Finite Group Databasebeta

← #26120042

Subgroups — #26120042

Order 12032.

398 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 93 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#20253187C211yesmaximal, characteristic
2#20243399C211yesmaximal, characteristic
3#20242046C211yesmaximal, characteristic, complemented
4#20241953C211yesmaximal, characteristic, complemented
5#20241708C211yesmaximal, characteristic
6#20241248C211yesmaximal, characteristic, complemented
7C47×#1570C211yesmaximal, characteristic, complemented
8#15689680C2211yescharacteristic
9C2×C4×D188C2211yescharacteristic
10#15689424—22no—
11#15689414—21no—
12#15689404—21no—
13#15689176C2211yescharacteristic
14#15689165—21no—
15#15689129—21no—
16C2×#13093946—21no—
17#15688989C2211yescharacteristic
18#15688886—21no—
19#15688871—21no—
20#15688868—21no—
21#15688861—22no—
22#15688846—22no—
23C47×#431C2211yescharacteristic
24C47×#423C2211yescharacteristic
25C2×C4⋊C8×C47C2211yescharacteristic
26C47×#334—22no—
27C22×D188D411yescharacteristic
28C22×C4×D47—21no—
29#13094024D411yescharacteristic
30#13094024D411yescharacteristic
31#13094016—21no—
32#13094003—21no—
33#13094003—21no—
34C4×Dic94—21no—
35C4×Dic94—21no—
36C4×D188—21no—
37C4×D188—21no—
38C4×D188—21no—
39C4×D188—21no—
40#13093983—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47D411yescharacteristic
43C2×(C2×D47)⋊C4—21no—
44#13093958—22no—
45#13093956—22no—
46C4⋊(C47⋊C8)—21no—
47#13093946—22no—
48#13093946—22no—
49#13093946—42no—
50C22⋊(C47⋊C8)—21no—
51C4×C47⋊C8—21no—
52C47⋊#95—21no—
53C47⋊#95—21no—
54C2×M16×C47—21no—
55C22×C8×C47—21no—
56C2×C42×C47C2311yescharacteristic
57C4⋊C8×C47—21no—
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C4⋊C8×C47D412yes—
62C22⋊C8×C47D412yes—
63C4×C8×C47—21no—
64C23×D47—21no—
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47#4111yescharacteristic
68C22×Dic47—21no—
69C22×Dic47—41no—
70C2×D188—21no—
71C2×D188—21no—
72C2×D188—21no—
73C2×D188—21no—
74C2×D188—21no—
75C2×D188SD1612yes—
76C2×C4×D47—41no—
77C2×C4×D47—41no—
78C2×C4×D47—82no—
79C2×Dic94—21no—
80C2×Dic94—21no—
81M16×C47—41no—
82M16×C47—41no—
83C2×C8×C47—22no—
84C2×C8×C47—22no—
85C2×C8×C47—41no—
86C2×C8×C47—41no—
87C2×C8×C47—41no—
88C42×C47C2×D411yescharacteristic
89C42×C47C2×D411yescharacteristic
90C42×C47#4111yescharacteristic
91C42×C47#4111yescharacteristic
92C22⋊Dic47—41no—
93(C2×D47)⋊C4—41no—
94(C2×D47)⋊C4—41no—
95C4⋊Dic47—21no—
96C4⋊Dic47—21no—
97C4⋊Dic47—21no—
98C4⋊Dic47—21no—
99C4⋊Dic47—21no—
100C4⋊Dic47—21no—
101C4⋊Dic47—41no—
102C4⋊Dic47—41no—
103Dic47⋊C4—41no—
104Dic47⋊C4—41no—
105C4×Dic47—41no—
106C2×C47⋊C8—41no—
107C23×C47#12111yescharacteristic
108C22×D47—22no—
109C22×D47—41no—
110C22×D47—41no—
111C22×D47—81no—
112C22×D47—81no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47#12011yescharacteristic
116C2×C4×C47#12111yescharacteristic
117C2×C4×C47#12111yescharacteristic
118C2×C4×C47#12311yescharacteristic
119C2×C4×C47#12311yescharacteristic
120C2×C4×C47#12411yescharacteristic
121C2×C4×C47C2×SD1611yescharacteristic
122C2×C4×C47#13511yescharacteristic
123C2×C4×C47#13711yescharacteristic
124C2×C4×C47#13711yescharacteristic
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—81no—
129C2×Dic47—81no—
130C2×Dic47—81no—
131C2×Dic47—81no—
132D188—22no—
133D188—22no—
134D188—41no—
135D188—41no—
136C4×D47—82no—
137C4×D47—82no—
138Dic94—81no—
139Dic94—81no—
140C8×C47—42no—
141C8×C47—42no—
142C8×C47—81no—
143C47⋊C8—81no—
144#10468—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—21no—
147C22×C47#39411yescharacteristic
148C22×C47#46511yescharacteristic
149C22×C47#47011yescharacteristic
150C2×D47—42no—
151C2×D47—82no—
152C2×D47—81no—
153C2×D47—81no—
154C2×D47—82no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47#44811yescharacteristic
160C4×C47#45211yescharacteristic
161C4×C47#48311yescharacteristic
162C4×C47#48511yescharacteristic
163Dic47—81no—
164Dic47—81no—
165Dic47—81no—
166Dic47—161no—
167#3077—471no—
168#1804—471no—
169#1570D4711yescharacteristic, complemented
