Finite Group Databasebeta

← #26113988

Subgroups — #26113988

Order 12032.

690 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 111 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#20252417C211yesmaximal, characteristic, complemented
2#20242102C212yesmaximal, complemented
3#20241729C211yesmaximal, characteristic, complemented
4#20241244C212yesmaximal, complemented
5C47×#2541C211yesmaximal, characteristic
6#15689195—24no—
7#15689180C2212yes—
8#15689180C2212yes—
9#15689087—21no—
10C2×C22⋊(C47⋊C8)—21no—
11#15688867—24no—
12#15688847—21no—
13#15688847—21no—
14C2×#127×C47C412yes—
15C2×#127×C47C2211yescharacteristic
16C2×#121×C47—21no—
17C2×#121×C47—21no—
18C2×C4×D4×C47C2211yescharacteristic
19C47×#392—21no—
20C47×#392—21no—
21C47×#390C412yes—
22C4×C4⋊C4×C47C2211yescharacteristic
23C2×#10625329—22no—
24C22×C47⋊C8—22no—
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)D414yescomplemented
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31#13093935—22no—
32#13093935—22no—
33C22×D4×C47—22no—
34C22×D4×C47—22no—
35C22×D4×C47—21no—
36C22×D4×C47—21no—
37C22×D4×C47C2×C411yescharacteristic
38C22×D4×C47C2×C412yes—
39C23×C4×C47D411yescharacteristic
40C23×C4×C47D411yescharacteristic
41#127×C47—22no—
42#127×C47—22no—
43#127×C47—44no—
44#121×C47—44no—
45#121×C47—44no—
46C4×D4×C47—22no—
47C4×D4×C47—22no—
48C2×C4⋊C4×C47—21no—
49C2×C4⋊C4×C47C2×C411yescharacteristic
50C2×C22⋊C4×C47—22no—
51C2×C22⋊C4×C47—22no—
52C2×C22⋊C4×C47D411yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C42×C47C2×C412yes—
55C2×C42×C47C2311yescharacteristic
56#95×C47—21no—
57C24×C47—22no—
58C24×C47C22⋊C411yescharacteristic
59C24×C47C22⋊C411yescharacteristic
60C2×D4×C47—22no—
61C2×D4×C47—22no—
62C2×D4×C47—24no—
63C2×D4×C47—41no—
64C2×D4×C47—42no—
65C2×D4×C47—42no—
66C2×D4×C47—44no—
67C2×D4×C47—44no—
68C2×D4×C47—42no—
69C2×D4×C47—42no—
70C2×D4×C47—44no—
71C2×D4×C47—44no—
72C2×D4×C47—41no—
73C2×D4×C47—42no—
74C2×D4×C47—41no—
75C2×D4×C47—82no—
76C2×D4×C47—82no—
77C2×D4×C47—82no—
78C22×C4×C47—22no—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—42no—
84C22×C4×C47—42no—
85C22×C4×C47C22⋊C411yescharacteristic
86C22×C4×C47C22⋊C411yescharacteristic
87C22×C4×C47C22×C411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C4⋊C4×C47—24no—
91C4⋊C4×C47—22no—
92C22⋊C4×C47—22no—
93C22⋊C4×C47—22no—
94C22⋊C4×C47—44no—
95C22⋊C4×C47—44no—
96C42×C47—42no—
97C42×C47C2×D412yes—
98C42×C47#4112yes—
99#10625329—82no—
100C2×C47⋊C8—24no—
101C2×C47⋊C8—42no—
102C2×C47⋊C8—82no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—42no—
113C23×C47—42no—
114C23×C47—42no—
115C23×C47—42no—
116C23×C47—42no—
117C23×C47—42no—
118C23×C47—42no—
119C23×C47#10012yes—
120C23×C47C2×C22⋊C411yescharacteristic
121D4×C47—44no—
122D4×C47—84no—
123D4×C47—82no—
124D4×C47—84no—
125D4×C47—88no—
126D4×C47—88no—
127D4×C47—82no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—22no—
131C2×C4×C47—24no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—41no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—81no—
142C2×C4×C47—81no—
143C2×C4×C47C4×D412yes—
144C2×C4×C47#12011yescharacteristic
145C2×C4×C47#12112yes—
146C2×C4×C47#12311yescharacteristic
147C47⋊C8—44no—
148C47⋊C8—82no—
149#9457—471nomaximal, Sylow
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—22no—
153C22×C47—42no—
154C22×C47—42no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47—41no—
160C22×C47—81no—
161C22×C47—81no—
162C22×C47—82no—
163C22×C47—82no—
164C22×C47—82no—
