Finite Group Databasebeta

← #26127433

Subgroups — #26127433

Order 12032.

372 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 95 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#20248750C211yesmaximal, characteristic
2#20243384C211yesmaximal, characteristic
3#20241419C212yesmaximal
4#20241416C212yesmaximal
5C47×#1461C211yesmaximal, characteristic
6C2×C4×Dic94C411yescharacteristic
7#15689147—21no—
8C2×#13093943—21no—
9#15689062C411yescharacteristic
10#15689062C2211yescharacteristic
11#15689043C411yescharacteristic
12#15689043C2211yescharacteristic
13#15689030—21no—
14#15689019—21no—
15#15689007—21no—
16#15689007—21no—
17#15688989C411yescharacteristic
18#15688892C2212yes—
19C2×C4×Q8×C47C2211yescharacteristic
20C47×#431—21no—
21C47×#324—21no—
22C47×#324C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C2×#10625329—22no—
25C4×Dic94—41no—
26C4×Dic94—41no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C4×Dic47—21no—
33C4⋊(C47⋊C8)—41no—
34C4⋊(C47⋊C8)—41no—
35#13093943—42no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95C2×C411yescharacteristic
44C47⋊#95C2×C411yescharacteristic
45C22×Q8×C47D411yescharacteristic
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—
55C22⋊C8×C47—22no—
56C4×C8×C47—41no—
57C2×Q8×C47—41no—
58C2×Q8×C47—41no—
59C2×Q8×C47SD1612yes—
60C2×Q8×C47Q1612yes—
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—
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—
84C4⋊Dic47—42no—
85C4⋊Dic47—41no—
86Dic47⋊C4—42no—
87Dic47⋊C4—42no—
88Dic47⋊C4—41no—
89Dic47⋊C4—41no—
90C4×Dic47—42no—
91#10625329—42no—
92#10625329—42no—
93C2×C47⋊C8—42no—
94C23×C47C2×C22⋊C411yescharacteristic
95Q8×C47—42no—
96Q8×C47—42no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47#9911yescharacteristic
111C2×C4×C47#9911yescharacteristic
112C2×C4×C47#10312yes—
113C2×C4×C47#10312yes—
114C2×C4×C47C2×SD1611yescharacteristic
115C2×C4×C47C2×Q1611yescharacteristic
116C2×Dic47—22no—
117C2×Dic47—41no—
118C2×Dic47—41no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—42no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127Dic94—42no—
128Dic94—42no—
129C8×C47—82no—
130C47⋊C8—82no—
131#11905—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47C2×#9911yescharacteristic
135C22×C47C2×#10311yescharacteristic
136C22×C47#42111yescharacteristic
137C4×C47—21no—
138C4×C47—21no—
139C4×C47—21no—
140C4×C47—22no—
141C4×C47—22no—
142C4×C47—21no—
143C4×C47—41no—
144C4×C47—41no—
145C4×C47—41no—
146Dic47—42no—
147Dic47—81no—
148Dic47—81no—
149Dic47—81no—
150#2581—471no—
151#1801—471no—
152#1461D4711yescharacteristic
153#1354—472no—
154#1353—472no—
155C2×C47—21no—
156C2×C47—21no—
157C2×C47#152011yescharacteristic
158C2×C47#212011yescharacteristic
159C2×C47#213311yescharacteristic
160C2×C4×Q8—471no—
161C2×C4×Q8C2×D4711yescharacteristic
162#431—21no—
163#423—941no—
164C2×#101—941no—
165#400—471no—
166#400—471no—
167#395—471no—
168#395—471no—
169#391—941no—
170#389—941no—
171#387—941no—
172#387—941no—
173#382—471no—
174#343—472no—
175#324—21no—
176#324C2×D4712yes—
177C47#1190511yescharacteristic, complemented, Sylow
178C22×Q8—471no—
179C22×Q8D18811yescharacteristic
180C2×M16—942no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—1881no—
184C4×Q8—1881no—
185C2×C4⋊C4—471no—
186C2×C4⋊C4—941no—
187C2×C4⋊C4—941no—
188C2×C4⋊C4—941no—
189C2×C4⋊C4—941no—
190C2×C4⋊C4C4×D4711yescharacteristic
191C2×C4⋊C4C22⋊D4711yescharacteristic
192C2×C4⋊C4C22×D4711yescharacteristic
193C2×C42—941no—
194C2×C42C4×D4711yescharacteristic
195C2×C42D18811yescharacteristic
196C2×C42C22⋊D4711yescharacteristic
197C4⋊C8—41no—
198C4⋊C8—1881no—
199C4⋊C8—1881no—
200#101—1882no—
201C22⋊C8—22no—
202C4×C8—41no—
203#95—471no—
204#95—471no—
205#95—941no—
206#95—941no—
207#95—941no—
208#95—941no—
209#95—941no—
210#95—941no—
211#95—941no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—1881no—
215C2×Q8—1882no—
216C2×Q8#1062532512yes—
217C2×Q8Dic18812yes—
218C22×C4—941no—
219C22×C4—941no—
220C22×C4—941no—
221C22×C4—941no—
222C22×C4(C2×D47)⋊C411yescharacteristic
223C22×C4(C2×D47)⋊C411yescharacteristic
224C22×C4(C2×D47)⋊C411yescharacteristic
225C22×C4(C2×D47)⋊C411yescharacteristic
226C22×C4C2×C4×D4711yescharacteristic
227C22×C4C2×D18811yescharacteristic
228C22×C4C2×C22⋊D4711yescharacteristic
229M16—1882no—
230M16—1882no—
231C2×C8—42no—
232C2×C8—1882no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—1881no—
237C4⋊C4—1882no—
238C4⋊C4—1882no—
239C4⋊C4—1882no—
240C4⋊C4—1881no—
241C4⋊C4—1881no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—1882no—
249C23C2×(C2×D47)⋊C411yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—1882no—
253Q8—1882no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—942no—
268C2×C4—1881no—
269C2×C4—1881no—
270C2×C4—1881no—
271C2×C4—1881no—
272C2×C4—1882no—
273C2×C4—1881no—
274C2×C4—1881no—
275C2×C4—1881no—
276C2×C4—1881no—
277C2×C4—1881no—
278C2×C4#1309393811yescharacteristic
279C2×C4#1309393811yescharacteristic
280C2×C4#1309394912yes—
281C2×C4#1309394912yes—
282C2×C4C2×#1062532511yescharacteristic
283C2×C4C2×Dic18811yescharacteristic
284C8—82no—
285C8—3762no—
286C22—21no—
287C22—21no—
288C22C2×#1309393811yescharacteristic
289C22C2×#1309394911yescharacteristic
290C22#1568913911yescharacteristic
291C4—21no—
292C4—22no—
293C4—22no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—3761no—
301C4—3761no—
302C4—3761no—
303C4—1882no—
304C2—21no—
305C2—21no—
306C2#2024196111yescharacteristic
307C2#2024502711yescharacteristic
308C2#2024507511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1461C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1461D47
hypocenterC47#11905
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
177. C47#11905semidirect

Maximal subgroups

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

subgroupindex
#202487502
#202433842
#202414192
#202414162
C47×#14612
#1190547