Finite Group Databasebeta

← #26126989

Subgroups — #26126989

Order 12032.

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

normal subgroups only

This group has 616 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
1C2×#15689059C211yesmaximal, characteristic
2#20241715C211yesmaximal, characteristic
3#20241411C214yesmaximal, complemented
4C47×#1819C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic, complemented
6C22×Dic47⋊C4—21no—
7C22×C22⋊Dic47—21no—
8#15689087—21no—
9#15689059—21no—
10#15689059—21no—
11#15689059C412yescomplemented
12#15689059C2212yes—
13C2×C47⋊#95C411yescharacteristic, complemented
14#15688847—21no—
15#15688847C2212yes—
16C22×C22⋊C4×C47C2211yescharacteristic
17C47×#379—21no—
18C47×#342—21no—
19C47×#342C2212yes—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C2×#10625329—41no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—41no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—22no—
29C2×Dic47⋊C4—44no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—44no—
33C22⋊(C47⋊C8)—22no—
34C22⋊(C47⋊C8)—41no—
35C47⋊#95—21no—
36C47⋊#95C2×C412yes—
37C25×C47D411yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C2×C22⋊C4×C47—22no—
41C2×C22⋊C4×C47—41no—
42C2×C22⋊C4×C47—41no—
43C2×C22⋊C4×C47—41no—
44C2×C22⋊C4×C47—41no—
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47D411yescharacteristic
48C2×C22⋊C4×C47C2311yescharacteristic
49C24×C47—22no—
50C24×C47—21no—
51C24×C47—21no—
52C24×C47—41no—
53C24×C47—42no—
54C24×C47—42no—
55C24×C47C22⋊C411yescharacteristic
56C24×C47C22⋊C411yescharacteristic
57C24×C47C2×D411yescharacteristic
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—24no—
70C22×Dic47—22no—
71C22×Dic47—22no—
72C22×Dic47—42no—
73C22×Dic47—42no—
74C22×Dic47—42no—
75C22×Dic47—42no—
76C22×Dic47—41no—
77C22×Dic47—41no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C22⋊C4×C47—42no—
83C22⋊C4×C47—42no—
84C22⋊C4×C47—82no—
85C22⋊C4×C47—82no—
86C22⋊Dic47—44no—
87C22⋊Dic47—44no—
88C4⋊Dic47—42no—
89C4⋊Dic47—42no—
90Dic47⋊C4—44no—
91Dic47⋊C4—44no—
92#10625329—82no—
93C2×C47⋊C8—42no—
94C23×C47—22no—
95C23×C47—22no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47—48no—
110C23×C47—42no—
111C23×C47—42no—
112C23×C47—81no—
113C23×C47—81no—
114C23×C47—84no—
115C23×C47#9911yescharacteristic
116C23×C47#9911yescharacteristic
117C23×C47C2×C22⋊C411yescharacteristic
118C23×C47#13712yes—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—81no—
131C2×C4×C47—82no—
132C2×C4×C47—82no—
133C2×C4×C47—82no—
134C2×Dic47—24no—
135C2×Dic47—48no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141C2×Dic47—82no—
142C2×Dic47—82no—
143C47⋊C8—82no—
144#11809—471nomaximal, Sylow
145C22×C47—24no—
146C22×C47—24no—
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—22no—
154C22×C47—42no—
155C22×C47—42no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47—41no—
160C22×C47—41no—
161C22×C47—81no—
162C22×C47—84no—
163C22×C47—84no—
164C22×C47—81no—
165C22×C47#35212yes—
166C22×C47#35312yes—
167C22×C47C2×#9911yescharacteristic
168C22×C47#41911yescharacteristic
169C22×C47#42111yescharacteristic
170C4×C47—41no—
171C4×C47—41no—
172C4×C47—41no—
173C4×C47—41no—
174C4×C47—162no—
175Dic47—44no—
176Dic47—82no—
177C2×#399—471no—
178#1819D4711yescharacteristic
179#1462—471no—
180#1352—474no—
181C2×C47—21no—
182C2×C47—21no—
183C2×C47—22no—
184C2×C47—22no—
185C2×C47—42no—
186C2×C47—81no—
