⋊C4—941no—
201C2×C4⋊C4C4×D4711yescharacteristic
202C2×C4⋊C4D18811yescharacteristic
203C2×C4⋊C4C22⋊D4711yescharacteristic
204C2×C4⋊C4C22×D4711yescharacteristic
205C2×C22⋊C4—941no—
206C2×C22⋊C4—941no—
207C2×C42—941no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—942no—
211C22⋊C8—1881no—
212#95—471no—
213#95—471no—
214#95—941no—
215#95—941no—
216#95—941no—
217#95—941no—
218#95—941no—
219#95—941no—
220#95—941no—
221C24(C2×D47)⋊C411yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—941no—
233C22×C4—941no—
234C22×C4—941no—
235C22×C4—941no—
236C22×C4(C2×D47)⋊C411yescharacteristic
237C22×C4(C2×D47)⋊C411yescharacteristic
238C22×C4(C2×D47)⋊C411yescharacteristic
239C22×C4C2×C4×D4711yescharacteristic
240C22×C4C2×D18811yescharacteristic
241C22×C4C2×C22⋊D4711yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—3762no—
246C2×C8—1882no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—1881no—
252C4⋊C4—1881no—
253C4⋊C4—1882no—
254C4⋊C4—1882no—
255C22⋊C4—1882no—
256C22⋊C4—1881no—
257C22⋊C4—1881no—
258C42—1882no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—41no—
263C23#1309393811yescharacteristic
264C23#1309393811yescharacteristic
265C23C2×(C2×D47)⋊C411yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1882no—
290C2×C4—1881no—
291C2×C4—1881no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4—1881no—
295C2×C4—1881no—
296C2×C4#1309404412yes—
297C8—82no—
298C8—3762no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22#1568891412yes—
309C22C2×#1309393811yescharacteristic
310C22#1568911711yescharacteristic
311C22#1568914111yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—3761no—
319C4—3761no—
320C4—3761no—
321C4—3761no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—41no—
326C2#2024194211yescharacteristic
327C2C2×#1568891411yescharacteristic
328C2#2024507511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1821C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1821D47
hypocenterC47#11850
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #20241414C2semidirect
4. #20241393C2semidirect
187. C47#11850semidirect

Maximal subgroups

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

subgroupindex
#202476752
C2×#156888492
#202414142
#202413932
C47×#18212
#1185047