Finite Group Databasebeta

← #26127092

Subgroups — #26127092

Order 12032.

384 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 83 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#20246679C211yesmaximal, characteristic
2#20241732C211yesmaximal, characteristic
3#20241417C212yesmaximal
4#20241393C212yesmaximal
5C47×#1801C211yesmaximal, characteristic
6C2×C4×Dic94C411yescharacteristic
7#15689180—21no—
8#15689054—21no—
9#15689053C411yescharacteristic
10#15689053C2211yescharacteristic
11#15689043C411yescharacteristic
12#15689043C2211yescharacteristic
13#15689042—21no—
14C4×Dic47⋊C4—21no—
15C4×Dic47⋊C4—21no—
16#15688969C411yescharacteristic
17#15688849—21no—
18#15688849C2212yes—
19C2×C4×Q8×C47C2211yescharacteristic
20C47×#423—21no—
21C2×#101×C47—21no—
22C47×#343C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C4×Dic94—41no—
25C4×Dic94—41no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C4×Dic47—21no—
33C2×C4×Dic47—21no—
34C4⋊(C47⋊C8)—41no—
35C22⋊(C47⋊C8)—22no—
36#13093935—41no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C22×Q8×C47D411yescharacteristic
45C2×M16×C47—22no—
46C4×Q8×C47—41no—
47C4×Q8×C47—41no—
48C2×C4⋊C4×C47C2×C411yescharacteristic
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C4⋊C4×C47C2311yescharacteristic
51C2×C42×C47C2×C411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C2×C42×C47D411yescharacteristic
54C4⋊C8×C47—41no—
55C4⋊C8×C47—41no—
56#101×C47—42no—
57C2×Q8×C47—21no—
58C2×Q8×C47—21no—
59C2×Q8×C47—41no—
60C2×Q8×C47—41no—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C22×Dic47—21no—
72C2×Dic94—41no—
73C2×Dic94—42no—
74M16×C47—42no—
75M16×C47—42no—
76C2×C8×C47—42no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47—21no—
85C42×C47—21no—
86C4⋊Dic47—41no—
87Dic47⋊C4—42no—
88Dic47⋊C4—42no—
89Dic47⋊C4—42no—
90Dic47⋊C4—42no—
91Dic47⋊C4—41no—
92Dic47⋊C4—41no—
93C4×Dic47—24no—
94C4×Dic47—41no—
95C4×Dic47—41no—
96C2×C47⋊C8—42no—
97C23×C47C2×C22⋊C411yescharacteristic
98Q8×C47—42no—
99Q8×C47—42no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
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—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×C47AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C47#13711yescharacteristic
119C2×Dic47—22no—
120C2×Dic47—22no—
121C2×Dic47—22no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131Dic94—42no—
132Dic94—42no—
133C8×C47—82no—
134C47⋊C8—82no—
135#11831—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47C2×#9911yescharacteristic
139C22×C47#41711yescharacteristic
140C22×C47#42111yescharacteristic
141C4×C47—21no—
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—41no—
145C4×C47—41no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47#35612yes—
150Dic47—42no—
151Dic47—42no—
152Dic47—42no—
153Dic47—81no—
154#2399—471no—
155#1801D4711yescharacteristic
156#1467—471no—
157#1354—472no—
158#1346—472no—
159C2×C47—21no—
160C2×C47—21no—
161C2×C47#151311yescharacteristic
162C2×C47#212011yescharacteristic
163C2×C47C2×#35611yescharacteristic
164C2×C4×Q8—471no—
165C2×C4×Q8C2×D4711yescharacteristic
166#432—941no—
167#423—21no—
168C2×#101—21no—
169#398—471no—
170#398—471no—
171#398—941no—
172#395—471no—
173#395—471no—
174#395—941no—
175C4×C4⋊C4—941no—
176C4×C4⋊C4—941no—
177#376—471no—
178#343C2×D4712yes—
179#324—472no—
180#324—941no—
181C47#1183111yescharacteristic, complemented, Sylow
182C22×Q8—471no—
183C22×Q8C22⋊D4711yescharacteristic
184C2×M16—22no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—1881no—
188C4×Q8—1881no—
189C2×C4⋊C4—471no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4—941no—
192C2×C4⋊C4—941no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4C4×D4711yescharacteristic
196C2×C4⋊C4D18811yescharacteristic
197C2×C4⋊C4C22×D4711yescharacteristic
198C2×C42—941no—
199C2×C42—941no—
200C2×C42C4×D4711yescharacteristic
201C2×C42D18811yescharacteristic
202C2×C42C22⋊D4711yescharacteristic
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—1881no—
206#101—42no—
207C22⋊C8—942no—
208#97—1881no—
209#95—471no—
210#95—471no—
211#95—941no—
212#95—941no—
213#95—941no—
214#95—941no—
215#95—941no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—1881no—
221C2×Q8—1882no—
222C22×C4—941no—
223C22×C4—941no—
224C22×C4—941no—
225C22×C4—941no—
226C22×C4(C2×D47)⋊C411yescharacteristic
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—42no—
234M16—42no—
235C2×C8—42no—
236C2×C8—1882no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—1881no—
241C4⋊C4—1882no—
242C4⋊C4—1882no—
243C4⋊C4—1882no—
244C4⋊C4—1882no—
245C4⋊C4—1881no—
246C4⋊C4—1881no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—944no—
254C42—1881no—
255C42—1881no—
256C23C2×(C2×D47)⋊C411yescharacteristic
257Q8—42no—
258Q8—42no—
259Q8—1882no—
260Q8—1882no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—942no—
277C2×C4—942no—
278C2×C4—942no—
279C2×C4—1881no—
280C2×C4—1881no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4#1309393811yescharacteristic
289C2×C4#1309393811yescharacteristic
290C2×C4#1309405011yescharacteristic
291C2×C4#1309405511yescharacteristic
292C8—82no—
293C8—3762no—
294C22—21no—
295C22—21no—
296C22C2×#1309393811yescharacteristic
297C22#1568912211yescharacteristic
298C22#1568914111yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—3761no—
308C4—1882no—
309C4—1882no—
310C4—1882no—
311C4#1568892112yes—
312C2—21no—
313C2—21no—
314C2#2024192911yescharacteristic
315C2#2024503011yescharacteristic
316C2C2×#1568892111yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1801C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1801D47
hypocenterC47#11831
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
181. C47#11831semidirect

Maximal subgroups

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

subgroupindex
#202466792
#202417322
#202414172
#202413932
C47×#18012
#1183147