Finite Group Databasebeta

← #26120682

Subgroups — #26120682

Order 12032.

604 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 121 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#20249502C211yesmaximal, characteristic
2#20243578C212yesmaximal
3#20241652C211yesmaximal, characteristic
4#20241268C212yesmaximal
5C47×#1280C211yesmaximal, characteristic
6C2×#13094003—21no—
7C22×C4⋊Dic47C2211yescharacteristic
8C22×Dic47⋊C4—21no—
9C2×#13093958—22no—
10#15689055—22no—
11#15689044—22no—
12#15688983—21no—
13#15688983C2212yes—
14C2×C47⋊#95—21no—
15C2×C47⋊#95C2211yescharacteristic
16#15688886—22no—
17C2×C22⋊C8×C47C2212yes—
18C2×#95×C47C2211yescharacteristic
19C22×Dic94—21no—
20C23×Dic47—21no—
21C23×Dic47—21no—
22#13094003—42no—
23#13094003—42no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—42no—
29C2×C4⋊Dic47D412yes—
30C2×C4⋊Dic47D412yes—
31C2×Dic47⋊C4—22no—
32C2×Dic47⋊C4—42no—
33C2×Dic47⋊C4—42no—
34C2×Dic47⋊C4—42no—
35C2×C22⋊Dic47—22no—
36#13093958—44no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95—21no—
40C47⋊#95—42no—
41C47⋊#95Q812yescomplemented
42C23×C4×C47D411yescharacteristic
43C23×C4×C47D411yescharacteristic
44C23×C4×C47C2311yescharacteristic
45C22×C8×C47—22no—
46C22⋊C8×C47—22no—
47C22⋊C8×C47Q814yescomplemented
48#95×C47—22no—
49C24×C47C2×D411yescharacteristic
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—41no—
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47—41no—
58C22×C4×C47—41no—
59C22×C4×C47—41no—
60C22×C4×C47D812yes—
61C22×C4×C47Q1612yes—
62C22×C4×C47C2×D412yes—
63C22×C4×C47C2×Q812yes—
64C22×C4×C47C2×Q811yescharacteristic
65C22×C4×C47#4111yescharacteristic
66C22×Dic47—22no—
67C22×Dic47—22no—
68C22×Dic47—22no—
69C22×Dic47—22no—
70C22×Dic47—22no—
71C22×Dic47—22no—
72C22×Dic47—41no—
73C22×Dic47—41no—
74C22×Dic47—41no—
75C22×Dic47—41no—
76C22×Dic47—41no—
77C22×Dic47—41no—
78C22×Dic47—41no—
79C22×Dic47—41no—
80C22×Dic47—41no—
81C22×Dic47—41no—
82C2×Dic94—42no—
83C2×Dic94—81no—
84C2×Dic94—81no—
85C2×C8×C47—24no—
86C2×C8×C47—42no—
87C2×C8×C47—42no—
88C22⋊Dic47—42no—
89C22⋊Dic47—42no—
90C4⋊Dic47—24no—
91C4⋊Dic47—42no—
92C4⋊Dic47—42no—
93C4⋊Dic47—42no—
94C4⋊Dic47—42no—
95C4⋊Dic47—81no—
96C4⋊Dic47—81no—
97Dic47⋊C4—44no—
98Dic47⋊C4—48no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47#12011yescharacteristic
104C23×C47C22⋊Q811yescharacteristic
105C23×C47C4⋊Q811yescharacteristic
106C23×C47C2×D811yescharacteristic
107C23×C47C2×Q1611yescharacteristic
108C23×C47AGL(1,ℤ/8)11yescharacteristic
109C23×C47#13711yescharacteristic
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—81no—
124C2×C4×C47—81no—
125C2×C4×C47C22⋊Q812yes—
126C2×C4×C47C4⋊Q812yes—
127C2×Dic47—24no—
128C2×Dic47—42no—
129C2×Dic47—42no—
130C2×Dic47—42no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141C2×Dic47—42no—
142C2×Dic47—42no—
143C2×Dic47—81no—
144C2×Dic47—81no—
145C2×Dic47—81no—
146C2×Dic47—81no—
147Dic94—84no—
148C8×C47—44no—
149#10567—471nomaximal, Sylow
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47#39511yescharacteristic
163C22×C47#44711yescharacteristic
164C22×C47#45111yescharacteristic
165C22×C47#45311yescharacteristic
166C22×C47#45611yescharacteristic
167C22×C47B(4,𝔽2)11yescharacteristic
168C22×C47#45811yescharacteristic
169C22×C47#47411yescharacteristic
170C22×C47#47711yescharacteristic
171C22×C47#50011yescharacteristic
172C22×C47#50111yescharacteristic
173C4×C47—22no—
174C4×C47—22no—
175C4×C47—42no—
176C4×C47—42no—
177Dic47—44no—
178Dic47—82no—
179Dic47—82no—
180Dic47—82no—
181#2659—471no—
182#1852—472no—
183#1444—471no—
184#1294—472no—
185#1280D4711yescharacteristic
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—21no—
189C2×C47—21no—
190C2×C47#159511yescharacteristic
191C2×C47#160511yescharacteristic
192C2×C47#161011yescharacteristic
193C2×C47#161411yescharacteristic
194C2×C47#205711yescharacteristic
