Finite Group Databasebeta

← #26108011

Subgroups — #26108011

Order 12032.

738 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 151 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#20243380C211yesmaximal, characteristic
2#20242060C212yesmaximal, complemented
3#20241817C212yesmaximal, complemented
4#20241709C211yesmaximal, characteristic
5C2×C47×#556C211yesmaximal, characteristic
6#15689176C2212yes—
7#15689147C2212yes—
8C2×#13093941—21no—
9#15688892—21no—
10#15688869—24no—
11Q8⋊(C47⋊C8)—24no—
12#15688847—22no—
13C47×#556—24no—
14C47×#556—24no—
15C2×C4⋊Q8×C47C2211yescharacteristic
16C2×#126×C47—22no—
17C2×#124×C47—21no—
18C2×C22⋊Q8×C47C412yes—
19C2×C4×Q8×C47—21no—
20C2×C4×Q8×C47C2211yescharacteristic
21C2×C4×D4×C47C2211yescharacteristic
22C2×#117×C47C412yes—
23C2×#10625329—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)—22no—
27#13093941—42no—
28C22⋊(C47⋊C8)—22no—
29C4×C47⋊C8—22no—
30C22×Q8×C47—21no—
31C22×Q8×C47C2×C411yescharacteristic
32C22×D4×C47D411yescharacteristic
33C23×C4×C47C2×C412yes—
34C4⋊Q8×C47—22no—
35C4⋊Q8×C47D414yes—
36#126×C47—216no—
37#124×C47—28no—
38C22⋊Q8×C47—24no—
39C22⋊Q8×C47—24no—
40C4×Q8×C47—28no—
41C4×Q8×C47—22no—
42C4×Q8×C47D414yes—
43C4×D4×C47—22no—
44C4×D4×C47—22no—
45#117×C47—24no—
46#117×C47—24no—
47C2×C4⋊C4×C47—22no—
48C2×C4⋊C4×C47—21no—
49C2×C4⋊C4×C47—21no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47C2×C412yes—
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C22⋊C4×C47—22no—
54C2×C22⋊C4×C47—22no—
55C2×C22⋊C4×C47D412yes—
56C2×C42×C47—22no—
57C2×C42×C47—21no—
58C2×C42×C47C2311yescharacteristic
59C24×C47C22⋊C412yes—
60C2×Q8×C47—21no—
61C2×Q8×C47—24no—
62C2×Q8×C47—42no—
63C2×Q8×C47—42no—
64C2×Q8×C47—41no—
65C2×Q8×C47—41no—
66C2×Q8×C47C22⋊C412yes—
67C2×D4×C47—22no—
68C2×D4×C47—22no—
69C2×D4×C47—21no—
70C2×D4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—22no—
74C22×C4×C47—22no—
75C22×C4×C47—22no—
76C22×C4×C47—44no—
77C22×C4×C47C22⋊C412yes—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C4⋊C4×C47—28no—
82C4⋊C4×C47—21no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—24no—
85C4⋊C4×C47—24no—
86C4⋊C4×C47—22no—
87C4⋊C4×C47C22⋊C412yes—
88C4⋊C4×C47C22⋊C414yes—
89C22⋊C4×C47—28no—
90C22⋊C4×C47—28no—
91C22⋊C4×C47—24no—
92C42×C47—28no—
93C42×C47—24no—
94C42×C47C2×D412yes—
95C42×C47C2×D412yes—
96#10625329—42no—
97#10625329—42no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—22no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47C2×C22⋊C411yescharacteristic
106Q8×C47—24no—
107Q8×C47—42no—
108Q8×C47—44no—
109Q8×C47—44no—
110D4×C47—42no—
111D4×C47—42no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—22no—
116C2×C4×C47—24no—
117C2×C4×C47—41no—
118C2×C4×C47—44no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—44no—
124C2×C4×C47—42no—
125C2×C4×C47C2×C22⋊C411yescharacteristic
126C2×C4×C47C2×C22⋊C411yescharacteristic
127C2×C4×C47#12012yes—
128C2×C4×C47#12012yes—
129C2×C4×C47#13512yes—
130C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C47#13711yescharacteristic
132C47⋊C8—82no—
133C47⋊C8—82no—
134#8389—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—42no—
138C22×C47—42no—
139C22×C47—42no—
140C22×C47—82no—
141C22×C47—82no—
142C22×C47#37911yescharacteristic
143C22×C47#41611yescharacteristic
144C22×C47#41711yescharacteristic
145C4×C47—22no—
146C4×C47—21no—
147C4×C47—42no—
148C4×C47—42no—
149C4×C47—42no—
150C4×C47—44no—
