Finite Group Databasebeta

← #26133687

Subgroups — #26133687

Order 12032.

470 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256491C211yesmaximal, characteristic, complemented
2C2×#15688936C211yesmaximal, characteristic, complemented
3#20245104C211yesmaximal, characteristic, complemented
4#20243442C211yesmaximal, characteristic, complemented
5#20241502C211yesmaximal, characteristic, complemented
6#20241491C211yesmaximal, characteristic, complemented
7C2×C47×#472C211yesmaximal, characteristic
8#15689540—22no—
9#15689540—22no—
10C2×#13093964—22no—
11C2×#13093964C2211yescharacteristic
12C2×#13093964C2211yescharacteristic
13C2×#13093960C2211yescharacteristic
14#15689161C2211yescharacteristic
15#15689140—21no—
16#15688936—22no—
17#15688924—22no—
18#15688895—21no—
19C2×AGL(1,ℤ/8)×C47—21no—
20C22×D8×C47C2211yescharacteristic
21C2×#131×C47C2211yescharacteristic
22C47×#472—22no—
23C47×#472C414yescomplemented
24C2×#108×C47C2211yescharacteristic
25C2×#100×C47—21no—
26C2×#10625329—21no—
27#13093983—21no—
28#13093964—24no—
29#13093964—21no—
30#13093964—21no—
31#13093964—21no—
32#13093964D412yes—
33C2×C47⋊C16—24no—
34C2×C47⋊C16—21no—
35C2×C47⋊C16D412yes—
36#13093960—21no—
37#13093960—21no—
38#13093958—21no—
39#13093956—21no—
40#13093951—42no—
41C2×#41×C47D411yescharacteristic
42C22×D4×C47—21no—
43AGL(1,ℤ/8)×C47—24no—
44AGL(1,ℤ/8)×C47—42no—
45C2×SD16×C47—41no—
46C2×D8×C47—21no—
47C2×D8×C47—21no—
48C2×D8×C47—21no—
49C2×D8×C47—21no—
50C2×D8×C47—21no—
51C2×D8×C47—41no—
52C2×D8×C47C2×C412yes—
53#131×C47—21no—
54#131×C47—21no—
55#131×C47C2×C412yes—
56#131×C47D412yes—
57C2×M16×C47—21no—
58C2×M16×C47—21no—
59C2×M16×C47D411yescharacteristic
60C22×C8×C47—21no—
61C22×C8×C47C2311yescharacteristic
62#108×C47—21no—
63#108×C47C2×C412yes—
64#100×C47—24no—
65C24×C47—41no—
66#41×C47—21no—
67#41×C47—21no—
68#41×C47C22⋊C412yes—
69#41×C47D812yes—
70C2×Q8×C47SD1611yescharacteristic
71C2×D4×C47—21no—
72C2×D4×C47—22no—
73C2×D4×C47—41no—
74C2×D4×C47—41no—
75C2×D4×C47—41no—
76C2×D4×C47—41no—
77C2×D4×C47—41no—
78C2×D4×C47SD1611yescharacteristic
79C22×C4×C47—21no—
80C22×C4×C47C2×D411yescharacteristic
81SD16×C47—44no—
82D8×C47—22no—
83D8×C47—22no—
84D8×C47—44no—
85D8×C47—41no—
86D8×C47—41no—
87M16×C47—21no—
88M16×C47—22no—
89M16×C47—22no—
90M16×C47—22no—
91M16×C47—41no—
92M16×C47C22⋊C412yes—
93C2×C8×C47—21no—
94C2×C8×C47—22no—
95C2×C8×C47—22no—
96C2×C8×C47—41no—
97C2×C8×C47C22×C411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C2×C8×C47#4111yescharacteristic
102C2×C8×C47#4111yescharacteristic
103C22⋊Dic47—41no—
104C4⋊Dic47—41no—
105C4×Dic47—41no—
106#10625329—41no—
107#10625329—41no—
108C2×C47⋊C8—41no—
109C47⋊C16—22no—
110C47⋊C16—22no—
111C23×C47—21no—
112C23×C47—42no—
113C23×C47—81no—
114C23×C47—81no—
115C23×C47—162no—
116C23×C47#13511yescharacteristic
117Q8×C47—21no—
118Q8×C47#10212yes—
119D4×C47—22no—
120D4×C47—22no—
121D4×C47—21no—
122D4×C47—42no—
123D4×C47—42no—
124D4×C47—81no—
125D4×C47—81no—
126D4×C47#10212yes—
127C2×C4×C47—21no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47C2×C22⋊C411yescharacteristic
131C2×C4×C47#12111yescharacteristic
132C2×C4×C47#12111yescharacteristic
133C2×C4×C47C2×D811yescharacteristic
134C2×C4×C47C2×SD1611yescharacteristic
135C2×C4×C47#13511yescharacteristic
136C2×Dic47—81no—
137C8×C47—22no—
138C8×C47—42no—
139C8×C47C4×D411yescharacteristic
140C8×C47C4×D411yescharacteristic
141C8×C47#12011yescharacteristic
142C8×C47#12311yescharacteristic
143C47⋊C8—81no—
144#13068—471nomaximal, Sylow
145C22×C47—22no—
146C22×C47—41no—
147C22×C47—81no—
148C22×C47—82no—
149C22×C47—162no—
150C22×C47—162no—
151C22×C47—162no—
152C22×C47#41611yescharacteristic
153C22×C47#46511yescharacteristic
154C22×C47#46611yescharacteristic
155C4×C47—22no—
156C4×C47#38611yescharacteristic
157C4×C47C2×#10211yescharacteristic
158C4×C47#46511yescharacteristic
159C4×C47#46611yescharacteristic
160Dic47—161no—
161#3407—471no—
162C2×#472D4711yescharacteristic
163C2×#361—471no—
164#2140—471no—