170#1542—471no—
171#1518—471no—
172#1460—471no—
173#1286—471no—
174C2×C47—21no—
175C2×C47—21no—
176C2×C47#203511yescharacteristic
177C2×C47#204811yescharacteristic
178C2×C47#205311yescharacteristic
179D47—82no—
180#533—471no—
181C2×C4×D4—471no—
182#478—942no—
183#476—941no—
184#474—941no—
185#431—471no—
186#431C2×D4711yescharacteristic
187#428—941no—
188#423C2×D4711yescharacteristic
189C2×C4⋊C8C2×D4711yescharacteristic
190#418—941no—
191C2×#102—941no—
192#382—471no—
193#340—941no—
194#334—22no—
195#333—941no—
196#332—941no—
197#328—942no—
198#323—942no—
199C47#1046811yescharacteristic, complemented, Sylow
200C22×D4—471no—
201C23×C4—941no—
202C2×M16—21no—
203C22×C8—21no—
204C4⋊Q8—471no—
205C4⋊Q8—471no—
206#125—941no—
207C22⋊Q8—941no—
208C22⋊Q8—941no—
209C4×Q8—941no—
210C4×Q8—941no—
211C4×D4—941no—
212C4×D4—941no—
213C4×D4—941no—
214C4×D4—941no—
215#117—941no—
216C2×C4⋊C4—471no—
217C2×C4⋊C4—941no—
218C2×C22⋊C4—941no—
219C2×C42C22×D4711yescharacteristic
220#107—942no—
221#106—942no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—941no—
227C4⋊C8C22⋊D4712yes—
228#102—942no—
229#102—942no—
230#102—1882no—
231C22⋊C8—941no—
232C22⋊C8D18812yes—
233C4×C8—21no—
234C4×C8—941no—
235#95—941no—
236#95—941no—
237C24—941no—
238C2×Q8—941no—
239C2×Q8—941no—
240C2×D4—472no—
241C2×D4—941no—
242C2×D4—941no—
243C2×D4—941no—
244C2×D4—941no—
245C2×D4—941no—
246C22×C4—941no—
247C22×C4—3762no—
248C22×C4—1881no—
249C22×C4—1881no—
250C22×C4—1881no—
251C22×C4C2×D18811yescharacteristic
252C22×C4D4×D4711yescharacteristic
253C22×C4#1062535411yescharacteristic
254M16—41no—
255M16—41no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—1881no—
262C4⋊C4—941no—
263C4⋊C4—941no—
264C4⋊C4—941no—
265C4⋊C4—941no—
266C4⋊C4—941no—
267C4⋊C4—941no—
268C4⋊C4—1881no—
269C4⋊C4—1881no—
270C4⋊C4—1881no—
271C4⋊C4—1881no—
272C22⋊C4—1881no—
273C22⋊C4—1881no—
274C22⋊C4—1881no—
275C42—1881no—
276C42#1062535011yescharacteristic
277C42D4×D4711yescharacteristic
278C42#1062535411yescharacteristic
279C42C2×C22⋊D4711yescharacteristic
280C23—942no—
281C23—3761no—
282C23—3761no—
283C23—1881no—
284C23—1881no—
285C23#1309399911yescharacteristic
286Q8—3761no—
287Q8—3761no—
288D4—942no—
289D4—942no—
290D4—1881no—
291D4—1881no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—3762no—
295C2×C4—3762no—
296C2×C4—3761no—
297C2×C4—3761no—
298C2×C4—3761no—
299C2×C4—3761no—
300C2×C4—1881no—
301C2×C4—1881no—
302C2×C4—1881no—
303C2×C4#1309399211yescharacteristic
304C2×C4#1309399811yescharacteristic
305C2×C4#1309399911yescharacteristic
306C2×C4#1309400911yescharacteristic
307C2×C4#1309400911yescharacteristic
308C2×C4#1309401311yescharacteristic
309C2×C4SD16×D4711yescharacteristic
310C2×C4#1309404511yescharacteristic
311C2×C4#1309405311yescharacteristic
312C2×C4#1309405311yescharacteristic
313C8—42no—
314C8—42no—
315C8—81no—
316C8—3761no—
317C22—21no—
318C22—21no—
319C22—3762no—
320C22—3762no—
321C22—3761no—
322C22—3761no—
323C22—1882no—
324C22#1568904011yescharacteristic
325C22#1568935311yescharacteristic
326C22#1568938211yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—3761no—
332C4—3761no—
333C4—3761no—
334C4—7521no—
335C4#1568926011yescharacteristic
336C4#1568928211yescharacteristic
337C4#1568945511yescharacteristic
338C4#1568946711yescharacteristic
339C2—21no—
340C2—21no—
341C2—3762no—
342C2#2024458611yescharacteristic
343C2#2024466311yescharacteristic
344C2#2024469411yescharacteristic
subgroupG/H
centerC22#15689040
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1570C2
socleC22×C47#394
prime radicalC22×D188D4
hypercenter#1570D47
hypocenterC47#10468
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
3. #20242046C2semidirect
4. #20241953C2semidirect
6. #20241248C2semidirect
7. C47×#1570C2semidirect
169. #1570D47semidirect
199. C47#10468semidirect

Maximal subgroups

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

subgroupindex
#202531872
#202433992
#202420462
#202419532
#202417082
#202412482
C47×#15702
#1046847