Finite Group Databasebeta

← #26125496

Subgroups — #26125496

Order 12032.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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#20252031C211yesmaximal, characteristic
2#20245188C211yesmaximal, characteristic, complemented
3#20245184C211yesmaximal, characteristic, complemented
4#20241330C211yesmaximal, characteristic, complemented
5#20241313C211yesmaximal, characteristic
6#20241313C211yesmaximal, characteristic
7C47×#1323C211yesmaximal, characteristic, complemented
8#15689591C411yescharacteristic
9#15689183—21no—
10#15689183—21no—
11#15689175—21no—
12#15689175—21no—
13#15689150C411yescharacteristic
14#15689091C411yescharacteristic
15#15689089C2211yescharacteristic
16M16×Dic47C411yescharacteristic
17C2×C8×Dic47C2211yescharacteristic
18Dic47⋊C16—21no—
19Dic47⋊C16—21no—
20C22⋊(C47⋊C16)C2211yescharacteristic
21C22⋊(C47⋊C16)C2211yescharacteristic
22#15688898—21no—
23C16×Dic47—21no—
24C47×#424C2211yescharacteristic
25C47×C22⋊C16C2211yescharacteristic
26C47×C22⋊C16C2211yescharacteristic
27C2×#10625350C2×C411yescharacteristic
28C2×#10625329C2×C411yescharacteristic
29C22×C47⋊C8C2×C411yescharacteristic
30#13093989C811yescharacteristic
31#13093989C811yescharacteristic
32#13093986C811yescharacteristic
33#13093986C811yescharacteristic
34(C22⋊D47)⋊C4—21no—
35(C22⋊D47)⋊C4—21no—
36#13093981—21no—
37C2×C4×Dic47C2×C411yescharacteristic
38C42×D47—21no—
39C2×C47⋊C16—21no—
40C2×C47⋊C16—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43(C2×D47)⋊C8—21no—
44(C2×D47)⋊C8—21no—
45(C2×D47)⋊C8—21no—
46(C2×D47)⋊C8—21no—
47#13093934—21no—
48C8×Dic47—21no—
49C8×Dic47D411yescharacteristic
50C8×Dic47D411yescharacteristic
51C8×Dic47D411yescharacteristic
52C8×Dic47D411yescharacteristic
53C2×M16×C47C2×C411yescharacteristic
54C22×C8×C47C2311yescharacteristic
55#117×C47C2×C411yescharacteristic
56C2×C16×C47—21no—
57C2×C16×C47—21no—
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C22×C4×C47C22×C411yescharacteristic
61C2×C22⋊D47—21no—
62C22×Dic47M1611yescharacteristic
63C22×Dic47M1611yescharacteristic
64#10625350—41no—
65#10625350—41no—
66C2×D188C2×C811yescharacteristic
67C2×C4×D47—21no—
68C2×C4×D47—21no—
69C2×C4×D47—21no—
70C2×Dic94C2×C811yescharacteristic
71M16×C47—41no—
72C2×C8×C47—21no—
73C2×C8×C47—41no—
74C2×C8×C47C2×D411yescharacteristic
75C2×C8×C47C2×D411yescharacteristic
76C4⋊C4×C47C2×C811yescharacteristic
77C4⋊C4×C47C2×C811yescharacteristic
78C22⋊C4×C47—21no—
79C42×C47—21no—
80(C2×D47)⋊C4—21no—
81(C2×D47)⋊C4—21no—
82Dic47⋊C4—21no—
83Dic47⋊C4—21no—
84C4×Dic47—21no—
85C4×Dic47—21no—
86C4×Dic47C22⋊C411yescharacteristic
87C4×Dic47C22⋊C411yescharacteristic
88C4×Dic47C2×C811yescharacteristic
89C4×Dic47C2×C811yescharacteristic
90#10625329—41no—
91C2×C47⋊C8—21no—
92C2×C47⋊C8—41no—
93C2×C47⋊C8C22⋊C411yescharacteristic
94C2×C47⋊C8C22⋊C411yescharacteristic
95C16×C47—41no—
96C16×C47—41no—
97C47⋊C16—41no—
98C47⋊C16—41no—
99C23×C47C2×M1611yescharacteristic
100C22×D47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47C2×C22⋊C411yescharacteristic
106C2×C4×C47C22×C811yescharacteristic
107C22⋊D47—42no—
108C2×Dic47—22no—
109C2×Dic47—21no—
110C2×Dic47—41no—
111C2×Dic47C22⋊C811yescharacteristic
112C2×Dic47C22⋊C811yescharacteristic
113C2×Dic47C22⋊C811yescharacteristic
114C2×Dic47C22⋊C811yescharacteristic
115D188—41no—
116C4×D47—42no—
117C4×D47—41no—
118C4×D47—41no—
119C4×D47—42no—
120Dic94—41no—
121C8×C47—21no—
122C8×C47—21no—
123C8×C47—41no—
124C47⋊C8—21no—
125C47⋊C8—21no—
126C47⋊C8—41no—
127#11464—471nomaximal, Sylow
128C22×C47—21no—
129C22×C47—21no—
130C22×C47—21no—
131C22×C47C2×C22⋊C811yescharacteristic
132C2×D47—41no—
133C2×D47—41no—
134C2×D47—41no—
135C4×C47—21no—
