Finite Group Databasebeta

← #26108723

Subgroups — #26108723

Order 12032.

736 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#15689873C211yesmaximal, characteristic
2#20244356C212yesmaximal, complemented
3C2×#15688864C212yesmaximal, complemented
4C2×#15688863C211yesmaximal, characteristic
5C47×#1959C211yesmaximal, characteristic
6#15689873—24no—
7#15689873—22no—
8#15689873C414yescomplemented
9C2×#13094022—21no—
10C2×#13094017—22no—
11C2×#13094016C2212yes—
12C2×#13094005—21no—
13C2×#13094001—22no—
14C2×#13093935C2212yes—
15#15688885—22no—
16#15688885—22no—
17#15688864—24no—
18#15688863—22no—
19C2×#124×C47C2211yescharacteristic
20C2×#103×C47—21no—
21C2×#102×C47—21no—
22C2×#97×C47C2211yescharacteristic
23C47×#384C2211yescharacteristic
24C22×Dic94—21no—
25C23×Dic47—21no—
26C22×C47⋊C8—22no—
27#13094022—24no—
28#13094022—24no—
29#13094017—28no—
30#13094017—28no—
31#13094016—22no—
32#13094016—22no—
33#13094016C2×C414yes—
34#13094016D414yes—
35#13094005—24no—
36#13094005—24no—
37#13094001—28no—
38#13094001—28no—
39C2×C4⋊Dic47D412yes—
40C2×C4⋊Dic47D412yes—
41C2×Dic47⋊C4—22no—
42C2×Dic47⋊C4—22no—
43C2×Dic47⋊C4—22no—
44C2×C22⋊Dic47—22no—
45C2×C22⋊Dic47—21no—
46C2×C4×Dic47—21no—
47#13093935—22no—
48#13093935—22no—
49C22×Q8×C47D411yescharacteristic
50C22×D4×C47D411yescharacteristic
51C22×C8×C47—21no—
52#124×C47—22no—
53#124×C47C2×C412yes—
54#124×C47D412yes—
55C2×C4⋊C4×C47—22no—
56C2×C4⋊C4×C47—21no—
57C2×C22⋊C4×C47—22no—
58C2×C42×C47C2311yescharacteristic
59#103×C47—42no—
60#102×C47—42no—
61#97×C47—22no—
62C24×C47—21no—
63C2×Q8×C47—21no—
64C2×Q8×C47—41no—
65C2×Q8×C47—41no—
66C2×Q8×C47C22⋊C412yes—
67C2×D4×C47—21no—
68C2×D4×C47—41no—
69C2×D4×C47—41no—
70C2×D4×C47C22⋊C412yes—
71C22×C4×C47—21no—
72C22×C4×C47—41no—
73C22×C4×C47C2×D412yes—
74C22×C4×C47C2×D411yescharacteristic
75C22×Dic47—22no—
76C22×Dic47—22no—
77C22×Dic47—24no—
78C22×Dic47—81no—
79C22×Dic47—81no—
80C2×Dic94—24no—
81C2×Dic94—81no—
82C2×Dic94—81no—
83C2×C8×C47—22no—
84C2×C8×C47—41no—
85C2×C8×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—44no—
88C4⋊C4×C47—41no—
89C22⋊C4×C47—24no—
90C22⋊C4×C47—24no—
91C42×C47C22×C411yescharacteristic
92C42×C47C2×D411yescharacteristic
93C42×C47C2×D412yes—
94C22⋊Dic47—28no—
95C22⋊Dic47—24no—
96C4⋊Dic47—22no—
97C4⋊Dic47—22no—
98C4⋊Dic47C22⋊C414yes—
99C4⋊Dic47C22⋊C414yes—
100Dic47⋊C4—28no—
101Dic47⋊C4—24no—
102Dic47⋊C4—24no—
103Dic47⋊C4—24no—
104Dic47⋊C4—24no—
105C4×Dic47—24no—
106C2×C47⋊C8—24no—
107C2×C47⋊C8—42no—
108C2×C47⋊C8—42no—
109C23×C47—22no—
110C23×C47—41no—
111C23×C47—42no—
112C23×C47—41no—
113C23×C47—42no—
114C23×C47#12011yescharacteristic
115Q8×C47—42no—
116Q8×C47—81no—
117D4×C47—42no—
118D4×C47—81no—
119C2×C4×C47—21no—
120C2×C4×C47—22no—
121C2×C4×C47—21no—
122C2×C4×C47—22no—
123C2×C4×C47—22no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—81no—
127C2×C4×C47—82no—
128C2×C4×C47C2×C22⋊C412yes—
129C2×C4×C47C2×C22⋊C411yescharacteristic
130C2×C4×C47#12011yescharacteristic
131C2×C4×C47#12012yes—
132C2×Dic47—24no—
133C2×Dic47—24no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—84no—
139Dic94—84no—
140C8×C47—42no—
141C47⋊C8—44no—
142#8507—471nomaximal, Sylow
143C22×C47—42no—
144C22×C47—44no—
145C22×C47—81no—
146C22×C47—81no—
147C22×C47—82no—
148C22×C47#37911yescharacteristic
149C22×C47#45311yescharacteristic
150C22×C47#45412yes—
151C22×C47#45511yescharacteristic
152C22×C47#45612yes—
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—42no—
158C4×C47—161no—
159Dic47—44no—
160Dic47—44no—
161C2×#563—471no—
162#1992—472no—
163#1959D4711yescharacteristic
164C2×#330—472no—
165C2×#329—471no—
166C2×C47—82no—
