Finite Group Databasebeta

← #26107757

Subgroups — #26107757

Order 12032.

814 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#15689679C211yesmaximal, characteristic
2#20243151C212yesmaximal
3C2×#15688865C211yesmaximal, characteristic, complemented
4C2×#15688864C212yesmaximal, complemented
5C2×C47×#330C211yesmaximal, characteristic, complemented
6#15689679—24no—
7#15689679—22no—
8#15689679—22no—
9C2×#13094023C412yes—
10C2×#13094016C2211yescharacteristic
11C2×#13094002—22no—
12C2×#13093989—22no—
13C2×#13093982C412yes—
14C2×C42×D47C2211yescharacteristic
15C2×(C2×D47)⋊C8—22no—
16C2×#13093935C2212yes—
17#15688877—22no—
18#15688865—22no—
19#15688864—24no—
20C2×#125×C47C2211yescharacteristic
21C2×#97×C47C2212yes—
22C47×#330—22no—
23C22×Dic94—22no—
24C22×C4×D47C2×C411yescharacteristic
25C22×C4×D47D412yes—
26C22×C47⋊C8—22no—
27#13094023—24no—
28#13094023—24no—
29#13094016—21no—
30#13094016—21no—
31#13094016D414yes—
32#13094002—28no—
33#13094002—28no—
34#13093989—28no—
35#13093989—28no—
36#13093982—24no—
37#13093982—24no—
38C2×C4⋊Dic47C2×C412yes—
39C2×Dic47⋊C4—22no—
40C2×Dic47⋊C4—22no—
41C2×(C2×D47)⋊C4—22no—
42C2×C4×Dic47—22no—
43C2×C4×Dic47C2×C411yescharacteristic
44C2×C4×Dic47D412yes—
45C42×D47—21no—
46C42×D47—22no—
47C42×D47—21no—
48(C2×D47)⋊C8—28no—
49#13093935—22no—
50#13093935—22no—
51C22×C8×C47—22no—
52#125×C47—22no—
53#125×C47D412yes—
54#125×C47D412yes—
55C2×C4⋊C4×C47—22no—
56C2×C4⋊C4×C47C2×C412yes—
57C2×C42×C47C2311yescharacteristic
58#97×C47—22no—
59#97×C47—22no—
60C23×D47C22⋊C411yescharacteristic
61C22×C4×C47—22no—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D412yes—
64C22×Dic47—22no—
65C22×Dic47C22⋊C411yescharacteristic
66C22×Dic47C22⋊C412yes—
67C2×C4×D47—22no—
68C2×C4×D47—21no—
69C2×C4×D47—21no—
70C2×C4×D47—22no—
71C2×C4×D47—22no—
72C2×C4×D47—22no—
73C2×C4×D47—24no—
74C2×C4×D47—42no—
75C2×Dic94—28no—
76C2×Dic94—44no—
77C2×Dic94—44no—
78C2×C8×C47—24no—
79C2×C8×C47—42no—
80C2×C8×C47—42no—
81C4⋊C4×C47—22no—
82C4⋊C4×C47—24no—
83C4⋊C4×C47—24no—
84C4⋊C4×C47C22⋊C414yes—
85C42×C47C2×D411yescharacteristic
86C42×C47C2×D412yes—
87C42×C47C2×D411yescharacteristic
88(C2×D47)⋊C4—28no—
89C4⋊Dic47—22no—
90C4⋊Dic47C22⋊C414yes—
91Dic47⋊C4—28no—
92Dic47⋊C4—24no—
93Dic47⋊C4—24no—
94C4×Dic47—28no—
95C4×Dic47—21no—
96C4×Dic47—21no—
97C4×Dic47—22no—
98C4×Dic47—22no—
99C2×C47⋊C8—24no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—42no—
102C23×C47C2×C22⋊C411yescharacteristic
103C22×D47—21no—
104C22×D47—21no—
105C22×D47—22no—
106C22×D47—22no—
107C22×D47#10112yes—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—24no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47C2×C22⋊C411yescharacteristic
116C2×C4×C47C2×C22⋊C411yescharacteristic
117C2×C4×C47#12012yes—
118C2×C4×C47#12012yes—
119C2×Dic47—21no—
120C2×Dic47—22no—
121C2×Dic47—21no—
122C2×Dic47—22no—
123C2×Dic47—24no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47#10112yes—
127C2×Dic47#10414yes—
128C4×D47—24no—
129C4×D47—24no—
130C4×D47—44no—
131C4×D47—44no—
132Dic94—416no—
133C8×C47—44no—
134C47⋊C8—44no—
135#8338—471nomaximal, Sylow
136C22×C47#37911yescharacteristic
137C22×C47C2×#10111yescharacteristic
138C22×C47C2×#10111yescharacteristic
139C22×C47C2×#10412yes—
140C22×C47#42112yes—
141C2×D47—22no—
142C2×D47—24no—
143C2×D47—41no—
144C2×D47—41no—
145C2×D47—42no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—22no—
149C4×C47—22no—
150C4×C47—44no—
151Dic47—22no—
152Dic47—24no—
153Dic47—44no—
154C2×#533—471no—
155#1749—472no—
156C2×#330—472no—
157C2×#330—471no—
