Finite Group Databasebeta

← #26110956

Subgroups — #26110956

Order 12032.

542 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 119 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#20252473C211yesmaximal, characteristic, complemented
2C4×#13094013C211yesmaximal, characteristic
3#20241780C212yesmaximal
4#20241220C212yesmaximal, complemented
5C47×#1461C211yesmaximal, characteristic, complemented
6C2×#13094013C411yescharacteristic
7C2×C4×D188C2211yescharacteristic
8#15689254—24no—
9#15689180—22no—
10#15689176C2212yes—
11#15689030C411yescharacteristic
12#15689024C411yescharacteristic
13#15689013—22no—
14#15688988C411yescharacteristic
15C4×(C2×D47)⋊C4—22no—
16C42×Dic47C2211yescharacteristic
17#15688852—24no—
18C2×C4×Q8×C47C2211yescharacteristic
19C47×#431C2212yes—
20C47×#324—22no—
21C22×D188C2×C411yescharacteristic
22C22×C4×D47—21no—
23#13094013—22no—
24#13094013—22no—
25C4×D188—22no—
26C4×D188—22no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×(C2×D47)⋊C4—22no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47—21no—
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47C2×C411yescharacteristic
35C4⋊(C47⋊C8)—24no—
36C4⋊(C47⋊C8)D414yes—
37C22⋊(C47⋊C8)—22no—
38#13093935—24no—
39C4×C47⋊C8—22no—
40C47⋊#95—22no—
41C47⋊#95—22no—
42C22×Q8×C47C2×C411yescharacteristic
43C4×Q8×C47—22no—
44C4×Q8×C47—22no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47D412yes—
47C2×C42×C47D412yes—
48C2×C42×C47C2311yescharacteristic
49C4⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C4×C8×C47—22no—
52C23×D47—21no—
53C2×Q8×C47—22no—
54C2×Q8×C47—21no—
55C2×Q8×C47—21no—
56C2×Q8×C47—22no—
57C22×C4×C47C22⋊C412yes—
58C22×C4×C47C22⋊C412yes—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—21no—
63C22×Dic47—21no—
64C22×Dic47—21no—
65C2×D188—22no—
66C2×D188—22no—
67C2×D188—21no—
68C2×D188M1612yes—
69C2×C4×D47—44no—
70C2×C4×D47—41no—
71C2×C4×D47—41no—
72C2×C8×C47—42no—
73C4⋊C4×C47—22no—
74C4⋊C4×C47—24no—
75C42×C47—22no—
76C42×C47—22no—
77C42×C47C2×D412yes—
78C42×C47#4112yes—
79(C2×D47)⋊C4—28no—
80(C2×D47)⋊C4—44no—
81(C2×D47)⋊C4—42no—
82C4⋊Dic47—22no—
83C4×Dic47—21no—
84C4×Dic47—22no—
85C4×Dic47—42no—
86C4×Dic47—42no—
87C4×Dic47M1612yes—
88C2×C47⋊C8—24no—
89C23×C47C2×C22⋊C411yescharacteristic
90C22×D47—22no—
91C22×D47—42no—
92C22×D47—42no—
93C22×D47—41no—
94C22×D47—41no—
95Q8×C47—42no—
96Q8×C47—42no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—41no—
103C2×C4×C47—42no—
104C2×C4×C47C4×D412yes—
105C2×C4×C47#12011yescharacteristic
106C2×C4×C47#12112yes—
107C2×C4×C47#12311yescharacteristic
108C2×C4×C47C2×M1611yescharacteristic
109C2×C4×C47C2×M1611yescharacteristic
110C2×C4×C47#13112yes—
111C2×Dic47—22no—
112C2×Dic47—22no—
113C2×Dic47—41no—
114C2×Dic47—41no—
115C2×Dic47—41no—
116C2×Dic47—41no—
117C2×Dic47—42no—
118C2×Dic47—41no—
119D188—24no—
120D188—42no—
121C4×D47—82no—
122C4×D47—82no—
123C8×C47—82no—
124C47⋊C8—44no—
125#8964—471nomaximal, Sylow
126C22×C47—21no—
127C22×C47—21no—
128C22×C47#38611yescharacteristic
129C22×C47#40711yescharacteristic
130C22×C47#40811yescharacteristic
131C2×D47—44no—
132C2×D47—42no—
133C2×D47—82no—
134C2×D47—81no—
135C2×D47—81no—
136C4×C47—22no—
137C4×C47—21no—
138C4×C47—42no—
139C4×C47—42no—
140C4×C47#43512yes—
141C4×C47#43612yes—
142C4×C47#45412yes—
143Dic47—42no—
144Dic47—42no—
145Dic47—81no—
146#2983—471no—
147C4×#124—471no—
148#1478—472no—
149#1461D4711yescharacteristic, complemented
