Finite Group Databasebeta

← #26104849

Subgroups — #26104849

Order 12032.

426 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 103 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#20243505C211yesmaximal, characteristic, complemented
3#20241212C212yesmaximal, complemented
4#20241208C212yesmaximal
5C47×#1745C211yesmaximal, characteristic, complemented
6C2×#13094018—21no—
7C2×C4⋊(C47⋊C8)—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×#97×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—
38C4⋊(C47⋊C8)—42no—
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
50#97×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—21no—
71C42×C47—22no—
72C42×C47—21no—
73C42×C47—41no—
74C42×C47—41no—
75C42×C47—41no—
76(C2×D47)⋊C4—42no—
77(C2×D47)⋊C4—42no—
78(C2×D47)⋊C4—42no—
79(C2×D47)⋊C4—42no—
80(C2×D47)⋊C4—41no—
81(C2×D47)⋊C4—42no—
82(C2×D47)⋊C4—41no—
83C4⋊Dic47—22no—
84C4⋊Dic47—41no—
85C4⋊Dic47—41no—
86Dic47⋊C4—42no—
87Dic47⋊C4—42no—
88Dic47⋊C4—42no—
89C2×C47⋊C8—42no—
90C2×C47⋊C8—42no—
91C2×C47⋊C8—42no—
92C23×C47C2×C22⋊C411yescharacteristic
93C22×D47—41no—
94C22×D47—41no—
95C22×D47—41no—
96C22×D47—41no—
97C22×D47—41no—
98C22×D47—41no—
99C22×D47—41no—
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—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47#10412yes—
118C2×C4×C47#13112yes—
119C2×C4×C47AGL(1,ℤ/8)12yes—
120C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C47#13711yescharacteristic
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131D188—82no—
132C8×C47—44no—
133C47⋊C8—82no—
134#7706—471nomaximal, Sylow
135C22×C47#40811yescharacteristic
136C22×C47#41011yescharacteristic
137C22×C47#41311yescharacteristic
138C22×C47#41711yescharacteristic
139C22×C47#41811yescharacteristic
140C22×C47C2×#10411yescharacteristic
141C22×C47#42111yescharacteristic
142C2×D47—81no—
143C2×D47—81no—
144C2×D47—81no—
145C2×D47—81no—
146C2×D47—81no—
147C2×D47—81no—
148C2×D47—81no—
149C4×C47—22no—
150C4×C47—22no—
151C4×C47—22no—
152C4×C47—21no—
153C4×C47—21no—
154C4×C47—41no—
155C4×C47—41no—
156C4×C47—41no—
157Dic47—81no—
158Dic47—81no—
159Dic47—81no—
160#2860—471no—
161#1831—471no—
162#1745D4711yescharacteristic, complemented
163#1271—472no—
164#1270—472no—
165C2×C47#147311yescharacteristic
166C2×C47#148311yescharacteristic
167C2×C47#149611yescharacteristic
168C2×C47#150511yescharacteristic
169C2×C47#151211yescharacteristic
170C2×C47#152911yescharacteristic
171C2×C47#155711yescharacteristic
172D47—161no—
173C2×#126—941no—
174C2×C4⋊C8—941no—
175C2×#97—21no—
176#390—471no—
177#390—471no—
178#390—471no—
179#389—471no—
180#389—471no—
181#389—471no—
182#388—941no—
183#388—941no—
184#388—941no—
185C43C2×D4711yescharacteristic
186#336—21no—
187#336—472no—
188#336—941no—
189#336C2×D4712yes—
190C47#770611yescharacteristic, complemented, Sylow
191C22×D4—471no—
192C22×C8—22no—
193C22×C8—942no—
194#126—1884no—
195C2×C4⋊C4—471no—
196C2×C4⋊C4—471no—
197C2×C4⋊C4—471no—
198C2×C4⋊C4—941no—
199C2×C4⋊C4—941no—
200C2×C4⋊C4—941no—
201C2×C22⋊C4—941no—
202C2×C22⋊C4—941no—
203C2×C22⋊C4—941no—
204C2×C22⋊C4—941no—
205C2×C22⋊C4—941no—
206C2×C22⋊C4—941no—
207C2×C42C4×D4711yescharacteristic
208C2×C42C4×D4711yescharacteristic
209C2×C42D18811yescharacteristic
210C2×C42D18811yescharacteristic
211C2×C42C22⋊D4711yescharacteristic
212C2×C42C22⋊D4711yescharacteristic
213C2×C42C22×D4711yescharacteristic
214C4⋊C8—1882no—
215#97—24no—
216#95—941no—
217#95—941no—
218#95—941no—
219C24—941no—
220C2×D4—942no—
221C2×D4—1881no—
222C2×D4—1881no—
223C22×C4—941no—
224C22×C4—941no—
225C22×C4—941no—
226C22×C4(C2×D47)⋊C411yescharacteristic
227C22×C4(C2×D47)⋊C411yescharacteristic
228C22×C4(C2×D47)⋊C411yescharacteristic
229C22×C4(C2×D47)⋊C411yescharacteristic
230C22×C4C2×C4×D4711yescharacteristic
231C22×C4C2×D18811yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233C2×C8—24no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—1882no—
237C2×C8—1882no—
238C2×C8—1882no—
239C4⋊C4—942no—
240C4⋊C4—1881no—
241C4⋊C4—1881no—
242C4⋊C4—1882no—
243C4⋊C4—1882no—
244C4⋊C4—1882no—
245C22⋊C4—1882no—
246C22⋊C4—1882no—
247C22⋊C4—1882no—
248C22⋊C4—1882no—
249C22⋊C4—1882no—
250C22⋊C4—1881no—
251C22⋊C4—1881no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—21no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42—41no—
260C23—1881no—
261C23—1881no—
262C23—1881no—
263C23—1881no—
264C23—1881no—
265C23—1881no—
266C23—1881no—
267C23C2×(C2×D47)⋊C411yescharacteristic
268D4—3762no—
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—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1881no—
290C2×C4—1881no—
291C2×C4—1881no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4—1881no—
295C2×C4#1309395212yes—
296C2×C4#1309403212yes—
297C2×C4#1309404912yes—
298C2×C4#1309405211yescharacteristic
299C2×C4#1309405711yescharacteristic
300C8—44no—
301C8—3762no—
302C22—3761no—
303C22—3761no—
304C22—3761no—
305C22—3761no—
306C22—3761no—
307C22—3761no—
308C22—3761no—
309C22#1568909111yescharacteristic
310C22#1568909611yescharacteristic
311C22#1568910511yescharacteristic
312C22#1568912511yescharacteristic
313C22#1568912711yescharacteristic
314C22C2×#1309395211yescharacteristic
315C22#1568913911yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—3761no—
325C4—3761no—
326C4—3761no—
327C2—7521no—
328C2#2024175811yescharacteristic
329C2#2024180211yescharacteristic
330C2#2024185611yescharacteristic
331C2#2024189511yescharacteristic
332C2#2024192711yescharacteristic
333C2#2024199311yescharacteristic
334C2#2024211111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1745C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1745D47
hypocenterC47#7706
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ #1745

Complements

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

NH ≅ G/Ntype
2. #20243505C2semidirect
3. #20241212C2semidirect
5. C47×#1745C2semidirect
162. #1745D47semidirect
190. C47#7706semidirect

Maximal subgroups

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

subgroupindex
#202517552
#202435052
#202412122
#202412082
C47×#17452
#770647