Finite Group Databasebeta

← #26103076

Subgroups — #26103076

Order 12032.

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

normal subgroups only

This group has 412 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#20241652C211yesmaximal, characteristic
2#20241231C211yesmaximal, characteristic
3C47×#1280C211yesmaximal, characteristic
4C22×C4⋊Dic47C411yescharacteristic
5C22×Dic47⋊C4—21no—
6C2×C22⋊(C47⋊C8)—21no—
7C2×C47⋊#95—21no—
8C2×C47⋊#95C411yescharacteristic
9C2×C22⋊C8×C47—21no—
10C2×#95×C47C2211yescharacteristic
11C23×Dic47—21no—
12C23×Dic47—21no—
13C22×C47⋊C8—41no—
14C2×C4⋊Dic47—41no—
15C2×C4⋊Dic47—42no—
16C2×Dic47⋊C4—42no—
17C2×Dic47⋊C4—42no—
18C2×Dic47⋊C4—41no—
19C2×Dic47⋊C4—41no—
20C22⋊(C47⋊C8)—22no—
21C22⋊(C47⋊C8)—41no—
22C47⋊#95—21no—
23C47⋊#95—42no—
24C47⋊#95C812yescomplemented
25C23×C4×C47C2×C411yescharacteristic
26C23×C4×C47D411yescharacteristic
27C23×C4×C47D411yescharacteristic
28C22×C8×C47—41no—
29C22⋊C8×C47—22no—
30C22⋊C8×C47—41no—
31#95×C47—41no—
32C24×C47C22⋊C411yescharacteristic
33C22×C4×C47—21no—
34C22×C4×C47—21no—
35C22×C4×C47—21no—
36C22×C4×C47—21no—
37C22×C4×C47—21no—
38C22×C4×C47—21no—
39C22×C4×C47—41no—
40C22×C4×C47—41no—
41C22×C4×C47—41no—
42C22×C4×C47—41no—
43C22×C4×C47—41no—
44C22×C4×C47—41no—
45C22×C4×C47C2×C811yescharacteristic
46C22×C4×C47M1611yescharacteristic
47C22×C4×C47D811yescharacteristic
48C22×C4×C47SD1611yescharacteristic
49C22×C4×C47SD1611yescharacteristic
50C22×C4×C47Q1611yescharacteristic
51C22×Dic47—22no—
52C22×Dic47—22no—
53C22×Dic47—22no—
54C22×Dic47—22no—
55C22×Dic47—41no—
56C22×Dic47—41no—
57C22×Dic47—41no—
58C22×Dic47—41no—
59C22×Dic47—41no—
60C22×Dic47—41no—
61C22×Dic47—41no—
62C22×Dic47—41no—
63C22×Dic47—41no—
64C22×Dic47—41no—
65C2×C8×C47—42no—
66C2×C8×C47—81no—
67C2×C8×C47—81no—
68C4⋊Dic47—42no—
69C4⋊Dic47—42no—
70Dic47⋊C4—44no—
71Dic47⋊C4—44no—
72C2×C47⋊C8—42no—
73C2×C47⋊C8—81no—
74C2×C47⋊C8—81no—
75C23×C47—21no—
76C23×C47—21no—
77C23×C47—21no—
78C23×C47—21no—
79C23×C47C22⋊C811yescharacteristic
80C23×C47#9911yescharacteristic
81C23×C47#10111yescharacteristic
82C23×C47#10211yescharacteristic
83C23×C47#10311yescharacteristic
84C23×C47#10411yescharacteristic
85C23×C47#10411yescharacteristic
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—42no—
93C2×C4×C47—41no—
94C2×C4×C47—41no—
95C2×C4×C47—41no—
96C2×C4×C47—41no—
97C2×C4×C47—41no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—81no—
106C2×C4×C47—81no—
107C2×Dic47—24no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—81no—
118C2×Dic47—81no—
119C2×Dic47—81no—
120C2×Dic47—81no—
121C8×C47—82no—
122C47⋊C8—82no—
123#7123—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47#32411yescharacteristic
137C22×C47D4⋊C811yescharacteristic
138C22×C47Q8⋊C811yescharacteristic
139C22×C47#32711yescharacteristic
140C22×C47#32811yescharacteristic
141C22×C47#33011yescharacteristic
142C22×C47#33211yescharacteristic
143C22×C47AΣL(1,𝔽16)11yescharacteristic
144C22×C47#35211yescharacteristic
145C22×C47#35311yescharacteristic
146C22×C47#35611yescharacteristic
147C4×C47—41no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47—41no—
153Dic47—44no—
154Dic47—82no—
155#1444—471no—
156#1280—471no—
157#1280D4711yescharacteristic
158C2×C47—21no—
159C2×C47—21no—
160C2×C47—21no—
161C2×C47—21no—
162C2×C47#127111yescharacteristic
163C2×C47#132511yescharacteristic
164C2×C47#132611yescharacteristic
