Finite Group Databasebeta

← C47×#29792 #26069490

Subgroups — C47×#29792 #26069490

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3095C212yesmaximal
2C2×C47×#470C211yesmaximal, characteristic
3C47×#2905C211yesmaximal, characteristic
4C47×#2090C212yesmaximal, complemented
5C47×#2064C214yesmaximal, complemented
6C47×#2047C212yesmaximal, complemented
7C47×#1901C212yesmaximal
8C47×#1811C211yesmaximal, characteristic
9C22×Q16×C47C2211yescharacteristic
10C22×M16×C47C2211yescharacteristic
11C22×M16×C47C2211yescharacteristic
12C2×C22⋊Q8×C47C2212yes—
13C2×#117×C47C2211yescharacteristic
14C47×#485—24no—
15C47×#484—22no—
16C47×#483—22no—
17C47×#470—22no—
18C47×#470—22no—
19C2×#108×C47C2212yes—
20C47×#428C2214yes—
21C2×#106×C47C2211yescharacteristic
22C47×#424—22no—
23C47×#424C2212yes—
24C47×#424C2212yes—
25C47×#423—22no—
26C47×#419C2214yes—
27C2×#103×C47C2212yes—
28C2×#101×C47C2214yes—
29C47×#407C2212yes—
30C47×#343C2214yes—
31C47×#341C2212yes—
32C22×Q8×C47C2312yes—
33C23×C4×C47C2311yescharacteristic
34C2×Q16×C47—22no—
35C2×Q16×C47—42no—
36C2×M16×C47—21no—
37C2×M16×C47—42no—
38C2×M16×C47—42no—
39C2×M16×C47D412yes—
40C2×M16×C47D412yes—
41C2×M16×C47C2312yes—
42C2×M16×C47C2314yes—
43C22×C8×C47—21no—
44C22×C8×C47C2311yescharacteristic
45C22×C8×C47C2311yescharacteristic
46C22⋊Q8×C47—22no—
47C22⋊Q8×C47—22no—
48C22⋊Q8×C47—22no—
49C22⋊Q8×C47—22no—
50#117×C47—21no—
51#117×C47—21no—
52#117×C47D412yes—
53#117×C47C2312yes—
54C2×C4⋊C4×C47—22no—
55C2×C4⋊C4×C47D412yes—
56C2×C4⋊C4×C47C2312yes—
57C2×C22⋊C4×C47—21no—
58C2×C22⋊C4×C47—22no—
59C2×C42×C47—21no—
60#108×C47—42no—
61#107×C47—24no—
62#106×C47—22no—
63#106×C47—24no—
64C4⋊C8×C47—24no—
65C4⋊C8×C47—24no—
66#103×C47—24no—
67#103×C47—24no—
68#103×C47—24no—
69#101×C47—44no—
70C22⋊C8×C47—22no—
71C22⋊C8×C47—22no—
72C24×C47#4111yescharacteristic
73C2×Q8×C47—22no—
74C2×Q8×C47—22no—
75C2×Q8×C47—24no—
76C2×Q8×C47—24no—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47#4111yescharacteristic
87C22×C4×C47C2411yescharacteristic
88Q16×C47—44no—
89M16×C47—24no—
90M16×C47—24no—
91M16×C47—24no—
92M16×C47—42no—
93M16×C47—42no—
94M16×C47—42no—
95C2×C8×C47—22no—
96C2×C8×C47—22no—
97C2×C8×C47—24no—
98C2×C8×C47—22no—
99C2×C8×C47—42no—
100C2×C8×C47C2×D412yes—
101C2×C8×C47C2×D412yes—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—42no—
106C4⋊C4×C47—42no—
107C4⋊C4×C47C2×D414yes—
108C22⋊C4×C47—22no—
109C22⋊C4×C47—22no—
110C22⋊C4×C47—42no—
111C22⋊C4×C47—42no—
112C42×C47—22no—
113C42×C47—22no—
114C23×C47—22no—
115C23×C47—42no—
116C23×C47#12111yescharacteristic
117C23×C47#12111yescharacteristic
118C23×C47C2×#4111yescharacteristic
119Q8×C47—42no—
120Q8×C47—42no—
121Q8×C47—44no—
122C2×C4×C47—22no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—24no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—44no—
133C2×C4×C47—82no—
134C2×C4×C47—82no—
135C2×C4×C47#12111yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C2×C4×C47C22×D411yescharacteristic
138C2×C4×C47C22×D411yescharacteristic
139C2×C4×C47H(5,𝔽2)12yes—
140C2×C4×C472−1+412yes—
141C8×C47—22no—
142C8×C47—22no—
143C8×C47—42no—
144C8×C47—44no—
145#29792C4711yesmaximal, characteristic, direct product factor, Sylow
146C22×C47—22no—
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—22no—
150C22×C47—22no—
151C22×C47—82no—
152C22×C47C2×#12111yescharacteristic
153C22×C47#53912yes—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—44no—
157C4×C47—82no—
158C4×C47—82no—
159C4×C47#53411yescharacteristic
160C4×C47#53611yescharacteristic
161C4×C47#53712yes—
162#3095C2×C4712yes—
163C2×#470C2×C4711yescharacteristic
164#2905C2×C4711yescharacteristic
165#2090C2×C4712yescomplemented
166#2064C2×C4714yescomplemented
