Finite Group Databasebeta

← #26134622

Subgroups — #26134622

Order 12032.

436 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 103 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#20245063C212yesmaximal
2#20245042C212yesmaximal
3#20243434C211yesmaximal, characteristic
4#20243188C211yesmaximal, characteristic
5C47×#3059C211yesmaximal, characteristic
6#15689161—21no—
7#15689107C2211yescharacteristic
8#15689097C2212yes—
9#15689097C2211yescharacteristic
10#15689078—21no—
11#15688918—24no—
12#15688913—24no—
13#15688894—21no—
14#15688880—21no—
15C22×Q16×C47C411yescharacteristic, complemented
16C2×#131×C47C2211yescharacteristic
17C47×#536C2212yes—
18C47×#470—21no—
19C47×#470—21no—
20C47×#467—21no—
21C47×#467—21no—
22C47×#427C411yescharacteristic, complemented
23C47×#419C412yescomplemented
24C2×#10625329—21no—
25#13093983—21no—
26#13093983—22no—
27#13093958—41no—
28#13093956—41no—
29#13093943—21no—
30#13093941D412yes—
31#13093939—21no—
32#13093939—22no—
33#13093939—22no—
34#13093939D412yes—
35#13093934—41no—
36C8×Dic47—41no—
37C2×#41×C47C2311yescharacteristic
38C22×Q8×C47C2×C412yes—
39C2×Q16×C47—21no—
40C2×Q16×C47—21no—
41C2×Q16×C47—41no—
42C2×Q16×C47—41no—
43#131×C47—41no—
44#131×C47—41no—
45C2×M16×C47C2×C411yescharacteristic
46C2×M16×C47D411yescharacteristic
47C2×M16×C47D411yescharacteristic
48C22×C8×C47C2×C411yescharacteristic
49C22×C8×C47D411yescharacteristic
50C22×C8×C47D411yescharacteristic
51C4⋊Q8×C47—22no—
52#124×C47—22no—
53#123×C47—22no—
54#123×C47—22no—
55C22⋊Q8×C47—22no—
56C22⋊Q8×C47—22no—
57#117×C47C2×C412yes—
58#107×C47—21no—
59#106×C47—21no—
60#103×C47—22no—
61#103×C47—22no—
62#41×C47—41no—
63#41×C47—41no—
64C2×Q8×C47—22no—
65C2×Q8×C47—22no—
66C2×Q8×C47—42no—
67C2×Q8×C47—42no—
68C2×Q8×C47#4111yescharacteristic
69C2×D4×C47C2×D411yescharacteristic
70C2×D4×C47C2×D411yescharacteristic
71C2×D4×C47#4111yescharacteristic
72C22×C4×C47—22no—
73C22×C4×C47C22×C411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76Q16×C47—42no—
77Q16×C47—41no—
78Q16×C47—41no—
79M16×C47—41no—
80M16×C47—41no—
81M16×C47—41no—
82C2×C8×C47—21no—
83C2×C8×C47—21no—
84C2×C8×C47—21no—
85C2×C8×C47—21no—
86C2×C8×C47—21no—
87C2×C8×C47—21no—
88C2×C8×C47C22⋊C411yescharacteristic
89C2×C8×C47C22⋊C411yescharacteristic
90C2×C8×C47C22⋊C411yescharacteristic
91C2×C8×C47C22⋊C411yescharacteristic
92C4⋊C4×C47—22no—
93C4⋊C4×C47—42no—
94C4⋊C4×C47—42no—
95C22⋊C4×C47—22no—
96C22⋊C4×C47—42no—
97C22⋊C4×C47—42no—
98C42×C47—22no—
99C22⋊Dic47—22no—
100C22⋊Dic47—42no—
101C4⋊Dic47—41no—
102C4⋊Dic47—42no—
103C4×Dic47—41no—
104C4×Dic47—42no—
105#10625329—22no—
106#10625329—41no—
107C2×C47⋊C8—41no—
108C23×C47C4×D411yescharacteristic
109C23×C47#12111yescharacteristic
110C23×C47#12711yescharacteristic
111Q8×C47—41no—
112Q8×C47—42no—
113Q8×C47—42no—
114Q8×C47—82no—
115D4×C47—41no—
116D4×C47—41no—
117D4×C47—41no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—82no—
127C2×C4×C47C2×C22⋊C411yescharacteristic
128C2×C4×C47C4×D411yescharacteristic
129C2×C4×C47#12111yescharacteristic
130C2×C4×C47#12411yescharacteristic
131C2×Dic47—42no—
132C2×Dic47—82no—
133C8×C47—21no—
134C8×C47—21no—
135C8×C47—21no—
136C8×C47#9911yescharacteristic
137C8×C47#9911yescharacteristic
138C47⋊C8—42no—
139#13230—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—21no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—41no—
146C22×C47#39011yescharacteristic
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—82no—
151C4×C47—82no—
152C4×C47C2×#9911yescharacteristic
153C4×C47#41011yescharacteristic
154Dic47—82no—
155Dic47—162no—
156#3059D4711yescharacteristic
157#2129—472no—
158#2123—472no—
159#1812—471no—
160#1757—471no—
161C2×C47—21no—
162C2×C47—41no—
163C2×C47—41no—
