Finite Group Databasebeta

← #26104782

Subgroups — #26104782

Order 12032.

444 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 107 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#20251755C211yesmaximal, characteristic
2#20243137C211yesmaximal, characteristic, complemented
3#20241212C212yesmaximal, complemented
4#20241208C212yesmaximal
5C47×#1744C211yesmaximal, characteristic, complemented
6C2×#13094018—21no—
7C2×#13093935—21no—
8#15689025C411yescharacteristic
9#15689025C411yescharacteristic
10#15689025C2211yescharacteristic
11#15689021C411yescharacteristic
12#15689021C411yescharacteristic
13#15689021C2211yescharacteristic
14#15689014—21no—
15#15689014—21no—
16#15689014—21no—
17#15688878—21no—
18#15688878C2212yes—
19C2×C4×C8×C47—21no—
20C43×C47C2211yescharacteristic
21C47×#336—21no—
22C47×#336C2212yes—
23C22×D188C2×C411yescharacteristic
24C22×C47⋊C8—22no—
25#13094018—44no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×(C2×D47)⋊C4—21no—
33C2×(C2×D47)⋊C4—21no—
34C2×(C2×D47)⋊C4—21no—
35C2×(C2×D47)⋊C4—21no—
36C2×(C2×D47)⋊C4—21no—
37C2×(C2×D47)⋊C4—21no—
38#13093935—24no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C22×C8×C47—22no—
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47C2311yescharacteristic
50C4×C8×C47—24no—
51C23×D47—21no—
52C22×C4×C47C22⋊C411yescharacteristic
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22×C411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×Dic47—21no—
60C22×Dic47—21no—
61C22×Dic47—21no—
62C2×D188—22no—
63C2×D188—41no—
64C2×D188—41no—
65C2×C8×C47—24no—
66C2×C8×C47—42no—
67C2×C8×C47—42no—
68C42×C47—22no—
69C42×C47—22no—
70C42×C47—22no—
71C42×C47—22no—
72C42×C47—22no—
73C42×C47—41no—
74C42×C47—41no—
75(C2×D47)⋊C4—42no—
76(C2×D47)⋊C4—42no—
77(C2×D47)⋊C4—42no—
78(C2×D47)⋊C4—42no—
79(C2×D47)⋊C4—42no—
80(C2×D47)⋊C4—42no—
81C4⋊Dic47—22no—
82C4⋊Dic47—41no—
83C4⋊Dic47—41no—
84Dic47⋊C4—42no—
85Dic47⋊C4—42no—
86Dic47⋊C4—42no—
87C2×C47⋊C8—24no—
88C2×C47⋊C8—42no—
89C2×C47⋊C8—42no—
90C23×C47C2×C22⋊C411yescharacteristic
91C22×D47—41no—
92C22×D47—41no—
93C22×D47—41no—
94C22×D47—41no—
95C22×D47—41no—
96C22×D47—41no—
97C22×D47—41no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
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#10412yes—
115C2×C4×C47#10412yes—
116C2×C4×C47#13112yes—
117C2×C4×C47#13512yes—
118C2×C4×C47AGL(1,ℤ/8)12yes—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128D188—82no—
129C8×C47—44no—
130C47⋊C8—44no—
131#7692—471nomaximal, Sylow
132C22×C47#40811yescharacteristic
133C22×C47#41011yescharacteristic
134C22×C47#41311yescharacteristic
135C22×C47#41611yescharacteristic
136C22×C47#41811yescharacteristic
137C22×C47C2×#10411yescharacteristic
138C22×C47C2×#10411yescharacteristic
139C2×D47—81no—
140C2×D47—81no—
141C2×D47—81no—
142C2×D47—81no—
143C2×D47—81no—
144C2×D47—81no—
145C2×D47—81no—
146C4×C47—22no—
147C4×C47—22no—
148C4×C47—22no—
149C4×C47—22no—
150C4×C47—22no—
151C4×C47—41no—
152C4×C47—41no—
153Dic47—81no—
154Dic47—81no—
155Dic47—81no—
156#2860—471no—
157#1745—471no—
158#1744D4711yescharacteristic, complemented
159#1271—472no—
160#1270—472no—
161C2×C47#147311yescharacteristic
162C2×C47#148411yescharacteristic
163C2×C47#149411yescharacteristic
164C2×C47#150011yescharacteristic
165C2×C47#150511yescharacteristic
166C2×C47#152411yescharacteristic
167C2×C47#155311yescharacteristic
168D47—161no—
169C2×#126—941no—
