Finite Group Databasebeta

← #26113655

Subgroups — #26113655

Order 12032.

470 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 117 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#20252468C211yesmaximal, characteristic
2#20248604C211yesmaximal, characteristic
3#20243381C211yesmaximal, characteristic
4#20242073C211yesmaximal, characteristic
5#20241290C211yesmaximal, characteristic
6#20241289C211yesmaximal, characteristic
7C47×#1549C211yesmaximal, characteristic
8C2×C4×Dic94C2211yescharacteristic
9C8×Dic94—22no—
10#15689247—22no—
11#15689176C2211yescharacteristic
12#15689173—22no—
13#15689147C2211yescharacteristic
14C2×#13093942—21no—
15#15689061C414yes—
16#15689030—22no—
17#15689028—21no—
18#15688988—21no—
19C4×C4⋊Dic47C2211yescharacteristic
20#15688891—21no—
21#15688869—22no—
22C2×C4⋊Q8×C47C2211yescharacteristic
23C47×#431C2211yescharacteristic
24C47×#423C2211yescharacteristic
25C47×#332—22no—
26C22×Dic94D411yescharacteristic
27C2×#10625329—21no—
28C4×Dic94—21no—
29C4×Dic94—21no—
30C4×Dic94—21no—
31C4×Dic94—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47D411yescharacteristic
34C2×Dic47⋊C4C2×C412yes—
35C2×C4×Dic47—21no—
36C2×C4×Dic47D412yes—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)D412yes—
40Dic47⋊C8—24no—
41#13093942—42no—
42C22⋊(C47⋊C8)—21no—
43C4×C47⋊C8—21no—
44C8×Dic47—24no—
45C47⋊#95—24no—
46C47⋊#95—24no—
47C47⋊#95C2×C412yes—
48C22×Q8×C47—21no—
49C2×M16×C47—21no—
50C4⋊Q8×C47—21no—
51C4⋊Q8×C47—21no—
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47C2×C412yes—
56C2×C42×C47C2311yescharacteristic
57C4⋊C8×C47—21no—
58C4⋊C8×C47—21no—
59C4⋊C8×C47D412yes—
60C22⋊C8×C47—21no—
61C4×C8×C47—21no—
62C2×Q8×C47—44no—
63C2×Q8×C47—41no—
64C2×Q8×C47—41no—
65C22×C4×C47—22no—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—22no—
70C22×Dic47C22⋊C412yes—
71C22×Dic47C22⋊C412yes—
72C2×Dic94—21no—
73C2×Dic94—21no—
74C2×Dic94—21no—
75C2×Dic94—21no—
76C2×Dic94SD1612yes—
77C2×Dic94Q1612yes—
78M16×C47—41no—
79M16×C47—41no—
80C2×C8×C47—22no—
81C2×C8×C47—41no—
82C4⋊C4×C47—22no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—41no—
86C42×C47C2×D411yescharacteristic
87C42×C47C2×D411yescharacteristic
88C42×C47#4111yescharacteristic
89C42×C47#4111yescharacteristic
90C4⋊Dic47—21no—
91C4⋊Dic47—21no—
92C4⋊Dic47—42no—
93Dic47⋊C4—22no—
94Dic47⋊C4—22no—
95C4×Dic47—22no—
96C4×Dic47—22no—
97C4×Dic47—41no—
98C4×Dic47—41no—
99#10625329—41no—
100#10625329—41no—
101C2×C47⋊C8—22no—
102C2×C47⋊C8—41no—
103C23×C47C2×C22⋊C411yescharacteristic
104Q8×C47—82no—
105Q8×C47—82no—
106C2×C4×C47—22no—
107C2×C4×C47—41no—
108C2×C4×C47—44no—
109C2×C4×C47—42no—
110C2×C4×C47C4×D411yescharacteristic
111C2×C4×C47C4×D411yescharacteristic
112C2×C4×C47#12011yescharacteristic
113C2×C4×C47#12111yescharacteristic
114C2×C4×C47#12111yescharacteristic
115C2×C4×C47#12311yescharacteristic
116C2×C4×C47C2×SD1611yescharacteristic
117C2×C4×C47C2×Q1611yescharacteristic
118C2×C4×C47#13511yescharacteristic
119C2×C4×C47#13511yescharacteristic
120C2×Dic47—22no—
121C2×Dic47—22no—
122C2×Dic47—22no—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47#10314yes—
128Dic94—22no—
129Dic94—22no—
130Dic94—22no—
131Dic94—22no—
132C8×C47—42no—
133C8×C47—81no—
134C47⋊C8—42no—
135C47⋊C8—81no—
136#9396—471nomaximal, Sylow
137C22×C47—21no—
138C22×C47—21no—
139C22×C47#38611yescharacteristic
140C22×C47C2×#10311yescharacteristic
141C22×C47#41611yescharacteristic
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—21no—
145C4×C47—82no—
146C4×C47#45412yes—
147C4×C47#46511yescharacteristic
148C4×C47#46511yescharacteristic
149C4×C47#46711yescharacteristic
150C4×C47#46711yescharacteristic
151Dic47—24no—
152Dic47—42no—
153Dic47—82no—
154#2982—471no—
