Finite Group Databasebeta

← #26127331

Subgroups — #26127331

Order 12032.

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

normal subgroups only

This group has 476 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#20248387C211yesmaximal, characteristic
2#20241712C211yesmaximal, characteristic, complemented
3#20241390C214yesmaximal, complemented
4C47×#1801C211yesmaximal, characteristic, complemented
5C2×#13094021C411yescharacteristic
6C2×#13094013—21no—
7#15689176—21no—
8#15689040C412yes—
9#15689040C2212yes—
10#15689025—21no—
11#15689025—21no—
12#15689024—21no—
13C4×C4⋊Dic47C411yescharacteristic
14#15688849—21no—
15#15688849C2212yes—
16C2×C4×Q8×C47C2211yescharacteristic
17C47×#423—21no—
18C2×#101×C47—21no—
19C47×#343C2212yes—
20C22×D188—21no—
21C22×D188—21no—
22C22×D188C2×C412yes—
23#13094021—41no—
24#13094021—41no—
25#13094013—42no—
26#13094013—42no—
27C2×C4⋊Dic47—21no—
28C2×(C2×D47)⋊C4—22no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C2×C4×Dic47—21no—
33C4⋊(C47⋊C8)—41no—
34C22⋊(C47⋊C8)—22no—
35C4×C47⋊C8—41no—
36C47⋊#95C2×C412yes—
37C22×Q8×C47D411yescharacteristic
38C2×M16×C47—22no—
39C4×Q8×C47—41no—
40C4×Q8×C47—41no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C4⋊C4×C47C2311yescharacteristic
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C4⋊C8×C47—41no—
48C4⋊C8×C47—41no—
49#101×C47—42no—
50C23×D47—22no—
51C2×Q8×C47—41no—
52C2×Q8×C47—41no—
53C2×Q8×C47D812yes—
54C2×Q8×C47SD1612yes—
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—22no—
63C2×D188—42no—
64C2×D188—42no—
65C2×D188—42no—
66C2×D188—42no—
67C2×D188—42no—
68C2×D188—41no—
69C2×D188—41no—
70C2×D188—42no—
71C2×D188—82no—
72M16×C47—42no—
73M16×C47—42no—
74C2×C8×C47—42no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C42×C47—21no—
79C42×C47—21no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84(C2×D47)⋊C4—44no—
85(C2×D47)⋊C4—42no—
86(C2×D47)⋊C4—42no—
87(C2×D47)⋊C4—82no—
88(C2×D47)⋊C4—82no—
89C4⋊Dic47—42no—
90C4×Dic47—42no—
91C2×C47⋊C8—42no—
92C23×C47C2×C22⋊C411yescharacteristic
93C22×D47—42no—
94C22×D47—42no—
95C22×D47—42no—
96C22×D47—42no—
97C22×D47—42no—
98C22×D47—42no—
99C22×D47—42no—
100Q8×C47—42no—
101Q8×C47—42no—
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—22no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47#9911yescharacteristic
116C2×C4×C47#9911yescharacteristic
117C2×C4×C47#10212yes—
118C2×C4×C47#10212yes—
119C2×C4×C47C2×D811yescharacteristic
120C2×C4×C47C2×SD1611yescharacteristic
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124D188—82no—
125D188—84no—
126D188—84no—
127D188—82no—
128C8×C47—82no—
129C47⋊C8—82no—
130#11882—471nomaximal, Sylow
131C22×C47—21no—
132C22×C47—21no—
133C22×C47C2×#9911yescharacteristic
134C22×C47C2×#10211yescharacteristic
135C22×C47#42111yescharacteristic
136C2×D47—82no—
137C2×D47—82no—
138C2×D47—82no—
139C2×D47—82no—
140C2×D47—82no—
141C2×D47—82no—
142C2×D47—82no—
143C4×C47—21no—
144C4×C47—21no—
145C4×C47—22no—
146C4×C47—22no—
147C4×C47—21no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47#35512yes—
152Dic47—82no—
153#2548—471no—
154#1801D4711yescharacteristic, complemented
155#1461—471no—
156#1345—474no—
157C2×C47—21no—
158C2×C47—21no—
159C2×C47#151911yescharacteristic
160C2×C47#212211yescharacteristic
161C2×C47C2×#35511yescharacteristic
162D47—162no—