187C2×C47#151111yescharacteristic
188C2×C47C2×#35211yescharacteristic
189C2×C47C2×#35311yescharacteristic
190C22×C4⋊C4—471no—
191C22×C4⋊C4—941no—
192C22×C22⋊C4—941no—
193C22×C22⋊C4C2×D4711yescharacteristic
194#407—941no—
195#399—472no—
196#399—472no—
197#399—941no—
198#399—941no—
199#379—21no—
200C2×#95—471no—
201#342—21no—
202#342C2×D4712yes—
203#323—472no—
204#323—941no—
205C47#1180911yescharacteristic, complemented, Sylow
206C25C22⋊D4711yescharacteristic
207C23×C4—941no—
208C23×C4—941no—
209C23×C4C4×D4711yescharacteristic
210C23×C4D18811yescharacteristic
211C2×M16—1881no—
212C2×C4⋊C4—472no—
213C2×C4⋊C4—941no—
214C2×C4⋊C4—942no—
215C2×C4⋊C4—942no—
216C2×C4⋊C4—1881no—
217C2×C4⋊C4—1881no—
218C2×C4⋊C4—1884no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—942no—
225C2×C22⋊C4—942no—
226C2×C22⋊C4—1884no—
227C2×C22⋊C4C4×D4711yescharacteristic
228C2×C22⋊C4D18811yescharacteristic
229C2×C22⋊C4C22⋊D4711yescharacteristic
230C2×C22⋊C4C22×D4711yescharacteristic
231C22⋊C8—942no—
232C22⋊C8—1881no—
233#95—472no—
234#95—941no—
235C24—22no—
236C24—21no—
237C24—21no—
238C24—41no—
239C24—42no—
240C24—42no—
241C24(C2×D47)⋊C411yescharacteristic
242C24(C2×D47)⋊C411yescharacteristic
243C24C2×C22⋊D4711yescharacteristic
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—944no—
252C22×C4—942no—
253C22×C4—942no—
254C22×C4—1882no—
255C22×C4—1882no—
256C22×C4—1881no—
257C22×C4—1881no—
258C22×C4—1882no—
259C22×C4—1882no—
260C22×C4(C2×D47)⋊C411yescharacteristic
261C22×C4(C2×D47)⋊C411yescharacteristic
262C22×C4C2×C4×D4711yescharacteristic
263C22×C4C2×D18811yescharacteristic
264M16—3762no—
265C2×C8—1882no—
266C4⋊C4—1882no—
267C4⋊C4—1882no—
268C4⋊C4—1884no—
269C4⋊C4—1884no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—82no—
277C22⋊C4—82no—
278C22⋊C4—1884no—
279C22⋊C4—1884no—
280C23—21no—
281C23—21no—
282C23—22no—
283C23—22no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—42no—
290C23—42no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—48no—
296C23—42no—
297C23—42no—
298C23—81no—
299C23—81no—
300C23—84no—
301C23#1309393811yescharacteristic
302C23#1309393811yescharacteristic
303C23C2×(C2×D47)⋊C411yescharacteristic
304C23#1309405412yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—944no—
321C2×C4—3762no—
322C2×C4—3762no—
323C2×C4—1882no—
324C2×C4—1882no—
325C2×C4—1882no—
326C2×C4—1888no—
327C2×C4—1882no—
328C2×C4—1882no—
329C8—3762no—
330C22—24no—
331C22—24no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—84no—
348C22—84no—
349C22—81no—
350C22#1568891212yes—
351C22#1568891512yes—
352C22C2×#1309393811yescharacteristic
353C22#1568913111yescharacteristic
354C22#1568914111yescharacteristic
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—162no—
360C4—3762no—
361C4—1884no—
362C2—21no—
363C2—21no—
364C2—22no—
365C2—22no—
366C2—42no—
367C2—81no—
368C2#2024192011yescharacteristic
369C2C2×#1568891211yescharacteristic
370C2C2×#1568891511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1819C2
socleC22×C47C2×#99
prime radicalC25×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1819D47
hypocenterC47#11809
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #20241411C2semidirect
5. C22×C4⋊Dic47C4semidirect
11. #15689059C4semidirect
13. C2×C47⋊#95C4semidirect
205. C47#11809semidirect

Maximal subgroups

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

subgroupindex
C2×#156890592
#202417152
#202414112
C47×#18192
#1180947