Finite Group Databasebeta

← #26109194

Subgroups — #26109194

Order 12032.

460 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 107 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×#15689194C211yesmaximal, characteristic
2#20243426C211yesmaximal, characteristic, complemented
3#20241724C212yesmaximal
4#20241264C212yesmaximal, complemented
5C47×#1810C211yesmaximal, characteristic, complemented
6C22×#10625324C411yescharacteristic
7C2×C4×C22⋊D47C2211yescharacteristic
8#15689194—21no—
9#15689194—21no—
10#15689194—21no—
11#15689194—21no—
12C2×#13093960—21no—
13C2×Dic47⋊C8C411yescharacteristic
14#15689087C2212yes—
15C2×C22⋊(C47⋊C8)C2211yescharacteristic
16C2×(C2×D47)⋊C8C411yescharacteristic
17C2×#13093934C411yescharacteristic
18#15688888—21no—
19#15688848—22no—
20#15688846—22no—
21C23×C8×C47C2211yescharacteristic
22C2×#108×C47—21no—
23C47×#407C2212yes—
24C47×#341—21no—
25C22×C22⋊D47Q811yescharacteristic
26C22×C4×D47C2×C411yescharacteristic
27C2×#10625329—22no—
28C2×#10625324—21no—
29C2×#10625324—21no—
30C2×#10625324—42no—
31C22×C47⋊C8C2×C411yescharacteristic
32C22×C47⋊C8C2×C411yescharacteristic
33C4×C22⋊D47—21no—
34C4×C22⋊D47—21no—
35C4×C22⋊D47—21no—
36C4×C22⋊D47—21no—
37C2×Dic47⋊C4D411yescharacteristic
38C2×C22⋊Dic47Q811yescharacteristic
39C2×(C2×D47)⋊C4D411yescharacteristic
40C2×C4×Dic47C2×C411yescharacteristic
41#13093960—42no—
42Dic47⋊C8—21no—
43Dic47⋊C8—21no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—21no—
46C22⋊(C47⋊C8)—22no—
47C22⋊(C47⋊C8)—22no—
48(C2×D47)⋊C8—21no—
49(C2×D47)⋊C8—21no—
50#13093934—21no—
51#13093934—21no—
52C23×C4×C47C2311yescharacteristic
53C2×M16×C47—22no—
54C22×C8×C47—42no—
55C22×C8×C47C2×C411yescharacteristic
56C22×C8×C47C2×C411yescharacteristic
57C22×C8×C47D411yescharacteristic
58C22×C8×C47D411yescharacteristic
59C22×C8×C47D411yescharacteristic
60C22×C8×C47D411yescharacteristic
61#108×C47—42no—
62C22⋊C8×C47—22no—
63C22⋊C8×C47—22no—
64C24×C47C2×Q811yescharacteristic
65C23×D47C4⋊C411yescharacteristic
66C22×C4×C47—41no—
67C22×C4×C47—41no—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47#4111yescharacteristic
73C22×C4×C47#4111yescharacteristic
74C2×C22⋊D47—21no—
75C2×C22⋊D47—21no—
76C2×C22⋊D47—42no—
77C22×Dic47C4⋊C411yescharacteristic
78C22×Dic47C4⋊C411yescharacteristic
79C22×Dic47C4⋊C411yescharacteristic
80C2×C4×D47—21no—
81C2×C4×D47—21no—
82C2×C4×D47—41no—
83C2×C4×D47—41no—
84M16×C47—42no—
85M16×C47—42no—
86C2×C8×C47—22no—
87C2×C8×C47—22no—
88C2×C8×C47—22no—
89C2×C8×C47—41no—
90C2×C8×C47—41no—
91C2×C8×C47—41no—
92C2×C8×C47—42no—
93C2×C8×C47C22⋊C411yescharacteristic
94C2×C8×C47C22⋊C411yescharacteristic
95C2×C8×C47C22⋊C411yescharacteristic
96C2×C8×C47C22⋊C411yescharacteristic
97C22⋊Dic47—21no—
98C22⋊Dic47—21no—
99(C2×D47)⋊C4—21no—
100(C2×D47)⋊C4—21no—
101Dic47⋊C4—21no—
102Dic47⋊C4—21no—
103C4×Dic47—21no—
104C4×Dic47—21no—
105#10625329—42no—
106#10625329—42no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—41no—
110C2×C47⋊C8—41no—
111C2×C47⋊C8—42no—
112#10625324—42no—
113#10625324—42no—
114C23×C47—22no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47C2×C4⋊C411yescharacteristic
118C23×C47C22⋊Q811yescharacteristic
119C23×C47C22⋊Q811yescharacteristic
120C22×D47—21no—
121C22×D47—22no—
122C22×D47—41no—
123C22×D47—41no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47C2×C22⋊C411yescharacteristic
134C2×C4×C47#11711yescharacteristic
135C2×C4×C47#12111yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C22⋊D47—42no—
138C22⋊D47—42no—
139C2×Dic47—21no—
140C2×Dic47—21no—
141C2×Dic47—21no—
142C2×Dic47—22no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C4×D47—42no—
146C4×D47—41no—
147C4×D47—41no—
148C8×C47—21no—
149C8×C47—21no—
150C8×C47—21no—
151C8×C47—82no—
152C8×C47#9912yes—
153C47⋊C8—41no—
154C47⋊C8—41no—
155C47⋊C8—82no—
156#8613—471nomaximal, Sylow
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—44no—
166C22×C47#38011yescharacteristic
167C22×C47#43012yes—
168C2×D47—41no—
169C2×D47—41no—
170C2×D47—41no—
171C2×D47—82no—
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—21no—
175C4×C47C2×#9911yescharacteristic
176C4×C47#41011yescharacteristic
177Dic47—41no—
