Finite Group Databasebeta

← #26105281

Subgroups — #26105281

Order 12032.

424 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#20244721C211yesmaximal, characteristic
2#20244716C211yesmaximal, characteristic
3#20244487C211yesmaximal, characteristic
4#20241214C211yesmaximal, characteristic
5#20241211C211yesmaximal, characteristic
6#20241211C211yesmaximal, characteristic
7C47×#2805C211yesmaximal, characteristic
8C22×#10625337—21no—
9C2×#13094024C2211yescharacteristic
10C2×#13093957—21no—
11C2×#13093955—21no—
12C2×#13093947—21no—
13#15688885—21no—
14#15688885—21no—
15#15688878C2211yescharacteristic
16#15688878C2211yescharacteristic
17#15688878C2211yescharacteristic
18C47×#400—21no—
19C47×#400—21no—
20C47×#395—21no—
21C47×#391C2211yescharacteristic
22C47×#389—21no—
23C47×#389C2211yescharacteristic
24C47×#389C2211yescharacteristic
25C47×#387—21no—
26C47×#387—21no—
27C22×Dic94—21no—
28C2×#10625337—42no—
29C2×#10625337—42no—
30C2×#10625337—42no—
31C22×C47⋊C8—21no—
32C22×C47⋊C8—21no—
33C22×C47⋊C8—21no—
34#13094024—42no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47—21no—
37#13093957—42no—
38#13093955—42no—
39#13093947—42no—
40C22×Q8×C47D411yescharacteristic
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C42×C47—21no—
51C2×C42×C47C2311yescharacteristic
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56#95×C47—21no—
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61C2×Q8×C47—21no—
62C2×Q8×C47—21no—
63C2×Q8×C47—41no—
64C2×Q8×C47—41no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×Dic47—41no—
75C2×Dic94—22no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×Dic94—41no—
80C2×Dic94—41no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—21no—
83C4⋊C4×C47—21no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—81no—
92C4⋊C4×C47—81no—
93C4⋊C4×C47Q1612yescomplemented
94C42×C47—41no—
95C42×C47—81no—
96#10625337—48no—
97C4⋊Dic47—22no—
98C4⋊Dic47—22no—
99C4⋊Dic47—41no—
100C4⋊Dic47—41no—
101C2×C47⋊C8—22no—
102C2×C47⋊C8—22no—
103C2×C47⋊C8—22no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C2×C47⋊C8—41no—
107C2×C47⋊C8—41no—
108C2×C47⋊C8—41no—
109C2×C47⋊C8—41no—
110C23×C47#12011yescharacteristic
111Q8×C47—42no—
112Q8×C47—81no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—22no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47C2×Q1611yescharacteristic
138C2×C4×C47#13511yescharacteristic
139C2×C4×C47#13511yescharacteristic
140C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C47#13711yescharacteristic
142C2×C4×C47#13711yescharacteristic
143C2×Dic47—42no—
144C2×Dic47—81no—
145C2×Dic47—81no—
146Dic94—42no—
147Dic94—42no—
148Dic94—81no—
149Dic94—81no—
150C47⋊C8—42no—
151C47⋊C8—42no—
152C47⋊C8—42no—
153#7783—471nomaximal, Sylow
154C22×C47#44911yescharacteristic
155C22×C47#45111yescharacteristic
156C22×C47#45211yescharacteristic
157C22×C47#45311yescharacteristic
158C22×C47#45611yescharacteristic
159C22×C47#45611yescharacteristic
160C22×C47B(4,𝔽2)11yescharacteristic
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—21no—
167C4×C47—42no—
168C4×C47—81no—
169C4×C47—81no—
170C4×C47—81no—
171C4×C47—81no—
172C4×C47—81no—
173Dic47—82no—
174#2805D4711yescharacteristic
175#2058—471no—
176#2057—471no—
177#2016—471no—
178#1272—471no—
179#1271—471no—
180#1271—471no—
181C2×C47#160011yescharacteristic
182C2×C47#160311yescharacteristic