151C4×C47#41312yes—
152C4×C47#44912yes—
153C4×C47#45212yes—
154C2×#556D4711yescharacteristic
155#1800—471no—
156#1546—472no—
157#1487—472no—
158#1460—471no—
159C2×C47—21no—
160C2×C47—21no—
161C2×C47—82no—
162C2×C47#178511yescharacteristic
163C2×C47#178811yescharacteristic
164C2×C47#179611yescharacteristic
165#556—24no—
166#556—24no—
167C2×C4⋊Q8C2×D4711yescharacteristic
168C2×#126—22no—
169C2×#124—21no—
170C2×C22⋊Q8Dic4712yes—
171C2×C4×Q8—21no—
172C2×C4×Q8C2×D4711yescharacteristic
173C2×C4×D4C2×D4711yescharacteristic
174C2×#117Dic4712yes—
175#431—472no—
176#423—472no—
177C2×#100—941no—
178#343—941no—
179#332—944no—
180Q8⋊C8—944no—
181#323—942no—
182C47#838911yescharacteristic, complemented, Sylow
183C22×Q8—21no—
184C22×Q8C2×Dic4711yescharacteristic
185C22×D4C22⋊D4711yescharacteristic
186C23×C4C2×Dic4712yes—
187C2×M16—942no—
188C4⋊Q8—22no—
189C4⋊Q8C22⋊D4714yes—
190#126—216no—
191#124—28no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C4×Q8—28no—
195C4×Q8—22no—
196C4×Q8C22⋊D4714yes—
197C4×D4—22no—
198C4×D4—22no—
199#117—24no—
200#117—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C2×Dic4711yescharacteristic
205C2×C4⋊C4C2×Dic4712yes—
206C2×C4⋊C4C22⋊D4711yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C22⋊D4712yes—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42C22×D4711yescharacteristic
213C4⋊C8—942no—
214C4⋊C8—942no—
215C4⋊C8—942no—
216#100—1882no—
217C22⋊C8—942no—
218C4×C8—942no—
219C24C22⋊Dic4712yes—
220C2×Q8—21no—
221C2×Q8—24no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8C22⋊Dic4712yes—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—44no—
237C22×C4C22⋊Dic4712yes—
238C22×C4C22×Dic4711yescharacteristic
239C22×C4C2×C22⋊D4711yescharacteristic
240C22×C4C2×C22⋊D4711yescharacteristic
241M16—1882no—
242M16—1882no—
243C2×C8—1882no—
244C2×C8—1882no—
245C4⋊C4—28no—
246C4⋊C4—21no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4C22⋊Dic4712yes—
252C4⋊C4C22⋊Dic4714yes—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C42—28no—
257C42—24no—
258C42C2×C22⋊D4712yes—
259C42C2×C22⋊D4712yes—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C2×C22⋊Dic4711yescharacteristic
266Q8—24no—
267Q8—42no—
268Q8—44no—
269Q8—44no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4C2×C22⋊Dic4711yescharacteristic
286C2×C4C2×C22⋊Dic4711yescharacteristic
287C2×C4#1309399312yes—
288C2×C4#1309399312yes—
289C2×C4#1309404412yes—
290C2×C4#1309404911yescharacteristic
291C2×C4#1309405411yescharacteristic
292C8—3762no—
293C8—3762no—
294C22—21no—
295C22—21no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22#1568897911yescharacteristic
302C22#1568911911yescharacteristic
303C22#1568912411yescharacteristic
304C4—21no—
305C4—22no—
306C4—42no—
307C4—42no—
308C4—44no—
309C4—42no—
310C4#1568910712yes—
311C4#1568926812yes—
312C4#1568928512yes—
313C2—21no—
314C2—21no—
315C2—82no—
316C2#2024331611yescharacteristic
317C2#2024332811yescharacteristic
318C2#2024336311yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#556C2
socleC22×C47#379
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenterC2×#556D47
hypocenterC47#8389
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. #20242060C2semidirect
3. #20241817C2semidirect
182. C47#8389semidirect

Maximal subgroups

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

subgroupindex
#202433802
#202420602
#202418172
#202417092
C2×C47×#5562
#838947