165#1814—471no—
166#1389—471no—
167#1385—471no—
168C2×C47—21no—
169C2×C47—21no—
170C2×C47—21no—
171C2×C47—42no—
172C2×C47—162no—
173C2×C47#189311yescharacteristic
174C2×AGL(1,ℤ/8)—21no—
175C22×D8C2×D4711yescharacteristic
176C2×#131C2×D4711yescharacteristic
177#504—942no—
178#504—942no—
179C2×M32—471no—
180C2×M32—471no—
181C2×M32—942no—
182#472—22no—
183#472Dic4714yescomplemented
184C2×#108—471no—
185C2×#108C2×D4711yescharacteristic
186#427—471no—
187#421—941no—
188C2×#100—21no—
189#361—942no—
190#357—942no—
191#344—941no—
192C47#1306811yescharacteristic, complemented, Sylow
193C2×#41C22⋊D4711yescharacteristic
194C22×D4—21no—
195AGL(1,ℤ/8)—24no—
196AGL(1,ℤ/8)—42no—
197C2×SD16—41no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8—21no—
201C2×D8—21no—
202C2×D8—21no—
203C2×D8—41no—
204C2×D8C2×Dic4712yes—
205#131—21no—
206#131—21no—
207#131C2×Dic4712yes—
208#131C22⋊D4712yes—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—941no—
212C2×M16C22⋊D4711yescharacteristic
213C22×C8—21no—
214C22×C8C22×D4711yescharacteristic
215#117—941no—
216M32—472no—
217M32—944no—
218M32—941no—
219M32—941no—
220M32—941no—
221C2×C16—472no—
222C2×C16—944no—
223C2×C16—941no—
224#108—21no—
225#108—941no—
226#108—941no—
227#108C2×Dic4712yes—
228#107—941no—
229#106—941no—
230#104—1882no—
231#100—24no—
232C24—41no—
233#41—21no—
234#41—21no—
235#41#1062533412yes—
236#41C22⋊Dic4712yes—
237C2×Q8#1062533511yescharacteristic
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4#1062533511yescharacteristic
246C22×C4—21no—
247C22×C4C2×C22⋊D4711yescharacteristic
248SD16—44no—
249D8—22no—
250D8—22no—
251D8—44no—
252D8—41no—
253D8—41no—
254M16—21no—
255M16—22no—
256M16—22no—
257M16—22no—
258M16—41no—
259M16—1881no—
260M16—1881no—
261M16C22⋊Dic4712yes—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—1881no—
267C2×C8D4×D4711yescharacteristic
268C2×C8D4×D4711yescharacteristic
269C2×C8#1062535211yescharacteristic
270C2×C8#1062535211yescharacteristic
271C2×C8C22×Dic4711yescharacteristic
272C2×C8C2×C22⋊D4711yescharacteristic
273C4⋊C4—1881no—
274C22⋊C4—1881no—
275C42—1881no—
276C16—942no—
277C16—942no—
278C23—21no—
279C23—42no—
280C23—81no—
281C23—81no—
282C23—162no—
283C23#1309404411yescharacteristic
284Q8—21no—
285Q8#1309394512yes—
286D4—22no—
287D4—22no—
288D4—21no—
289D4—42no—
290D4—42no—
291D4—81no—
292D4—81no—
293D4#1309394512yes—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—3761no—
298C2×C4C2×C22⋊Dic4711yescharacteristic
299C2×C4#1309399611yescharacteristic
300C2×C4#1309399611yescharacteristic
301C2×C4C2×#1062533411yescharacteristic
302C2×C4C2×#1062533511yescharacteristic
303C2×C4#1309404411yescharacteristic
304C8—22no—
305C8—42no—
306C8—3761no—
307C8D4×Dic4711yescharacteristic
308C8D4×Dic4711yescharacteristic
309C8#1309399211yescharacteristic
310C8#1309400511yescharacteristic
311C22—22no—
312C22—41no—
313C22—81no—
314C22—82no—
315C22—162no—
316C22—162no—
317C22—162no—
318C22#1568911911yescharacteristic
319C22#1568935011yescharacteristic
320C22#1568935711yescharacteristic
321C4—22no—
322C4—7521no—
323C4#1568900311yescharacteristic
324C4C2×#1309394511yescharacteristic
325C4#1568935011yescharacteristic
326C4#1568935711yescharacteristic
327C2—21no—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—162no—
332C2#2024381711yescharacteristic
subgroupG/H
centerC2#20243817
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
Fitting, prime radicalC2×C47×#472C2
socleC2×C47#1893
cosocleC22×C8×C47C23
hypercenterC2×#472D47
hypocenterC47#13068
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. #20256491C2semidirect
2. C2×#15688936C2semidirect
3. #20245104C2semidirect
4. #20243442C2semidirect
5. #20241502C2semidirect
6. #20241491C2semidirect
23. C47×#472C4semidirect
183. #472Dic47semidirect
192. C47#13068semidirect

Maximal subgroups

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

subgroupindex
#202564912
C2×#156889362
#202451042
#202434422
#202415022
#202414912
C2×C47×#4722
#1306847