167C2×C47#165711yescharacteristic
168C2×C47#166011yescharacteristic
169C2×C47#166412yes—
170C2×C47#179211yescharacteristic
171C2×C47#179312yes—
172#563—474no—
173#563—944no—
174#563—942no—
175C2×C4⋊Q8—941no—
176C2×#126—942no—
177C2×#125—472no—
178C2×#124C2×D4711yescharacteristic
179C2×#123—941no—
180C2×C22⋊Q8—942no—
181C2×#103—21no—
182C2×#102—21no—
183C2×#97—472no—
184C2×#97C2×D4711yescharacteristic
185#384C2×D4711yescharacteristic
186#340—942no—
187#340—942no—
188#330—944no—
189#329—942no—
190C47#850711yescharacteristic, complemented, Sylow
191C22×Q8—941no—
192C22×Q8C22⋊D4711yescharacteristic
193C22×D4C22⋊D4711yescharacteristic
194C23×C4—941no—
195C22×C8—21no—
196C22×C8—942no—
197C4⋊Q8—944no—
198C4⋊Q8—944no—
199#126—948no—
200#126—948no—
201#125—474no—
202#125—474no—
203#125—942no—
204#125—942no—
205#124—22no—
206#124C4×D4712yes—
207#124D18812yes—
208#123—944no—
209#123—944no—
210C22⋊Q8—948no—
211C22⋊Q8—948no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—472no—
215C2×C4⋊C4—472no—
216C2×C4⋊C4—942no—
217C2×C4⋊C4—942no—
218C2×C4⋊C4—942no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—942no—
221C2×C22⋊C4—941no—
222C2×C42—941no—
223C2×C42C22×D4711yescharacteristic
224#103—42no—
225#102—42no—
226#97—22no—
227#97—942no—
228#97—942no—
229C24—21no—
230C2×Q8—21no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—944no—
234C2×Q8—3761no—
235C2×Q8—3761no—
236C2×Q8(C2×D47)⋊C412yes—
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4(C2×D47)⋊C412yes—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—944no—
244C22×C4—942no—
245C22×C4—942no—
246C22×C4—3761no—
247C22×C4—3761no—
248C22×C4D4×D4712yes—
249C22×C4C2×C22⋊D4711yescharacteristic
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—944no—
254C2×C8—1882no—
255C2×C8—1882no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—44no—
259C4⋊C4—474no—
260C4⋊C4—474no—
261C4⋊C4—948no—
262C4⋊C4—944no—
263C4⋊C4—944no—
264C4⋊C4—944no—
265C4⋊C4—944no—
266C4⋊C4—942no—
267C4⋊C4—942no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—948no—
271C22⋊C4—944no—
272C42—944no—
273C42C2×C4×D4711yescharacteristic
274C42C2×D18811yescharacteristic
275C42D4×D4712yes—
276C23—22no—
277C23—41no—
278C23—42no—
279C23—41no—
280C23—42no—
281C23#1309399211yescharacteristic
282Q8—42no—
283Q8—81no—
284Q8—3764no—
285D4—42no—
286D4—81no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—944no—
297C2×C4—944no—
298C2×C4—3764no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4—1882no—
302C2×C4—1882no—
303C2×C4C22⋊C4×D4712yes—
304C2×C4C2×(C2×D47)⋊C411yescharacteristic
305C2×C4#1309399211yescharacteristic
306C2×C4#1309399412yes—
307C8—42no—
308C8—1884no—
309C22—42no—
310C22—44no—
311C22—81no—
312C22—82no—
313C22—81no—
314C22#1568897811yescharacteristic
315C22#1568929011yescharacteristic
316C22#1568929412yes—
317C22#1568930011yescharacteristic
318C22#1568930412yes—
319C4—21no—
320C4—21no—
321C4—22no—
322C4—22no—
323C4—42no—
324C4—161no—
325C4—1884no—
326C4—1884no—
327C2—82no—
328C2#2024268411yescharacteristic
329C2#2024269711yescharacteristic
330C2#2024272112yes—
331C2#2024334111yescharacteristic
332C2#2024334512yes—
subgroupG/H
centerC23#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1959C2
socleC23×C47#120
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#1959D47
hypocenterC47#8507
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. #20244356C2semidirect
3. C2×#15688864C2semidirect
8. #15689873C4semidirect
190. C47#8507semidirect

Maximal subgroups

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

subgroupindex
C2×#156898732
#202443562
C2×#156888642
C2×#156888632
C47×#19592
#850747