Finite Group Databasebeta

← #26109890

Subgroups — #26109890

Order 12032.

446 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 95 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#20246328C211yesmaximal, characteristic
2#20241729C211yesmaximal, characteristic
3#20241282C212yesmaximal, complemented
4#20241253C212yesmaximal, complemented
5C4×#100×C47C211yesmaximal, characteristic
6#15689180C2212yes—
7#15689016—21no—
8#15689009—21no—
9#15689003—21no—
10#15688991C2211yescharacteristic
11#15688986C411yescharacteristic
12#15688983C411yescharacteristic
13C22⋊C4×Dic47—21no—
14C4×C22⋊Dic47C411yescharacteristic
15#15688969C2211yescharacteristic
16C2×C47⋊#95C411yescharacteristic
17#15688847—21no—
18#15688847—21no—
19C2×C4×D4×C47C2211yescharacteristic
20C2×#100×C47—21no—
21C4×M16×C47C2212yes—
22C47×#343—21no—
23C23×Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C22⋊Dic47—21no—
26C2×C22⋊Dic47—21no—
27C2×C22⋊Dic47—21no—
28C2×C4×Dic47—21no—
29C2×C4×Dic47—21no—
30C4⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32#13093935—22no—
33C47⋊#95—21no—
34C47⋊#95—21no—
35C47⋊#95—21no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95C2×C411yescharacteristic
39C47⋊#95C2×C411yescharacteristic
40C47⋊#95C2×C411yescharacteristic
41C47⋊#95C2×C411yescharacteristic
42C22×D4×C47D411yescharacteristic
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47C2×C411yescharacteristic
45C2×M16×C47—22no—
46C4×D4×C47—22no—
47C4×D4×C47—22no—
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C22⋊C4×C47D411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52#100×C47—24no—
53#97×C47—22no—
54C4×C8×C47—22no—
55C24×C47C22⋊C411yescharacteristic
56C24×C47C22⋊C411yescharacteristic
57C2×D4×C47—22no—
58C2×D4×C47—22no—
59C2×D4×C47—41no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—81no—
65C22×C4×C47—81no—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—42no—
76C22×Dic47—42no—
77C22×Dic47—81no—
78M16×C47—24no—
79M16×C47—42no—
80C2×C8×C47—42no—
81C4⋊C4×C47—22no—
82C22⋊C4×C47—22no—
83C22⋊C4×C47—22no—
84C42×C47#4112yes—
85C42×C47#4112yes—
86C22⋊Dic47—42no—
87C22⋊Dic47—42no—
88C22⋊Dic47—81no—
89C4⋊Dic47—81no—
90C4×Dic47—42no—
91C4×Dic47—81no—
92C2×C47⋊C8—42no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47#10412yes—
103C23×C47C2×C22⋊C411yescharacteristic
104D4×C47—44no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—81no—
114C2×C4×C47—81no—
115C2×C4×C47—81no—
116C2×C4×C47—81no—
117C2×C4×C47#11711yescharacteristic
118C2×C4×C47#11711yescharacteristic
119C2×C4×C47#12312yes—
120C2×C4×C47#12312yes—
121C2×Dic47—22no—
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—
133C2×Dic47—84no—
134C8×C47—44no—
135C47⋊C8—82no—
136#8773—471nomaximal, Sylow
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—22no—
140C22×C47—42no—
141C22×C47—42no—
142C22×C47—41no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—41no—
146C22×C47—41no—
147C22×C47—81no—
148C22×C47—81no—
149C22×C47#38111yescharacteristic
150C22×C47C2×#10411yescharacteristic
151C22×C47#42111yescharacteristic
152C4×C47—22no—
153C4×C47—22no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—41no—
157C4×C47—41no—
158Dic47—42no—
159Dic47—81no—
160Dic47—81no—
161Dic47—81no—
162#2353—471no—
