Finite Group Databasebeta

← #26111729

Subgroups — #26111729

Order 12032.

536 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252429C211yesmaximal, characteristic, complemented
2C2×#15689233C211yesmaximal, characteristic
3#20241778C211yesmaximal, characteristic, complemented
4#20241708C211yesmaximal, characteristic, complemented
5#20241232C211yesmaximal, characteristic, complemented
6#20241218C211yesmaximal, characteristic, complemented
7C47×#1478C211yesmaximal, characteristic, complemented
8C22×#10625334C411yescharacteristic
9C2×C4×D188C2211yescharacteristic
10#15689233—22no—
11#15689233—22no—
12#15689201—22no—
13#15689201—22no—
14#15689176C2211yescharacteristic
15#15689176C2211yescharacteristic
16C2×#13093955C411yescharacteristic
17C2×#13093945C411yescharacteristic
18C2×#13093944C411yescharacteristic
19#15689087—22no—
20C2×#13093935C2211yescharacteristic
21#15688850—22no—
22#15688846—22no—
23C2×C4×D4×C47C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C47×#431C2211yescharacteristic
26C47×D4⋊C8—22no—
27C22×D188C2×C411yescharacteristic
28C22×C4×D47—21no—
29C2×#10625334—22no—
30C2×#10625334—22no—
31C2×#10625334—22no—
32C2×#10625329—24no—
33C22×C47⋊C8C2×C411yescharacteristic
34C22×C47⋊C8C2×C411yescharacteristic
35C4×D188—21no—
36C4×D188—21no—
37C4×D188—21no—
38C4×D188—21no—
39C2×C4⋊Dic47C2×C411yescharacteristic
40C2×(C2×D47)⋊C4—21no—
41#13093955—22no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44#13093945—22no—
45#13093944—22no—
46C22⋊(C47⋊C8)—24no—
47C22⋊(C47⋊C8)—21no—
48C22⋊(C47⋊C8)D412yes—
49#13093935—21no—
50#13093935—21no—
51C4×C47⋊C8—21no—
52C4×C47⋊C8D412yes—
53C22×D4×C47C2×C411yescharacteristic
54C23×C4×C47—21no—
55C4×D4×C47—21no—
56C4×D4×C47—21no—
57C4×D4×C47D412yes—
58C4×D4×C47D412yes—
59C2×C4⋊C4×C47C2×C411yescharacteristic
60C2×C22⋊C4×C47—21no—
61C2×C42×C47C2311yescharacteristic
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C22⋊C8×C47—21no—
65C22⋊C8×C47—21no—
66C4×C8×C47—21no—
67C4×C8×C47—21no—
68C24×C47—21no—
69C23×D47—21no—
70C2×D4×C47—21no—
71C2×D4×C47—21no—
72C2×D4×C47—21no—
73C2×D4×C47—21no—
74C2×D4×C47—21no—
75C2×D4×C47C22⋊C412yes—
76C22×C4×C47—21no—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47—82no—
80C22×C4×C47C22×C411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47#4111yescharacteristic
83C22×Dic47—21no—
84C2×D188—21no—
85C2×D188—21no—
86C2×D188—21no—
87C2×D188—21no—
88C2×D188—21no—
89C2×D188M1612yescomplemented
90C2×C4×D47—41no—
91C2×C4×D47—41no—
92C2×C4×D47—82no—
93C2×C8×C47—41no—
94C2×C8×C47—41no—
95C4⋊C4×C47—21no—
96C4⋊C4×C47C22⋊C412yes—
97C22⋊C4×C47—22no—
98C22⋊C4×C47—22no—
99C42×C47C2×D411yescharacteristic
100C42×C47C2×D411yescharacteristic
101C42×C47C2×D411yescharacteristic
102C42×C47#4111yescharacteristic
103#10625334—28no—
104(C2×D47)⋊C4—41no—
105(C2×D47)⋊C4—41no—
106C4⋊Dic47—21no—
107C4⋊Dic47—21no—
108#10625329—84no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8—21no—
111C2×C47⋊C8—21no—
112C2×C47⋊C8—22no—
113C2×C47⋊C8—22no—
114C2×C47⋊C8—21no—
115C2×C47⋊C8—21no—