164C2×C47—41no—
165C2×C47#196411yescharacteristic
166C22×Q16Dic4711yescharacteristic, complemented
167C2×#131C2×D4711yescharacteristic
168#536C2×D4712yes—
169#470—21no—
170#470—21no—
171#467—21no—
172#467—21no—
173#427—941no—
174#427Dic4711yescharacteristic, complemented
175#419Dic4712yescomplemented
176#413—471no—
177#410—471no—
178#410—472no—
179#405—941no—
180#354—944no—
181#352—944no—
182#344—941no—
183H(3,ℤ/4)—941no—
184C47#1323011yescharacteristic, complemented, Sylow
185C2×#41C22×D4711yescharacteristic
186C22×Q8C2×Dic4712yes—
187C2×Q16—21no—
188C2×Q16—21no—
189C2×Q16—41no—
190C2×Q16—41no—
191#131—41no—
192#131—41no—
193C2×M16—941no—
194C2×M16C2×Dic4711yescharacteristic
195C2×M16C22⋊D4711yescharacteristic
196C2×M16C22⋊D4711yescharacteristic
197C22×C8C2×Dic4711yescharacteristic
198C22×C8C22⋊D4711yescharacteristic
199C22×C8C22⋊D4711yescharacteristic
200C4⋊Q8—22no—
201#124—22no—
202#123—22no—
203#123—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206#117—941no—
207#117—942no—
208#117C2×Dic4712yes—
209#107—21no—
210#107—1881no—
211#106—21no—
212#106—1881no—
213#103—22no—
214#103—22no—
215#101—941no—
216#100—472no—
217#99—472no—
218#99—941no—
219#99—942no—
220#99—942no—
221#97—1881no—
222C4×C8—1881no—
223#41—41no—
224#41—41no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8#1062535211yescharacteristic
230C2×D4D4×D4711yescharacteristic
231C2×D4D4×D4711yescharacteristic
232C2×D4#1062535211yescharacteristic
233C22×C4—22no—
234C22×C4C22×Dic4711yescharacteristic
235C22×C4C2×C22⋊D4711yescharacteristic
236C22×C4C2×C22⋊D4711yescharacteristic
237Q16—42no—
238Q16—41no—
239Q16—41no—
240M16—41no—
241M16—41no—
242M16—41no—
243M16—942no—
244M16—1881no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—1881no—
252C2×C8C22⋊Dic4711yescharacteristic
253C2×C8C22⋊Dic4711yescharacteristic
254C2×C8C22⋊Dic4711yescharacteristic
255C2×C8C22⋊Dic4711yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—1882no—
260C4⋊C4—1881no—
261C22⋊C4—22no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—942no—
265C22⋊C4—1882no—
266C42—22no—
267C42—1882no—
268C42—1881no—
269C23D4×Dic4711yescharacteristic
270C23#1309399511yescharacteristic
271C23#1309402011yescharacteristic
272Q8—41no—
273Q8—42no—
274Q8—42no—
275Q8—82no—
276D4—41no—
277D4—41no—
278D4—41no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—82no—
288C2×C4—3762no—
289C2×C4—1882no—
290C2×C4C2×C22⋊Dic4711yescharacteristic
291C2×C4D4×Dic4711yescharacteristic
292C2×C4#1309399511yescharacteristic
293C2×C4#1309401111yescharacteristic
294C8—21no—
295C8—21no—
296C8—21no—
297C8—1882no—
298C8#1309393911yescharacteristic
299C8#1309393911yescharacteristic
300C22—21no—
301C22—21no—
302C22—21no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22#1568902611yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—82no—
311C4—82no—
312C4—3762no—
313C4—7522no—
314C4C2×#1309393911yescharacteristic
315C4#1568909711yescharacteristic
316C2—21no—
317C2—41no—
318C2—41no—
319C2—41no—
320C2#2024422511yescharacteristic
subgroupG/H
centerC2#20244225
commutatorC22×C4×C47C22×C4
FrattiniC2×#41C22×D47
FittingC47×#3059C2
socleC2×C47#1964
prime radical, cosocleC2×#41×C47C23
hypercenter#3059D47
hypocenterC47#13230
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #3059

Complements

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

NH ≅ G/Ntype
15. C22×Q16×C47C4semidirect
22. C47×#427C4semidirect
23. C47×#419C4semidirect
166. C22×Q16Dic47semidirect
174. #427Dic47semidirect
175. #419Dic47semidirect
184. C47#13230semidirect

Maximal subgroups

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

subgroupindex
#202450632
#202450422
#202434342
#202431882
C47×#30592
#1323047