Finite Group Databasebeta

← #26109916

Subgroups — #26109916

Order 12032.

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

normal subgroups only

This group has 466 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#20246361C211yesmaximal, characteristic
2C4×#13093939C211yesmaximal, characteristic
3#20241273C212yesmaximal, complemented
4#20241242C212yesmaximal, complemented
5C47×#1465C211yesmaximal, characteristic
6C2×#13093939—21no—
7#15689026—21no—
8#15689016—21no—
9#15689016—21no—
10#15688989C411yescharacteristic, complemented
11#15688986C411yescharacteristic, complemented
12#15688986C2211yescharacteristic
13C4⋊C4×Dic47—21no—
14C4×C4⋊Dic47C411yescharacteristic, complemented
15C4×C22⋊Dic47C411yescharacteristic, complemented
16C4×C22⋊Dic47C2212yes—
17C4×C22⋊Dic47C2211yescharacteristic
18#15688890—21no—
19C2×C4×D4×C47C2211yescharacteristic
20C47×#407C2212yes—
21C47×#324—21no—
22C47×#323—21no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C22⋊Dic47—22no—
26C2×C22⋊Dic47—21no—
27C2×C22⋊Dic47—21no—
28C2×C22⋊Dic47—21no—
29C2×C22⋊Dic47—21no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47—21no—
33C2×C4×Dic47—22no—
34#13093939—24no—
35C47⋊#95—21no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—22no—
39C47⋊#95C2×C411yescharacteristic
40C47⋊#95C2×C411yescharacteristic
41C47⋊#95C2×C411yescharacteristic
42C47⋊#95C2×C411yescharacteristic
43C22×D4×C47D411yescharacteristic
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47C2311yescharacteristic
46C2×M16×C47—22no—
47C4×D4×C47—22no—
48C4×D4×C47—22no—
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C22⋊C4×C47D411yescharacteristic
51C2×C22⋊C4×C47D411yescharacteristic
52C2×C42×C47C2×C411yescharacteristic
53C22⋊C8×C47—22no—
54C22⋊C8×C47—22no—
55C24×C47C22⋊C411yescharacteristic
56C24×C47C2×D411yescharacteristic
57C2×D4×C47—22no—
58C2×D4×C47—22no—
59C2×D4×C47—41no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—81no—
63C22×C4×C47—81no—
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47#4112yes—
70C22×C4×C47#4112yes—
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—42no—
76M16×C47—82no—
77C2×C8×C47—42no—
78C4⋊C4×C47—22no—
79C22⋊C4×C47—22no—
80C22⋊C4×C47—22no—
81C42×C47—21no—
82C42×C47—21no—
83C22⋊Dic47—24no—
84C22⋊Dic47—42no—
85C22⋊Dic47—42no—
86C22⋊Dic47—42no—
87C22⋊Dic47—81no—
88C22⋊Dic47—81no—
89C4⋊Dic47—42no—
90C4⋊Dic47—42no—
91C4×Dic47—42no—
92C4×Dic47—81no—
93C4×Dic47—81no—
94C4×Dic47—82no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47C2×C22⋊C411yescharacteristic
105C23×C47#12312yes—
106D4×C47—44no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—41no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—81no—
118C2×C4×C47—81no—
119C2×C4×C47#10412yes—
120C2×C4×C47#11711yescharacteristic
121C2×C4×C47#11711yescharacteristic
122C2×C4×C47#12312yes—
123C2×Dic47—22no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—44no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—82no—
137C2×Dic47—82no—
138C8×C47—82no—
139#8779—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—22no—
143C22×C47—22no—
144C22×C47—22no—
145C22×C47—41no—
146C22×C47—41no—
147C22×C47—41no—
148C22×C47—81no—
149C22×C47—81no—
150C22×C47—81no—
151C22×C47—81no—
152C22×C47#38111yescharacteristic
153C22×C47C2×#10411yescharacteristic
154C22×C47#42111yescharacteristic
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—41no—
160C4×C47—41no—
161Dic47—42no—
162Dic47—81no—
163Dic47—81no—
164Dic47—81no—
165Dic47—84no—
166#2357—471no—
167C4×#99—471no—
168#1465D4711yescharacteristic
169#1297—472no—
