Finite Group Databasebeta

← #26126673

Subgroups — #26126673

Order 12032.

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

normal subgroups only

This group has 420 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#20249305C211yesmaximal, characteristic
2C4×#13093941C211yesmaximal, characteristic
3#20241419C212yesmaximal, complemented
4#20241395C212yesmaximal, complemented
5C47×#1459C211yesmaximal, characteristic
6C2×#13094003C411yescharacteristic
7C2×#13093941—21no—
8C4×#10625329—21no—
9#15689053C2211yescharacteristic
10#15689051—21no—
11#15689048C411yescharacteristic
12#15689043C2211yescharacteristic
13#15689037—21no—
14#15689035C411yescharacteristic
15#15689007—21no—
16#15688996—21no—
17#15688986C411yescharacteristic
18#15688892C2212yes—
19C2×C4×D4×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C47×#324C2212yes—
22C47×#323—21no—
23C22×Dic94C2×C411yescharacteristic
24C2×#10625329—22no—
25#13094003—41no—
26#13094003—41no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35#13093941—24no—
36#13093935—41no—
37C4×C47⋊C8—41no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95C2×C411yescharacteristic
44C47⋊#95C2×C411yescharacteristic
45C22×D4×C47D411yescharacteristic
46C23×C4×C47C2×C411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C22×C8×C47—41no—
49C4×D4×C47—41no—
50C4×D4×C47—41no—
51C2×C4⋊C4×C47C2311yescharacteristic
52C2×C22⋊C4×C47C2×C411yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C22⋊C8×C47—22no—
56C22⋊C8×C47—41no—
57C24×C47C22⋊C411yescharacteristic
58C24×C47C22⋊C411yescharacteristic
59C2×D4×C47—22no—
60C2×D4×C47—22no—
61C2×D4×C47—41no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—81no—
67C22×C4×C47—81no—
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—
77C2×Dic94—41no—
78C2×Dic94—81no—
79C2×C8×C47—42no—
80C2×C8×C47—82no—
81C4⋊C4×C47—41no—
82C22⋊C4×C47—41no—
83C22⋊C4×C47—41no—
84C42×C47—21no—
85C42×C47—21no—
86C22⋊Dic47—42no—
87C22⋊Dic47—42no—
88C22⋊Dic47—81no—
89C22⋊Dic47—81no—
90C4⋊Dic47—41no—
91Dic47⋊C4—41no—
92Dic47⋊C4—41no—
93Dic47⋊C4—42no—
94Dic47⋊C4—81no—
95#10625329—24no—
96#10625329—42no—
97C2×C47⋊C8—42no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47#10412yes—
108C23×C47C2×C22⋊C411yescharacteristic
109D4×C47—44no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—41no—
116C2×C4×C47—42no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—81no—
121C2×C4×C47—81no—
122C2×C4×C47—81no—
123C2×C4×C47#9911yescharacteristic
124C2×C4×C47#9911yescharacteristic
125C2×C4×C47#13512yes—
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—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138Dic94—82no—
139C8×C47—82no—
140C47⋊C8—44no—
141#11739—471nomaximal, Sylow
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—22no—
145C22×C47—42no—
146C22×C47—41no—
147C22×C47—41no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—81no—
152C22×C47—81no—
153C22×C47—81no—
154C22×C47C2×#9911yescharacteristic
155C22×C47#41611yescharacteristic
156C22×C47C2×#10411yescharacteristic
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—22no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47—41no—
163Dic47—81no—
164Dic47—81no—
165Dic47—81no—
166Dic47—81no—
167#2638—471no—
168C4×#100—471no—
169#1459D4711yescharacteristic
170#1354—472no—
171#1346—472no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47—42no—
175C2×C47—81no—
176C2×C47#150111yescharacteristic
177C2×C47#212111yescharacteristic
178C2×C47#213511yescharacteristic
179C2×C22⋊Q8—471no—
180C2×C4×D4C2×D4711yescharacteristic
181C2×#100—941no—
182C2×C22⋊C8—21no—
183C4×M16—941no—
184#398—471no—
185#397—471no—
186#397—941no—
187#395—471no—
188#394—941no—
189#393—471no—
190#387—941no—
191#385—941no—
192#381—471no—
193#343—472no—
194#324C2×D4712yes—
195#323—21no—
196C47#1173911yescharacteristic, complemented, Sylow
197C22×Q8—471no—
198C22×D4D18811yescharacteristic
199C23×C4C4×D4711yescharacteristic
200C23×C4C22⋊D4711yescharacteristic
201C2×M16—942no—
202C22×C8—41no—
203C22⋊Q8—1881no—
204C22⋊Q8—1881no—
205C4×D4—41no—
206C4×D4—41no—
207C2×C4⋊C4—471no—
208C2×C4⋊C4—941no—
209C2×C4⋊C4—941no—
210C2×C4⋊C4—941no—
211C2×C4⋊C4C22×D4711yescharacteristic
212C2×C22⋊C4—941no—
213C2×C22⋊C4—941no—
214C2×C22⋊C4—941no—
215C2×C22⋊C4—941no—
216C2×C22⋊C4C4×D4711yescharacteristic
217C2×C22⋊C4C22⋊D4711yescharacteristic
218C2×C42D18811yescharacteristic
219#100—944no—
220C22⋊C8—22no—
221C22⋊C8—41no—
222#97—1881no—
223C4×C8—1881no—
224#95—471no—
225#95—471no—
226#95—941no—
227#95—941no—
228#95—941no—
229#95—941no—
230#95—941no—
231C24(C2×D47)⋊C411yescharacteristic
232C24(C2×D47)⋊C411yescharacteristic
233C2×Q8—3761no—
234C2×Q8—1881no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—941no—
245C22×C4—941no—
246C22×C4—941no—
247C22×C4—941no—
248C22×C4(C2×D47)⋊C411yescharacteristic
249C22×C4(C2×D47)⋊C411yescharacteristic
250C22×C4C2×C4×D4711yescharacteristic
251C22×C4C2×D18811yescharacteristic
252C22×C4C2×C22⋊D4711yescharacteristic
253M16—944no—
254M16—1882no—
255C2×C8—42no—
256C2×C8—82no—
257C2×C8—1882no—
258C4⋊C4—41no—
259C4⋊C4—3761no—
260C4⋊C4—1881no—
261C4⋊C4—1881no—
262C4⋊C4—1881no—
263C4⋊C4—1882no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—3761no—
267C22⋊C4—3761no—
268C22⋊C4—1882no—
269C22⋊C4—1882no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23#1309395012yes—
282C23C2×(C2×D47)⋊C411yescharacteristic
283Q8—3762no—
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
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—1881no—
308C2×C4—1881no—
309C2×C4—1881no—
310C2×C4#1309393811yescharacteristic
311C2×C4#1309393811yescharacteristic
312C2×C4#1309404612yes—
313C8—82no—
314C8—1884no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22C2×#1309393811yescharacteristic
328C22#1568912011yescharacteristic
329C22C2×#1309395011yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—3761no—
337C4—3761no—
338C4—3761no—
339C4—3761no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#2024188011yescharacteristic
345C2#2024503211yescharacteristic
346C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1459C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C4⋊C4×C47C23
hypercenter#1459D47
hypocenterC47#11739
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20241419C2semidirect
4. #20241395C2semidirect
196. C47#11739semidirect

Maximal subgroups

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

subgroupindex
#202493052
C4×#130939412
#202414192
#202413952
C47×#14592
#1173947