Finite Group Databasebeta

← #26136694

Subgroups — #26136694

Order 12032.

316 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 75 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#20245083C211yesmaximal, characteristic, complemented
2#20245077C211yesmaximal, characteristic, complemented
3#20241561C212yesmaximal, complemented
4#20241555C212yesmaximal, complemented
5C47×#3221C211yesmaximal, characteristic
6#15689107—21no—
7C2×#13093943—21no—
8#15688922—21no—
9#15688922C2212yes—
10#15688922C2212yes—
11#15688920—21no—
12C47×#540C411yescharacteristic
13C47×#536C2211yescharacteristic
14C47×#496C411yescharacteristic
15C47×#492C2211yescharacteristic
16C47×#491C2211yescharacteristic
17C47×#488C411yescharacteristic
18C47×#463—21no—
19C47×#461—21no—
20C47×#449—21no—
21C47×#448—21no—
22C47×#433C411yescharacteristic
23C2×#10625329—41no—
24C2×#10625329—41no—
25#13093943—22no—
26#13093943—22no—
27#13093943—41no—
28#13093941—41no—
29C2×#41×C47D411yescharacteristic
30C22×Q8×C47D411yescharacteristic
31C2×SD16×C47—21no—
32C2×SD16×C47—21no—
33C2×SD16×C47—21no—
34C2×D8×C47—21no—
35C4⋊Q8×C47D411yescharacteristic
36C4⋊Q8×C47C2311yescharacteristic
37#127×C47C2×C411yescharacteristic
38#125×C47C2×C411yescharacteristic
39#124×C47C2×C411yescharacteristic
40#124×C47D411yescharacteristic
41#123×C47—41no—
42#123×C47—41no—
43C22⋊Q8×C47—41no—
44#121×C47—21no—
45#121×C47—41no—
46C4×D4×C47—21no—
47#117×C47C2×C411yescharacteristic
48C4⋊C8×C47—21no—
49#103×C47—21no—
50#103×C47—21no—
51#103×C47—21no—
52#102×C47—21no—
53C22⋊C8×C47—21no—
54#97×C47C2×C411yescharacteristic
55C4×C8×C47C2×C411yescharacteristic
56#41×C47—81no—
57C2×Q8×C47—21no—
58C2×Q8×C47—81no—
59C2×Q8×C47C22⋊C411yescharacteristic
60C2×Q8×C47C2×D411yescharacteristic
61C2×Q8×C47C2×D411yescharacteristic
62C2×D4×C47—21no—
63C2×D4×C47—21no—
64C2×D4×C47—41no—
65C2×D4×C47—41no—
66C2×D4×C47C22⋊C411yescharacteristic
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—41no—
70C22×C4×C47C22⋊C411yescharacteristic
71SD16×C47—81no—
72SD16×C47—81no—
73SD16×C47—81no—
74D8×C47—81no—
75C2×C8×C47—21no—
76C2×C8×C47—21no—
77C4⋊C4×C47—21no—
78C4⋊C4×C47—21no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C22⋊C4×C47—21no—
83C22⋊C4×C47—41no—
84C22⋊C4×C47—41no—
85C22⋊C4×C47—41no—
86C22⋊C4×C47—41no—
87C42×C47C22⋊C411yescharacteristic
88C42×C47C22×C411yescharacteristic
89#10625329—42no—
90#10625329—42no—
91#10625329—81no—
92#10625329—81no—
93C2×C47⋊C8—81no—
94C2×C47⋊C8—81no—
95C23×C47—21no—
96C23×C47—41no—
97C23×C47#9911yescharacteristic
98Q8×C47—41no—
99Q8×C47—41no—
100Q8×C47—41no—
101Q8×C47—81no—
102D4×C47—41no—
103D4×C47—41no—
104D4×C47—41no—
105D4×C47—81no—
106D4×C47—161no—
107D4×C47—161no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—161no—
120C2×C4×C47#9911yescharacteristic
121C2×C4×C47C2×C22⋊C411yescharacteristic
122C8×C47—41no—
123C8×C47—41no—
124C47⋊C8—82no—
125C47⋊C8—82no—
126#13611—471nomaximal, Sylow
127C22×C47—21no—
128C22×C47—41no—
129C22×C47—41no—
130C22×C47—81no—
131C22×C47—81no—
132C22×C47—81no—
133C22×C47—81no—
134C22×C47C2×#9911yescharacteristic
135C4×C47—21no—
136C4×C47—21no—
137C4×C47—41no—
138C4×C47—41no—
139C4×C47—41no—
140C4×C47—41no—
141C4×C47—41no—
142C4×C47—81no—
143#3221D4711yescharacteristic