158C2×#330D4711yescharacteristic, complemented
159C2×C47#152611yescharacteristic
160C2×C47#153112yes—
161C2×C47#153711yescharacteristic
162C2×C47#178411yescharacteristic
163C2×C47#178912yes—
164D47—42no—
165#533—944no—
166#533—942no—
167#533—942no—
168C2×C4⋊Q8—472no—
169C2×#125—471no—
170C2×#125C2×D4711yescharacteristic
171C2×C22⋊Q8—942no—
172C2×C4×Q8—942no—
173C2×#117—472no—
174C22×C42—471no—
175C2×C22⋊C8—942no—
176C2×#97—472no—
177C2×#97C2×D4712yes—
178#336—942no—
179#330—22no—
180#330—944no—
181#330—942no—
182C47#833811yescharacteristic, complemented, Sylow
183C22×Q8—942no—
184C23×C4—471no—
185C23×C4—472no—
186C22×C8—22no—
187C22×C8—942no—
188C4⋊Q8—944no—
189C4⋊Q8—944no—
190#125—22no—
191#125—474no—
192#125—941no—
193#125—941no—
194#125D18812yes—
195#125C22⋊D4712yes—
196C22⋊Q8—948no—
197C22⋊Q8—948no—
198C4×Q8—948no—
199C4×Q8—948no—
200#117—944no—
201#117—944no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—472no—
204C2×C4⋊C4—942no—
205C2×C4⋊C4—942no—
206C2×C4⋊C4C4×D4712yes—
207C2×C22⋊C4—942no—
208C2×C42—471no—
209C2×C42—472no—
210C2×C42—942no—
211C2×C42—941no—
212C2×C42—942no—
213C2×C42—941no—
214C2×C42C22×D4711yescharacteristic
215C22⋊C8—948no—
216#97—22no—
217#97—22no—
218#97—942no—
219#97—942no—
220C24—471no—
221C2×Q8—948no—
222C2×Q8—1884no—
223C2×Q8—1884no—
224C22×C4—22no—
225C22×C4—471no—
226C22×C4—472no—
227C22×C4—942no—
228C22×C4—944no—
229C22×C4—942no—
230C22×C4—941no—
231C22×C4—941no—
232C22×C4—942no—
233C22×C4—942no—
234C22×C4—942no—
235C22×C4—1882no—
236C22×C4C2×C4×D4711yescharacteristic
237C22×C4D4×D4712yes—
238C2×C8—24no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—944no—
242C2×C8—1882no—
243C2×C8—1882no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—474no—
248C4⋊C4—948no—
249C4⋊C4—942no—
250C4⋊C4—944no—
251C4⋊C4—944no—
252C4⋊C4(C2×D47)⋊C414yes—
253C22⋊C4—948no—
254C42—948no—
255C42—941no—
256C42—941no—
257C42—942no—
258C42—942no—
259C42C2×D18811yescharacteristic
260C42D4×D4712yes—
261C42C2×C22⋊D4711yescharacteristic
262C23—472no—
263C23—941no—
264C23—941no—
265C23—942no—
266C23—942no—
267C23C22⋊C4×D4711yescharacteristic
268Q8—18816no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—472no—
277C2×C4—474no—
278C2×C4—944no—
279C2×C4—944no—
280C2×C4—941no—
281C2×C4—942no—
282C2×C4—941no—
283C2×C4—942no—
284C2×C4—944no—
285C2×C4—1884no—
286C2×C4—1884no—
287C2×C4—1882no—
288C2×C4—1882no—
289C2×C4C22⋊C4×D4711yescharacteristic
290C2×C4C2×(C2×D47)⋊C411yescharacteristic
291C2×C4#1309399212yes—
292C2×C4#1309399412yes—
293C8—44no—
294C8—1884no—
295C22—944no—
296C22—942no—
297C22—1881no—
298C22—1881no—
299C22—1882no—
300C22#1568897811yescharacteristic
301C22#101×D4711yescharacteristic
302C22#101×D4711yescharacteristic
303C22#104×D4712yes—
304C22#1568913812yes—
305C4—21no—
306C4—21no—
307C4—22no—
308C4—22no—
309C4—44no—
310C4—942no—
311C4—944no—
312C4—1884no—
313C2—1882no—
314C2#2024198111yescharacteristic
315C2#2024200112yes—
316C2#2024202411yescharacteristic
317C2#2024331011yescharacteristic
318C2#2024333012yes—
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#330C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenterC2×#330D47
hypocenterC47#8338
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
3. C2×#15688865C2semidirect
4. C2×#15688864C2semidirect
5. C2×C47×#330C2semidirect
158. C2×#330D47semidirect
182. C47#8338semidirect

Maximal subgroups

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

subgroupindex
C2×#156896792
#202431512
C2×#156888652
C2×#156888642
C2×C47×#3302
#833847