Finite Group Databasebeta

← #26110414

Subgroups — #26110414

Order 12032.

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

normal subgroups only

This group has 364 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#20252325C211yesmaximal, characteristic
2#20244109C211yesmaximal, characteristic
3#20241888C211yesmaximal, characteristic
4#20241790C211yesmaximal, characteristic
5#20241289C211yesmaximal, characteristic
6#20241232C211yesmaximal, characteristic
7C47×#1571C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689252—22no—
10#15689252—22no—
11#15689176C2211yescharacteristic
12#15689151—21no—
13C2×Dic47⋊C8—21no—
14#15689081—21no—
15C2×#13093935C2211yescharacteristic
16C4×C4⋊Dic47C2211yescharacteristic
17#15688877—21no—
18#15688861C412yes—
19#15688861C412yes—
20#15688853—22no—
21C47×#431C2211yescharacteristic
22C47×#423C2211yescharacteristic
23C2×C4⋊C8×C47C2211yescharacteristic
24C47×#335—22no—
25C22×C47⋊C8D411yescharacteristic
26C22×C47⋊C8D411yescharacteristic
27#13094024Q811yescharacteristic
28#13094024Q811yescharacteristic
29#13094016—21no—
30#13094003—21no—
31#13094003C2×C412yes—
32C4×Dic94—21no—
33C4×Dic94D412yes—
34#13093983—21no—
35C2×C4⋊Dic47—21no—
36C2×C4×Dic47—21no—
37C4⋊(C47⋊C8)—21no—
38Dic47⋊C8—24no—
39Dic47⋊C8—42no—
40C22⋊(C47⋊C8)—21no—
41#13093935—21no—
42#13093935—21no—
43#13093934—22no—
44C4×C47⋊C8—21no—
45C8×Dic47—22no—
46C47⋊#95C2×C411yescharacteristic
47C47⋊#95C2×C411yescharacteristic
48C2×M16×C47—21no—
49C22×C8×C47—21no—
50C2×C42×C47C2311yescharacteristic
51C4⋊C8×C47—21no—
52C4⋊C8×C47—21no—
53C4⋊C8×C47—21no—
54C4⋊C8×C47—21no—
55C4⋊C8×C47D412yes—
56C22⋊C8×C47C2×C412yes—
57C4×C8×C47—21no—
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47#4111yescharacteristic
61C22×Dic47—21no—
62C22×Dic47—21no—
63C2×Dic94—21no—
64C2×Dic94C4⋊C412yes—
65M16×C47—41no—
66M16×C47—41no—
67C2×C8×C47—22no—
68C2×C8×C47—22no—
69C2×C8×C47—41no—
70C2×C8×C47—41no—
71C2×C8×C47—41no—
72C42×C47C2×D411yescharacteristic
73C42×C47C2×D411yescharacteristic
74C42×C47C2×Q811yescharacteristic
75C42×C47#4111yescharacteristic
76C22⋊Dic47—22no—
77C4⋊Dic47—21no—
78C4⋊Dic47—21no—
79C4⋊Dic47—21no—
80C4⋊Dic47—42no—
81C4⋊Dic47C4⋊C412yes—
82Dic47⋊C4—22no—
83Dic47⋊C4—22no—
84C4×Dic47—22no—
85C4×Dic47—41no—
86C4×Dic47—41no—
87C2×C47⋊C8—21no—
88C2×C47⋊C8—21no—
89C2×C47⋊C8—21no—
90C2×C47⋊C8—21no—
91C2×C47⋊C8—21no—
92C2×C47⋊C8—41no—
93C2×C47⋊C8Q1612yes—
94C23×C47C4×D411yescharacteristic
95C2×C4×C47—41no—
96C2×C4×C47—41no—
97C2×C4×C47C2×C4⋊C411yescharacteristic
98C2×C4×C47C4×D411yescharacteristic
99C2×C4×C47#12011yescharacteristic
100C2×C4×C47C22⋊Q811yescharacteristic
101C2×C4×C47C22⋊Q811yescharacteristic
102C2×C4×C47#12311yescharacteristic
103C2×C4×C47C2×Q1611yescharacteristic
104C2×C4×C47#13511yescharacteristic
105C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
106C2×C4×C47#13711yescharacteristic
107C2×Dic47—22no—
108C2×Dic47—22no—
109C2×Dic47—22no—
110C2×Dic47—41no—
111C2×Dic47—41no—
112C2×Dic47—41no—
113C2×Dic47—41no—
114C2×Dic47—41no—
115Dic94—42no—
116Dic94—81no—
117C8×C47—42no—
118C8×C47—42no—
119C8×C47—81no—
120C47⋊C8—22no—
121C47⋊C8—41no—
122C47⋊C8—81no—
123#8879—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47#38511yescharacteristic
127C22×C47C4×Q1611yescharacteristic
128C22×C47#44011yescharacteristic
129C4×C47—21no—
130C4×C47—21no—
131C4×C47—21no—
132C4×C47—21no—
133C4×C47#44911yescharacteristic
134C4×C47#45111yescharacteristic
135C4×C47#47711yescharacteristic
136C4×C47#47911yescharacteristic