167#2047C2×C4712yescomplemented
168#1901C2×C4712yes—
169#1811C2×C4711yescharacteristic
170C2×C47—22no—
171C2×C47—41no—
172C2×C47—41no—
173C2×C47#261811yescharacteristic
174C2×C47#306912yes—
175C22×Q16C22×C4711yescharacteristic
176C22×M16C22×C4711yescharacteristic
177C22×M16C22×C4711yescharacteristic
178C2×C22⋊Q8C22×C4712yes—
179C2×#117C22×C4711yescharacteristic
180#485—24no—
181#484—22no—
182#483—22no—
183#470—22no—
184#470—22no—
185C2×#108C22×C4712yes—
186#428C22×C4714yes—
187C2×#106C22×C4711yescharacteristic
188#424—22no—
189#424C22×C4712yes—
190#424C22×C4712yes—
191#423—22no—
192#419C22×C4714yes—
193C2×#103C22×C4712yes—
194C2×#101C22×C4714yes—
195#407C22×C4712yes—
196#343C22×C4714yes—
197#341C22×C4712yes—
198C47#2979211yescharacteristic, direct product factor, Sylow
199C22×Q8C23×C4712yes—
200C23×C4C23×C4711yescharacteristic
201C2×Q16—22no—
202C2×Q16—42no—
203C2×M16—21no—
204C2×M16—42no—
205C2×M16—42no—
206C2×M16D4×C4712yes—
207C2×M16D4×C4712yes—
208C2×M16C23×C4712yes—
209C2×M16C23×C4714yes—
210C22×C8—21no—
211C22×C8C23×C4711yescharacteristic
212C22×C8C23×C4711yescharacteristic
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217#117—21no—
218#117—21no—
219#117D4×C4712yes—
220#117C23×C4712yes—
221C2×C4⋊C4—22no—
222C2×C4⋊C4D4×C4712yes—
223C2×C4⋊C4C23×C4712yes—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C42—21no—
227#108—42no—
228#107—24no—
229#106—22no—
230#106—24no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233#103—24no—
234#103—24no—
235#103—24no—
236#101—44no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C24#41×C4711yescharacteristic
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4C2×D4×C4711yescharacteristic
250C22×C4C2×D4×C4711yescharacteristic
251C22×C4C2×D4×C4711yescharacteristic
252C22×C4C2×D4×C4711yescharacteristic
253C22×C4#41×C4711yescharacteristic
254C22×C4C24×C4711yescharacteristic
255Q16—44no—
256M16—24no—
257M16—24no—
258M16—24no—
259M16—42no—
260M16—42no—
261M16—42no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—24no—
265C2×C8—22no—
266C2×C8—42no—
267C2×C8C2×D4×C4712yes—
268C2×C8C2×D4×C4712yes—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4C2×D4×C4714yes—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C42—22no—
280C42—22no—
281C23—22no—
282C23—42no—
283C23#121×C4711yescharacteristic
284C23#121×C4711yescharacteristic
285C23C2×#41×C4711yescharacteristic
286Q8—42no—
287Q8—42no—
288Q8—44no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4#121×C4711yescharacteristic
303C2×C4#121×C4711yescharacteristic
304C2×C4C22×D4×C4711yescharacteristic
305C2×C4C22×D4×C4711yescharacteristic
306C2×C4H(5,𝔽2)×C4712yes—
307C2×C42−1+4×C4712yes—
308C8—22no—
309C8—22no—
310C8—42no—
311C8—44no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—21no—
316C22—21no—
317C22—82no—
318C22C2×#121×C4711yescharacteristic
319C22C47×#53912yes—
320C4—21no—
321C4—21no—
322C4—44no—
323C4—82no—
324C4—82no—
325C4C47×#53411yescharacteristic
326C4C47×#53611yescharacteristic
327C4C47×#53712yes—
328C2—22no—
329C2—41no—
330C2—41no—
331C2C47×#261811yescharacteristic
332C2C47×#306912yes—
subgroupG/H
center, socleC22×C47C2×#121
commutator, Frattini, cosocleC22×C4C24×C47
prime radicalC24×C47#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C47 ◁ C23×C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
4. C47×#2090C2semidirect
5. C47×#2064C2semidirect
6. C47×#2047C2semidirect
145. #29792C47direct (N × H)
165. #2090C2×C47semidirect
166. #2064C2×C47semidirect
167. #2047C2×C47semidirect
198. C47#29792direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#30952
C2×C47×#4702
C47×#29052
C47×#20902
C47×#20642
C47×#20472
C47×#19012
C47×#18112
#2979247