Finite Group Databasebeta

← #26126953

Subgroups — #26126953

Order 12032.

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

normal subgroups only

This group has 502 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#20243465C211yesmaximal, characteristic, complemented
3#20241386C214yesmaximal, complemented
4C47×#1464C211yesmaximal, characteristic, complemented
5C2×#13094021C411yescharacteristic, complemented
6C2×#13093998—21no—
7#15689040—21no—
8#15689040C412yescomplemented
9#15689032—21no—
10#15689032C2212yes—
11#15689025—21no—
12#15688989C411yescharacteristic, complemented
13#15688986—21no—
14#15688890—21no—
15#15688890C2212yes—
16C2×#117×C47C2211yescharacteristic
17C47×#432—21no—
18C47×#324—21no—
19C47×#323C2212yes—
20C22×D188—21no—
21C22×D188—21no—
22C22×D188C2×C412yes—
23C22×C22⋊D47—22no—
24#13094021—41no—
25#13094021—41no—
26#13093998—44no—
27C2×C4⋊Dic47—21no—
28C2×C22⋊Dic47—22no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C47⋊#95—21no—
33C47⋊#95—41no—
34C47⋊#95—41no—
35C47⋊#95C2×C412yes—
36C23×C4×C47D411yescharacteristic
37#117×C47—41no—
38#117×C47—41no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C22⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47C2311yescharacteristic
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C4⋊C8×C47—41no—
46C22⋊C8×C47—22no—
47#97×C47—41no—
48C24×C47C2×D411yescharacteristic
49C23×D47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—41no—
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C2×C22⋊D47—44no—
61C2×C22⋊D47—82no—
62C2×C22⋊D47—82no—
63C22×Dic47—22no—
64C22×Dic47—42no—
65C2×D188—42no—
66C2×D188—42no—
67C2×D188—42no—
68C2×D188—42no—
69C2×D188—41no—
70C2×D188—41no—
71C2×D188—82no—
72C2×D188—82no—
73C2×C8×C47—42no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—41no—
78C42×C47—21no—
79C42×C47—21no—
80C42×C47—21no—
81C42×C47—21no—
82C22⋊Dic47—42no—
83C22⋊Dic47—42no—
84(C2×D47)⋊C4—44no—
85(C2×D47)⋊C4—82no—
86(C2×D47)⋊C4—82no—
87C4⋊Dic47—42no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47C2×C22⋊C411yescharacteristic
93C23×C47AGL(1,ℤ/8)12yes—
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—21no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47#9911yescharacteristic
117C2×C4×C47#9911yescharacteristic
118C22⋊D47—88no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—82no—
123C2×Dic47—82no—
124C2×Dic47—82no—
125D188—82no—
126D188—82no—
127D188—84no—
128D188—84no—
129C8×C47—82no—
130#11800—471nomaximal, Sylow
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—22no—
134C22×C47—42no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47—81no—
138C22×C47C2×#9911yescharacteristic
139C22×C47#41811yescharacteristic
140C22×C47#42111yescharacteristic
141C2×D47—82no—
142C2×D47—82no—
143C2×D47—82no—
144C2×D47—82no—
145C2×D47—82no—
146C2×D47—82no—
147C2×D47—82no—
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—41no—
154C4×C47—41no—
155Dic47—82no—
156Dic47—162no—
157#2619—471no—
158#1820—471no—
159#1464D4711yescharacteristic, complemented
160#1343—474no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47—42no—
164C2×C47#151011yescharacteristic
165C2×C47#212211yescharacteristic
166C2×C47#212811yescharacteristic
167D47—162no—
168C2×#127—471no—
169C2×#121—941no—
170C2×#117C2×D4711yescharacteristic
171#432—21no—
172#394—472no—
173#394—941no—
174#392—472no—
175#392—941no—
176#390—941no—
177#382—471no—
178#381—941no—
179#342—472no—
180#342—941no—
181#324—21no—
182#323C2×D4712yes—
183C47#1180011yescharacteristic, complemented, Sylow
184C22×D4—472no—
185C22×D4—942no—
186C22×D4—941no—
187C22×D4—941no—
188C23×C4D18811yescharacteristic
189#127—1881no—
190#127—1881no—
191#121—1884no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4C4×D4711yescharacteristic
196C2×C4⋊C4C22⋊D4711yescharacteristic
197C2×C22⋊C4—942no—
198C2×C22⋊C4—942no—
199C2×C22⋊C4—942no—
200C2×C22⋊C4—942no—
201C2×C22⋊C4D18811yescharacteristic
202C2×C22⋊C4C22×D4711yescharacteristic
203C2×C42C4×D4711yescharacteristic
204C2×C42C22⋊D4711yescharacteristic
205C4⋊C8—41no—
206C22⋊C8—22no—
207#97—41no—
208#95—472no—
209#95—941no—
210#95—1881no—
211#95—1881no—
212C24—942no—
213C24C2×D18811yescharacteristic
214C2×D4—3762no—
215C2×D4—3762no—
216C2×D4—3762no—
217C2×D4—3762no—
218C2×D4—1882no—
219C2×D4—1882no—
220C2×D4—1882no—
221C2×D4—1882no—
222C2×D4—1884no—
223C2×D4—1881no—
224C2×D4—1881no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—942no—
230C22×C4—1882no—
231C22×C4(C2×D47)⋊C411yescharacteristic
232C22×C4(C2×D47)⋊C411yescharacteristic
233C22×C4(C2×D47)⋊C411yescharacteristic
234C22×C4(C2×D47)⋊C411yescharacteristic
235C22×C4C2×C4×D4711yescharacteristic
236C22×C4C2×C22⋊D4711yescharacteristic
237C2×C8—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—1882no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—3762no—
244C22⋊C4—3762no—
245C22⋊C4—1884no—
246C22⋊C4—1882no—
247C22⋊C4—1882no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—1882no—
257C23—1882no—
258C23—1882no—
259C23—1882no—
260C23—1882no—
261C23—1882no—
262C23—1882no—
263C23C2×(C2×D47)⋊C411yescharacteristic
264C23#1309405212yes—
265D4—3762no—
266D4—3762no—
267D4—3764no—
268D4—3764no—
269D4—3768no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—3762no—
286C2×C4—3762no—
287C2×C4—3762no—
288C2×C4—1882no—
289C2×C4—1882no—
290C2×C4—1882no—
291C2×C4#1309393811yescharacteristic
292C2×C4#1309393811yescharacteristic
293C8—82no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—3762no—
302C22—3762no—
303C22—3762no—
304C22—3762no—
305C22—3762no—
306C22—3762no—
307C22—3762no—
308C22C2×#1309393811yescharacteristic
309C22#1568912911yescharacteristic
310C22#1568913911yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—3762no—
319C4—7522no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—7522no—
324C2#2024191811yescharacteristic
325C2#2024503911yescharacteristic
326C2#2024505911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1464C2
socleC22×C47C2×#99
prime radical#20249138C2
cosocleC2×C22⋊C4×C47C23
hypercenter#1464D47
hypocenterC47#11800
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. #20243465C2semidirect
3. #20241386C2semidirect
4. C47×#1464C2semidirect
5. C2×#13094021C4semidirect
8. #15689040C4semidirect
12. #15688989C4semidirect
159. #1464D47semidirect
183. C47#11800semidirect

Maximal subgroups

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

subgroupindex
#202491382
#202434652
#202413862
C47×#14642
#1180047