116C2×C47⋊C8M1612yescomplemented
117C23×C47—22no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47—81no—
121C23×C47—81no—
122C23×C47C4×D411yescharacteristic
123C22×D47—22no—
124C22×D47—41no—
125C22×D47—41no—
126C22×D47—81no—
127C22×D47—81no—
128D4×C47—22no—
129D4×C47—22no—
130D4×C47—41no—
131D4×C47—41no—
132C2×C4×C47—22no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—44no—
136C2×C4×C47—82no—
137C2×C4×C47—81no—
138C2×C4×C47C2×C22⋊C411yescharacteristic
139C2×C4×C47C4×D411yescharacteristic
140C2×C4×C47#12011yescharacteristic
141C2×C4×C47#12111yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×C4×C47#12311yescharacteristic
144C2×C4×C47C2×M1611yescharacteristic
145C2×C4×C47C2×M1611yescharacteristic
146C2×C4×C47#13111yescharacteristic
147C2×C4×C47#13111yescharacteristic
148C2×Dic47—41no—
149C2×Dic47—81no—
150D188—22no—
151D188—22no—
152D188—41no—
153D188—41no—
154C4×D47—82no—
155C4×D47—82no—
156C8×C47—81no—
157C8×C47—81no—
158C47⋊C8—22no—
159C47⋊C8—42no—
160C47⋊C8—42no—
161C47⋊C8—41no—
162#9093—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—42no—
166C22×C47—42no—
167C22×C47—42no—
168C22×C47—82no—
169C22×C47—82no—
170C22×C47#38611yescharacteristic
171C22×C47#43511yescharacteristic
172C22×C47#43611yescharacteristic
173C2×D47—42no—
174C2×D47—82no—
175C2×D47—81no—
176C2×D47—81no—
177C2×D47—82no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—42no—
183C4×C47#40711yescharacteristic
184C4×C47#40811yescharacteristic
185C4×C47#43511yescharacteristic
186C4×C47#43611yescharacteristic
187Dic47—81no—
188#2975—471no—
189C2×#442—471no—
190#1478—471no—
191#1478D4711yescharacteristic, complemented
192#1460—471no—
193#1281—471no—
194#1275—471no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47—82no—
198C2×C47#187011yescharacteristic
199C2×C47#190311yescharacteristic
200C2×C47#190411yescharacteristic
201D47—82no—
202C22×D8—471no—
203C2×C4×D4—471no—
204C2×C4×D4C2×D4711yescharacteristic
205#442—942no—
206#442—942no—
207#436—942no—
208#436—942no—
209#431—471no—
210#431—471no—
211#431C2×D4711yescharacteristic
212#431C2×D4711yescharacteristic
213C2×#106—471no—
214C2×#102—471no—
215C2×#102—471no—
216#407—942no—
217C2×#97—471no—
218D4⋊C8—22no—
219D4⋊C8—942no—
220#323—942no—
221C47#909311yescharacteristic, complemented, Sylow
222C22×D4—471no—
223C22×D4C4×D4711yescharacteristic
224C23×C4—21no—
225C23×C4—941no—
226C2×D8—942no—
227C2×D8—942no—
228C2×D8—942no—
229C2×M16—944no—
230C22×C8—471no—
231C22×C8—471no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—941no—
235C4×D4—941no—
236C4×D4—941no—
237C4×D4—941no—
238C4×D4D18812yes—
239C4×D4C22⋊D4712yes—
240C2×C4⋊C4—471no—
241C2×C4⋊C4C4×D4711yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—941no—
244C2×C42C22×D4711yescharacteristic
245#106—942no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—941no—
249C4⋊C8—941no—
250#102—942no—
251#102—942no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—472no—
255C22⋊C8—944no—
256C22⋊C8—941no—
257#97—941no—
258#97—941no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—472no—
