Finite Group Databasebeta

← #26248214

Subgroups — #26248214

Order 12032.

1286 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1286 conjugacy classes of subgroups — too many to draw. Only its 207 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
1C2×#15689261C212yesmaximal, complemented
2#20245972C211yesmaximal, characteristic, complemented
3#20244536C212yesmaximal, complemented
4#20244476C212yesmaximal, complemented
5#20243722C214yesmaximal, complemented
6#20243711C212yesmaximal, complemented
7#20243459C211yesmaximal, characteristic, complemented
8C2×C47×#540C211yesmaximal, characteristic
9C23×Dic94C2211yescharacteristic
10C22×#10625335C2214yes—
11C2×#13093983C2211yescharacteristic
12#15689261—28no—
13C2×#13093947C2214yes—
14C2×#13093945C2214yes—
15C2×C22⋊(C47⋊C8)C2212yes—
16#15689029C2212yes—
17#15689029C2212yes—
18#15689028—22no—
19#15688984—22no—
20#15688885C2214yes—
21#15688878C2212yes—
22#15688878C2212yes—
23C47×#540—28no—
24C2×#127×C47C2211yescharacteristic
25C2×#125×C47C2211yescharacteristic
26C2×#123×C47—22no—
27C2×#121×C47—22no—
28C2×#121×C47C2212yes—
29C2×C4×D4×C47C2212yes—
30C2×#117×C47C2211yescharacteristic
31C22×Dic94—21no—
32C22×Dic94—24no—
33C22×Dic94—24no—
34C22×Dic94—42no—
35C22×Dic94C2312yes—
36C23×Dic47—21no—
37C2×#10625335—28no—
38C2×#10625335—28no—
39C2×#10625335—44no—
40C22×C47⋊C8C2314yes—
41#13093983—42no—
42C2×C4⋊Dic47C2312yes—
43C2×C22⋊Dic47—21no—
44C2×C4×Dic47—21no—
45#13093947—28no—
46#13093945—44no—
47C22⋊(C47⋊C8)—24no—
48C47⋊#95—24no—
49C47⋊#95—24no—
50C22×D4×C47—22no—
51C22×D4×C47—22no—
52C22×D4×C47C2312yes—
53C23×C4×C47—22no—
54C23×C4×C47C2311yescharacteristic
55#127×C47—24no—
56#125×C47—24no—
57#123×C47—216no—
58#121×C47—216no—
59#121×C47—24no—
60#121×C47D418yes—
61C4×D4×C47—24no—
62C4×D4×C47—24no—
63#117×C47—24no—
64C2×C4⋊C4×C47—22no—
65C2×C4⋊C4×C47D412yes—
66C2×C4⋊C4×C47C2312yes—
67C2×C22⋊C4×C47—22no—
68C2×C22⋊C4×C47—22no—
69C2×C22⋊C4×C47D412yes—
70C2×C42×C47C2311yescharacteristic
71C2×C42×C47C2311yescharacteristic
72C24×C47—22no—
73C24×C47C2×D411yescharacteristic
74C2×D4×C47—28no—
75C2×D4×C47—28no—
76C2×D4×C47—22no—
77C2×D4×C47—24no—
78C2×D4×C47—24no—
79C2×D4×C47—84no—
80C2×D4×C47—84no—
81C2×D4×C47C2×D414yes—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—22no—
85C22×C4×C47—28no—
86C22×C4×C47—84no—
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2×D412yes—
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D414yes—
91C22×C4×C47C2411yescharacteristic
92C22×Dic47—24no—
93C22×Dic47—22no—
94C22×Dic47—22no—
95C22×Dic47—42no—
96C22×Dic47—42no—
97C2×Dic94—24no—
98C2×Dic94—24no—
99C2×Dic94—24no—
100C2×Dic94—22no—
101C2×Dic94—416no—
102C2×Dic94—42no—
103C2×Dic94—42no—
104C2×Dic94#4114yes—
105C4⋊C4×C47—28no—
106C4⋊C4×C47—24no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47C2×D414yes—
109C22⋊C4×C47—28no—
110C22⋊C4×C47—24no—
111C22⋊C4×C47—28no—
112C42×C47—22no—
113C42×C47—22no—
114C22⋊Dic47—42no—
115#10625335—416no—
116C4⋊Dic47—24no—
117C4×Dic47—42no—
118C2×C47⋊C8—24no—
119C2×C47⋊C8—24no—
120C2×C47⋊C8—24no—
121C23×C47—21no—
122C23×C47—21no—
123C23×C47—24no—
124C23×C47—24no—
125C23×C47—42no—
126C23×C47—42no—
127C23×C47—44no—
128C23×C47—44no—
129C23×C47#12012yes—
130C23×C47C22×D411yescharacteristic
131D4×C47—28no—
132D4×C47—44no—
133D4×C47—88no—
134D4×C47—88no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—28no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—24no—
141C2×C4×C47—24no—
142C2×C4×C47—24no—
143C2×C4×C47—44no—
144C2×C4×C47—88no—
145C2×C4×C47#12012yes—
146C2×C4×C47C22×D412yes—
147C2×C4×C47C2×#4112yes—
148C2×C4×C47H(5,𝔽2)12yes—
149C2×Dic47—24no—
150C2×Dic47—48no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—44no—
155Dic94—28no—
156Dic94—48no—
157Dic94—44no—
158C47⋊C8—44no—
159#23940—471nomaximal, Sylow
160C22×C47—22no—
161C22×C47—24no—
162C22×C47—42no—
163C22×C47—42no—
164C22×C47—44no—
165C22×C47—44no—
166C22×C47—44no—
167C22×C47—84no—
168C22×C47—84no—
169C22×C47C2×#12011yescharacteristic
170C22×C47#52511yescharacteristic
171C22×C47#53511yescharacteristic
172C22×C47#57714yes—
173C4×C47—22no—
174C4×C47—42no—
175C4×C47—42no—