144#2135—471no—
145#2133—471no—
146#1413—472no—
147#1410—472no—
148C2×C47—21no—
149C2×C47—41no—
150C2×C47—81no—
151C2×C47—161no—
152C2×C47#212711yescharacteristic
153#540Dic4711yescharacteristic
154#536C2×D4711yescharacteristic
155#496Dic4711yescharacteristic
156#492C2×D4711yescharacteristic
157#491C2×D4711yescharacteristic
158#488Dic4711yescharacteristic
159#463—21no—
160#461—21no—
161#449—21no—
162#448—21no—
163#433Dic4711yescharacteristic
164#413—941no—
165C2×#101—941no—
166#356—472no—
167#356—472no—
168#356—941no—
169#355—941no—
170C47#1361111yescharacteristic, complemented, Sylow
171C2×#41C22⋊D4711yescharacteristic
172C22×Q8C22⋊D4711yescharacteristic
173C2×SD16—21no—
174C2×SD16—21no—
175C2×SD16—21no—
176C2×D8—21no—
177C2×M16—1881no—
178C2×M16—1881no—
179C4⋊Q8C22⋊D4711yescharacteristic
180C4⋊Q8C22×D4711yescharacteristic
181#127C2×Dic4711yescharacteristic
182#125C2×Dic4711yescharacteristic
183#124C2×Dic4711yescharacteristic
184#124C22⋊D4711yescharacteristic
185#123—41no—
186#123—41no—
187C22⋊Q8—41no—
188#121—21no—
189#121—41no—
190C4×D4—21no—
191#117C2×Dic4711yescharacteristic
192C4⋊C8—21no—
193#103—21no—
194#103—21no—
195#103—21no—
196#102—21no—
197#101—942no—
198#101—942no—
199#101—1881no—
200#100—1881no—
201C22⋊C8—21no—
202#97C2×Dic4711yescharacteristic
203C4×C8C2×Dic4711yescharacteristic
204#41—81no—
205C2×Q8—21no—
206C2×Q8—81no—
207C2×Q8C22⋊Dic4711yescharacteristic
208C2×Q8C2×C22⋊D4711yescharacteristic
209C2×Q8C2×C22⋊D4711yescharacteristic
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4C22⋊Dic4711yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4C22⋊Dic4711yescharacteristic
219SD16—81no—
220SD16—81no—
221SD16—81no—
222D8—81no—
223M16—3761no—
224M16—3761no—
225M16—1882no—
226M16—1882no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—3761no—
230C2×C8—3761no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—21no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C42C22⋊Dic4711yescharacteristic
242C42C22×Dic4711yescharacteristic
243C23—21no—
244C23—41no—
245C23#1309393911yescharacteristic
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—81no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—81no—
254D4—161no—
255D4—161no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—161no—
268C2×C4#1309393911yescharacteristic
269C2×C4C2×C22⋊Dic4711yescharacteristic
270C8—41no—
271C8—41no—
272C8—3762no—
273C8—3762no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22C2×#1309393911yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—81no—
290C2—21no—
291C2—41no—
292C2—81no—
293C2—161no—
294C2#2024505711yescharacteristic
subgroupG/H
centerC2#20245057
commutatorC42×C47C22×C4
FrattiniC4⋊Q8C22×D47
Fitting, prime radicalC47×#3221C2
socleC2×C47#2127
cosocleC4⋊Q8×C47C23
hypercenter#3221D47
hypocenterC47#13611
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #3221

Complements

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

NH ≅ G/Ntype
1. #20245083C2semidirect
2. #20245077C2semidirect
3. #20241561C2semidirect
4. #20241555C2semidirect
170. C47#13611semidirect

Maximal subgroups

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

subgroupindex
#202450832
#202450772
#202415612
#202415552
C47×#32212
#1361147