262C4×C8—941no—
263C24—21no—
264C24—941no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—472no—
271C2×D4—941no—
272C2×D4—941no—
273C2×D4—941no—
274C2×D4—941no—
275C2×D4—941no—
276C2×D4(C2×D47)⋊C412yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—82no—
281C22×C4—941no—
282C22×C4—3762no—
283C22×C4—1881no—
284C22×C4—1881no—
285C22×C4C2×C4×D4711yescharacteristic
286C22×C4D4×D4711yescharacteristic
287C22×C4#1062535211yescharacteristic
288D8—948no—
289M16—3764no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—472no—
293C2×C8—941no—
294C2×C8—941no—
295C2×C8—941no—
296C2×C8—942no—
297C2×C8—942no—
298C2×C8—941no—
299C2×C8—941no—
300C4⋊C4—21no—
301C4⋊C4—941no—
302C4⋊C4—941no—
303C4⋊C4(C2×D47)⋊C412yes—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—1881no—
307C22⋊C4—1881no—
308C42C2×D18811yescharacteristic
309C42D4×D4711yescharacteristic
310C42#1062535211yescharacteristic
311C42C2×C22⋊D4711yescharacteristic
312C23—22no—
313C23—41no—
314C23—41no—
315C23—81no—
316C23—81no—
317C23—942no—
318C23—3761no—
319C23—3761no—
320C23—1881no—
321C23—1881no—
322C23(C22⋊D47)⋊C411yescharacteristic
323D4—22no—
324D4—22no—
325D4—41no—
326D4—41no—
327D4—942no—
328D4—942no—
329D4—1881no—
330D4—1881no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—3762no—
338C2×C4—3762no—
339C2×C4—3761no—
340C2×C4—1881no—
341C2×C4C2×(C2×D47)⋊C411yescharacteristic
342C2×C4(C22⋊D47)⋊C411yescharacteristic
343C2×C4#1309399411yescharacteristic
344C2×C4#1309399611yescharacteristic
345C2×C4#1309399611yescharacteristic
346C2×C4#1309400811yescharacteristic
347C2×C4M16×D4711yescharacteristic
348C2×C4M16×D4711yescharacteristic
349C2×C4#1309403011yescharacteristic
350C2×C4#1309403011yescharacteristic
351C8—81no—
352C8—81no—
353C8—942no—
354C8—1882no—
355C8—1882no—
356C8—1881no—
357C22—21no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—3762no—
365C22—3762no—
366C22—3761no—
367C22—3761no—
368C22—1882no—
369C22#1568900011yescharacteristic
370C22#1568919711yescharacteristic
371C22#1568919811yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—3761no—
378C4#1568908611yescharacteristic
379C4#1568908911yescharacteristic
380C4#1568919711yescharacteristic
381C4#1568919811yescharacteristic
382C2—21no—
383C2—21no—
384C2—82no—
385C2—3762no—
386C2#2024368311yescharacteristic
387C2#2024386611yescharacteristic
388C2#2024387311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1478C2
socleC22×C47#386
prime radicalC22×#10625334C4
cosocleC2×C42×C47C23
hypercenter#1478D47
hypocenterC47#9093
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
1. #20252429C2semidirect
3. #20241778C2semidirect
4. #20241708C2semidirect
5. #20241232C2semidirect
6. #20241218C2semidirect
7. C47×#1478C2semidirect
89. C2×D188M16semidirect
116. C2×C47⋊C8M16semidirect
191. #1478D47semidirect
221. C47#9093semidirect

Maximal subgroups

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

subgroupindex
#202524292
C2×#156892332
#202417782
#202417082
#202412322
#202412182
C47×#14782
#909347