Finite Group Databasebeta

← #26104507

Subgroups — #26104507

Order 12032.

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

normal subgroups only

This group has 510 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#20245852C211yesmaximal, characteristic
2#20243137C211yesmaximal, characteristic, complemented
3#20241210C212yesmaximal, complemented
4#20241208C212yesmaximal
5C47×#1767C211yesmaximal, characteristic, complemented
6C2×C4×D188C411yescharacteristic
7C2×#13093981—21no—
8C2×#13093935—21no—
9#15689025C411yescharacteristic
10#15689025C2211yescharacteristic
11#15689021C411yescharacteristic
12#15689021C2211yescharacteristic
13#15689014—21no—
14#15689014—21no—
15C4×C4⋊Dic47C411yescharacteristic
16C4×(C2×D47)⋊C4—21no—
17#15688878—21no—
18#15688878C2212yes—
19C2×C4⋊C8×C47—21no—
20C2×#97×C47—21no—
21C43×C47C2211yescharacteristic
22C47×#336C2212yes—
23C22×D188C2×C411yescharacteristic
24C22×C4×D47—21no—
25C22×C47⋊C8—22no—
26C4×D188—22no—
27C4×D188—22no—
28#13093981—44no—
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×C4⋊Dic47C2×C411yescharacteristic
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×(C2×D47)⋊C4—21no—
35C2×(C2×D47)⋊C4—21no—
36C2×(C2×D47)⋊C4—21no—
37C2×(C2×D47)⋊C4—21no—
38C2×(C2×D47)⋊C4—21no—
39C2×C4×Dic47—21no—
40#13093935—24no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C22×C8×C47—22no—
45C2×C42×C47C2×C411yescharacteristic
46C2×C42×C47C2×C411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52C4⋊C8×C47—24no—
53#97×C47—42no—
54C23×D47—21no—
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—21no—
63C22×Dic47—21no—
64C22×Dic47—21no—
65C2×D188—21no—
66C2×D188—41no—
67C2×D188—41no—
68C2×D188M1612yes—
69C2×C4×D47—24no—
70C2×C4×D47—42no—
71C2×C4×D47—42no—
72C2×C8×C47—24no—
73C2×C8×C47—42no—
74C2×C8×C47—42no—
75C42×C47—22no—
76C42×C47—22no—
77C42×C47—21no—
78C42×C47—21no—
79C42×C47—22no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47—41no—
83C42×C47—41no—
84(C2×D47)⋊C4—24no—
85(C2×D47)⋊C4—42no—
86(C2×D47)⋊C4—42no—
87(C2×D47)⋊C4—41no—
88(C2×D47)⋊C4—42no—
89(C2×D47)⋊C4—41no—
90C4⋊Dic47—21no—
91C4⋊Dic47—41no—
92C4⋊Dic47—41no—
93C4⋊Dic47M1612yes—
94Dic47⋊C4—42no—
95Dic47⋊C4—42no—
96C4×Dic47—42no—
97C2×C47⋊C8—24no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C23×C47C2×C22⋊C411yescharacteristic
101C22×D47—22no—
102C22×D47—41no—
103C22×D47—41no—
104C22×D47—41no—
105C22×D47—41no—
106C22×D47—41no—
107C22×D47—41no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
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#10412yes—
125C2×C4×C47#10412yes—
126C2×C4×C47C2×M1611yescharacteristic
127C2×C4×C47C2×M1611yescharacteristic
128C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C47AGL(1,ℤ/8)12yes—
130C2×C4×C47#13711yescharacteristic
131C2×Dic47—22no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140D188—42no—
141D188—42no—
142C4×D47—48no—
143C8×C47—44no—
144C47⋊C8—44no—
145#7628—471nomaximal, Sylow
146C22×C47#40711yescharacteristic
147C22×C47#41011yescharacteristic
148C22×C47#41311yescharacteristic
149C22×C47#41711yescharacteristic
150C22×C47#41811yescharacteristic
151C22×C47C2×#10411yescharacteristic
152C22×C47C2×#10411yescharacteristic
153C2×D47—42no—
154C2×D47—42no—
155C2×D47—42no—
156C2×D47—81no—
157C2×D47—81no—
158C2×D47—81no—
159C2×D47—81no—
160C4×C47—21no—
161C4×C47—21no—
162C4×C47—22no—
163C4×C47—22no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—22no—
167C4×C47—41no—
168C4×C47—41no—
169Dic47—42no—
170Dic47—81no—
171Dic47—81no—
172#2293—471no—
173#1767D4711yescharacteristic, complemented
174#1745—471no—
175#1271—472no—
176#1270—472no—
177C2×C47#146411yescharacteristic
178C2×C47#148911yescharacteristic
179C2×C47#149011yescharacteristic
180C2×C47#150511yescharacteristic
181C2×C47#150911yescharacteristic
182C2×C47#152411yescharacteristic
