Finite Group Databasebeta

← C47×#9337 #26049035

Subgroups — C47×#9337 #26049035

Order 12032.

412 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3218C211yesmaximal, characteristic
2C47×#2995C211yesmaximal, characteristic
3C47×#1997C211yesmaximal, characteristic, complemented
4C47×#1558C211yesmaximal, characteristic, complemented
5C47×#1529C211yesmaximal, characteristic
6C47×#1307C211yesmaximal, characteristic, complemented
7C47×#1291C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C47C2211yescharacteristic
9C47×#529C2211yescharacteristic
10C47×#497—21no—
11C47×#491—21no—
12C47×#462—21no—
13C47×#461—21no—
14C47×#451C412yes—
15C47×#448C412yes—
16C47×#446—21no—
17C47×#436—21no—
18C47×#435—21no—
19C47×#435—21no—
20C47×#433—21no—
21C47×#432C2211yescharacteristic
22C47×#432C2211yescharacteristic
23C47×#432C2211yescharacteristic
24C47×#424—21no—
25C47×#423C2211yescharacteristic
26C4×M16×C47C2211yescharacteristic
27C47×#343—21no—
28C47×#343—21no—
29C47×#333—21no—
30C47×#333—21no—
31C47×#330—21no—
32C47×#329—21no—
33C22×Q8×C47C2×C412yes—
34C2×Q16×C47—22no—
35C2×SD16×C47—22no—
36C2×M16×C47—21no—
37C2×M16×C47—21no—
38C2×M16×C47D411yescharacteristic
39C2×M16×C47D411yescharacteristic
40C22×C8×C47—21no—
41C4⋊Q8×C47—21no—
42C4⋊Q8×C47—21no—
43C4⋊Q8×C47—21no—
44C4⋊Q8×C47—21no—
45#126×C47—21no—
46#125×C47D411yescharacteristic
47#124×C47D411yescharacteristic
48#123×C47—21no—
49C22⋊Q8×C47C2×C411yescharacteristic
50#121×C47C2×C411yescharacteristic
51C4×Q8×C47D411yescharacteristic
52C4×D4×C47—21no—
53C4×D4×C47D411yescharacteristic
54#117×C47—21no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C4⋊C8×C47D411yescharacteristic
65C4⋊C8×C47D411yescharacteristic
66#103×C47—22no—
67#103×C47—22no—
68C22⋊C8×C47—21no—
69C22⋊C8×C47—21no—
70C22⋊C8×C47—21no—
71C22⋊C8×C47—21no—
72C22⋊C8×C47C2×C412yes—
73#97×C47—21no—
74#97×C47—21no—
75#97×C47—21no—
76#97×C47—21no—
77#97×C47—21no—
78C4×C8×C47—21no—
79C4×C8×C47—21no—
80C2×Q8×C47—22no—
81C2×Q8×C47—22no—
82C2×Q8×C47—82no—
83C2×Q8×C47C22⋊C411yescharacteristic
84C2×D4×C47—21no—
85C2×D4×C47C22⋊C411yescharacteristic
86C22×C4×C47—21no—
87C22×C4×C47—22no—
88C22×C4×C47C22×C411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47#4111yescharacteristic
91Q16×C47—82no—
92SD16×C47—82no—
93M16×C47—22no—
94M16×C47—41no—
95M16×C47—41no—
96M16×C47—41no—
97M16×C47—81no—
98C2×C8×C47—22no—
99C2×C8×C47—21no—
100C2×C8×C47—21no—
101C2×C8×C47—21no—
102C2×C8×C47—21no—
103C2×C8×C47—41no—
104C2×C8×C47—41no—
105C2×C8×C47—81no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—41no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47—41no—
113C4⋊C4×C47—41no—
114C4⋊C4×C47C22⋊C411yescharacteristic
115C4⋊C4×C47C22⋊C411yescharacteristic
116C22⋊C4×C47—21no—
117C22⋊C4×C47—21no—
118C22⋊C4×C47—21no—
119C22⋊C4×C47—21no—
120C22⋊C4×C47—21no—
121C42×C47—21no—
122C42×C47C2×D411yescharacteristic
123C42×C47C2×D411yescharacteristic
124C42×C47C2×D411yescharacteristic
125C42×C47#4111yescharacteristic
126C23×C47—21no—
127C23×C47C4×D411yescharacteristic
128Q8×C47—41no—
129Q8×C47—42no—
130Q8×C47—42no—
131Q8×C47—82no—
132D4×C47—41no—
133D4×C47—81no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—42no—
144C2×C4×C47—82no—
145C2×C4×C47C2×C22⋊C411yescharacteristic
146C2×C4×C47C4×D411yescharacteristic
147C2×C4×C47#12011yescharacteristic
148C2×C4×C47#12111yescharacteristic
149C2×C4×C47#12111yescharacteristic
150C2×C4×C47#12311yescharacteristic
151C8×C47—42no—
152C8×C47—42no—
153C8×C47—41no—
154C8×C47—41no—
155C8×C47—41no—
156C8×C47—41no—
157C8×C47—81no—
158#9337C4711yesmaximal, characteristic, direct product factor, Sylow
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—41no—
163C22×C47—41no—
164C22×C47—41no—
165C22×C47#38611yescharacteristic
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—41no—
172C4×C47—41no—
173C4×C47—41no—
174C4×C47—82no—
175C4×C47#45511yescharacteristic
176C4×C47#45611yescharacteristic
177#3218C2×C4711yescharacteristic
178#2995C2×C4711yescharacteristic
179#1997C2×C4711yescharacteristic, complemented
