Finite Group Databasebeta

← #26126979

Subgroups — #26126979

Order 12032.

488 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 79 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#20249138C211yesmaximal, characteristic
2#20243471C211yesmaximal, characteristic, complemented
3#20241392C214yesmaximal, complemented
4C47×#1464C211yesmaximal, characteristic, complemented
5C2×#13094021C411yescharacteristic
6C2×#13093998—21no—
7#15689087—21no—
8#15689040—21no—
9#15689040C2212yes—
10#15689032—21no—
11#15689032C412yes—
12#15689025—21no—
13#15688989C411yescharacteristic
14#15688892—21no—
15#15688892C2212yes—
16C2×#117×C47C2211yescharacteristic
17C47×#432—21no—
18C47×#324C2212yes—
19C47×#323—21no—
20C22×D188—21no—
21C22×D188—21no—
22C22×D188C2×C412yes—
23C22×C22⋊D47—22no—
24C2×#10625329—22no—
25#13094021—41no—
26#13094021—41no—
27#13093998—44no—
28C2×C4⋊Dic47—21no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C22⋊(C47⋊C8)—41no—
33C22⋊(C47⋊C8)—41no—
34C47⋊#95—21no—
35C47⋊#95C2×C412yes—
36C23×C4×C47D411yescharacteristic
37#117×C47—41no—
38#117×C47—41no—
39C2×C4⋊C4×C47D411yescharacteristic
40C2×C4⋊C4×C47C2311yescharacteristic
41C2×C22⋊C4×C47C2×C411yescharacteristic
42C2×C22⋊C4×C47D411yescharacteristic
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C4⋊C8×C47—41no—
46C22⋊C8×C47—22no—
47#97×C47—41no—
48C24×C47C22⋊C411yescharacteristic
49C23×D47—22no—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—22no—
53C22×C4×C47—41no—
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22×C411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C2×C22⋊D47—44no—
61C2×C22⋊D47—82no—
62C2×C22⋊D47—82no—
63C22×Dic47—22no—
64C2×D188—42no—
65C2×D188—42no—
66C2×D188—42no—
67C2×D188—42no—
68C2×D188—41no—
69C2×D188—41no—
70C2×D188—82no—
71C2×D188—82no—
72C2×C8×C47—42no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C22⋊C4×C47—41no—
76C22⋊C4×C47—41no—
77C42×C47—21no—
78C42×C47—21no—
79C42×C47—21no—
80C42×C47—21no—
81(C2×D47)⋊C4—44no—
82(C2×D47)⋊C4—82no—
83(C2×D47)⋊C4—82no—
84C4⋊Dic47—42no—
85#10625329—42no—
86#10625329—42no—
87C2×C47⋊C8—42no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47C2×C22⋊C411yescharacteristic
94C22×D47—42no—
95C22×D47—42no—
96C22×D47—42no—
97C22×D47—42no—
98C22×D47—42no—
99C22×D47—42no—
100C22×D47—42no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47#9911yescharacteristic
116C2×C4×C47#9911yescharacteristic
117C2×C4×C47AGL(1,ℤ/8)12yes—
118C22⋊D47—88no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122D188—82no—
123D188—82no—
124D188—84no—
125D188—84no—
126C8×C47—82no—
127C47⋊C8—82no—
128#11806—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—42no—
132C22×C47—41no—
133C22×C47—41no—
134C22×C47—41no—
135C22×C47—81no—
136C22×C47C2×#9911yescharacteristic
137C22×C47#41811yescharacteristic
138C22×C47#42111yescharacteristic
139C2×D47—82no—
140C2×D47—82no—
141C2×D47—82no—
142C2×D47—82no—
143C2×D47—82no—
144C2×D47—82no—
145C2×D47—82no—
146C4×C47—22no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47—41no—
153Dic47—82no—
154#2619—471no—
155#1822—471no—
156#1464D4711yescharacteristic, complemented
157#1345—474no—
158C2×C47—21no—
159C2×C47—21no—
160C2×C47—81no—
161C2×C47#151011yescharacteristic
