Finite Group Databasebeta

← #26113680

Subgroups — #26113680

Order 12032.

534 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 109 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#20252428C211yesmaximal, characteristic
2#20248384C211yesmaximal, characteristic
3#20243377C211yesmaximal, characteristic
4#20242060C211yesmaximal, characteristic, complemented
5#20241290C211yesmaximal, characteristic, complemented
6#20241289C211yesmaximal, characteristic, complemented
7C47×#1546C211yesmaximal, characteristic, complemented
8C2×#13094013—21no—
9C2×C4×D188C2211yescharacteristic
10#15689202—22no—
11#15689200—22no—
12#15689176C2211yescharacteristic
13#15689147C2211yescharacteristic
14C2×#13093940—21no—
15#15689085—22no—
16#15689049C414yes—
17#15689030—21no—
18#15688980—22no—
19C4×C4⋊Dic47C2211yescharacteristic
20#15688891—21no—
21#15688869—22no—
22C2×C4⋊Q8×C47C2211yescharacteristic
23C47×#431C2211yescharacteristic
24C47×#423C2211yescharacteristic
25C47×#332—22no—
26C22×D188D411yescharacteristic
27C22×C4×D47D412yes—
28C2×#10625329—21no—
29C2×#10625324—24no—
30#13094013—42no—
31#13094013—42no—
32C4×D188—21no—
33C4×D188—21no—
34C4×D188—21no—
35C4×D188—21no—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47D411yescharacteristic
38C2×(C2×D47)⋊C4—24no—
39C2×(C2×D47)⋊C4C2×C412yes—
40C2×C4×Dic47—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)D412yes—
44#13093940—42no—
45C22⋊(C47⋊C8)—21no—
46(C2×D47)⋊C8—24no—
47C4×C47⋊C8—21no—
48C47⋊#95—24no—
49C47⋊#95C2×C412yes—
50C22×Q8×C47—21no—
51C2×M16×C47—21no—
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47—21no—
54C4⋊Q8×C47—21no—
55C4⋊Q8×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47C2×C412yes—
58C2×C42×C47C2311yescharacteristic
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C4⋊C8×C47D412yes—
62C22⋊C8×C47—21no—
63C4×C8×C47—21no—
64C23×D47C22⋊C412yes—
65C2×Q8×C47—41no—
66C2×Q8×C47—41no—
67C2×Q8×C47—41no—
68C2×Q8×C47—41no—
69C2×Q8×C47—41no—
70C2×Q8×C47—41no—
71C22×C4×C47—22no—
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×Dic47—22no—
76C22×Dic47C22⋊C412yes—
77C2×D188—21no—
78C2×D188—21no—
79C2×D188—21no—
80C2×D188—21no—
81C2×D188—21no—
82C2×D188—21no—
83C2×C4×D47—22no—
84C2×C4×D47—22no—
85C2×C4×D47—44no—
86M16×C47—41no—
87M16×C47—41no—
88C2×C8×C47—22no—
89C2×C8×C47—41no—
90C4⋊C4×C47—22no—
91C4⋊C4×C47—22no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—41no—
94C42×C47C2×D411yescharacteristic
95C42×C47C2×D411yescharacteristic
96C42×C47#4111yescharacteristic
97C42×C47#4111yescharacteristic
98(C2×D47)⋊C4—22no—
99(C2×D47)⋊C4—22no—
100(C2×D47)⋊C4—48no—
101C4⋊Dic47—21no—
102C4⋊Dic47—21no—
103C4⋊Dic47—42no—
104C4×Dic47—41no—
105C4×Dic47—41no—
106#10625329—41no—
107#10625329—41no—
108C2×C47⋊C8—22no—
109C2×C47⋊C8—41no—
110#10625324—84no—
111C23×C47C2×C22⋊C411yescharacteristic
112C22×D47—22no—
113C22×D47—22no—
114C22×D47—22no—
115C22×D47—42no—
116C22×D47—42no—
117Q8×C47—81no—
118Q8×C47—81no—
119Q8×C47—81no—
120Q8×C47—81no—
121C2×C4×C47—22no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47C4×D411yescharacteristic
127C2×C4×C47C4×D411yescharacteristic
128C2×C4×C47#12011yescharacteristic
129C2×C4×C47#12111yescharacteristic
130C2×C4×C47#12111yescharacteristic
131C2×C4×C47#12311yescharacteristic
132C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C47#13711yescharacteristic
136C2×Dic47—22no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141D188—41no—
142D188—41no—
143D188—41no—
144D188—41no—
145C4×D47—44no—
146C4×D47—82no—
147C8×C47—42no—
148C8×C47—81no—
149C47⋊C8—42no—
150C47⋊C8—81no—
151#9401—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47#38611yescharacteristic
155C22×C47#41711yescharacteristic
156C22×C47#41811yescharacteristic
157C2×D47—44no—
158C2×D47—42no—
159C2×D47—42no—
160C2×D47—42no—
161C2×D47—82no—
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—82no—
166C4×C47#45512yes—
167C4×C47#46811yescharacteristic
168C4×C47#46811yescharacteristic
169C4×C47#46911yescharacteristic
170C4×C47#46911yescharacteristic
171Dic47—42no—
172Dic47—82no—
173#2975—471no—
174#2547—471no—
