Finite Group Databasebeta

← #26111599

Subgroups — #26111599

Order 12032.

530 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 107 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#20253814C211yesmaximal, characteristic
2#20241980C211yesmaximal, characteristic, complemented
3#20241804C212yesmaximal, complemented
4#20241224C212yesmaximal, complemented
5C47×#2979C211yesmaximal, characteristic, complemented
6C2×#13094044—22no—
7#15689673C2211yescharacteristic
8#15689505—22no—
9#15689502C411yescharacteristic
10#15689499C411yescharacteristic
11#15689497C411yescharacteristic
12#15689495C411yescharacteristic
13C4×#10625329C2212yes—
14C2×C4×C47⋊C8C2211yescharacteristic
15#15688863—22no—
16#15688850—22no—
17#15688850—22no—
18C47×#532C2211yescharacteristic
19C47×#436—22no—
20C8×D4×C47—22no—
21C47×#431C2212yes—
22C47×#408—22no—
23C2×#10625350—21no—
24#13094044—44no—
25#13094044—44no—
26C2×#10625337—22no—
27C2×#10625335—22no—
28C2×#10625336—22no—
29C2×#10625334—22no—
30C2×#10625329—22no—
31C22×C47⋊C8—21no—
32#13094024C2×C411yescharacteristic
33#13094021C2×C411yescharacteristic
34#13094012—21no—
35#13093998—22no—
36C4×D188—22no—
37#13093935—22no—
38C4×C47⋊C8—21no—
39C4×C47⋊C8—22no—
40C4×C47⋊C8C2×C411yescharacteristic
41C4×C47⋊C8C2×C411yescharacteristic
42C2×#41×C47D412yes—
43C2×M16×C47—22no—
44C22×C8×C47—22no—
45C4⋊Q8×C47C2×C411yescharacteristic
46#127×C47C2×C411yescharacteristic
47#124×C47D412yes—
48#121×C47—22no—
49C4×D4×C47—22no—
50C2×C42×C47C2311yescharacteristic
51C4⋊C8×C47D412yes—
52C4⋊C8×C47D412yes—
53C22⋊C8×C47—24no—
54C22⋊C8×C47—22no—
55C4×C8×C47—22no—
56#41×C47—42no—
57#41×C47—42no—
58C2×Q8×C47C22⋊C412yes—
59C2×D4×C47—22no—
60C2×D4×C47—22no—
61C2×D4×C47C22⋊C412yes—
62C22×C4×C47—22no—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47#4112yes—
65C2×C22⋊D47—41no—
66#10625350—42no—
67#10625350—42no—
68C2×D188—21no—
69C2×D188—22no—
70C2×C4×D47—41no—
71C2×Dic94—21no—
72M16×C47—82no—
73C2×C8×C47—22no—
74C2×C8×C47—22no—
75C2×C8×C47—82no—
76C4⋊C4×C47—22no—
77C22⋊C4×C47—24no—
78C42×C47C22×C411yescharacteristic
79C42×C47C2×D411yescharacteristic
80C42×C47C2×D412yes—
81#10625337—44no—
82#10625336—44no—
83#10625335—44no—
84#10625334—44no—
85(C2×D47)⋊C4—42no—
86C4⋊Dic47—22no—
87#10625329—24no—
88#10625329—42no—
89C2×C47⋊C8—22no—
90C2×C47⋊C8—22no—
91C2×C47⋊C8—21no—
92C2×C47⋊C8—21no—
93C2×C47⋊C8—42no—
94C23×C47—22no—
95C23×C47#12311yescharacteristic
96C22×D47—41no—
97Q8×C47—42no—
98D4×C47—42no—
99D4×C47—42no—
100D4×C47—42no—
101D4×C47—82no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47C2×C22⋊C411yescharacteristic
110C2×C4×C47C4×D412yes—
111C2×C4×C47#12011yescharacteristic
112C2×C4×C47#12112yes—
113C22⋊D47—82no—
114C2×Dic47—41no—
115D188—42no—
116D188—42no—
117D188—42no—
118C4×D47—81no—
119C4×D47—81no—
120Dic94—42no—
121C8×C47—42no—
122C8×C47—42no—
123C47⋊C8—22no—
124C47⋊C8—22no—
125C47⋊C8—44no—
126#9069—471nomaximal, Sylow
127C22×C47—22no—
128C22×C47—21no—
129C22×C47—44no—
130C22×C47—42no—
131C22×C47#38611yescharacteristic
132C2×D47—82no—
133C2×D47—81no—
134C4×C47—22no—
135C4×C47—21no—
136C4×C47—42no—
137C4×C47#44312yes—
138C4×C47#44412yes—
139C4×C47#45311yescharacteristic
140C4×C47#45311yescharacteristic
141Dic47—81no—
142#3147—471no—