162C2×C47#212211yescharacteristic
163C2×C47#213411yescharacteristic
164D47—162no—
165C2×#127—471no—
166C2×#121—941no—
167C2×#117C2×D4711yescharacteristic
168#432—21no—
169#407—941no—
170#394—472no—
171#394—941no—
172#392—472no—
173#392—941no—
174#390—941no—
175#382—471no—
176#343—472no—
177#343—941no—
178#324C2×D4712yes—
179#323—21no—
180C47#1180611yescharacteristic, complemented, Sylow
181C22×D4—472no—
182C22×D4—942no—
183C22×D4—941no—
184C22×D4—941no—
185C23×C4D18811yescharacteristic
186C2×M16—942no—
187#127—1881no—
188#127—1881no—
189#121—1884no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—941no—
193C2×C4⋊C4D18811yescharacteristic
194C2×C4⋊C4C22×D4711yescharacteristic
195C2×C22⋊C4—942no—
196C2×C22⋊C4—942no—
197C2×C22⋊C4—942no—
198C2×C22⋊C4C4×D4711yescharacteristic
199C2×C22⋊C4C22⋊D4711yescharacteristic
200C2×C42C4×D4711yescharacteristic
201C2×C42C22⋊D4711yescharacteristic
202C4⋊C8—41no—
203C22⋊C8—22no—
204C22⋊C8—1881no—
205C22⋊C8—1881no—
206#97—41no—
207#95—472no—
208#95—941no—
209C24—942no—
210C24(C2×D47)⋊C411yescharacteristic
211C2×D4—3762no—
212C2×D4—3762no—
213C2×D4—3762no—
214C2×D4—3762no—
215C2×D4—1882no—
216C2×D4—1882no—
217C2×D4—1882no—
218C2×D4—1882no—
219C2×D4—1884no—
220C2×D4—1881no—
221C2×D4—1881no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4—942no—
227C22×C4(C2×D47)⋊C411yescharacteristic
228C22×C4(C2×D47)⋊C411yescharacteristic
229C22×C4(C2×D47)⋊C411yescharacteristic
230C22×C4C2×C4×D4711yescharacteristic
231C22×C4C2×D18811yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233M16—1882no—
234M16—1882no—
235C2×C8—42no—
236C2×C8—1882no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—1882no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—3762no—
243C22⋊C4—3762no—
244C22⋊C4—1884no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—1882no—
255C23—1882no—
256C23—1882no—
257C23—1882no—
258C23—1882no—
259C23—1882no—
260C23—1882no—
261C23C2×(C2×D47)⋊C411yescharacteristic
262D4—3762no—
263D4—3762no—
264D4—3764no—
265D4—3764no—
266D4—3768no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—1882no—
282C2×C4—1882no—
283C2×C4—1882no—
284C2×C4#1309393811yescharacteristic
285C2×C4#1309393811yescharacteristic
286C2×C4#1309405212yes—
287C8—82no—
288C8—3762no—
289C22—21no—
290C22—21no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—3762no—
297C22—3762no—
298C22—3762no—
299C22—3762no—
300C22—3762no—
301C22—3762no—
302C22—3762no—
303C22C2×#1309393811yescharacteristic
304C22#1568912911yescharacteristic
305C22#1568913911yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—3762no—
314C2—21no—
315C2—21no—
316C2—81no—
317C2—7522no—
318C2#2024191811yescharacteristic
319C2#2024503711yescharacteristic
320C2#2024507911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1464C2
socleC22×C47C2×#99
prime radical#20249138C2
cosocleC2×C4⋊C4×C47C23
hypercenter#1464D47
hypocenterC47#11806
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #20243471C2semidirect
3. #20241392C2semidirect
4. C47×#1464C2semidirect
156. #1464D47semidirect
180. C47#11806semidirect

Maximal subgroups

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

subgroupindex
#202491382
#202434712
#202413922
C47×#14642
#1180647