175#1800—471no—
176#1546—471no—
177#1546D4711yescharacteristic, complemented
178#1303—471no—
179#1303—471no—
180C2×C47—21no—
181C2×C47—21no—
182C2×C47#188511yescharacteristic
183C2×C47#189511yescharacteristic
184C2×C47#191511yescharacteristic
185D47—82no—
186C2×C4⋊Q8C2×D4711yescharacteristic
187C2×#124—941no—
188C2×C4×D4—471no—
189#436—942no—
190#436—942no—
191#431—471no—
192#431C2×D4711yescharacteristic
193#423—471no—
194#423C2×D4711yescharacteristic
195C2×#100—941no—
196#407—942no—
197#397—474no—
198#391—941no—
199#380—942no—
200C4×C4⋊C4—471no—
201#343—941no—
202#332—22no—
203#332—942no—
204C47#940111yescharacteristic, complemented, Sylow
205C22×Q8—21no—
206C22×D4—471no—
207C23×C4—472no—
208C2×M16—21no—
209C2×M16—944no—
210C2×M16—941no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215#124—1882no—
216#124—1882no—
217C4×D4—941no—
218C4×D4—941no—
219C4×D4—941no—
220C4×D4—941no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—471no—
223C2×C4⋊C4—941no—
224C2×C4⋊C4C4×D4712yes—
225C2×C22⋊C4—472no—
226C2×C22⋊C4—944no—
227C2×C42—941no—
228C2×C42C22×D4711yescharacteristic
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—472no—
232C4⋊C8—941no—
233C4⋊C8—941no—
234C4⋊C8C22⋊D4712yes—
235#100—1882no—
236C22⋊C8—21no—
237C22⋊C8—944no—
238C22⋊C8—941no—
239C4×C8—21no—
240C4×C8—941no—
241#95—472no—
242#95—944no—
243C24—472no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—941no—
251C2×D4—941no—
252C2×D4—941no—
253C2×D4—941no—
254C2×D4—941no—
255C2×D4—941no—
256C22×C4—22no—
257C22×C4—472no—
258C22×C4—942no—
259C22×C4—942no—
260C22×C4—942no—
261C22×C4—1884no—
262C22×C4C2×C4×D4711yescharacteristic
263C22×C4D4×D4711yescharacteristic
264C22×C4D4×D4711yescharacteristic
265M16—41no—
266M16—41no—
267M16—3764no—
268M16—1881no—
269M16—1881no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—942no—
273C2×C8—1881no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—941no—
279C4⋊C4—941no—
280C4⋊C4—1882no—
281C22⋊C4—942no—
282C22⋊C4—942no—
283C22⋊C4—1888no—
284C42—1881no—
285C42—1881no—
286C42#1062535011yescharacteristic
287C42D4×D4711yescharacteristic
288C42#1062535211yescharacteristic
289C42C2×C22⋊D4711yescharacteristic
290C23—942no—
291C23—942no—
292C23—942no—
293C23—1882no—
294C23—1882no—
295C23C22⋊C4×D4711yescharacteristic
296Q8—81no—
297Q8—81no—
298Q8—81no—
299Q8—81no—
300D4—1881no—
301D4—1881no—
302D4—1881no—
303D4—1881no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—942no—
310C2×C4—3762no—
311C2×C4—1882no—
312C2×C4—1882no—
313C2×C4—1882no—
314C2×C4—1884no—
315C2×C4—1882no—
316C2×C4(C22⋊D47)⋊C411yescharacteristic
317C2×C4C4×C22⋊D4711yescharacteristic
318C2×C4#1309399211yescharacteristic
319C2×C4#1309399511yescharacteristic
320C2×C4#1309399711yescharacteristic
321C2×C4#1309400611yescharacteristic
322C2×C4#1309404811yescharacteristic
323C2×C4#1309404811yescharacteristic
324C2×C4#1309405111yescharacteristic
325C2×C4#1309405311yescharacteristic
326C8—42no—
327C8—81no—
328C8—3761no—
329C8—1882no—
330C22—21no—
331C22—21no—
332C22—3762no—
333C22—1884no—
334C22—1882no—
335C22—1882no—
336C22—1882no—
337C22#1568899911yescharacteristic
338C22#1568912111yescharacteristic
339C22#1568912611yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—82no—
344C4—3762no—
345C4—1882no—
346C4#1568929912yes—
347C4#1568936611yescharacteristic
348C4#1568936811yescharacteristic
349C4#1568937311yescharacteristic
350C4#1568937511yescharacteristic
351C2—21no—
352C2—21no—
353C2—3762no—
354C2#2024377311yescharacteristic
355C2#2024382311yescharacteristic
356C2#2024393711yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1546C2
socleC22×C47#386
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1546D47
hypocenterC47#9401
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
4. #20242060C2semidirect
5. #20241290C2semidirect
6. #20241289C2semidirect
7. C47×#1546C2semidirect
177. #1546D47semidirect
204. C47#9401semidirect

Maximal subgroups

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

subgroupindex
#202524282
#202483842
#202433772
#202420602
#202412902
#202412892
C47×#15462
#940147