155#2568—471no—
156#1801—471no—
157#1549—471no—
158#1549D4711yescharacteristic
159#1303—471no—
160#1303—471no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47#188811yescharacteristic
164C2×C47#189211yescharacteristic
165C2×C47#191511yescharacteristic
166C2×C4⋊Q8C2×D4711yescharacteristic
167C2×C4×Q8—471no—
168C8×Q8—942no—
169C8×Q8—942no—
170#431—471no—
171#431—942no—
172#431C2×D4711yescharacteristic
173#423—471no—
174#423C2×D4711yescharacteristic
175C2×#101—941no—
176#400—474no—
177#391—942no—
178#391—941no—
179#382—941no—
180C4×C4⋊C4—471no—
181#343—941no—
182#332—22no—
183#332—942no—
184C47#939611yescharacteristic, complemented, Sylow
185C22×Q8—21no—
186C22×Q8—471no—
187C2×M16—21no—
188C2×M16—941no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4×Q8—941no—
194C4×Q8—941no—
195C4×Q8—941no—
196C4×Q8—941no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—472no—
199C2×C4⋊C4—471no—
200C2×C4⋊C4—941no—
201C2×C4⋊C4C4×D4712yes—
202C2×C42—472no—
203C2×C42—941no—
204C2×C42C22×D4711yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—472no—
208C4⋊C8—944no—
209C4⋊C8—941no—
210C4⋊C8—941no—
211C4⋊C8C22⋊D4712yes—
212#101—1882no—
213C22⋊C8—21no—
214C22⋊C8—941no—
215C4×C8—21no—
216C4×C8—944no—
217C4×C8—941no—
218#95—472no—
219#95—944no—
220#95—944no—
221C2×Q8—44no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—472no—
225C2×Q8—472no—
226C2×Q8—941no—
227C2×Q8—941no—
228C2×Q8—941no—
229C2×Q8—941no—
230C22×C4—22no—
231C22×C4—472no—
232C22×C4—472no—
233C22×C4—942no—
234C22×C4C2×C4×D4711yescharacteristic
235C22×C4D4×D4711yescharacteristic
236C22×C4D4×D4711yescharacteristic
237M16—41no—
238M16—41no—
239M16—1881no—
240M16—1881no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—942no—
244C2×C8—1881no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—942no—
250C4⋊C4—942no—
251C4⋊C4—941no—
252C4⋊C4—941no—
253C4⋊C4—1882no—
254C42—942no—
255C42—942no—
256C42—1881no—
257C42—1881no—
258C42#1062535011yescharacteristic
259C42D4×D4711yescharacteristic
260C42#1062535211yescharacteristic
261C42C2×C22⋊D4711yescharacteristic
262C23C22⋊C4×D4711yescharacteristic
263Q8—82no—
264Q8—82no—
265Q8—942no—
266Q8—942no—
267Q8—942no—
268Q8—942no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—474no—
274C2×C4—942no—
275C2×C4—942no—
276C2×C4—942no—
277C2×C4—1882no—
278C2×C4—1882no—
279C2×C4—1882no—
280C2×C4—1882no—
281C2×C4(C22⋊D47)⋊C411yescharacteristic
282C2×C4C4×C22⋊D4711yescharacteristic
283C2×C4#1309399211yescharacteristic
284C2×C4#1309399511yescharacteristic
285C2×C4#1309399711yescharacteristic
286C2×C4#1309400611yescharacteristic
287C2×C4SD16×D4711yescharacteristic
288C2×C4Q16×D4711yescharacteristic
289C2×C4#1309404511yescharacteristic
290C2×C4#1309404711yescharacteristic
291C8—42no—
292C8—81no—
293C8—3761no—
294C8—1882no—
295C22—21no—
296C22—21no—
297C22#1568899911yescharacteristic
298C22#103×D4711yescharacteristic
299C22#1568911811yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—82no—
304C4—944no—
305C4—3762no—
306C4—1882no—
307C4#1568929412yes—
308C4#1568934911yescharacteristic
309C4#1568935511yescharacteristic
310C4#1568936111yescharacteristic
311C4#1568936311yescharacteristic
312C2—21no—
313C2—21no—
314C2#2024378811yescharacteristic
315C2#2024380811yescharacteristic
316C2#2024393711yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1549C2
socleC22×C47#386
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1549D47
hypocenterC47#9396
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
184. C47#9396semidirect

Maximal subgroups

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

subgroupindex
#202524682
#202486042
#202433812
#202420732
#202412902
#202412892
C47×#15492
#939647