Finite Group Databasebeta

← C47×#7726 #26047424

Subgroups — C47×#7726 #26047424

Order 12032.

400 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#2860C211yesmaximal, characteristic
2C47×#1839C211yesmaximal, characteristic, complemented
3C47×#1839C211yesmaximal, characteristic, complemented
4C47×#1271C212yesmaximal, complemented
5C47×#1270C212yesmaximal
6C2×#126×C47—21no—
7C47×#390C411yescharacteristic
8C47×#390C411yescharacteristic
9C47×#390C2211yescharacteristic
10C47×#389C411yescharacteristic
11C47×#389C411yescharacteristic
12C47×#389C2211yescharacteristic
13C47×#388—21no—
14C47×#388—21no—
15C47×#388—21no—
16C43×C47C2211yescharacteristic
17C47×#336—21no—
18C47×#336—21no—
19C47×#336—21no—
20C47×#336—21no—
21C47×#336C2212yes—
22C47×#336C2212yes—
23C22×D4×C47C2×C411yescharacteristic
24C22×C8×C47—22no—
25C22×C8×C47—22no—
26#126×C47—44no—
27C2×C4⋊C4×C47—21no—
28C2×C4⋊C4×C47—21no—
29C2×C4⋊C4×C47—21no—
30C2×C4⋊C4×C47C2×C411yescharacteristic
31C2×C4⋊C4×C47C2×C411yescharacteristic
32C2×C4⋊C4×C47C2×C411yescharacteristic
33C2×C22⋊C4×C47—21no—
34C2×C22⋊C4×C47—21no—
35C2×C22⋊C4×C47—21no—
36C2×C22⋊C4×C47—21no—
37C2×C22⋊C4×C47—21no—
38C2×C22⋊C4×C47—21no—
39C2×C42×C47C2×C411yescharacteristic
40C2×C42×C47C2×C411yescharacteristic
41C2×C42×C47D411yescharacteristic
42C2×C42×C47D411yescharacteristic
43C2×C42×C47D411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C2×C42×C47C2311yescharacteristic
46#95×C47—21no—
47#95×C47—21no—
48#95×C47—21no—
49C24×C47—21no—
50C2×D4×C47—22no—
51C2×D4×C47—41no—
52C2×D4×C47—41no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C2×C8×C47—42no—
64C2×C8×C47—42no—
65C2×C8×C47—42no—
66C2×C8×C47—42no—
67C2×C8×C47—42no—
68C2×C8×C47—42no—
69C4⋊C4×C47—22no—
70C4⋊C4×C47—42no—
71C4⋊C4×C47—42no—
72C4⋊C4×C47—41no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—42no—
75C22⋊C4×C47—42no—
76C22⋊C4×C47—42no—
77C22⋊C4×C47—42no—
78C22⋊C4×C47—42no—
79C22⋊C4×C47—42no—
80C22⋊C4×C47—42no—
81C42×C47—22no—
82C42×C47—22no—
83C42×C47—22no—
84C42×C47—41no—
85C42×C47—41no—
86C42×C47—41no—
87C42×C47—41no—
88C23×C47—41no—
89C23×C47—41no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47—41no—
94C23×C47—41no—
95C23×C47C2×C22⋊C411yescharacteristic
96D4×C47—82no—
97C2×C4×C47—21no—
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—21no—
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—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47#13112yes—
125C2×C4×C47#13512yes—
126C2×C4×C47AGL(1,ℤ/8)12yes—
127C8×C47—82no—
128C8×C47—82no—
129#7726C4711yesmaximal, characteristic, direct product factor, Sylow
130C22×C47—81no—
131C22×C47—81no—
132C22×C47—81no—
133C22×C47—81no—
134C22×C47—81no—
135C22×C47—81no—
136C22×C47—81no—
137C22×C47#40811yescharacteristic
138C22×C47#41011yescharacteristic
139C22×C47#41311yescharacteristic
140C22×C47#41611yescharacteristic
141C22×C47#41811yescharacteristic
142C22×C47#42111yescharacteristic
143C22×C47#42111yescharacteristic
144C4×C47—22no—
145C4×C47—22no—
146C4×C47—22no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—81no—
152C4×C47—81no—
153C4×C47—81no—
154#2860C2×C4711yescharacteristic
155#1839C2×C4711yescharacteristic, complemented
156#1839C2×C4711yescharacteristic, complemented
