Finite Group Databasebeta

← #26107758

Subgroups — #26107758

Order 12032.

814 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 139 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×#15689680C211yesmaximal, characteristic
2#20243150C211yesmaximal, characteristic
3C2×#15688864C214yesmaximal, complemented
4C47×#1749C211yesmaximal, characteristic
5#15689680—28no—
6C2×#13094024C411yescharacteristic
7C2×#13094024C411yescharacteristic
8C2×#13094016C2212yes—
9C2×#13094003—21no—
10C2×#13094003—21no—
11C2×C4×Dic94—21no—
12C2×C4×Dic94—21no—
13C2×#13093983C412yes—
14C2×C22⋊(C47⋊C8)—21no—
15C2×#13093935C2212yes—
16#15688878—21no—
17#15688864—28no—
18C22×C42×C47C2211yescharacteristic
19C2×C22⋊C8×C47—21no—
20C2×#97×C47C2212yes—
21C47×#336—21no—
22C22×Dic94—21no—
23C22×Dic94—21no—
24C22×C47⋊C8—22no—
25#13094024—24no—
26#13094024—24no—
27#13094016—24no—
28#13094016D418yes—
29#13094003—28no—
30#13094003—28no—
31C4×Dic94—28no—
32C4×Dic94—28no—
33#13093983—24no—
34#13093983—24no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47C2×C412yes—
38C2×C4⋊Dic47C2×C412yes—
39C2×Dic47⋊C4—22no—
40C2×Dic47⋊C4—22no—
41C2×C22⋊Dic47—22no—
42C2×C4×Dic47—22no—
43C22⋊(C47⋊C8)—24no—
44#13093935—24no—
45C23×C4×C47C2×C411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C22×C8×C47—22no—
49C2×C42×C47—24no—
50C2×C42×C47C2×C411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C2×C42×C47C2311yescharacteristic
54C22⋊C8×C47—24no—
55#97×C47—24no—
56C24×C47C22⋊C411yescharacteristic
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—41no—
65C22×C4×C47—41no—
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—22no—
73C22×Dic47—22no—
74C2×Dic94—24no—
75C2×Dic94—24no—
76C2×Dic94—42no—
77C2×Dic94—42no—
78C2×Dic94—42no—
79C2×Dic94—42no—
80C2×C8×C47—24no—
81C2×C8×C47—42no—
82C2×C8×C47—42no—
83C42×C47—22no—
84C42×C47—22no—
85C42×C47—22no—
86C42×C47C2×D414yes—
87C22⋊Dic47—28no—
88C4⋊Dic47—24no—
89C4⋊Dic47—24no—
90C4⋊Dic47—22no—
91C4⋊Dic47—22no—
92C4⋊Dic47C22⋊C414yes—
93C4⋊Dic47C22⋊C414yes—
94Dic47⋊C4—28no—
95Dic47⋊C4—28no—
96C4×Dic47—28no—
97C2×C47⋊C8—24no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47#10112yes—
107C23×C47C2×C22⋊C411yescharacteristic
108C2×C4×C47—28no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—44no—
122C2×C4×C47—44no—
123C2×C4×C47#10112yes—
124C2×C4×C47#10412yes—
125C2×C4×C47#10412yes—
126C2×C4×C47C2×C22⋊C412yes—
127C2×C4×C47#12012yes—
128C2×C4×C47#12012yes—
129C2×Dic47—24no—
130C2×Dic47—24no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135Dic94—48no—
136Dic94—48no—
137C8×C47—44no—
138C47⋊C8—44no—
139#8338—471nomaximal, Sylow
140C22×C47—22no—
141C22×C47—22no—
142C22×C47—22no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—41no—
146C22×C47—41no—
147C22×C47#37911yescharacteristic
148C22×C47C2×#10111yescharacteristic
149C22×C47C2×#10111yescharacteristic
150C22×C47C2×#10411yescharacteristic
151C22×C47C2×#10411yescharacteristic
152C22×C47#42111yescharacteristic
153C22×C47#42111yescharacteristic
154C4×C47—22no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—22no—
158C4×C47—22no—
159C4×C47—22no—
160Dic47—44no—
161Dic47—44no—
162C2×#533—471no—
163#1749—471no—
164#1749D4711yescharacteristic
165C2×#330—474no—
166C2×C47—42no—
167C2×C47#152611yescharacteristic
168C2×C47#153111yescharacteristic
169C2×C47#153111yescharacteristic
170C2×C47#153711yescharacteristic
171C2×C47#178411yescharacteristic
172C2×C47#178911yescharacteristic