170C2×#97—941no—
171C2×C4×C8—21no—
172#390—471no—
173#390—471no—
174#390—471no—
175#389—471no—
176#389—471no—
177#389—471no—
178#388—941no—
179#388—941no—
180#388—941no—
181C43C2×D4711yescharacteristic
182#336—21no—
183#336—472no—
184#336—941no—
185#336C2×D4712yes—
186C47#769211yescharacteristic, complemented, Sylow
187C22×D4—471no—
188C22×C8—22no—
189C22×C8—942no—
190#126—1884no—
191C2×C4⋊C4—471no—
192C2×C4⋊C4—471no—
193C2×C4⋊C4—471no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4—941no—
197C2×C22⋊C4—941no—
198C2×C22⋊C4—941no—
199C2×C22⋊C4—941no—
200C2×C22⋊C4—941no—
201C2×C22⋊C4—941no—
202C2×C22⋊C4—941no—
203C2×C42C4×D4711yescharacteristic
204C2×C42C4×D4711yescharacteristic
205C2×C42D18811yescharacteristic
206C2×C42D18811yescharacteristic
207C2×C42C22⋊D4711yescharacteristic
208C2×C42C22⋊D4711yescharacteristic
209C2×C42C22×D4711yescharacteristic
210#97—944no—
211C4×C8—24no—
212#95—941no—
213#95—941no—
214#95—941no—
215C24—941no—
216C2×D4—942no—
217C2×D4—1881no—
218C2×D4—1881no—
219C22×C4—941no—
220C22×C4—941no—
221C22×C4—941no—
222C22×C4(C2×D47)⋊C411yescharacteristic
223C22×C4(C2×D47)⋊C411yescharacteristic
224C22×C4(C2×D47)⋊C411yescharacteristic
225C22×C4(C2×D47)⋊C411yescharacteristic
226C22×C4C2×C4×D4711yescharacteristic
227C22×C4C2×D18811yescharacteristic
228C22×C4C2×C22⋊D4711yescharacteristic
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—944no—
233C2×C8—1882no—
234C2×C8—1882no—
235C4⋊C4—942no—
236C4⋊C4—1881no—
237C4⋊C4—1881no—
238C4⋊C4—1882no—
239C4⋊C4—1882no—
240C4⋊C4—1882no—
241C22⋊C4—1882no—
242C22⋊C4—1882no—
243C22⋊C4—1882no—
244C22⋊C4—1882no—
245C22⋊C4—1882no—
246C22⋊C4—1882no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—41no—
253C42—41no—
254C23—1881no—
255C23—1881no—
256C23—1881no—
257C23—1881no—
258C23—1881no—
259C23—1881no—
260C23—1881no—
261C23C2×(C2×D47)⋊C411yescharacteristic
262D4—3762no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—1881no—
280C2×C4—1881no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4#1309395012yes—
289C2×C4#1309395212yes—
290C2×C4#1309403012yes—
291C2×C4#1309404612yes—
292C2×C4#1309404912yes—
293C8—44no—
294C8—1884no—
295C22—3761no—
296C22—3761no—
297C22—3761no—
298C22—3761no—
299C22—3761no—
300C22—3761no—
301C22—3761no—
302C22#1568908911yescharacteristic
303C22#1568909611yescharacteristic
304C22#1568910511yescharacteristic
305C22#1568912011yescharacteristic
306C22#1568912711yescharacteristic
307C22C2×#1309395011yescharacteristic
308C22C2×#1309395211yescharacteristic
309C4—22no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—3761no—
317C4—3761no—
318C4—3761no—
319C2—7521no—
320C2#2024175511yescharacteristic
321C2#2024180411yescharacteristic
322C2#2024184911yescharacteristic
323C2#2024187511yescharacteristic
324C2#2024189511yescharacteristic
325C2#2024197411yescharacteristic
326C2#2024209111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1744C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1744D47
hypocenterC47#7692
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ #1744

Complements

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

NH ≅ G/Ntype
2. #20243137C2semidirect
3. #20241212C2semidirect
5. C47×#1744C2semidirect
158. #1744D47semidirect
186. C47#7692semidirect

Maximal subgroups

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

subgroupindex
#202517552
#202431372
#202412122
#202412082
C47×#17442
#769247