165C2×C47#134311yescharacteristic
166C2×C47#134611yescharacteristic
167C2×C47#135211yescharacteristic
168C2×C47#135411yescharacteristic
169C22×C4⋊C4—471no—
170C22×C4⋊C4—941no—
171C2×C22⋊C8—21no—
172C2×C22⋊C8—941no—
173C2×#95—471no—
174C2×#95—941no—
175C2×#95C2×D4711yescharacteristic
176C47#712311yescharacteristic, complemented, Sylow
177C23×C4—941no—
178C23×C4—941no—
179C23×C4C4×D4711yescharacteristic
180C23×C4D18811yescharacteristic
181C23×C4C22⋊D4711yescharacteristic
182C22×C8—41no—
183C22×C8—1881no—
184C2×C4⋊C4—1881no—
185C2×C4⋊C4—1882no—
186C2×C4⋊C4—1882no—
187C2×C4⋊C4—1881no—
188C2×C4⋊C4—1881no—
189C2×C4⋊C4—1882no—
190C22⋊C8—22no—
191C22⋊C8—41no—
192C22⋊C8—942no—
193C22⋊C8—1881no—
194#95—41no—
195#95—472no—
196#95—941no—
197#95—1882no—
198C24(C2×D47)⋊C411yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—942no—
212C22×C4—942no—
213C22×C4—942no—
214C22×C4—942no—
215C22×C4—1881no—
216C22×C4—1881no—
217C22×C4—1881no—
218C22×C4—1881no—
219C22×C4—1881no—
220C22×C4—1881no—
221C22×C4—1881no—
222C22×C4—1881no—
223C22×C4—1881no—
224C22×C4—1881no—
225C22×C4C8×D4711yescharacteristic
226C22×C4#1062532411yescharacteristic
227C22×C4#1062532511yescharacteristic
228C22×C4Dic18811yescharacteristic
229C22×C4#1062533411yescharacteristic
230C22×C4#1062533611yescharacteristic
231C2×C8—42no—
232C2×C8—81no—
233C2×C8—81no—
234C2×C8—3761no—
235C2×C8—3761no—
236C2×C8—1882no—
237C4⋊C4—1882no—
238C4⋊C4—1884no—
239C4⋊C4—1884no—
240C4⋊C4—1882no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23(C2×D47)⋊C811yescharacteristic
246C23#1309393811yescharacteristic
247C23#1309394211yescharacteristic
248C23#1309394411yescharacteristic
249C23#1309394911yescharacteristic
250C23#1309395011yescharacteristic
251C23#1309395211yescharacteristic
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—944no—
274C2×C4—3761no—
275C2×C4—3761no—
276C2×C4—3761no—
277C2×C4—3761no—
278C2×C4—1882no—
279C2×C4—1882no—
280C2×C4—1882no—
281C2×C4—1882no—
282C2×C4—1882no—
283C2×C4—1882no—
284C2×C4—1882no—
285C2×C4—1882no—
286C2×C4—1882no—
287C8—82no—
288C8—3762no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22#1568884811yescharacteristic
302C22#1568885011yescharacteristic
303C22#1568885511yescharacteristic
304C22#1568885611yescharacteristic
305C22#1568886111yescharacteristic
306C22#1568886411yescharacteristic
307C22#1568887011yescharacteristic
308C22#1568890911yescharacteristic
309C22#1568891411yescharacteristic
310C22#1568891511yescharacteristic
311C22#1568892111yescharacteristic
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—3762no—
319C4—1884no—
320C2—21no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2#2024121211yescharacteristic
325C2#2024133311yescharacteristic
326C2#2024133511yescharacteristic
327C2#2024138411yescharacteristic
328C2#2024139311yescharacteristic
329C2#2024141311yescharacteristic
330C2#2024141911yescharacteristic
subgroupG/H
centerC23#13093938
commutatorC22×C4×C47C2×C8
FrattiniC2×#95C2×D47
FittingC47×#1280C2
socleC23×C47#99
prime radicalC24×C47C22⋊C4
cosocleC2×#95×C47C22
hypercenter#1280D47
hypocenterC47#7123
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ #1280

Complements

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

NH ≅ G/Ntype
24. C47⋊#95C8semidirect
176. C47#7123semidirect

Maximal subgroups

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

subgroupindex
#202416522
#202412312
C47×#12802
#712347