Finite Group Databasebeta

← #26108415

Subgroups — #26108415

Order 12032.

1038 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 147 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#20243312C211yesmaximal, characteristic, complemented
2#20241892C214yesmaximal, complemented
3#20241659C211yesmaximal, characteristic, complemented
4C47×#3486C211yesmaximal, characteristic
5C2×#13093943—22no—
6#15689087C2212yes—
7C2×C47⋊#95—22no—
8C2×C47⋊#95C2212yes—
9#15688860—24no—
10#15688857—24no—
11C2×2−1+4×C47C411yescharacteristic, complemented
12C23×Q8×C47C2211yescharacteristic
13D4×Q8×C47—24no—
14C47×#548—24no—
15C47×#536—22no—
16C47×#536C412yescomplemented
17C47×#525—22no—
18C47×#525C2212yes—
19C47×#519C411yescharacteristic, complemented
20C23×Dic47—22no—
21C2×#10625329—22no—
22#13093943—44no—
23C22⋊(C47⋊C8)—24no—
24C47⋊#95—24no—
25C47⋊#95—44no—
262−1+4×C47—44no—
27C2×#41×C47—22no—
28C2×#41×C47—22no—
29C2×#41×C47C2×C412yes—
30C22×Q8×C47—22no—
31C22×Q8×C47—24no—
32C22×Q8×C47—44no—
33C22×Q8×C47C2×C411yescharacteristic
34C22×Q8×C47C2×C411yescharacteristic
35C22×Q8×C47D412yes—
36C23×C4×C47D412yes—
37C23×C4×C47C2311yescharacteristic
38C4⋊Q8×C47—24no—
39C4⋊Q8×C47—22no—
40#124×C47—24no—
41#124×C47—22no—
42#123×C47—24no—
43#123×C47—22no—
44C22⋊Q8×C47—24no—
45C22⋊Q8×C47—28no—
46C22⋊Q8×C47—24no—
47C22⋊Q8×C47D414yes—
48#121×C47—24no—
49#121×C47D414yes—
50#120×C47—22no—
51C4×Q8×C47—24no—
52C4×D4×C47—28no—
53C4×D4×C47—24no—
54#117×C47—22no—
55#117×C47C2×C412yes—
56C24×C47C2×D411yescharacteristic
57#41×C47—44no—
58#41×C47—44no—
59#41×C47—48no—
60#41×C47—44no—
61C2×Q8×C47—22no—
62C2×Q8×C47—24no—
63C2×Q8×C47—24no—
64C2×Q8×C47—22no—
65C2×Q8×C47—22no—
66C2×Q8×C47—24no—
67C2×Q8×C47—42no—
68C2×Q8×C47—42no—
69C2×Q8×C47—416no—
70C2×Q8×C47—48no—
71C2×Q8×C47C22⋊C412yes—
72C2×D4×C47—22no—
73C2×D4×C47—22no—
74C2×D4×C47—24no—
75C2×D4×C47C22⋊C412yes—
76C22×C4×C47—24no—
77C22×C4×C47—24no—
78C22×C4×C47—24no—
79C22×C4×C47—22no—
80C22×C4×C47—42no—
81C22×C4×C47—42no—
82C22×C4×C47C22⋊C412yes—
83C22×C4×C47C22⋊C412yes—
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D414yes—
87C22×Dic47—24no—
88C22×Dic47—44no—
89C22×Dic47—42no—
90C22×Dic47—44no—
91C22×Dic47—42no—
92C4⋊C4×C47—24no—
93C4⋊C4×C47—28no—
94C4⋊C4×C47—24no—
95C4⋊C4×C47C22⋊C414yes—
96C22⋊C4×C47—28no—
97C22⋊C4×C47—22no—
98C22⋊C4×C47—24no—
99C22⋊C4×C47—24no—
100C42×C47—24no—
101C42×C47—22no—
102#10625329—44no—
103C2×C47⋊C8—42no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—22no—
107C23×C47—22no—
108C23×C47C2×C22⋊C411yescharacteristic
109C23×C47#12012yes—
110Q8×C47—28no—
111Q8×C47—48no—
112Q8×C47—42no—
113Q8×C47—44no—
114Q8×C47—44no—
115D4×C47—42no—
116D4×C47—44no—
117D4×C47—44no—
118D4×C47—44no—
119D4×C47—44no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—22no—
124C2×C4×C47—24no—
125C2×C4×C47—22no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—44no—
130C2×C4×C47—48no—
131C2×C4×C47—48no—
132C2×C4×C47—44no—
133C2×C4×C47#9914yes—
134C2×C4×C47C2×C22⋊C412yes—
135C2×C4×C47#12012yes—
136C2×Dic47—48no—
137C2×Dic47—44no—
138C2×Dic47—84no—
139C2×Dic47—82no—
140C2×Dic47—82no—
141C47⋊C8—82no—
142#8455—471nomaximal, Sylow
143C22×C47—24no—
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—22no—
147C22×C47—22no—
148C22×C47—22no—
149C22×C47—44no—
150C22×C47—42no—
151C22×C47#37911yescharacteristic
152C22×C47C2×#9912yes—
153C22×C47#45412yes—
154C22×C47#45612yes—
155C4×C47—22no—
156C4×C47—24no—
157C4×C47—22no—
158C4×C47—44no—
159C4×C47—42no—
160C4×C47—42no—
161Dic47—84no—
162#3486D4711yescharacteristic
