Finite Group Databasebeta

← #26111372

Subgroups — #26111372

Order 12032.

482 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 115 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#20241816C212yesmaximal, complemented
3#20241702C211yesmaximal, characteristic
4#20241289C212yesmaximal, complemented
5C47×#2921C211yesmaximal, characteristic
6C2×C4×Dic94C2211yescharacteristic
7#15689176C2212yes—
8#15689062C411yescharacteristic
9#15689059C411yescharacteristic
10#15689056—21no—
11#15689046—21no—
12#15689043C411yescharacteristic
13#15689035C411yescharacteristic
14C4×Dic47⋊C4—21no—
15C4×C4⋊Dic47C2211yescharacteristic
16C4×C22⋊Dic47—21no—
17#15688853—24no—
18#15688849—21no—
19#15688847—21no—
20C2×#117×C47C2211yescharacteristic
21C47×#433—24no—
22C47×#423C2212yes—
23C47×#407—21no—
24C4×M16×C47—21no—
25C22×Dic94C2×C411yescharacteristic
26C4×Dic94—22no—
27C4×Dic94—22no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35C2×C4×Dic47—21no—
36C2×C4×Dic47—21no—
37C4⋊(C47⋊C8)—22no—
38C22⋊(C47⋊C8)—22no—
39C4×C47⋊C8—22no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95C2×C411yescharacteristic
45C47⋊#95C2×C411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C2×M16×C47—22no—
48#117×C47—22no—
49#117×C47—22no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47C2×C411yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—22no—
57C4⋊C8×C47D414yes—
58C22⋊C8×C47—24no—
59#97×C47—22no—
60C4×C8×C47—22no—
61C24×C47C22⋊C411yescharacteristic
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76C2×Dic94—22no—
77C2×Dic94—42no—
78M16×C47—42no—
79M16×C47—42no—
80C2×C8×C47—24no—
81C4⋊C4×C47—22no—
82C4⋊C4×C47—22no—
83C22⋊C4×C47—22no—
84C22⋊C4×C47—22no—
85C42×C47—21no—
86C42×C47—21no—
87C42×C47C2×D412yes—
88C42×C47#4112yes—
89C22⋊Dic47—42no—
90C22⋊Dic47—42no—
91C4⋊Dic47—22no—
92C4⋊Dic47—42no—
93Dic47⋊C4—42no—
94Dic47⋊C4—42no—
95Dic47⋊C4—42no—
96C4×Dic47—41no—
97C4×Dic47—41no—
98C4×Dic47—42no—
99C2×C47⋊C8—42no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47C2×C22⋊C411yescharacteristic
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47C4×D412yes—
118C2×C4×C47#12011yescharacteristic
119C2×C4×C47#12112yes—
120C2×C4×C47#12311yescharacteristic
121C2×C4×C47#13512yes—
122C2×C4×C47#13712yes—
123C2×Dic47—22no—
124C2×Dic47—22no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135Dic94—24no—
136Dic94—42no—
137C8×C47—44no—
138C47⋊C8—82no—
139#9034—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—42no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—41no—
146C22×C47—81no—
147C22×C47#38611yescharacteristic
148C22×C47#41611yescharacteristic
149C22×C47#41911yescharacteristic
150C4×C47—22no—
151C4×C47—22no—
152C4×C47—41no—
153C4×C47—41no—
154C4×C47—41no—
155C4×C47#45212yes—
156C4×C47#48512yes—
157Dic47—42no—
158Dic47—42no—
159Dic47—81no—
160Dic47—81no—
161#2921D4711yescharacteristic
162#2396—471no—
163#1487—472no—
164#1458—471no—
165#1303—472no—
166C2×C47—21no—
167C2×C47—21no—
168C2×C47—81no—
169C2×C47#187311yescharacteristic
170C2×C47#188411yescharacteristic
171C2×C47#189111yescharacteristic
172C2×C4×Q8—471no—
173C2×#117C2×D4711yescharacteristic
174#433—24no—
175#431—472no—
176#423C2×D4712yes—
177#407—21no—
178C4×M16—21no—
179#400—471no—
180#399—471no—
181#398—941no—
182#396—941no—
183#395—471no—
184#393—471no—
185C4×C4⋊C4—471no—
186C4×C4⋊C4—941no—
187C4×C22⋊C4—941no—
188Q8⋊C8—944no—
189#324—941no—
190#323—941no—
191C47#903411yescharacteristic, complemented, Sylow
192C22×Q8—471no—
193C23×C4C22⋊D4711yescharacteristic
194C2×M16—22no—
195C4×Q8—942no—
196C4×Q8—942no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—471no—
200C2×C4⋊C4—941no—
201C2×C4⋊C4—941no—
202C2×C4⋊C4—941no—
203C2×C4⋊C4—941no—
204C2×C4⋊C4C4×D4711yescharacteristic
205C2×C4⋊C4D18811yescharacteristic
206C2×C22⋊C4—941no—
207C2×C22⋊C4—941no—
208C2×C22⋊C4C4×D4711yescharacteristic
209C2×C22⋊C4D18811yescharacteristic
210C2×C42—941no—
211C2×C42—941no—
212C2×C42C22⋊D4711yescharacteristic
213C2×C42C22×D4711yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—942no—
216C4⋊C8C22⋊D4714yes—
217C22⋊C8—24no—
218C22⋊C8—942no—
219#97—22no—
220C4×C8—22no—
221C4×C8—942no—
222#95—471no—
223#95—471no—
224#95—941no—
225#95—941no—
226#95—941no—
227#95—941no—
228C24(C2×D47)⋊C411yescharacteristic
229C2×Q8—942no—
230C2×Q8—1882no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—941no—
236C22×C4—941no—
237C22×C4—941no—
238C22×C4—941no—
239C22×C4(C2×D47)⋊C411yescharacteristic
240C22×C4(C2×D47)⋊C411yescharacteristic
241C22×C4(C2×D47)⋊C411yescharacteristic
242C22×C4C2×C4×D4711yescharacteristic
243C22×C4C2×D18811yescharacteristic
244C22×C4C2×C22⋊D4711yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—24no—
248C2×C8—1882no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—942no—
252C4⋊C4—1882no—
253C4⋊C4—1882no—
254C4⋊C4—1882no—
255C4⋊C4—1882no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—1882no—
259C22⋊C4—1882no—
260C42—21no—
261C42—21no—
262C42—1881no—
263C42—1881no—
264C42—1882no—
265C42#1062535012yes—
266C42C2×C22⋊D4712yes—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23C2×(C2×D47)⋊C411yescharacteristic
273Q8—944no—
274Q8—1882no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
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×C4C4×C22⋊D4712yes—
299C2×C4#1309399311yescharacteristic
300C2×C4#1309399812yes—
301C2×C4#1309400711yescharacteristic
302C2×C4#1309404412yes—
303C2×C4#1309405412yes—
304C8—44no—
305C8—3762no—
306C22—21no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—81no—
313C22#1568900211yescharacteristic
314C22#1568911711yescharacteristic
315C22#1568913111yescharacteristic
316C4—22no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—3761no—
322C4—3761no—
323C4—1882no—
324C4—1882no—
325C4#1568928512yes—
326C4#1568946412yes—
327C2—21no—
328C2—21no—
329C2—81no—
330C2#2024370011yescharacteristic
331C2#2024377111yescharacteristic
332C2#2024380611yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2921C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2921D47
hypocenterC47#9034
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #20241816C2semidirect
4. #20241289C2semidirect
191. C47#9034semidirect

Maximal subgroups

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

subgroupindex
#202466562
#202418162
#202417022
#202412892
C47×#29212
#903447