163C4×#100D4711yescharacteristic
164#1466—471no—
165#1300—472no—
166#1288—472no—
167C2×C47—21no—
168C2×C47—21no—
169C2×C47—42no—
170C2×C47—81no—
171C2×C47#182611yescharacteristic
172C2×C47#182711yescharacteristic
173C2×C47#182911yescharacteristic
174C2×C4×D4C2×D4711yescharacteristic
175#432—472no—
176C2×#100—21no—
177C4×M16C2×D4712yes—
178#388—941no—
179#387—941no—
180#386—941no—
181#383—471no—
182#381—471no—
183#380—471no—
184C4×C22⋊C4—471no—
185C4×C22⋊C4—941no—
186#376—471no—
187C2×#95—471no—
188#343—21no—
189#323—941no—
190#323—941no—
191C47#877311yescharacteristic, complemented, Sylow
192C22×D4C22⋊D4711yescharacteristic
193C23×C4—941no—
194C23×C4C4×D4711yescharacteristic
195C23×C4C4×D4711yescharacteristic
196C2×M16—22no—
197C4×D4—22no—
198C4×D4—22no—
199C2×C4⋊C4—941no—
200C2×C4⋊C4C22⋊D4711yescharacteristic
201C2×C22⋊C4—941no—
202C2×C22⋊C4—941no—
203C2×C22⋊C4—941no—
204C2×C22⋊C4D18811yescharacteristic
205C2×C22⋊C4D18811yescharacteristic
206C2×C42—941no—
207C2×C42—941no—
208C2×C42C22×D4711yescharacteristic
209C4⋊C8—942no—
210#100—24no—
211C22⋊C8—942no—
212#97—22no—
213#97—942no—
214C4×C8—22no—
215#95—471no—
216#95—471no—
217#95—471no—
218#95—471no—
219#95—941no—
220#95—941no—
221#95—941no—
222#95—941no—
223#95—941no—
224C24(C2×D47)⋊C411yescharacteristic
225C24(C2×D47)⋊C411yescharacteristic
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—941no—
236C22×C4—941no—
237C22×C4—941no—
238C22×C4—941no—
239C22×C4—3761no—
240C22×C4—1882no—
241C22×C4—1882no—
242C22×C4(C2×D47)⋊C411yescharacteristic
243C22×C4(C2×D47)⋊C411yescharacteristic
244C22×C4C2×C4×D4711yescharacteristic
245C22×C4C2×D18811yescharacteristic
246C22×C4C2×C22⋊D4711yescharacteristic
247M16—24no—
248M16—42no—
249C2×C8—42no—
250C2×C8—1882no—
251C4⋊C4—22no—
252C4⋊C4—3761no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—3761no—
256C22⋊C4—1882no—
257C22⋊C4—1882no—
258C42—3761no—
259C42—1882no—
260C42#1062535212yes—
261C42#1062535212yes—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#1309395212yes—
272C23C2×(C2×D47)⋊C411yescharacteristic
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—942no—
287C2×C4—3764no—
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—1881no—
299C2×C4#1309398011yescharacteristic
300C2×C4#1309398011yescharacteristic
301C2×C4#1309400512yes—
302C2×C4#1309400812yes—
303C8—44no—
304C8—3762no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22#1568898411yescharacteristic
318C22C2×#1309395211yescharacteristic
319C22#1568914111yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—3761no—
327C4—3761no—
328C4—3761no—
329C4—1882no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—81no—
334C2#2024348811yescharacteristic
335C2#2024349311yescharacteristic
336C2#2024349711yescharacteristic
subgroupG/H
centerC22#15688984
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC4×#100×C47C2
socleC22×C47#381
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenterC4×#100D47
hypocenterC47#8773
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
3. #20241282C2semidirect
4. #20241253C2semidirect
191. C47#8773semidirect

Maximal subgroups

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

subgroupindex
#202463282
#202417292
#202412822
#202412532
C4×#100×C472
#877347