165C22×C47—82no—
166C22×C47—82no—
167C22×C47—82no—
168C22×C47—82no—
169C22×C47#38611yescharacteristic
170C22×C47C2×#10011yescharacteristic
171C22×C47#41311yescharacteristic
172C4×C47—21no—
173C4×C47—22no—
174C4×C47—22no—
175C4×C47—44no—
176C4×C47—41no—
177C4×C47—41no—
178C4×C47#45312yes—
179#2973—471no—
180#2541D4711yescharacteristic
181#1555—472no—
182#1466—471no—
183#1285—472no—
184C2×C47—21no—
185C2×C47—21no—
186C2×C47—42no—
187C2×C47—81no—
188C2×C47—162no—
189C2×C47#188711yescharacteristic
190C2×C47#190011yescharacteristic
191C2×C47#191011yescharacteristic
192C2×#127Dic4712yes—
193C2×#127C2×D4711yescharacteristic
194C2×#121—21no—
195C2×#121—21no—
196C2×C4×D4C2×D4711yescharacteristic
197#435—944no—
198#432—472no—
199#432—472no—
200#407—941no—
201C2×C22⋊C8—941no—
202#392—21no—
203#392—21no—
204#390Dic4712yes—
205C4×C4⋊C4C2×D4711yescharacteristic
206#331—944no—
207#323—941no—
208#323—941no—
209C47#945711yescharacteristic, complemented, Sylow
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4C2×Dic4711yescharacteristic
215C22×D4C2×Dic4712yes—
216C23×C4C22⋊D4711yescharacteristic
217C23×C4C22⋊D4711yescharacteristic
218C2×M16—942no—
219C22×C8—942no—
220#127—22no—
221#127—22no—
222#127—44no—
223#121—44no—
224#121—44no—
225C4×D4—22no—
226C4×D4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C2×Dic4711yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4C22⋊D4711yescharacteristic
232C2×C22⋊C4C22⋊D4711yescharacteristic
233C2×C42C2×Dic4712yes—
234C2×C42C22×D4711yescharacteristic
235C4⋊C8—474no—
236C4⋊C8—942no—
237C22⋊C8—942no—
238C22⋊C8—942no—
239C22⋊C8—942no—
240C22⋊C8—942no—
241#97—942no—
242#97—942no—
243#95—21no—
244C24—22no—
245C24C22⋊Dic4711yescharacteristic
246C24C22⋊Dic4711yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—24no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4C22⋊Dic4711yescharacteristic
273C22×C4C22⋊Dic4711yescharacteristic
274C22×C4C22×Dic4711yescharacteristic
275C22×C4C2×C22⋊D4711yescharacteristic
276C22×C4C2×C22⋊D4711yescharacteristic
277M16—3762no—
278C2×C8—944no—
279C2×C8—3762no—
280C2×C8—1882no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C42—42no—
288C42D4×D4712yes—
289C42#1062535212yes—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23#1309394112yes—
307C23C2×C22⋊Dic4711yescharacteristic
308D4—44no—
309D4—84no—
310D4—82no—
311D4—84no—
312D4—88no—
313D4—88no—
314D4—82no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4D4×Dic4712yes—
331C2×C4#1309399211yescharacteristic
332C2×C4#1309399612yes—
333C2×C4#1309400511yescharacteristic
334C8—3762no—
335C8—1884no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22—82no—
353C22—82no—
354C22—82no—
355C22#1568900311yescharacteristic
356C22C2×#1309394111yescharacteristic
357C22#1568910711yescharacteristic
358C4—21no—
359C4—22no—
360C4—22no—
361C4—44no—
362C4—41no—
363C4—41no—
364C4#1568929012yes—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—81no—
369C2—162no—
370C2#2024378711yescharacteristic
371C2#2024385211yescharacteristic
372C2#2024391611yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#2541C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#2541D47
hypocenterC47#9457
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. #20252417C2semidirect
2. #20242102C2semidirect
3. #20241729C2semidirect
4. #20241244C2semidirect
26. C4⋊(C47⋊C8)D4semidirect
209. C47#9457semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202421022
#202417292
#202412442
C47×#25412
#945747