195C2×C47#206011yescharacteristic
196C2×C47#206211yescharacteristic
197C2×C22⋊Q8—941no—
198C22×C4⋊C4—471no—
199C22×C4⋊C4—941no—
200C2×#107—942no—
201C2×C22⋊C8C2×D4712yes—
202#398—942no—
203#395—942no—
204#380—472no—
205#380—941no—
206C2×#95—471no—
207C2×#95—941no—
208C2×#95C2×D4711yescharacteristic
209#340—942no—
210C47#1056711yescharacteristic, complemented, Sylow
211C22×Q8—941no—
212C23×C4—941no—
213C23×C4—941no—
214C23×C4D18811yescharacteristic
215C23×C4D18811yescharacteristic
216C23×C4C22×D4711yescharacteristic
217C22×C8—22no—
218C22⋊Q8—1882no—
219C22⋊Q8—1882no—
220C2×C4⋊C4—472no—
221C2×C4⋊C4—472no—
222C2×C4⋊C4—942no—
223C2×C4⋊C4—941no—
224C2×C4⋊C4—941no—
225C2×C4⋊C4—942no—
226C2×C4⋊C4—941no—
227C2×C4⋊C4—1882no—
228C2×C4⋊C4—1882no—
229C2×C4⋊C4—1882no—
230C2×C4⋊C4—1882no—
231C2×C22⋊C4—942no—
232#107—1884no—
233C22⋊C8—22no—
234C22⋊C8Dic9414yescomplemented
235#95—22no—
236#95—472no—
237#95—942no—
238#95—942no—
239#95—941no—
240#95—1882no—
241C24C2×D18811yescharacteristic
242C2×Q8—3761no—
243C2×Q8—3761no—
244C2×Q8—1882no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—942no—
256C22×C4—942no—
257C22×C4—942no—
258C22×C4—942no—
259C22×C4—942no—
260C22×C4—942no—
261C22×C4—1881no—
262C22×C4—1881no—
263C22×C4—1881no—
264C22×C4—1881no—
265C22×C4—1881no—
266C22×C4—1881no—
267C22×C4—1881no—
268C22×C4—1881no—
269C22×C4—1881no—
270C22×C4—1881no—
271C22×C4D37612yes—
272C22×C4Dic18812yes—
273C22×C4C2×Dic9412yes—
274C22×C4D4×D4712yes—
275C22×C4#1062535211yescharacteristic
276C22×C4Q8×D4711yescharacteristic
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C4⋊C4—944no—
281C4⋊C4—3761no—
282C4⋊C4—3761no—
283C4⋊C4—1882no—
284C4⋊C4—1882no—
285C4⋊C4—1884no—
286C4⋊C4—1882no—
287C4⋊C4—1882no—
288C4⋊C4—1888no—
289C22⋊C4—1882no—
290C22⋊C4—1882no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1309399411yescharacteristic
296C23#1309400411yescharacteristic
297C23#1309402411yescharacteristic
298C23C2×D37611yescharacteristic
299C23C2×Dic18811yescharacteristic
300C23#1309405211yescharacteristic
301C23#1309405711yescharacteristic
302Q8—3764no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—944no—
319C2×C4—3761no—
320C2×C4—3761no—
321C2×C4—3761no—
322C2×C4—3761no—
323C2×C4—1882no—
324C2×C4—1882no—
325C2×C4—1882no—
326C2×C4—1882no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1882no—
330C2×C4—1882no—
331C2×C4—1882no—
332C2×C4—1882no—
333C2×C4—1882no—
334C2×C4—1882no—
335C2×C4—1882no—
336C2×C4—1882no—
337C2×C4—1882no—
338C2×C4#1309400112yes—
339C2×C4#1309402312yes—
340C8—44no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#1568904411yescharacteristic
354C22#1568925911yescharacteristic
355C22#1568928111yescharacteristic
356C22#1568929311yescharacteristic
357C22#1568930811yescharacteristic
358C22#1568931111yescharacteristic
359C22#1568931411yescharacteristic
360C22#1568940411yescharacteristic
361C22#1568941911yescharacteristic
362C22#1568952811yescharacteristic
363C22#1568953211yescharacteristic
364C4—22no—
365C4—22no—
366C4—42no—
367C4—42no—
368C4—3762no—
369C4—3762no—
370C4—3762no—
371C4—1884no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#2024230811yescharacteristic
377C2#2024236411yescharacteristic
378C2#2024239311yescharacteristic
379C2#2024241511yescharacteristic
380C2#2024471711yescharacteristic
381C2#2024473411yescharacteristic
382C2#2024474211yescharacteristic
subgroupG/H
centerC23#13093994
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#1280C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenter#1280D47
hypocenterC47#10567
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
41. C47⋊#95Q8semidirect
47. C22⋊C8×C47Q8semidirect
210. C47#10567semidirect
234. C22⋊C8Dic94semidirect

Maximal subgroups

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

subgroupindex
#202495022
#202435782
#202416522
#202412682
C47×#12802
#1056747