163C2×#127—471no—
164C2×#124—941no—
165C2×C4×Q8C2×D4711yescharacteristic
166#431—941no—
167#423—21no—
168C2×#101—21no—
169#394—472no—
170#394—472no—
171#390—941no—
172#390—941no—
173#390—941no—
174C4×C4⋊C4—471no—
175#343C2×D4712yes—
176#324—472no—
177#324—941no—
178C47#1188211yescharacteristic, complemented, Sylow
179C22×Q8C22⋊D4711yescharacteristic
180C22×D4—472no—
181C22×D4—941no—
182C22×D4—941no—
183C2×M16—22no—
184#127—1881no—
185#127—1881no—
186#124—1882no—
187#124—1882no—
188C4×Q8—41no—
189C4×Q8—41no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4C4×D4711yescharacteristic
192C2×C4⋊C4D18811yescharacteristic
193C2×C4⋊C4C22×D4711yescharacteristic
194C2×C22⋊C4—942no—
195C2×C22⋊C4—942no—
196C2×C22⋊C4—942no—
197C2×C22⋊C4—942no—
198C2×C42—941no—
199C2×C42C4×D4711yescharacteristic
200C2×C42D18811yescharacteristic
201C2×C42C22⋊D4711yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—1881no—
205#101—42no—
206C22⋊C8—942no—
207C4×C8—1881no—
208#95—472no—
209C24—942no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#1062533412yes—
213C2×Q8#1062533612yes—
214C2×D4—3762no—
215C2×D4—1882no—
216C2×D4—1882no—
217C2×D4—1882no—
218C2×D4—1882no—
219C2×D4—1882no—
220C2×D4—1881no—
221C2×D4—1881no—
222C2×D4—1882no—
223C22×C4—942no—
224C22×C4(C2×D47)⋊C411yescharacteristic
225C22×C4(C2×D47)⋊C411yescharacteristic
226C22×C4(C2×D47)⋊C411yescharacteristic
227C22×C4(C2×D47)⋊C411yescharacteristic
228C22×C4C2×C4×D4711yescharacteristic
229C22×C4C2×D18811yescharacteristic
230C22×C4C2×C22⋊D4711yescharacteristic
231M16—42no—
232M16—42no—
233C2×C8—42no—
234C2×C8—1882no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—1882no—
239C22⋊C4—3762no—
240C22⋊C4—3762no—
241C22⋊C4—1884no—
242C22⋊C4—1882no—
243C22⋊C4—1882no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—1882no—
251C23—1882no—
252C23—1882no—
253C23—1882no—
254C23—1882no—
255C23—1882no—
256C23—1882no—
257C23—1882no—
258C23C2×(C2×D47)⋊C411yescharacteristic
259Q8—42no—
260Q8—42no—
261D4—3764no—
262D4—3762no—
263D4—3764no—
264D4—3762no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—1882no—
279C2×C4—1882no—
280C2×C4—1882no—
281C2×C4#1309393811yescharacteristic
282C2×C4#1309393811yescharacteristic
283C2×C4#1309394412yes—
284C2×C4#1309394412yes—
285C2×C4C2×#1062533411yescharacteristic
286C2×C4C2×#1062533611yescharacteristic
287C8—82no—
288C8—3762no—
289C22—21no—
290C22—21no—
291C22—3762no—
292C22—3762no—
293C22—3762no—
294C22—3762no—
295C22—3762no—
296C22—3762no—
297C22—3762no—
298C22C2×#1309393811yescharacteristic
299C22C2×#1309394411yescharacteristic
300C22#1568914111yescharacteristic
301C4—21no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—3762no—
310C4#1568891912yes—
311C2—21no—
312C2—21no—
313C2—7522no—
314C2#2024195511yescharacteristic
315C2#2024503911yescharacteristic
316C2C2×#1568891911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1801C2
socleC22×C47C2×#99
prime radicalC2×#13094021C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1801D47
hypocenterC47#11882
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #20241712C2semidirect
3. #20241390C2semidirect
4. C47×#1801C2semidirect
154. #1801D47semidirect
178. C47#11882semidirect

Maximal subgroups

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

subgroupindex
#202483872
#202417122
#202413902
C47×#18012
#1188247