143#2979D4711yescharacteristic, complemented
144#1525—471no—
145#1484—472no—
146#1278—472no—
147C2×C47—41no—
148C2×C47—82no—
149C2×C47#187612yes—
150C2×C47#188711yescharacteristic
151D47—161no—
152C2×#135—942no—
153#532—471no—
154#532C2×D4711yescharacteristic
155#495—942no—
156#494—471no—
157#493—471no—
158#492—471no—
159#492—471no—
160#436—22no—
161C8×D4—22no—
162#431C2×D4712yes—
163#408—22no—
164C4×M16—472no—
165C2×C4×C8—471no—
166#329—942no—
167D4⋊C8—942no—
168D4⋊C8—942no—
169C47#906911yescharacteristic, complemented, Sylow
170C2×#41—941no—
171C2×#41C22⋊D4712yes—
172#135—1884no—
173#135—1884no—
174C2×Q16—942no—
175C2×SD16—942no—
176C2×SD16—942no—
177C2×D8—942no—
178C2×M16—22no—
179C2×M16—942no—
180C22×C8—22no—
181C22×C8—941no—
182C4⋊Q8—471no—
183C4⋊Q8C4×D4711yescharacteristic
184#127—471no—
185#127C4×D4711yescharacteristic
186#124—941no—
187#124D18812yes—
188#121—22no—
189#121—942no—
190C4×D4—22no—
191C4×D4—942no—
192C2×C42C22×D4711yescharacteristic
193C4⋊C8C22⋊D4712yes—
194C4⋊C8C22⋊D4712yes—
195C22⋊C8—24no—
196C22⋊C8—22no—
197#97—942no—
198C4×C8—22no—
199C4×C8—471no—
200C4×C8—471no—
201C4×C8—941no—
202C4×C8—942no—
203#41—42no—
204#41—42no—
205#41—1882no—
206#41—1882no—
207C2×Q8—941no—
208C2×Q8(C2×D47)⋊C412yes—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—941no—
212C2×D4—942no—
213C2×D4—1881no—
214C2×D4(C2×D47)⋊C412yes—
215C22×C4—22no—
216C22×C4—1881no—
217C22×C4#1062535012yes—
218C22×C4C2×C22⋊D4711yescharacteristic
219Q16—1884no—
220SD16—1884no—
221SD16—1884no—
222D8—1884no—
223M16—82no—
224M16—944no—
225M16—1882no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—82no—
229C2×C8—942no—
230C2×C8—942no—
231C2×C8—941no—
232C2×C8—941no—
233C2×C8—1882no—
234C4⋊C4—22no—
235C4⋊C4—942no—
236C22⋊C4—24no—
237C22⋊C4—1882no—
238C42C2×C4×D4711yescharacteristic
239C42C2×D18811yescharacteristic
240C42C2×C22⋊D4712yes—
241C23—22no—
242C23—1881no—
243C23#1309400711yescharacteristic
244Q8—42no—
245Q8—1882no—
246D4—42no—
247D4—42no—
248D4—42no—
249D4—82no—
250D4—3762no—
251D4—1882no—
252D4—1882no—
253D4—1882no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—3761no—
262C2×C4—3761no—
263C2×C4—1881no—
264C2×C4C2×(C2×D47)⋊C411yescharacteristic
265C2×C4C4×C22⋊D4712yes—
266C2×C4#1309399311yescharacteristic
267C2×C4#1309399812yes—
268C8—42no—
269C8—42no—
270C8—942no—
271C8—942no—
272C8—1884no—
273C22—22no—
274C22—21no—
275C22—44no—
276C22—42no—
277C22—3762no—
278C22—3761no—
279C22#1568900211yescharacteristic
280C4—22no—
281C4—21no—
282C4—42no—
283C4—3761no—
284C4#1568923812yes—
285C4#1568924312yes—
286C4#1568929111yescharacteristic
287C4#1568929111yescharacteristic
288C2—41no—
289C2—82no—
290C2—7521no—
291C2#2024371512yes—
292C2#2024378611yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2979C2
socleC22×C47#386
prime radical#20253814C2
cosocleC2×C42×C47C23
hypercenter#2979D47
hypocenterC47#9069
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
2. #20241980C2semidirect
3. #20241804C2semidirect
4. #20241224C2semidirect
5. C47×#2979C2semidirect
143. #2979D47semidirect
169. C47#9069semidirect

Maximal subgroups

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

subgroupindex
#202538142
#202419802
#202418042
#202412242
C47×#29792
#906947