Finite Group Databasebeta

← #26105572

Subgroups — #26105572

Order 12032.

404 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 87 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#20244951C211yesmaximal, characteristic, complemented
2#20244951C211yesmaximal, characteristic, complemented
3#20244921C211yesmaximal, characteristic
4#20241211C211yesmaximal, characteristic, complemented
5#20241211C211yesmaximal, characteristic, complemented
6#20241209C211yesmaximal, characteristic
7C47×#2727C211yesmaximal, characteristic
8C2×#13093955—21no—
9C2×#13093945—21no—
10#15688993C2211yescharacteristic
11#15688885—21no—
12#15688885—21no—
13#15688885—21no—
14#15688885—21no—
15#15688878C2211yescharacteristic
16#15688878C2211yescharacteristic
17#15688878C2211yescharacteristic
18C2×#126×C47—21no—
19C47×#399—21no—
20C47×#396—21no—
21C47×#394—21no—
22C47×#390C2211yescharacteristic
23C47×#389—21no—
24C47×#389C2211yescharacteristic
25C47×#389C2211yescharacteristic
26C47×#388—21no—
27C22×C47⋊C8—21no—
28C22×C47⋊C8—21no—
29C22×C47⋊C8—21no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47—21no—
33#13093955—42no—
34#13093945—42no—
35C22×D4×C47D411yescharacteristic
36#126×C47—42no—
37#126×C47—42no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C42×C47—21no—
54C2×C42×C47C2311yescharacteristic
55#95×C47—21no—
56#95×C47—21no—
57#95×C47—21no—
58C24×C47—21no—
59C2×D4×C47—21no—
60C2×D4×C47—21no—
61C2×D4×C47—41no—
62C2×D4×C47—41no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—41no—
72C4⋊C4×C47—21no—
73C4⋊C4×C47—21no—
74C4⋊C4×C47—21no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—81no—
84C4⋊C4×C47SD1612yes—
85C22⋊C4×C47—42no—
86C22⋊C4×C47—42no—
87C22⋊C4×C47—41no—
88C22⋊C4×C47—41no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—81no—
92C22⋊C4×C47—81no—
93C42×C47—41no—
94C42×C47—42no—
95C4⋊Dic47—41no—
96C4⋊Dic47—41no—
97C4⋊Dic47—42no—
98C4⋊Dic47—42no—
99C2×C47⋊C8—22no—
100C2×C47⋊C8—41no—
101C2×C47⋊C8—41no—
102C2×C47⋊C8—41no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C2×C47⋊C8—41no—
107C2×C47⋊C8—41no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47#12011yescharacteristic
116D4×C47—42no—
117D4×C47—81no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—22no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47C2×SD1611yescharacteristic
140C2×C4×C47#13511yescharacteristic
141C2×C4×C47#13511yescharacteristic
142C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C47#13711yescharacteristic
144C2×C4×C47#13711yescharacteristic
145C2×Dic47—81no—
146C2×Dic47—81no—
147C2×Dic47—81no—
148C47⋊C8—42no—
149C47⋊C8—81no—
150C47⋊C8—81no—
151#7828—471nomaximal, Sylow
152C22×C47—81no—
153C22×C47—81no—
154C22×C47—81no—
155C22×C47—81no—
156C22×C47—81no—
157C22×C47—81no—
158C22×C47—81no—
159C22×C47#45011yescharacteristic
160C22×C47#45211yescharacteristic
161C22×C47#45211yescharacteristic
162C22×C47#45311yescharacteristic
163C22×C47#45411yescharacteristic
164C22×C47#45411yescharacteristic
165C22×C47#45811yescharacteristic
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—42no—
173C4×C47—81no—
174C4×C47—81no—
175C4×C47—81no—
176C4×C47—81no—
177Dic47—161no—
178#2727D4711yescharacteristic
179#2100—471no—
180#2100—471no—
181#2094—471no—
182#1271—471no—
183#1271—471no—
184#1270—471no—
185C2×C47—161no—
186C2×C47#161511yescharacteristic
187C2×C47#161611yescharacteristic
188C2×C47#161811yescharacteristic
189C2×C47#163111yescharacteristic
190C2×C47#163511yescharacteristic
191C2×C47#163811yescharacteristic
192C2×C47#165711yescharacteristic
193C2×#126—21no—
194C2×#106—941no—
195C2×#102—941no—
196#399—21no—
197#396—21no—
198#394—21no—
199#390C2×D4711yescharacteristic
200#389—21no—
201#389C2×D4711yescharacteristic
202#389C2×D4711yescharacteristic
203#388—21no—
204#384—471no—
205#340—941no—
206#340—941no—
207#340—941no—
208#340—941no—
209#336—471no—
210#336—471no—
211#336—471no—
212C47#782811yescharacteristic, complemented, Sylow
213C22×D4C22⋊D4711yescharacteristic
214C22×C8—941no—
215C22×C8—941no—
216C22×C8—941no—
217#126—42no—
218#126—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—941no—
224C2×C4⋊C4—941no—
225C2×C4⋊C4—941no—
226C2×C4⋊C4C22⋊D4711yescharacteristic
227C2×C4⋊C4C22⋊D4711yescharacteristic
228C2×C4⋊C4C22⋊D4711yescharacteristic
229C2×C4⋊C4C22⋊D4711yescharacteristic
230C2×C4⋊C4C22⋊D4711yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42C22×D4711yescharacteristic
239#106—1882no—
240#102—1882no—
241#95—21no—
242#95—21no—
243#95—21no—
244C24—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—1881no—
255C22×C4C2×C22⋊D4711yescharacteristic
256C22×C4C2×C22⋊D4711yescharacteristic
257C22×C4C2×C22⋊D4711yescharacteristic
258C2×C8—942no—
259C2×C8—1881no—
260C2×C8—1881no—
261C2×C8—1881no—
262C2×C8—1881no—
263C2×C8—1881no—
264C2×C8—1881no—
265C2×C8—1881no—
266C2×C8—1881no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—81no—
279C4⋊C4—1881no—
280C4⋊C4—1881no—
281C4⋊C4—1882no—
282C4⋊C4—1882no—
283C4⋊C4#1062533512yes—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C42—41no—
293C42—42no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23#1309399311yescharacteristic
302D4—42no—
303D4—81no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—3761no—
326C2×C4—3761no—
327C2×C4—3761no—
328C2×C4C2×#1062533511yescharacteristic
329C2×C4#1309404411yescharacteristic
330C2×C4#1309404411yescharacteristic
331C2×C4#1309405011yescharacteristic
332C2×C4#1309405411yescharacteristic
333C2×C4#1309405411yescharacteristic
334C8—3761no—
335C8—3761no—
336C8—1882no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22#1568927411yescharacteristic
345C22#1568928511yescharacteristic
346C22#1568928511yescharacteristic
347C22#1568929111yescharacteristic
348C22#1568929611yescharacteristic
349C22#1568929611yescharacteristic
350C22#1568931311yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—7521no—
363C2—161no—
364C2#2024241911yescharacteristic
365C2#2024242511yescharacteristic
366C2#2024243611yescharacteristic
367C2#2024251511yescharacteristic
368C2#2024253911yescharacteristic
369C2#2024256011yescharacteristic
370C2#2024268611yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2727C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2727D47
hypocenterC47#7828
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
1. #20244951C2semidirect
2. #20244951C2semidirect
4. #20241211C2semidirect
5. #20241211C2semidirect
212. C47#7828semidirect

Maximal subgroups

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

subgroupindex
#202449512
#202449212
#202412112
#202412092
C47×#27272
#782847