183C2×C47#160511yescharacteristic
184C2×C47#163211yescharacteristic
185C2×C47#164811yescharacteristic
186C2×C47#165211yescharacteristic
187C2×C47#166011yescharacteristic
188C22×Q16—941no—
189C2×C4⋊Q8—471no—
190C2×#107—941no—
191C2×#106—941no—
192C2×#103—941no—
193#400—21no—
194#400—21no—
195#395—21no—
196#391C2×D4711yescharacteristic
197#389—21no—
198#389C2×D4711yescharacteristic
199#389C2×D4711yescharacteristic
200#387—21no—
201#387—21no—
202#340—941no—
203#340—941no—
204#336—471no—
205#336—471no—
206#336—471no—
207C47#778311yescharacteristic, complemented, Sylow
208C22×Q8—941no—
209C22×Q8C22⋊D4711yescharacteristic
210C2×Q16—1882no—
211C2×Q16—1882no—
212C2×Q16—1882no—
213C22×C8—941no—
214C22×C8—941no—
215C22×C8—941no—
216C4⋊Q8—1882no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—941no—
222C2×C4⋊C4—941no—
223C2×C4⋊C4C22⋊D4711yescharacteristic
224C2×C4⋊C4C22⋊D4711yescharacteristic
225C2×C4⋊C4C22⋊D4711yescharacteristic
226C2×C4⋊C4C22⋊D4711yescharacteristic
227C2×C4⋊C4C22⋊D4711yescharacteristic
228C2×C42—21no—
229C2×C42C22×D4711yescharacteristic
230#107—1882no—
231#106—1882no—
232#103—1882no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—942no—
247C2×Q8—1881no—
248C2×Q8—1881no—
249C2×Q8—1881no—
250C2×Q8—1881no—
251C2×Q8—1881no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—1881no—
259C22×C4C2×C22⋊D4711yescharacteristic
260C22×C4C2×C22⋊D4711yescharacteristic
261C22×C4C2×C22⋊D4711yescharacteristic
262Q16—1888no—
263C2×C8—942no—
264C2×C8—942no—
265C2×C8—942no—
266C2×C8—1881no—
267C2×C8—1881no—
268C2×C8—1881no—
269C2×C8—1881no—
270C2×C8—1881no—
271C2×C8—1881no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C4⋊C4—81no—
284C4⋊C4—942no—
285C4⋊C4—942no—
286C4⋊C4—1881no—
287C4⋊C4—1881no—
288C4⋊C4#1062533712yescomplemented
289C42—41no—
290C42—81no—
291C23#1309399311yescharacteristic
292Q8—42no—
293Q8—81no—
294Q8—3761no—
295Q8—3761no—
296Q8—1882no—
297Q8—1882no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—3761no—
323C2×C4—3761no—
324C2×C4—1882no—
325C2×C4C2×#1062533711yescharacteristic
326C2×C4#1309404411yescharacteristic
327C2×C4#1309404411yescharacteristic
328C2×C4#1309404911yescharacteristic
329C2×C4#1309405411yescharacteristic
330C2×C4#1309405511yescharacteristic
331C8—1882no—
332C8—1882no—
333C8—1882no—
334C22#1568926811yescharacteristic
335C22#1568927911yescharacteristic
336C22#1568928511yescharacteristic
337C22#1568929111yescharacteristic
338C22#1568930611yescharacteristic
339C22#1568930611yescharacteristic
340C22#1568931011yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—3762no—
354C2#2024233411yescharacteristic
355C2#2024235111yescharacteristic
356C2#2024236211yescharacteristic
357C2#2024252211yescharacteristic
358C2#2024263011yescharacteristic
359C2#2024265811yescharacteristic
360C2#2024269911yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2805C2
socle, prime radicalC23×C47#120
hypercenter#2805D47
hypocenterC47#7783
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2805

Complements

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

NH ≅ G/Ntype
93. C4⋊C4×C47Q16semidirect
207. C47#7783semidirect
288. C4⋊C4#10625337semidirect

Maximal subgroups

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

subgroupindex
#202447212
#202447162
#202444872
#202412142
#202412112
C47×#28052
#778347