183C2×C47#155711yescharacteristic
184D47—82no—
185C2×C4×D4—471no—
186C2×#117—941no—
187C2×C4⋊C8—21no—
188C2×#97—21no—
189C2×#97—941no—
190#390—471no—
191#390—471no—
192#389—471no—
193#389—471no—
194#388—941no—
195#388—941no—
196C4×C4⋊C4—471no—
197C4×C22⋊C4—941no—
198C43C2×D4711yescharacteristic
199#336—472no—
200#336—941no—
201#336C2×D4712yes—
202C47#762811yescharacteristic, complemented, Sylow
203C22×D4—471no—
204C23×C4—941no—
205C22×C8—22no—
206C22×C8—942no—
207C4×D4—942no—
208C4×D4—942no—
209#117—1884no—
210C2×C4⋊C4—471no—
211C2×C4⋊C4—471no—
212C2×C4⋊C4—471no—
213C2×C4⋊C4—941no—
214C2×C4⋊C4—941no—
215C2×C22⋊C4—941no—
216C2×C22⋊C4—941no—
217C2×C22⋊C4—941no—
218C2×C22⋊C4—941no—
219C2×C22⋊C4—941no—
220C2×C42—941no—
221C2×C42C4×D4711yescharacteristic
222C2×C42C4×D4711yescharacteristic
223C2×C42D18811yescharacteristic
224C2×C42D18811yescharacteristic
225C2×C42C22⋊D4711yescharacteristic
226C2×C42C22⋊D4711yescharacteristic
227C2×C42C22×D4711yescharacteristic
228C4⋊C8—24no—
229#97—42no—
230#97—944no—
231#95—941no—
232#95—941no—
233#95—941no—
234C24—941no—
235C2×D4—472no—
236C2×D4—941no—
237C2×D4—1881no—
238C2×D4—1881no—
239C22×C4—944no—
240C22×C4—941no—
241C22×C4—941no—
242C22×C4—941no—
243C22×C4—1882no—
244C22×C4—1882no—
245C22×C4(C2×D47)⋊C411yescharacteristic
246C22×C4(C2×D47)⋊C411yescharacteristic
247C22×C4(C2×D47)⋊C411yescharacteristic
248C22×C4(C2×D47)⋊C411yescharacteristic
249C22×C4C2×C4×D4711yescharacteristic
250C22×C4C2×D18811yescharacteristic
251C22×C4C2×C22⋊D4711yescharacteristic
252C2×C8—24no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—944no—
256C2×C8—1882no—
257C2×C8—1882no—
258C4⋊C4—472no—
259C4⋊C4—941no—
260C4⋊C4—1881no—
261C4⋊C4—1881no—
262C4⋊C4—1882no—
263C4⋊C4—1882no—
264C22⋊C4—944no—
265C22⋊C4—1882no—
266C22⋊C4—1882no—
267C22⋊C4—1881no—
268C22⋊C4—1882no—
269C22⋊C4—1881no—
270C42—22no—
271C42—22no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—22no—
276C42—21no—
277C42—41no—
278C42—41no—
279C42—1882no—
280C23—942no—
281C23—1881no—
282C23—1881no—
283C23—1881no—
284C23—1881no—
285C23—1881no—
286C23—1881no—
287C23C2×(C2×D47)⋊C411yescharacteristic
288D4—1882no—
289D4—1882no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—942no—
307C2×C4—1888no—
308C2×C4—1881no—
309C2×C4—1881no—
310C2×C4—1881no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4—1881no—
314C2×C4—1881no—
315C2×C4—1881no—
316C2×C4#1309395012yes—
317C2×C4#1309395212yes—
318C2×C4M16×D4711yescharacteristic
319C2×C4M16×D4711yescharacteristic
320C2×C4#1309405011yescharacteristic
321C2×C4#1309405212yes—
322C2×C4#1309405511yescharacteristic
323C8—44no—
324C8—1884no—
325C22—3761no—
326C22—3761no—
327C22—3761no—
328C22—3761no—
329C22—1882no—
330C22—1882no—
331C22—1882no—
332C22#1568908611yescharacteristic
333C22#1568909611yescharacteristic
334C22#1568910511yescharacteristic
335C22#1568912211yescharacteristic
336C22#1568912911yescharacteristic
337C22C2×#1309395011yescharacteristic
338C22C2×#1309395211yescharacteristic
339C4—22no—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—3761no—
349C4—3761no—
350C4—1882no—
351C2—3762no—
352C2#2024172011yescharacteristic
353C2#2024182511yescharacteristic
354C2#2024183211yescharacteristic
355C2#2024189711yescharacteristic
356C2#2024191111yescharacteristic
357C2#2024197411yescharacteristic
358C2#2024210811yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1767C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1767D47
hypocenterC47#7628
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
2. #20243137C2semidirect
3. #20241210C2semidirect
5. C47×#1767C2semidirect
173. #1767D47semidirect
202. C47#7628semidirect

Maximal subgroups

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

subgroupindex
#202458522
#202431372
#202412102
#202412082
C47×#17672
#762847