178Dic47—41no—
179Dic47—41no—
180C2×#435—471no—
181#1810—471no—
182#1810D4711yescharacteristic, complemented
183#1465—472no—
184#1293—472no—
185C2×C47—22no—
186C2×C47—41no—
187C2×C47—41no—
188C2×C47#179911yescharacteristic
189C2×C47#181111yescharacteristic
190C2×C47#181311yescharacteristic
191D47—81no—
192C22×M16—471no—
193C23×C8C2×D4711yescharacteristic
194C2×C4×D4—471no—
195#435—941no—
196#435—941no—
197#435—941no—
198#435—941no—
199C2×#108—21no—
200C2×#108—941no—
201C2×C4⋊C8—471no—
202#407—472no—
203#407C2×D4712yes—
204C2×C22⋊C8—471no—
205C2×C22⋊C8—471no—
206C2×#97—471no—
207#341—21no—
208#341—941no—
209#324—942no—
210#323—942no—
211C47#861311yescharacteristic, complemented, Sylow
212C22×D4—471no—
213C23×C4—471no—
214C23×C4C22×D4711yescharacteristic
215C2×M16—22no—
216C2×M16—941no—
217C2×M16—941no—
218C2×M16—942no—
219C2×M16—1882no—
220C22×C8—42no—
221C22×C8—471no—
222C22×C8—471no—
223C22×C8C4×D4711yescharacteristic
224C22×C8C4×D4711yescharacteristic
225C22×C8D18811yescharacteristic
226C22×C8D18811yescharacteristic
227C22×C8C22⋊D4711yescharacteristic
228C22×C8C22⋊D4711yescharacteristic
229C4×D4—941no—
230C4×D4—941no—
231C4×D4—941no—
232C4×D4—941no—
233C2×C4⋊C4—471no—
234C2×C22⋊C4—471no—
235C2×C22⋊C4—471no—
236C2×C42—471no—
237#108—42no—
238#108—1882no—
239C4⋊C8—941no—
240C4⋊C8—941no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—941no—
244C22⋊C8—941no—
245C22⋊C8—941no—
246C22⋊C8—941no—
247C22⋊C8—942no—
248C22⋊C8—942no—
249#97—941no—
250#97—941no—
251C24—471no—
252C24Q8×D4711yescharacteristic
253C2×D4—941no—
254C2×D4—941no—
255C2×D4—1882no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—471no—
259C22×C4—471no—
260C22×C4—471no—
261C22×C4—941no—
262C22×C4—941no—
263C22×C4—1881no—
264C22×C4—1881no—
265C22×C4C2×C4×D4711yescharacteristic
266C22×C4C2×D18811yescharacteristic
267C22×C4D4×D4711yescharacteristic
268C22×C4#1062535211yescharacteristic
269C22×C4#1062535411yescharacteristic
270C22×C4C2×C22⋊D4711yescharacteristic
271M16—42no—
272M16—42no—
273M16—1882no—
274M16—1882no—
275M16—1882no—
276M16—1882no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—941no—
285C2×C8—941no—
286C2×C8—1882no—
287C2×C8—1881no—
288C2×C8—1881no—
289C2×C8(C2×D47)⋊C411yescharacteristic
290C2×C8(C2×D47)⋊C411yescharacteristic
291C2×C8(C2×D47)⋊C411yescharacteristic
292C2×C8(C2×D47)⋊C411yescharacteristic
293C4⋊C4—941no—
294C4⋊C4—941no—
295C22⋊C4—941no—
296C22⋊C4—941no—
297C22⋊C4—941no—
298C22⋊C4—941no—
299C42—941no—
300C42—941no—
301C23—22no—
302C23—41no—
303C23—41no—
304C23—942no—
305C23—941no—
306C23—1881no—
307C23—1881no—
308C23C4⋊C4×D4711yescharacteristic
309C23#1309400011yescharacteristic
310C23#1309400411yescharacteristic
311D4—1882no—
312D4—1882no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—941no—
323C2×C4—941no—
324C2×C4—941no—
325C2×C4—942no—
326C2×C4—1882no—
327C2×C4—1881no—
328C2×C4—1881no—
329C2×C4—1881no—
330C2×C4—1881no—
331C2×C4C2×(C2×D47)⋊C411yescharacteristic
332C2×C4#1309398211yescharacteristic
333C2×C4#1309399511yescharacteristic
334C2×C4#1309399911yescharacteristic
335C8—21no—
336C8—21no—
337C8—21no—
338C8—82no—
339C8—3762no—
340C8—1881no—
341C8—1881no—
342C8#1309393812yes—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22—3762no—
353C22—1881no—
354C22—1881no—
355C22—1881no—
356C22#1568898011yescharacteristic
357C22#1568917012yes—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—1881no—
362C4—1881no—
363C4—1881no—
364C4C2×#1309393811yescharacteristic
365C4#1568909611yescharacteristic
366C2—22no—
367C2—41no—
368C2—41no—
369C2—3761no—
370C2#2024337311yescharacteristic
371C2#2024342711yescharacteristic
372C2#2024343511yescharacteristic
subgroupG/H
centerC22#15688980
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1810C2
socleC22×C47#380
prime radicalC22×C22⋊D47Q8
cosocleC23×C4×C47C23
hypercenter#1810D47
hypocenterC47#8613
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
2. #20243426C2semidirect
4. #20241264C2semidirect
5. C47×#1810C2semidirect
182. #1810D47semidirect
211. C47#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#156891942
#202434262
#202417242
#202412642
C47×#18102
#861347