180#1558C2×C4711yescharacteristic, complemented
181#1529C2×C4711yescharacteristic
182#1307C2×C4711yescharacteristic, complemented
183#1291C2×C4711yescharacteristic, complemented
184C2×C47—41no—
185C2×C47—81no—
186C2×C47#188211yescharacteristic
187C2×C47#190211yescharacteristic
188C2×C47#191411yescharacteristic
189C2×C4⋊Q8C22×C4711yescharacteristic
190#529C22×C4711yescharacteristic
191#497—21no—
192#491—21no—
193#462—21no—
194#461—21no—
195#451C4×C4712yes—
196#448C4×C4712yes—
197#446—21no—
198#436—21no—
199#435—21no—
200#435—21no—
201#433—21no—
202#432C22×C4711yescharacteristic
203#432C22×C4711yescharacteristic
204#432C22×C4711yescharacteristic
205#424—21no—
206#423C22×C4711yescharacteristic
207C4×M16C22×C4711yescharacteristic
208#343—21no—
209#343—21no—
210#333—21no—
211#333—21no—
212#330—21no—
213#329—21no—
214C47#933711yescharacteristic, direct product factor, Sylow
215C22×Q8C2×C4×C4712yes—
216C2×Q16—22no—
217C2×SD16—22no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16D4×C4711yescharacteristic
221C2×M16D4×C4711yescharacteristic
222C22×C8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227#126—21no—
228#125D4×C4711yescharacteristic
229#124D4×C4711yescharacteristic
230#123—21no—
231C22⋊Q8C2×C4×C4711yescharacteristic
232#121C2×C4×C4711yescharacteristic
233C4×Q8D4×C4711yescharacteristic
234C4×D4—21no—
235C4×D4D4×C4711yescharacteristic
236#117—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C42C23×C4711yescharacteristic
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8D4×C4711yescharacteristic
247C4⋊C8D4×C4711yescharacteristic
248#103—22no—
249#103—22no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8C2×C4×C4712yes—
255#97—21no—
256#97—21no—
257#97—21no—
258#97—21no—
259#97—21no—
260C4×C8—21no—
261C4×C8—21no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—82no—
265C2×Q8C22⋊C4×C4711yescharacteristic
266C2×D4—21no—
267C2×D4C22⋊C4×C4711yescharacteristic
268C22×C4—21no—
269C22×C4—22no—
270C22×C4C22×C4×C4711yescharacteristic
271C22×C4C2×D4×C4711yescharacteristic
272C22×C4#41×C4711yescharacteristic
273Q16—82no—
274SD16—82no—
275M16—22no—
276M16—41no—
277M16—41no—
278M16—41no—
279M16—81no—
280C2×C8—22no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—81no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4C22⋊C4×C4711yescharacteristic
297C4⋊C4C22⋊C4×C4711yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C42—21no—
304C42C2×D4×C4711yescharacteristic
305C42C2×D4×C4711yescharacteristic
306C42C2×D4×C4711yescharacteristic
307C42#41×C4711yescharacteristic
308C23—21no—
309C23C4×D4×C4711yescharacteristic
310Q8—41no—
311Q8—42no—
312Q8—42no—
313Q8—82no—
314D4—41no—
315D4—81no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4C2×C22⋊C4×C4711yescharacteristic
328C2×C4C4×D4×C4711yescharacteristic
329C2×C4#120×C4711yescharacteristic
330C2×C4#121×C4711yescharacteristic
331C2×C4#121×C4711yescharacteristic
332C2×C4#123×C4711yescharacteristic
333C8—42no—
334C8—42no—
335C8—41no—
336C8—41no—
337C8—41no—
338C8—41no—
339C8—81no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22C47×#38611yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—82no—
356C4C47×#45511yescharacteristic
357C4C47×#45611yescharacteristic
358C2—41no—
359C2—81no—
360C2C47×#188211yescharacteristic
361C2C47×#190211yescharacteristic
362C2C47×#191411yescharacteristic
subgroupG/H
center, socleC22×C47#386
commutatorC22×C4C22×C4×C47
Frattini, cosocleC2×C42C23×C47
prime radical#121×C47C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C47 ◁ C2×C42×C47 ◁ G

Complements

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

NH ≅ G/Ntype
3. C47×#1997C2semidirect
4. C47×#1558C2semidirect
6. C47×#1307C2semidirect
7. C47×#1291C2semidirect
158. #9337C47direct (N × H)
179. #1997C2×C47semidirect
180. #1558C2×C47semidirect
182. #1307C2×C47semidirect
183. #1291C2×C47semidirect
214. C47#9337direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#32182
C47×#29952
C47×#19972
C47×#15582
C47×#15292
C47×#13072
C47×#12912
#933747