176C4×C47—44no—
177C4×C47#54614yes—
178Dic47—44no—
179Dic47—82no—
180C2×#540D4711yescharacteristic
181C2×#448—472no—
182#2310—471no—
183#2026—472no—
184#2014—472no—
185#1877—474no—
186#1875—472no—
187#1818—471no—
188C2×C47—42no—
189C2×C47—84no—
190C2×C47#240411yescharacteristic
191C2×C47#300912yes—
192C2×C47#312612yes—
193C2×C47#322112yes—
194C23×Q8—471no—
195C22×SD16—474no—
196#540—28no—
197C2×#127C2×D4711yescharacteristic
198C2×#125C2×D4711yescharacteristic
199C2×#123—22no—
200C2×#121—22no—
201C2×#121C2×D4712yes—
202C2×C4×D4C2×D4712yes—
203C2×#117—471no—
204C2×#117C2×D4711yescharacteristic
205#448—948no—
206C2×#103—474no—
207C2×#102—474no—
208C2×C22⋊C8—472no—
209#391—472no—
210#391—472no—
211#391—942no—
212#381—942no—
213#340—474no—
214#336—472no—
215#336—472no—
216C47#2394011yescharacteristic, complemented, Sylow
217C22×Q8—472no—
218C22×Q8—944no—
219C22×Q8—941no—
220C22×Q8—944no—
221C22×Q8—1882no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4C22×D4712yes—
225C23×C4—22no—
226C23×C4—941no—
227C23×C4C22×D4711yescharacteristic
228C2×SD16—948no—
229C2×SD16—948no—
230C2×SD16—1884no—
231C22×C8—474no—
232#127—24no—
233#125—24no—
234#123—216no—
235#121—216no—
236#121—24no—
237#121C22⋊D4718yes—
238C4×D4—24no—
239C4×D4—24no—
240#117—24no—
241#117—1882no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—472no—
244C2×C4⋊C4C22⋊D4712yes—
245C2×C4⋊C4C22×D4712yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—941no—
249C2×C22⋊C4C22⋊D4712yes—
250C2×C42—941no—
251C2×C42C22×D4711yescharacteristic
252C2×C42C22×D4711yescharacteristic
253#103—948no—
254#102—1884no—
255C22⋊C8—944no—
256#95—944no—
257#95—944no—
258C24—22no—
259C24C2×C22⋊D4711yescharacteristic
260C2×Q8—474no—
261C2×Q8—944no—
262C2×Q8—942no—
263C2×Q8—944no—
264C2×Q8—944no—
265C2×Q8—18816no—
266C2×Q8—1882no—
267C2×Q8—1882no—
268C2×D4—28no—
269C2×D4—28no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—84no—
274C2×D4—84no—
275C2×D4C2×C22⋊D4714yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—28no—
280C22×C4—84no—
281C22×C4—944no—
282C22×C4—942no—
283C22×C4—942no—
284C22×C4—1882no—
285C22×C4—1882no—
286C22×C4C2×C22⋊D4712yes—
287C22×C4C2×C22⋊D4712yes—
288C22×C4C2×C22⋊D4711yescharacteristic
289C22×C4C2×C22⋊D4714yes—
290C22×C4C23×D4711yescharacteristic
291SD16—18816no—
292C2×C8—944no—
293C2×C8—944no—
294C2×C8—944no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—944no—
299C4⋊C4C2×C22⋊D4714yes—
300C22⋊C4—28no—
301C22⋊C4—24no—
302C22⋊C4—28no—
303C22⋊C4—1882no—
304C42—22no—
305C42—22no—
306C42—1882no—
307C23—21no—
308C23—24no—
309C23—21no—
310C23—24no—
311C23—42no—
312C23—44no—
313C23—44no—
314C23—42no—
315C23#1309399312yes—
316C23C22×C22⋊D4711yescharacteristic
317Q8—948no—
318Q8—1888no—
319Q8—1884no—
320D4—28no—
321D4—44no—
322D4—88no—
323D4—88no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—44no—
333C2×C4—88no—
334C2×C4—944no—
335C2×C4—1884no—
336C2×C4—1882no—
337C2×C4—1888no—
338C2×C4—1882no—
339C2×C4—1882no—
340C2×C4#1309399312yes—
341C2×C4C22×C22⋊D4712yes—
342C2×C4#41×D4712yes—
343C2×C4#1309407012yes—
344C8—1884no—
345C22—22no—
346C22—24no—
347C22—42no—
348C22—42no—
349C22—44no—
350C22—44no—
351C22—44no—
352C22—84no—
353C22—84no—
354C22C2×#1309399311yescharacteristic
355C22#1568963011yescharacteristic
356C22#1568968711yescharacteristic
357C22#1568993414yes—
358C4—22no—
359C4—42no—
360C4—42no—
361C4—44no—
362C4—3762no—
363C4—1884no—
364C4#1568976114yes—
365C2—42no—
366C2—84no—
367C2#2024670311yescharacteristic
368C2#2025266512yes—
369C2#2025367412yes—
370C2#2025444312yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#540C2
socleC23×C47C22×D4
hypercenterC2×#540D47
hypocenterC47#23940
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×#15689261C2semidirect
2. #20245972C2semidirect
3. #20244536C2semidirect
4. #20244476C2semidirect
5. #20243722C2semidirect
6. #20243711C2semidirect
7. #20243459C2semidirect
216. C47#23940semidirect

Maximal subgroups

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

subgroupindex
C2×#156892612
#202459722
#202445362
#202444762
#202437222
#202437112
#202434592
C2×C47×#5402
#2394047