150#1276—472no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47#187011yescharacteristic
154C2×C47#188411yescharacteristic
155C2×C47#188811yescharacteristic
156D47—82no—
157C2×#124—471no—
158C2×C4×Q8C2×D4711yescharacteristic
159C2×C4×D4—471no—
160#446—944no—
161#432—942no—
162#431—472no—
163#431C2×D4712yes—
164#391—471no—
165#390—471no—
166#388—942no—
167#382—471no—
168C4×C22⋊C4—942no—
169C43—471no—
170D4⋊C8—944no—
171#324—22no—
172C47#896411yescharacteristic, complemented, Sylow
173C22×Q8C4×D4711yescharacteristic
174C22×D4—471no—
175C23×C4—941no—
176#124—942no—
177#124—942no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×D4—942no—
181C4×D4—942no—
182C2×C4⋊C4—471no—
183C2×C4⋊C4C4×D4711yescharacteristic
184C2×C4⋊C4C22⋊D4712yes—
185C2×C22⋊C4—942no—
186C2×C22⋊C4—942no—
187C2×C22⋊C4—941no—
188C2×C42—471no—
189C2×C42—471no—
190C2×C42—941no—
191C2×C42—941no—
192C2×C42D18812yes—
193C2×C42C22×D4711yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—474no—
196C4⋊C8—944no—
197C22⋊C8—22no—
198C22⋊C8—942no—
199#97—944no—
200C4×C8—22no—
201C4×C8—942no—
202#95—942no—
203#95—942no—
204C24—941no—
205C2×Q8—22no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×D4—472no—
210C2×D4—942no—
211C2×D4—941no—
212C2×D4—942no—
213C22×C4—941no—
214C22×C4—941no—
215C22×C4—941no—
216C22×C4—1884no—
217C22×C4—1881no—
218C22×C4—1881no—
219C22×C4(C2×D47)⋊C412yes—
220C22×C4(C2×D47)⋊C412yes—
221C22×C4C2×C4×D4711yescharacteristic
222C22×C4C2×D18811yescharacteristic
223C22×C4C2×C22⋊D4711yescharacteristic
224C2×C8—42no—
225C2×C8—944no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—942no—
229C22⋊C4—948no—
230C22⋊C4—1884no—
231C22⋊C4—1882no—
232C42—22no—
233C42—22no—
234C42—472no—
235C42—941no—
236C42—942no—
237C42—1882no—
238C42—1882no—
239C42D4×D4712yes—
240C42#1062535212yes—
241C23—942no—
242C23—1882no—
243C23—1882no—
244C23—1881no—
245C23—1881no—
246C23C2×(C2×D47)⋊C411yescharacteristic
247Q8—42no—
248Q8—42no—
249D4—944no—
250D4—1882no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—942no—
259C2×C4—942no—
260C2×C4—3762no—
261C2×C4—3762no—
262C2×C4—1882no—
263C2×C4—1881no—
264C2×C4—1881no—
265C2×C4—1881no—
266C2×C4—1881no—
267C2×C4—1881no—
268C2×C4(C22⋊D47)⋊C412yes—
269C2×C4#1309399411yescharacteristic
270C2×C4#1309399612yes—
271C2×C4#1309400811yescharacteristic
272C2×C4M16×D4711yescharacteristic
273C2×C4M16×D4711yescharacteristic
274C2×C4#1309403012yes—
275C8—82no—
276C8—1884no—
277C22—21no—
278C22—21no—
279C22—3762no—
280C22—3761no—
281C22—3761no—
282C22—1884no—
283C22—1882no—
284C22#1568900011yescharacteristic
285C22#1568908611yescharacteristic
286C22#1568908911yescharacteristic
287C4—22no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—3761no—
292C4—1882no—
293C4—1882no—
294C4#1568919712yes—
295C4#1568919812yes—
296C4#1568929812yes—
297C2—21no—
298C2—21no—
299C2—3762no—
300C2#2024368311yescharacteristic
301C2#2024376911yescharacteristic
302C2#2024378911yescharacteristic
subgroupG/H
centerC2×C4#13093994
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1461C2
socleC22×C47#386
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1461D47
hypocenterC47#8964
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #20252473C2semidirect
4. #20241220C2semidirect
5. C47×#1461C2semidirect
149. #1461D47semidirect
172. C47#8964semidirect

Maximal subgroups

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

subgroupindex
#202524732
C4×#130940132
#202417802
#202412202
C47×#14612
#896447