163#1784—471no—
164#1504—474no—
165#1446—471no—
166C2×C47—22no—
167C2×C47—22no—
168C2×C47—22no—
169C2×C47—82no—
170C2×C47#178111yescharacteristic
171C2×C47#179312yes—
172C2×2−1+4Dic4711yescharacteristic, complemented
173C23×Q8C2×D4711yescharacteristic
174D4×Q8—24no—
175#548—24no—
176#536—22no—
177#536Dic4712yescomplemented
178#525—22no—
179#525C2×D4712yes—
180#519Dic4711yescharacteristic, complemented
181C2×#101—942no—
182#407—472no—
183C2×#95—472no—
184C2×#95—942no—
185#328—944no—
186#327—944no—
187C47#845511yescharacteristic, complemented, Sylow
1882−1+4—44no—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41C2×Dic4712yes—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—44no—
195C22×Q8C2×Dic4711yescharacteristic
196C22×Q8C2×Dic4711yescharacteristic
197C22×Q8C22⋊D4712yes—
198C23×C4—942no—
199C23×C4C22⋊D4712yes—
200C23×C4C22×D4711yescharacteristic
201C2×M16—942no—
202C4⋊Q8—24no—
203C4⋊Q8—22no—
204#124—24no—
205#124—22no—
206#123—24no—
207#123—22no—
208C22⋊Q8—24no—
209C22⋊Q8—28no—
210C22⋊Q8—24no—
211C22⋊Q8C22⋊D4714yes—
212#121—24no—
213#121C22⋊D4714yes—
214#120—22no—
215C4×Q8—24no—
216C4×D4—28no—
217C4×D4—24no—
218#117—22no—
219#117C2×Dic4712yes—
220#101—1884no—
221C22⋊C8—944no—
222#95—944no—
223#95—1884no—
224C24C2×C22⋊D4711yescharacteristic
225#41—44no—
226#41—44no—
227#41—48no—
228#41—44no—
229C2×Q8—22no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—416no—
238C2×Q8—48no—
239C2×Q8C22⋊Dic4712yes—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4C22⋊Dic4712yes—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—944no—
251C22×C4—1884no—
252C22×C4—1882no—
253C22×C4—1884no—
254C22×C4—1882no—
255C22×C4C22⋊Dic4712yes—
256C22×C4C22⋊Dic4712yes—
257C22×C4C22×Dic4711yescharacteristic
258C22×C4C2×C22⋊D4711yescharacteristic
259C22×C4C2×C22⋊D4714yes—
260M16—1884no—
261C2×C8—1882no—
262C4⋊C4—24no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4C22⋊Dic4714yes—
266C22⋊C4—28no—
267C22⋊C4—22no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C42—24no—
271C42—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23C2×C22⋊Dic4711yescharacteristic
277C23#1309399312yes—
278Q8—28no—
279Q8—48no—
280Q8—42no—
281Q8—44no—
282Q8—44no—
283D4—42no—
284D4—44no—
285D4—44no—
286D4—44no—
287D4—44no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—3764no—
302C2×C4—3762no—
303C2×C4—3762no—
304C2×C4—1888no—
305C2×C4—1884no—
306C2×C4#1309393914yes—
307C2×C4C2×C22⋊Dic4712yes—
308C2×C4#1309399312yes—
309C8—3762no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—42no—
317C22—44no—
318C22#1568897911yescharacteristic
319C22C2×#1309393912yes—
320C22#1568929612yes—
321C22#1568930612yes—
322C4—22no—
323C4—24no—
324C4—22no—
325C4—42no—
326C4—42no—
327C4—44no—
328C4—3764no—
329C2—22no—
330C2—22no—
331C2—22no—
332C2—82no—
333C2#2024330311yescharacteristic
334C2#2024334712yes—
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC47×#3486C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenter#3486D47
hypocenterC47#8455
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3486

Complements

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

NH ≅ G/Ntype
1. #20243312C2semidirect
2. #20241892C2semidirect
3. #20241659C2semidirect
11. C2×2−1+4×C47C4semidirect
16. C47×#536C4semidirect
19. C47×#519C4semidirect
172. C2×2−1+4Dic47semidirect
177. #536Dic47semidirect
180. #519Dic47semidirect
187. C47#8455semidirect

Maximal subgroups

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

subgroupindex
#202433122
#202418922
#202416592
C47×#34862
#845547