173C2×C47#178911yescharacteristic
174#533—948no—
175C2×C4⋊Q8—471no—
176C2×C4⋊Q8—471no—
177C2×#125—472no—
178C2×C22⋊Q8—941no—
179C2×C22⋊Q8—941no—
180C2×C4×Q8—941no—
181C2×C4×Q8—941no—
182C2×#117—472no—
183C22×C42C2×D4711yescharacteristic
184C2×C22⋊C8—21no—
185C2×C22⋊C8—941no—
186C2×#97—472no—
187C2×#97C2×D4712yes—
188#336—21no—
189#336—941no—
190#330—948no—
191C47#833811yescharacteristic, complemented, Sylow
192C22×Q8—941no—
193C22×Q8—941no—
194C23×C4C4×D4711yescharacteristic
195C23×C4D18811yescharacteristic
196C23×C4C22⋊D4711yescharacteristic
197C22×C8—22no—
198C22×C8—942no—
199C4⋊Q8—944no—
200C4⋊Q8—944no—
201#125—478no—
202#125—944no—
203C22⋊Q8—948no—
204C22⋊Q8—948no—
205C4×Q8—948no—
206C4×Q8—948no—
207#117—944no—
208#117—944no—
209C2×C4⋊C4—472no—
210C2×C4⋊C4—472no—
211C2×C4⋊C4—941no—
212C2×C4⋊C4—941no—
213C2×C4⋊C4—942no—
214C2×C4⋊C4—942no—
215C2×C22⋊C4—942no—
216C2×C42—24no—
217C2×C42—942no—
218C2×C42C4×D4711yescharacteristic
219C2×C42D18811yescharacteristic
220C2×C42C22⋊D4711yescharacteristic
221C2×C42C22×D4711yescharacteristic
222C22⋊C8—24no—
223C22⋊C8—944no—
224#97—24no—
225#97—944no—
226C24(C2×D47)⋊C411yescharacteristic
227C2×Q8—944no—
228C2×Q8—944no—
229C2×Q8—1882no—
230C2×Q8—1882no—
231C2×Q8—1882no—
232C2×Q8—1882no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—942no—
243C22×C4—942no—
244C22×C4(C2×D47)⋊C411yescharacteristic
245C22×C4(C2×D47)⋊C411yescharacteristic
246C22×C4(C2×D47)⋊C411yescharacteristic
247C22×C4C2×C4×D4711yescharacteristic
248C22×C4C2×D18811yescharacteristic
249C22×C4C2×C22⋊D4711yescharacteristic
250C2×C8—24no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—944no—
254C2×C8—1882no—
255C2×C8—1882no—
256C4⋊C4—474no—
257C4⋊C4—474no—
258C4⋊C4—944no—
259C4⋊C4—944no—
260C4⋊C4—942no—
261C4⋊C4—942no—
262C4⋊C4—948no—
263C4⋊C4—948no—
264C22⋊C4—948no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—948no—
269C42D4×D4714yes—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23#1309394212yes—
277C23C2×(C2×D47)⋊C411yescharacteristic
278Q8—1888no—
279Q8—1888no—
280C2×C4—28no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—944no—
296C2×C4—944no—
297C2×C4—1882no—
298C2×C4—1882no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4#1309394212yes—
302C2×C4#1309395012yes—
303C2×C4#1309395212yes—
304C2×C4C22⋊C4×D4712yes—
305C2×C4#1309399212yes—
306C2×C4#1309399412yes—
307C8—44no—
308C8—1884no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22#1568897811yescharacteristic
317C22C2×#1309394211yescharacteristic
318C22C2×#1309394211yescharacteristic
319C22C2×#1309395011yescharacteristic
320C22C2×#1309395211yescharacteristic
321C22#1568913911yescharacteristic
322C22#1568914111yescharacteristic
323C4—22no—
324C4—22no—
325C4—22no—
326C4—22no—
327C4—22no—
328C4—22no—
329C4—1884no—
330C4—1884no—
331C2—42no—
332C2#2024198211yescharacteristic
333C2#2024200211yescharacteristic
334C2#2024200411yescharacteristic
335C2#2024202511yescharacteristic
336C2#2024331111yescharacteristic
337C2#2024333111yescharacteristic
338C2#2024333311yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1749C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1749D47
hypocenterC47#8338
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. C2×#15688864C2semidirect
191. C47#8338semidirect

Maximal subgroups

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

subgroupindex
C2×#156896802
#202431502
C2×#156888642
C47×#17492
#833847