136C4×C47—41no—
137C4×C47—41no—
138C4×C47#41011yescharacteristic
139C4×C47#41311yescharacteristic
140Dic47—21no—
141Dic47—21no—
142Dic47—21no—
143Dic47—41no—
144Dic47#35012yes—
145#2922—471no—
146#2163—471no—
147#2162—471no—
148#1323—471no—
149#1323D4711yescharacteristic, complemented
150#1315—471no—
151#1315—471no—
152C2×C47—41no—
153C2×C47#145811yescharacteristic
154C2×C47C2×#35011yescharacteristic
155C2×C47#211711yescharacteristic
156D47—81no—
157C4×#41—471no—
158#433—941no—
159#433—941no—
160#431—941no—
161#431—941no—
162#424—471no—
163#424C2×D4711yescharacteristic
164#408—471no—
165#408—471no—
166C4×M16—471no—
167C2×C4×C8—471no—
168C4⋊C16—941no—
169C4⋊C16—941no—
170C22⋊C16—471no—
171C22⋊C16—471no—
172C22⋊C16C2×D4711yescharacteristic
173C22⋊C16C2×D4711yescharacteristic
174#346—941no—
175C4×C16—941no—
176C47#1146411yescharacteristic, complemented, Sylow
177C2×#41—471no—
178C2×M16—471no—
179C2×M16C4×D4711yescharacteristic
180C22×C8—471no—
181C22×C8C22×D4711yescharacteristic
182C4×Q8—471no—
183C4×Q8—471no—
184C4×D4—471no—
185C4×D4—471no—
186C4×D4—941no—
187C4×D4—941no—
188#117—941no—
189#117C4×D4711yescharacteristic
190C2×C42—471no—
191C2×C42—941no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—941no—
195C2×C16—941no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—941no—
199C4⋊C8—941no—
200C22⋊C8—941no—
201C22⋊C8—941no—
202C22⋊C8—941no—
203C22⋊C8—941no—
204#97—941no—
205C4×C8—471no—
206C4×C8—471no—
207C4×C8—471no—
208C4×C8—471no—
209C4×C8—941no—
210#41—1881no—
211#41—1881no—
212C2×Q8—471no—
213C2×D4—471no—
214C2×D4—941no—
215C22×C4—471no—
216C22×C4—471no—
217C22×C4—941no—
218C22×C4—941no—
219C22×C4—941no—
220C22×C4C2×C4×D4711yescharacteristic
221M16—41no—
222M16—1881no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—471no—
226C2×C8—471no—
227C2×C8—941no—
228C2×C8—1881no—
229C2×C8D4×D4711yescharacteristic
230C2×C8D4×D4711yescharacteristic
231C4⋊C4—941no—
232C4⋊C4—941no—
233C4⋊C4C8×D4711yescharacteristic
234C4⋊C4C8×D4711yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—941no—
237C22⋊C4—941no—
238C42—21no—
239C42—471no—
240C42—471no—
241C42—471no—
242C42—471no—
243C42—941no—
244C42—941no—
245C16—41no—
246C16—41no—
247C16—1881no—
248C16—1881no—
249C23—941no—
250C23M16×D4711yescharacteristic
251Q8—1881no—
252D4—1881no—
253D4—1882no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—471no—
259C2×C4—471no—
260C2×C4—471no—
261C2×C4—471no—
262C2×C4—942no—
263C2×C4—941no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4—1881no—
267C2×C4—1881no—
268C2×C4—1881no—
269C2×C4C22⋊C4×D4711yescharacteristic
270C2×C4C2×C8×D4711yescharacteristic
271C8—21no—
272C8—21no—
273C8—41no—
274C8—941no—
275C8—941no—
276C8—1881no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—1881no—
281C22—1881no—
282C22—1881no—
283C22C22⋊C8×D4711yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—472no—
288C4—941no—
289C4—941no—
290C4—941no—
291C4—1881no—
292C4#1568909511yescharacteristic
293C4#1568910611yescharacteristic
294C2—41no—
295C2—3761no—
296C2#2024170111yescharacteristic
297C2#350×D4711yescharacteristic
298C2#2024501411yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC22×C8C22×D47
FittingC47×#1323C2
socleC22×C47C2×C22⋊C8
prime radicalC2×#10625350C2×C4
cosocleC22×C8×C47C23
hypercenter#1323D47
hypocenterC47#11464
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #20245188C2semidirect
3. #20245184C2semidirect
4. #20241330C2semidirect
7. C47×#1323C2semidirect
149. #1323D47semidirect
176. C47#11464semidirect

Maximal subgroups

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

subgroupindex
#202520312
#202451882
#202451842
#202413302
#202413132
C47×#13232
#1146447