Finite Group Databasebeta

← #26120047

Subgroups — #26120047

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#20253185C211yesmaximal, characteristic
2#20242162C211yesmaximal, characteristic, complemented
3#20242046C211yesmaximal, characteristic, complemented
4#20241951C211yesmaximal, characteristic, complemented
5#20241709C211yesmaximal, characteristic
6#20241246C211yesmaximal, characteristic, complemented
7C47×#1804C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689423—22no—
10#15689413—21no—
11#15689403—21no—
12#15689176C2211yescharacteristic
13#15689176C2211yescharacteristic
14#15689163—21no—
15#15689147C2211yescharacteristic
16#15689128—21no—
17#15688875—22no—
18#15688871—21no—
19#15688868—21no—
20#15688859—22no—
21#15688847—22no—
22C2×C4×D4×C47C2211yescharacteristic
23C2×C4⋊C8×C47C2211yescharacteristic
24C2×#102×C47—21no—
25C47×#382C2211yescharacteristic
26C47×#340—21no—
27C2×#10625329—21no—
28#13094024D411yescharacteristic
29#13094024D411yescharacteristic
30#13094016—21no—
31#13094003—21no—
32#13094003—21no—
33C4×Dic94—21no—
34C4×Dic94—21no—
35#13093983—21no—
36#13093957—22no—
37#13093955—22no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)D412yes—
42#13093945—22no—
43#13093945—22no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)D412yes—
46C4×C47⋊C8—21no—
47C4×C47⋊C8—21no—
48C22×D4×C47D411yescharacteristic
49C23×C4×C47—21no—
50C22×C8×C47—21no—
51C4×D4×C47—21no—
52C4×D4×C47—21no—
53C4×D4×C47—21no—
54C4×D4×C47—21no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47D411yescharacteristic
57C2×C22⋊C4×C47—21no—
58C2×C42×C47C2311yescharacteristic
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61#102×C47—42no—
62#95×C47—21no—
63#95×C47—21no—
64C24×C47—21no—
65C2×D4×C47—21no—
66C2×D4×C47—21no—
67C2×D4×C47—21no—
68C2×D4×C47—21no—
69C2×D4×C47—21no—
70C2×D4×C47Q1612yes—
71C22×C4×C47—21no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47—82no—
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47#4111yescharacteristic
79C2×Dic94—21no—
80C2×Dic94—21no—
81C2×C8×C47—41no—
82C2×C8×C47—41no—
83C2×C8×C47—41no—
84C4⋊C4×C47—21no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C22⋊C4×C47—41no—
89C22⋊C4×C47—41no—
90C42×C47C2×D411yescharacteristic
91C42×C47C2×D411yescharacteristic
92C42×C47#4111yescharacteristic
93C42×C47#4111yescharacteristic
94C22⋊Dic47—41no—
95C4⋊Dic47—21no—
96C4⋊Dic47—21no—
97C4⋊Dic47—21no—
98C4⋊Dic47—21no—
99Dic47⋊C4—41no—
100Dic47⋊C4—41no—
101C4×Dic47—41no—
102#10625329—41no—
103#10625329—41no—
104C2×C47⋊C8—22no—
105C2×C47⋊C8—22no—
106C2×C47⋊C8—41no—
107C23×C47—22no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47—81no—
111C23×C47—81no—
112C23×C47#12111yescharacteristic
113D4×C47—22no—
114D4×C47—22no—
115D4×C47—41no—
116D4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—82no—
121C2×C4×C47—82no—
122C2×C4×C47—81no—
123C2×C4×C47—81no—
124C2×C4×C47—81no—
125C2×C4×C47—81no—
126C2×C4×C47#12011yescharacteristic
127C2×C4×C47#12111yescharacteristic
128C2×C4×C47#12111yescharacteristic
129C2×C4×C47#12311yescharacteristic
130C2×C4×C47#12311yescharacteristic
131C2×C4×C47#12411yescharacteristic
132C2×C4×C47C2×Q1611yescharacteristic
133C2×C4×C47#13511yescharacteristic
134C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C47#13711yescharacteristic
136C2×Dic47—41no—
137C2×Dic47—41no—
138Dic94—81no—
139Dic94—81no—
140C8×C47—81no—
141C47⋊C8—42no—
142C47⋊C8—42no—
143C47⋊C8—81no—
144#10469—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—21no—
147C22×C47—42no—
148C22×C47—82no—
149C22×C47—82no—
150C22×C47—81no—
151C22×C47—81no—
152C22×C47#39411yescharacteristic
153C22×C47#46711yescharacteristic
154C22×C47#46811yescharacteristic
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—81no—
160C4×C47—161no—
161C4×C47#44911yescharacteristic
162C4×C47#45111yescharacteristic
163C4×C47#48411yescharacteristic
164C4×C47#48511yescharacteristic
