Finite Group Databasebeta

← #26104828

Subgroups — #26104828

Order 12032.

470 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 107 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#20251727C211yesmaximal, characteristic
2#20243368C211yesmaximal, characteristic
3#20241213C212yesmaximal, complemented
4#20241210C212yesmaximal, complemented
5C47×#1749C211yesmaximal, characteristic
6C2×#13094003C411yescharacteristic
7C2×C4⋊(C47⋊C8)—21no—
8C2×C22⋊(C47⋊C8)—21no—
9#15689029C411yescharacteristic
10#15689029C2211yescharacteristic
11#15689021C411yescharacteristic
12#15689021C2211yescharacteristic
13#15689007—21no—
14#15689007—21no—
15#15688996—21no—
16#15688986—21no—
17#15688983C411yescharacteristic
18#15688878C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C2×#97×C47—21no—
22C47×#336C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C22×C47⋊C8—22no—
25#13094003—42no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C4⋊(C47⋊C8)—42no—
35C22⋊(C47⋊C8)—42no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C22×C8×C47—22no—
47C2×C42×C47—42no—
48C2×C42×C47C2×C411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52C22⋊C8×C47—24no—
53#97×C47—42no—
54C24×C47C22⋊C411yescharacteristic
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—22no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47C22⋊C411yescharacteristic
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—
75C2×Dic94—22no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×C8×C47—24no—
79C2×C8×C47—42no—
80C2×C8×C47—42no—
81C42×C47—22no—
82C42×C47—22no—
83C42×C47—41no—
84C42×C47—41no—
85C22⋊Dic47—42no—
86C22⋊Dic47—42no—
87C4⋊Dic47—22no—
88C4⋊Dic47—41no—
89C4⋊Dic47—41no—
90Dic47⋊C4—42no—
91Dic47⋊C4—42no—
92Dic47⋊C4—41no—
93Dic47⋊C4—41no—
94C2×C47⋊C8—42no—
95C2×C47⋊C8—42no—
96C2×C47⋊C8—42no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47#10412yes—
104C23×C47C2×C22⋊C411yescharacteristic
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
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—44no—
120C2×C4×C47—44no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47#9912yes—
124C2×C4×C47#13112yes—
125C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C47#13712yes—
127C2×C4×C47#13711yescharacteristic
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—
138C2×Dic47—41no—
139C2×Dic47—41no—
140Dic94—82no—
141C8×C47—44no—
142C47⋊C8—82no—
143#7701—471nomaximal, Sylow
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—22no—
147C22×C47—41no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47#40811yescharacteristic
152C22×C47C2×#9911yescharacteristic
153C22×C47#41311yescharacteristic
154C22×C47#41711yescharacteristic
155C22×C47#41911yescharacteristic
156C22×C47C2×#10411yescharacteristic
157C22×C47#42111yescharacteristic
158C4×C47—22no—
159C4×C47—22no—
160C4×C47—22no—
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—41no—
164C4×C47—41no—
165Dic47—81no—
166Dic47—81no—
167Dic47—81no—
168Dic47—81no—
169#2855—471no—
170#1797—471no—
171#1749D4711yescharacteristic
172#1272—472no—
173#1271—472no—
174C2×C47—42no—
175C2×C47#146811yescharacteristic
176C2×C47#148311yescharacteristic
177C2×C47#149711yescharacteristic
178C2×C47#150811yescharacteristic
179C2×C47#151311yescharacteristic
180C2×C47#153211yescharacteristic
181C2×C47#155911yescharacteristic
182C2×C22⋊Q8—471no—
183C22×C42C2×D4711yescharacteristic
184C2×C4⋊C8—941no—
185C2×C22⋊C8—21no—
186C2×C22⋊C8—941no—
187C2×#97—21no—
188#391—471no—
189#391—471no—
190#389—471no—
191#389—471no—
192#387—941no—
193#387—941no—
194#385—941no—
195#381—941no—
196#380—471no—
197#336—472no—
198#336C2×D4712yes—
199C47#770111yescharacteristic, complemented, Sylow
200C22×Q8—471no—
201C23×C4C4×D4711yescharacteristic
202C23×C4D18811yescharacteristic
203C23×C4C22⋊D4711yescharacteristic
204C22×C8—22no—
205C22×C8—942no—
206C22⋊Q8—1882no—
207C2×C4⋊C4—471no—
208C2×C4⋊C4—471no—
209C2×C4⋊C4—471no—
210C2×C4⋊C4—941no—
211C2×C4⋊C4—941no—
212C2×C4⋊C4—941no—
213C2×C22⋊C4—941no—
214C2×C22⋊C4—941no—
215C2×C42—42no—
216C2×C42C4×D4711yescharacteristic
217C2×C42D18811yescharacteristic
218C2×C42C22⋊D4711yescharacteristic
219C2×C42C22×D4711yescharacteristic
220C4⋊C8—1882no—
221C22⋊C8—24no—
222C22⋊C8—1882no—
223#97—42no—
224#95—941no—
225#95—941no—
226#95—941no—
227#95—941no—
228#95—941no—
229#95—941no—
230#95—941no—
231C24(C2×D47)⋊C411yescharacteristic
232C2×Q8—942no—
233C2×Q8—1881no—
234C2×Q8—1881no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—941no—
246C22×C4—941no—
247C22×C4—941no—
248C22×C4—941no—
249C22×C4(C2×D47)⋊C411yescharacteristic
250C22×C4(C2×D47)⋊C411yescharacteristic
251C22×C4(C2×D47)⋊C411yescharacteristic
252C22×C4C2×C4×D4711yescharacteristic
253C22×C4C2×D18811yescharacteristic
254C22×C4C2×C22⋊D4711yescharacteristic
255C2×C8—24no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—1882no—
259C2×C8—1882no—
260C2×C8—1882no—
261C4⋊C4—942no—
262C4⋊C4—1881no—
263C4⋊C4—1881no—
264C4⋊C4—1882no—
265C4⋊C4—1882no—
266C4⋊C4—1881no—
267C4⋊C4—1881no—
268C22⋊C4—1882no—
269C22⋊C4—1882no—
270C42—22no—
271C42—22no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23#1309395212yes—
281C23C2×(C2×D47)⋊C411yescharacteristic
282Q8—3762no—
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—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
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—1881no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4#1309393812yes—
314C2×C4#1309403212yes—
315C2×C4#1309405211yescharacteristic
316C2×C4#1309405412yes—
317C2×C4#1309405711yescharacteristic
318C8—44no—
319C8—3762no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22#1568909111yescharacteristic
328C22C2×#1309393811yescharacteristic
329C22#1568910511yescharacteristic
330C22#1568912511yescharacteristic
331C22#1568913111yescharacteristic
332C22C2×#1309395211yescharacteristic
333C22#1568913911yescharacteristic
334C4—21no—
335C4—21no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—3761no—
342C4—3761no—
343C4—3761no—
344C4—3761no—
345C2—42no—
346C2#2024173711yescharacteristic
347C2#2024180211yescharacteristic
348C2#2024186011yescharacteristic
349C2#2024190711yescharacteristic
350C2#2024193211yescharacteristic
351C2#2024200611yescharacteristic
352C2#2024211911yescharacteristic
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#7701
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. #20241213C2semidirect
4. #20241210C2semidirect
199. C47#7701semidirect

Maximal subgroups

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

subgroupindex
#202517272
#202433682
#202412132
#202412102
C47×#17492
#770147