Finite Group Databasebeta

← #26127327

Subgroups — #26127327

Order 12032.

460 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 99 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#20247696C211yesmaximal, characteristic
2#20241706C211yesmaximal, characteristic
3#20241393C214yesmaximal, complemented
4C47×#1800C211yesmaximal, characteristic
5C2×#13094005—21no—
6C22×C4⋊Dic47C411yescharacteristic
7C2×C22⋊(C47⋊C8)—21no—
8#15689053C2212yes—
9#15689048C412yes—
10#15689021—21no—
11#15688984—21no—
12#15688983—21no—
13#15688983C411yescharacteristic
14#15688849C2212yes—
15#15688847—21no—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#423—21no—
18C2×#100×C47—21no—
19C47×#343C2212yes—
20C23×Dic47—21no—
21C22×C47⋊C8—41no—
22#13094005—42no—
23#13094005—42no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—41no—
27C2×C4⋊Dic47—41no—
28C2×C4⋊Dic47C2×C412yes—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—21no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—41no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95C2×C412yes—
37C22×D4×C47D411yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C2×M16×C47—22no—
41C4×D4×C47—41no—
42C4×D4×C47—41no—
43C2×C4⋊C4×C47C2311yescharacteristic
44C2×C22⋊C4×C47C2×C411yescharacteristic
45C2×C22⋊C4×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C4⋊C8×C47—41no—
48C4⋊C8×C47—41no—
49#100×C47—42no—
50C24×C47C22⋊C411yescharacteristic
51C24×C47C22⋊C411yescharacteristic
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C2×D4×C47D812yes—
55C2×D4×C47SD1612yes—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×Dic47—22no—
68C22×Dic47—22no—
69C22×Dic47—42no—
70C22×Dic47—42no—
71C22×Dic47—42no—
72M16×C47—42no—
73M16×C47—42no—
74C2×C8×C47—42no—
75C4⋊C4×C47—41no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—41no—
78C42×C47—21no—
79C42×C47—21no—
80C22⋊Dic47—42no—
81C22⋊Dic47—44no—
82C4⋊Dic47—42no—
83C4⋊Dic47—41no—
84C4⋊Dic47—41no—
85C4⋊Dic47—42no—
86Dic47⋊C4—42no—
87Dic47⋊C4—42no—
88C2×C47⋊C8—42no—
89C2×C47⋊C8—82no—
90C23×C47—21no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47#10212yes—
99C23×C47#10212yes—
100C23×C47C2×C22⋊C411yescharacteristic
101D4×C47—42no—
102D4×C47—42no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—22no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—81no—
116C2×C4×C47#9911yescharacteristic
117C2×C4×C47#9911yescharacteristic
118C2×C4×C47C2×D811yescharacteristic
119C2×C4×C47C2×SD1611yescharacteristic
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—84no—
127C8×C47—82no—
128C47⋊C8—82no—
129#11881—471nomaximal, Sylow
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—22no—
133C22×C47—22no—
134C22×C47—42no—
135C22×C47—42no—
136C22×C47—42no—
137C22×C47—41no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—81no—
142C22×C47C2×#9911yescharacteristic
143C22×C47C2×#10211yescharacteristic
144C22×C47#42111yescharacteristic
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47#35212yes—
152Dic47—82no—
153Dic47—82no—
154#2486—471no—
155#1800D4711yescharacteristic
156#1459—471no—
157#1346—474no—
158C2×C47—21no—
159C2×C47—21no—
160C2×C47—42no—
161C2×C47—42no—
162C2×C47#151911yescharacteristic
163C2×C47C2×#35211yescharacteristic
164C2×C47#213511yescharacteristic
165C2×#123—941no—
166C2×C4×D4C2×D4711yescharacteristic
167C22×C4⋊C4—471no—
168#423—21no—
169C2×#100—21no—
170C2×C22⋊C8—941no—
171#398—472no—
172#397—472no—
173#389—941no—
174#381—941no—
175#380—471no—
176#380—941no—
177#343C2×D4712yes—
178#324—472no—
179#323—941no—
180C47#1188111yescharacteristic, complemented, Sylow
181C22×D4C22⋊D4711yescharacteristic
182C23×C4—941no—
183C23×C4C4×D4711yescharacteristic
184C23×C4D18811yescharacteristic
185C2×M16—22no—
186C22×C8—1881no—
187#123—1882no—
188#123—1882no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—472no—
192C2×C4⋊C4—941no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4—942no—
195C2×C4⋊C4—1881no—
196C2×C4⋊C4—1881no—
197C2×C4⋊C4C22×D4711yescharacteristic
198C2×C22⋊C4—941no—
199C2×C22⋊C4—942no—
200C2×C22⋊C4C4×D4711yescharacteristic
201C2×C22⋊C4D18811yescharacteristic
202C2×C42C22⋊D4711yescharacteristic
203C4⋊C8—41no—
204C4⋊C8—41no—
205#100—42no—
206C22⋊C8—942no—
207C22⋊C8—1881no—
208#95—472no—
209#95—942no—
210#95—942no—
211C24(C2×D47)⋊C411yescharacteristic
212C24(C2×D47)⋊C411yescharacteristic
213C2×D4—41no—
214C2×D4—41no—
215C2×D4#1062533412yes—
216C2×D4#1062533612yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—942no—
224C22×C4—942no—
225C22×C4—1882no—
226C22×C4—1882no—
227C22×C4—1882no—
228C22×C4(C2×D47)⋊C411yescharacteristic
229C22×C4(C2×D47)⋊C411yescharacteristic
230C22×C4C2×C4×D4711yescharacteristic
231C22×C4C2×D18811yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233M16—42no—
234M16—42no—
235C2×C8—42no—
236C2×C8—3762no—
237C2×C8—1882no—
238C4⋊C4—41no—
239C4⋊C4—1882no—
240C4⋊C4—1882no—
241C4⋊C4—1881no—
242C4⋊C4—1881no—
243C4⋊C4—1882no—
244C4⋊C4—1882no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—1882no—
248C22⋊C4—1884no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#1309394412yes—
260C23#1309394412yes—
261C23C2×(C2×D47)⋊C411yescharacteristic
262D4—42no—
263D4—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—3764no—
278C2×C4—1882no—
279C2×C4—1882no—
280C2×C4—1882no—
281C2×C4—1882no—
282C2×C4—1882no—
283C2×C4—1882no—
284C2×C4#1309393811yescharacteristic
285C2×C4#1309393811yescharacteristic
286C2×C4C2×#1062533411yescharacteristic
287C2×C4C2×#1062533611yescharacteristic
288C8—82no—
289C8—3762no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#1309393811yescharacteristic
303C22C2×#1309394411yescharacteristic
304C22#1568914111yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—3762no—
312C4—3762no—
313C4#1568891412yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2024195511yescharacteristic
319C2C2×#1568891411yescharacteristic
320C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1800C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C4⋊C4×C47C23
hypercenter#1800D47
hypocenterC47#11881
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #20241393C2semidirect
180. C47#11881semidirect

Maximal subgroups

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

subgroupindex
#202476962
#202417062
#202413932
C47×#18002
#1188147