Finite Group Databasebeta

← #26126655

Subgroups — #26126655

Order 12032.

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

normal subgroups only

This group has 402 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#20246679C211yesmaximal, characteristic
2#20241712C211yesmaximal, characteristic
3#20241417C212yesmaximal
4#20241393C212yesmaximal
5C4×#101×C47C211yesmaximal, characteristic
6C2×C4×Dic94C411yescharacteristic
7#15689176—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
20C2×#101×C47—21no—
21C4×M16×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—
36C4×C47⋊C8—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
54#101×C47—24no—
55#97×C47—41no—
56C4×C8×C47—41no—
57C2×Q8×C47—22no—
58C2×Q8×C47—22no—
59C2×Q8×C47—41no—
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×Dic47—21no—
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C2×Dic94—41no—
72C2×Dic94—42no—
73M16×C47—24no—
74M16×C47—42no—
75C2×C8×C47—42no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C42×C47—21no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47—21no—
85C4⋊Dic47—41no—
86Dic47⋊C4—42no—
87Dic47⋊C4—42no—
88Dic47⋊C4—42no—
89Dic47⋊C4—42no—
90Dic47⋊C4—42no—
91C4×Dic47—24no—
92C4×Dic47—41no—
93C4×Dic47—41no—
94C2×C47⋊C8—42no—
95C23×C47C2×C22⋊C411yescharacteristic
96Q8×C47—44no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
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—41no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47#9911yescharacteristic
112C2×C4×C47#9911yescharacteristic
113C2×C4×C47#10412yes—
114C2×C4×C47#13512yes—
115C2×Dic47—22no—
116C2×Dic47—22no—
117C2×Dic47—22no—
118C2×Dic47—41no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127Dic94—42no—
128Dic94—42no—
129C8×C47—44no—
130C47⋊C8—82no—
131#11736—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47C2×#9911yescharacteristic
135C22×C47#41611yescharacteristic
136C22×C47C2×#10411yescharacteristic
137C4×C47—21no—
138C4×C47—22no—
139C4×C47—22no—
140C4×C47—41no—
141C4×C47—41no—
142C4×C47—41no—
143C4×C47—41no—
144C4×C47#35612yes—
145Dic47—42no—
146Dic47—42no—
147Dic47—42no—
148Dic47—81no—
149#2399—471no—
150C4×#101D4711yescharacteristic
151#1461—471no—
152#1354—472no—
153#1346—472no—
154C2×C47—21no—
155C2×C47—21no—
156C2×C47#150111yescharacteristic
157C2×C47#212011yescharacteristic
158C2×C47C2×#35611yescharacteristic
159C2×C4×Q8—471no—
160C2×C4×Q8C2×D4711yescharacteristic
161#431—941no—
162C2×#101—21no—
163C4×M16—21no—
164#398—471no—
165#398—471no—
166#398—941no—
167#395—471no—
168#395—471no—
169#395—941no—
170C4×C4⋊C4—941no—
171C4×C4⋊C4—941no—
172#376—471no—
173#343C2×D4712yes—
174#324—472no—
175#324—941no—
176C47#1173611yescharacteristic, complemented, Sylow
177C22×Q8—471no—
178C22×Q8C22⋊D4711yescharacteristic
179C2×M16—22no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—1881no—
183C4×Q8—1881no—
184C2×C4⋊C4—471no—
185C2×C4⋊C4—941no—
186C2×C4⋊C4—941no—
187C2×C4⋊C4—941no—
188C2×C4⋊C4—941no—
189C2×C4⋊C4—941no—
190C2×C4⋊C4C4×D4711yescharacteristic
191C2×C4⋊C4D18811yescharacteristic
192C2×C4⋊C4C22×D4711yescharacteristic
193C2×C42—941no—
194C2×C42—941no—
195C2×C42C4×D4711yescharacteristic
196C2×C42D18811yescharacteristic
197C2×C42C22⋊D4711yescharacteristic
198C4⋊C8—1881no—
199#101—24no—
200C22⋊C8—942no—
201#97—41no—
202C4×C8—41no—
203C4×C8—1881no—
204#95—471no—
205#95—471no—
206#95—941no—
207#95—941no—
208#95—941no—
209#95—941no—
210#95—941no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—1881no—
215C2×Q8—1882no—
216C22×C4—941no—
217C22×C4—941no—
218C22×C4—941no—
219C22×C4—941no—
220C22×C4(C2×D47)⋊C411yescharacteristic
221C22×C4(C2×D47)⋊C411yescharacteristic
222C22×C4(C2×D47)⋊C411yescharacteristic
223C22×C4(C2×D47)⋊C411yescharacteristic
224C22×C4C2×C4×D4711yescharacteristic
225C22×C4C2×D18811yescharacteristic
226C22×C4C2×C22⋊D4711yescharacteristic
227M16—24no—
228M16—42no—
229C2×C8—42no—
230C2×C8—1882no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—1881no—
235C4⋊C4—1882no—
236C4⋊C4—1882no—
237C4⋊C4—1882no—
238C4⋊C4—1882no—
239C4⋊C4—1882no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—944no—
247C42—1881no—
248C42—1881no—
249C23C2×(C2×D47)⋊C411yescharacteristic
250Q8—44no—
251Q8—1882no—
252Q8—1882no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—942no—
268C2×C4—942no—
269C2×C4—942no—
270C2×C4—1881no—
271C2×C4—1881no—
272C2×C4—1881no—
273C2×C4—1881no—
274C2×C4—1881no—
275C2×C4—1881no—
276C2×C4—1881no—
277C2×C4—1881no—
278C2×C4—1881no—
279C2×C4#1309393811yescharacteristic
280C2×C4#1309393811yescharacteristic
281C2×C4#1309395212yes—
282C2×C4#1309404412yes—
283C8—44no—
284C8—3762no—
285C22—21no—
286C22—21no—
287C22C2×#1309393811yescharacteristic
288C22#1568911711yescharacteristic
289C22C2×#1309395211yescharacteristic
290C4—21no—
291C4—22no—
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—3761no—
298C4—1882no—
299C4—1882no—
300C4—1882no—
301C4#1568892112yes—
302C2—21no—
303C2—21no—
304C2#2024187711yescharacteristic
305C2#2024503011yescharacteristic
306C2C2×#1568892111yescharacteristic
subgroupG/H
centerC2×C4#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC4×#101×C47C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenterC4×#101D47
hypocenterC47#11736
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
176. C47#11736semidirect

Maximal subgroups

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

subgroupindex
#202466792
#202417122
#202414172
#202413932
C4×#101×C472
#1173647