170#1284—472no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—42no—
174C2×C47—81no—
175C2×C47#182611yescharacteristic
176C2×C47#182711yescharacteristic
177C2×C47#182811yescharacteristic
178C2×C4×D4C2×D4711yescharacteristic
179C2×#99—941no—
180#407C2×D4712yes—
181#390—941no—
182#388—941no—
183#388—941no—
184#382—471no—
185#381—471no—
186#381—471no—
187C4×C4⋊C4—471no—
188C4×C4⋊C4—941no—
189C4×C22⋊C4—471no—
190C4×C22⋊C4—471no—
191C4×C22⋊C4—472no—
192#342—941no—
193#324—21no—
194#323—21no—
195C47#877911yescharacteristic, complemented, Sylow
196C22×D4D18811yescharacteristic
197C23×C4C4×D4711yescharacteristic
198C23×C4C22×D4711yescharacteristic
199C2×M16—22no—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—941no—
203C2×C4⋊C4—941no—
204C2×C4⋊C4C22⋊D4711yescharacteristic
205C2×C22⋊C4—942no—
206C2×C22⋊C4—941no—
207C2×C22⋊C4—941no—
208C2×C22⋊C4—941no—
209C2×C22⋊C4—941no—
210C2×C22⋊C4D18811yescharacteristic
211C2×C22⋊C4C22⋊D4711yescharacteristic
212C2×C42—941no—
213C2×C42—941no—
214C2×C42—942no—
215C2×C42—941no—
216C2×C42C4×D4711yescharacteristic
217#99—944no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220#95—471no—
221#95—471no—
222#95—471no—
223#95—471no—
224#95—941no—
225#95—941no—
226#95—941no—
227#95—942no—
228C24(C2×D47)⋊C411yescharacteristic
229C24C2×D18811yescharacteristic
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—941no—
238C22×C4—941no—
239C22×C4—941no—
240C22×C4—941no—
241C22×C4—1882no—
242C22×C4(C2×D47)⋊C411yescharacteristic
243C22×C4(C2×D47)⋊C411yescharacteristic
244C22×C4(C2×D47)⋊C411yescharacteristic
245C22×C4C2×C4×D4711yescharacteristic
246C22×C4#1062535212yes—
247C22×C4#1062535212yes—
248C22×C4C2×C22⋊D4711yescharacteristic
249M16—82no—
250C2×C8—42no—
251C4⋊C4—22no—
252C4⋊C4—1882no—
253C4⋊C4—1882no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—944no—
257C22⋊C4—3761no—
258C22⋊C4—3761no—
259C22⋊C4—1882no—
260C22⋊C4—1882no—
261C22⋊C4—1882no—
262C42—21no—
263C42—21no—
264C42—3762no—
265C42—3761no—
266C42—3761no—
267C42—1882no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×(C2×D47)⋊C411yescharacteristic
278C23#1309400812yes—
279D4—44no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—942no—
293C2×C4—3762no—
294C2×C4—3762no—
295C2×C4—1881no—
296C2×C4—1881no—
297C2×C4—1884no—
298C2×C4—1881no—
299C2×C4—1881no—
300C2×C4—1881no—
301C2×C4—1881no—
302C2×C4—1881no—
303C2×C4—1881no—
304C2×C4—1881no—
305C2×C4—1881no—
306C2×C4—1881no—
307C2×C4#1309395012yes—
308C2×C4#1309398011yescharacteristic
309C2×C4#1309398011yescharacteristic
310C2×C4#1309400512yes—
311C8—82no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1568898411yescharacteristic
325C22C2×#1309395011yescharacteristic
326C22#1568913911yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—3761no—
334C4—3761no—
335C4—3761no—
336C4—3764no—
337C4—1882no—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#2024348511yescharacteristic
343C2#2024349311yescharacteristic
344C2#2024349411yescharacteristic
subgroupG/H
centerC22#15688984
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1465C2
socleC22×C47#381
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenter#1465D47
hypocenterC47#8779
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20241273C2semidirect
4. #20241242C2semidirect
10. #15688989C4semidirect
11. #15688986C4semidirect
14. C4×C4⋊Dic47C4semidirect
15. C4×C22⋊Dic47C4semidirect
195. C47#8779semidirect

Maximal subgroups

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

subgroupindex
#202463612
C4×#130939392
#202412732
#202412422
C47×#14652
#877947