165Dic47—81no—
166Dic47—81no—
167#3077—471no—
168#1804D4711yescharacteristic
169#1571—471no—
170#1542—471no—
171#1518—471no—
172#1460—471no—
173#1286—471no—
174C2×C47—21no—
175C2×C47—21no—
176C2×C47—82no—
177C2×C47#203611yescharacteristic
178C2×C47#204811yescharacteristic
179C2×C47#205311yescharacteristic
180#533—471no—
181C2×C4×D4C2×D4711yescharacteristic
182#478—942no—
183#476—941no—
184#474—941no—
185#431—471no—
186#431—471no—
187#428—941no—
188#423—471no—
189C2×C4⋊C8C2×D4711yescharacteristic
190#418—941no—
191C2×#102—21no—
192#382C2×D4711yescharacteristic
193#340—21no—
194#335—942no—
195#333—941no—
196#332—941no—
197#328—942no—
198#323—942no—
199C47#1046911yescharacteristic, complemented, Sylow
200C22×D4C22⋊D4711yescharacteristic
201C23×C4—21no—
202C2×M16—941no—
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—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215#117—941no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C22⋊D4711yescharacteristic
218C2×C22⋊C4—21no—
219C2×C42C22×D4711yescharacteristic
220#107—942no—
221#106—942no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—472no—
225C4⋊C8—941no—
226C4⋊C8—941no—
227C4⋊C8—941no—
228#102—42no—
229#102—942no—
230#102—942no—
231C22⋊C8—472no—
232C22⋊C8—941no—
233C4×C8—941no—
234C4×C8—941no—
235#95—21no—
236#95—21no—
237C24—21no—
238C2×Q8—941no—
239C2×Q8—941no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4#1062533712yes—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—82no—
251C22×C4D4×D4711yescharacteristic
252C22×C4#1062535211yescharacteristic
253C22×C4C2×C22⋊D4711yescharacteristic
254M16—1881no—
255M16—1881no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—942no—
260C2×C8—942no—
261C2×C8—1881no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—941no—
267C4⋊C4—941no—
268C4⋊C4—941no—
269C4⋊C4—941no—
270C4⋊C4—1881no—
271C4⋊C4—1881no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—1881no—
275C42—1881no—
276C42#1062535011yescharacteristic
277C42#1062535011yescharacteristic
278C42D4×D4711yescharacteristic
279C42D4×D4711yescharacteristic
280C23—22no—
281C23—41no—
282C23—41no—
283C23—81no—
284C23—81no—
285C23#1309399611yescharacteristic
286Q8—3761no—
287Q8—3761no—
288D4—22no—
289D4—22no—
290D4—41no—
291D4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—1881no—
302C2×C4—1881no—
303C2×C4#1309399211yescharacteristic
304C2×C4#1309399711yescharacteristic
305C2×C4#1309399711yescharacteristic
306C2×C4#1309400611yescharacteristic
307C2×C4#1309400711yescharacteristic
308C2×C4#1309401011yescharacteristic
309C2×C4C2×#1062533711yescharacteristic
310C2×C4#1309404411yescharacteristic
311C2×C4#1309405011yescharacteristic
312C2×C4#1309405411yescharacteristic
313C8—81no—
314C8—3761no—
315C8—1882no—
316C8—1882no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—82no—
321C22—82no—
322C22—81no—
323C22—81no—
324C22#1568903811yescharacteristic
325C22#1568936211yescharacteristic
326C22#1568936711yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—81no—
332C4—161no—
333C4—3761no—
334C4—3761no—
335C4#1568926611yescharacteristic
336C4#1568927811yescharacteristic
337C4#1568945811yescharacteristic
338C4#1568946411yescharacteristic
339C2—21no—
340C2—21no—
341C2—82no—
342C2#2024459111yescharacteristic
343C2#2024465911yescharacteristic
344C2#2024469211yescharacteristic
subgroupG/H
centerC22#15689038
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1804C2
socleC22×C47#394
prime radicalC22×D4×C47D4
hypercenter#1804D47
hypocenterC47#10469
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. #20242162C2semidirect
3. #20242046C2semidirect
4. #20241951C2semidirect
6. #20241246C2semidirect
199. C47#10469semidirect

Maximal subgroups

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

subgroupindex
#202531852
#202421622
#202420462
#202419512
#202417092
#202412462
C47×#18042
#1046947