137Dic47—42no—
138Dic47—42no—
139Dic47—42no—
140Dic47—81no—
141#2961—471no—
142#1940—471no—
143#1571D4711yescharacteristic
144#1503—471no—
145#1481—471no—
146#1303—471no—
147#1281—471no—
148C2×C47—21no—
149C2×C47—21no—
150C2×C47#186111yescharacteristic
151C2×C47#186311yescharacteristic
152C2×C47#186811yescharacteristic
153#533—471no—
154#446—942no—
155#446—942no—
156#431—471no—
157#431C2×D4711yescharacteristic
158#424—941no—
159#423C2×D4711yescharacteristic
160C2×C4⋊C8—941no—
161C2×C4⋊C8C2×D4711yescharacteristic
162#405—941no—
163C2×#97—471no—
164C4×C4⋊C4—471no—
165#336—941no—
166#335—22no—
167#328—472no—
168#328—472no—
169Q8⋊C8—942no—
170C47#887911yescharacteristic, complemented, Sylow
171C2×M16—21no—
172C22×C8—21no—
173C22×C8—471no—
174C22×C8—471no—
175C4⋊Q8—471no—
176C4⋊Q8—471no—
177#125—941no—
178C22⋊Q8—472no—
179C22⋊Q8—941no—
180C4×Q8—472no—
181C4×Q8—941no—
182#117—941no—
183C2×C4⋊C4—941no—
184C2×C42—941no—
185C2×C42C22×D4711yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—944no—
191C4⋊C8—941no—
192C4⋊C8—1882no—
193C4⋊C8C22⋊D4712yes—
194C22⋊C8—941no—
195C22⋊C8C4×D4712yes—
196#97—942no—
197#97—941no—
198#97—941no—
199C4×C8—21no—
200C4×C8—942no—
201C4×C8—941no—
202#95—471no—
203#95—471no—
204C2×Q8—472no—
205C2×Q8—941no—
206C22×C4—941no—
207C22×C4—941no—
208C22×C4C2×C4×D4711yescharacteristic
209C22×C4D4×D4711yescharacteristic
210C22×C4#1062535411yescharacteristic
211M16—41no—
212M16—41no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—472no—
219C2×C8—941no—
220C2×C8—941no—
221C2×C8—941no—
222C2×C8—941no—
223C2×C8—941no—
224C2×C8—1881no—
225C4⋊C4—472no—
226C4⋊C4—942no—
227C4⋊C4—941no—
228C4⋊C4—941no—
229C4⋊C4—941no—
230C4⋊C4—942no—
231C4⋊C4—1882no—
232C22⋊C4—942no—
233C42—942no—
234C42—1881no—
235C42—1881no—
236C42#1062535011yescharacteristic
237C42D4×D4711yescharacteristic
238C42Q8×D4711yescharacteristic
239C42C2×C22⋊D4711yescharacteristic
240C23#1309398611yescharacteristic
241Q8—3761no—
242Q8—1882no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—942no—
246C2×C4—942no—
247C2×C4—942no—
248C2×C4—1881no—
249C2×C4—1881no—
250C2×C4—1881no—
251C2×C4—1881no—
252C2×C4—1881no—
253C2×C4C4⋊C4×D4711yescharacteristic
254C2×C4C4×C22⋊D4711yescharacteristic
255C2×C4#1309399211yescharacteristic
256C2×C4#1309400011yescharacteristic
257C2×C4#1309400211yescharacteristic
258C2×C4#1309400911yescharacteristic
259C2×C4Q16×D4711yescharacteristic
260C2×C4#1309404311yescharacteristic
261C2×C4#1309404811yescharacteristic
262C2×C4#1309405611yescharacteristic
263C8—42no—
264C8—42no—
265C8—81no—
266C8—942no—
267C8—3761no—
268C8—1881no—
269C22—21no—
270C22—21no—
271C22#1568899411yescharacteristic
272C22#1568921711yescharacteristic
273C22#1568922211yescharacteristic
274C4—21no—
275C4—21no—
276C4—21no—
277C4—21no—
278C4—3761no—
279C4—1882no—
280C4—1882no—
281C4—1882no—
282C4#1568926511yescharacteristic
283C4#1568927711yescharacteristic
284C4#1568941711yescharacteristic
285C4#1568942911yescharacteristic
286C2—21no—
287C2—21no—
288C2#2024362511yescharacteristic
289C2#2024363911yescharacteristic
290C2#2024367211yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1571C2
socleC22×C47#385
prime radicalC23×C47C4×D4
cosocleC2×C42×C47C23
hypercenter#1571D47
hypocenterC47#8879
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
170. C47#8879semidirect

Maximal subgroups

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

subgroupindex
#202523252
#202441092
#202418882
#202417902
#202412892
#202412322
C47×#15712
#887947