157#1271C2×C4712yescomplemented
158#1270C2×C4712yes—
159C2×C47—161no—
160C2×C47#147311yescharacteristic
161C2×C47#148211yescharacteristic
162C2×C47#149611yescharacteristic
163C2×C47#151411yescharacteristic
164C2×C47#152111yescharacteristic
165C2×C47#153511yescharacteristic
166C2×C47#155311yescharacteristic
167C2×#126—21no—
168#390C4×C4711yescharacteristic
169#390C4×C4711yescharacteristic
170#390C22×C4711yescharacteristic
171#389C4×C4711yescharacteristic
172#389C4×C4711yescharacteristic
173#389C22×C4711yescharacteristic
174#388—21no—
175#388—21no—
176#388—21no—
177C43C22×C4711yescharacteristic
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336C22×C4712yes—
183#336C22×C4712yes—
184C47#772611yescharacteristic, direct product factor, Sylow
185C22×D4C2×C4×C4711yescharacteristic
186C22×C8—22no—
187C22×C8—22no—
188#126—44no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C2×C4×C4711yescharacteristic
193C2×C4⋊C4C2×C4×C4711yescharacteristic
194C2×C4⋊C4C2×C4×C4711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C42C2×C4×C4711yescharacteristic
202C2×C42C2×C4×C4711yescharacteristic
203C2×C42D4×C4711yescharacteristic
204C2×C42D4×C4711yescharacteristic
205C2×C42D4×C4711yescharacteristic
206C2×C42D4×C4711yescharacteristic
207C2×C42C23×C4711yescharacteristic
208#95—21no—
209#95—21no—
210#95—21no—
211C24—21no—
212C2×D4—22no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4C22⋊C4×C4711yescharacteristic
219C22×C4C22⋊C4×C4711yescharacteristic
220C22×C4C22⋊C4×C4711yescharacteristic
221C22×C4C22⋊C4×C4711yescharacteristic
222C22×C4C22×C4×C4711yescharacteristic
223C22×C4C2×D4×C4711yescharacteristic
224C22×C4C2×D4×C4711yescharacteristic
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊C4×C4711yescharacteristic
258D4—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
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—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4#131×C4712yes—
287C2×C4#135×C4712yes—
288C2×C4AGL(1,ℤ/8)×C4712yes—
289C8—82no—
290C8—82no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22C47×#40811yescharacteristic
299C22C47×#41011yescharacteristic
300C22C47×#41311yescharacteristic
301C22C47×#41611yescharacteristic
302C22C47×#41811yescharacteristic
303C22C47×#42111yescharacteristic
304C22C47×#42111yescharacteristic
305C4—22no—
306C4—22no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4—81no—
315C2—161no—
316C2C47×#147311yescharacteristic
317C2C47×#148211yescharacteristic
318C2C47×#149611yescharacteristic
319C2C47×#151411yescharacteristic
320C2C47×#152111yescharacteristic
321C2C47×#153511yescharacteristic
322C2C47×#155311yescharacteristic
subgroupG/H
center, socleC23×C47C2×C22⋊C4
commutatorC22×C4C22×C4×C47
Frattini, cosocleC2×C42C23×C47
prime radicalC22×D4×C47C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C47 ◁ C43×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C47×#1839C2semidirect
3. C47×#1839C2semidirect
4. C47×#1271C2semidirect
129. #7726C47direct (N × H)
155. #1839C2×C47semidirect
156. #1839C2×C47semidirect
157. #1271C2×C47semidirect
184. C47#7726direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#28602
C47×#18392
C47×#12712
C47×#12702
#772647