Finite Group Databasebeta

← #26113650

Subgroups — #26113650

Order 12032.

594 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 117 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#20252411C211yesmaximal, characteristic
2#20248342C211yesmaximal, characteristic
3#20243377C211yesmaximal, characteristic
4#20242073C211yesmaximal, characteristic, complemented
5#20241290C211yesmaximal, characteristic, complemented
6#20241289C211yesmaximal, characteristic, complemented
7C47×#1549C211yesmaximal, characteristic, complemented
8C2×#13094000—22no—
9C2×C4×D188C2211yescharacteristic
10C8×D188—22no—
11#15689188—22no—
12#15689176C2211yescharacteristic
13#15689147C2211yescharacteristic
14C2×#13093940—21no—
15C22⋊C8×D47—22no—
16#15689058C414yes—
17#15689024—21no—
18#15688992—21no—
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×C8×D47—24no—
30#13094000—48no—
31C4×D188—21no—
32C4×D188—21no—
33C4×D188—21no—
34C4×D188—21no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47D411yescharacteristic
37C2×Dic47⋊C4—24no—
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⋊#95C2×C412yes—
49C22×Q8×C47—21no—
50C2×M16×C47—21no—
51C4⋊Q8×C47—21no—
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47—21no—
54C4⋊Q8×C47—21no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47C2×C412yes—
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C4⋊C8×C47D412yes—
61C22⋊C8×C47—21no—
62C4×C8×C47—21no—
63C23×D47C22⋊C412yes—
64C2×Q8×C47—44no—
65C2×Q8×C47—41no—
66C2×Q8×C47—41no—
67C22×C4×C47—22no—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—22no—
72C22×Dic47C22⋊C412yes—
73C2×D188—21no—
74C2×D188—21no—
75C2×D188—21no—
76C2×D188—21no—
77C2×D188SD1612yes—
78C2×D188Q1612yes—
79C2×C4×D47—22no—
80C2×C4×D47—22no—
81C2×C4×D47—44no—
82M16×C47—41no—
83M16×C47—41no—
84C2×C8×C47—22no—
85C2×C8×C47—41no—
86C4⋊C4×C47—22no—
87C4⋊C4×C47—22no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C42×C47C2×D411yescharacteristic
91C42×C47C2×D411yescharacteristic
92C42×C47#4111yescharacteristic
93C42×C47#4111yescharacteristic
94(C2×D47)⋊C4—22no—
95(C2×D47)⋊C4—22no—
96(C2×D47)⋊C4—48no—
97C4⋊Dic47—24no—
98C4⋊Dic47—21no—
99C4⋊Dic47—21no—
100Dic47⋊C4—48no—
101C4×Dic47—41no—
102C4×Dic47—41no—
103#10625329—41no—
104#10625329—41no—
105C2×C47⋊C8—22no—
106C2×C47⋊C8—41no—
107C8×D47—84no—
108C23×C47C2×C22⋊C411yescharacteristic
109C22×D47—22no—
110C22×D47—22no—
111C22×D47—42no—
112C22×D47—42no—
113C22×D47#10314yes—
114Q8×C47—82no—
115Q8×C47—82no—
116C2×C4×C47—22no—
117C2×C4×C47—41no—
118C2×C4×C47—44no—
119C2×C4×C47—42no—
120C2×C4×C47C4×D411yescharacteristic
121C2×C4×C47C4×D411yescharacteristic
122C2×C4×C47#12011yescharacteristic
123C2×C4×C47#12111yescharacteristic
124C2×C4×C47#12111yescharacteristic
125C2×C4×C47#12311yescharacteristic
126C2×C4×C47C2×SD1611yescharacteristic
127C2×C4×C47C2×Q1611yescharacteristic
128C2×C4×C47#13511yescharacteristic
129C2×C4×C47#13511yescharacteristic
130C2×Dic47—24no—
131C2×Dic47—22no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135D188—22no—
136D188—22no—
137D188—22no—
138D188—22no—
139C4×D47—44no—
140C4×D47—82no—
141C8×C47—42no—
142C8×C47—81no—
143C47⋊C8—42no—
144C47⋊C8—81no—
145#9395—471nomaximal, Sylow
146C22×C47—21no—
147C22×C47—21no—
148C22×C47#38611yescharacteristic
149C22×C47C2×#10311yescharacteristic
150C22×C47#41611yescharacteristic
151C2×D47—24no—
152C2×D47—42no—
153C2×D47—42no—
154C2×D47—44no—
155C2×D47—82no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—82no—
160C4×C47#45312yes—
161C4×C47#46511yescharacteristic
162C4×C47#46511yescharacteristic
163C4×C47#46711yescharacteristic
164C4×C47#46711yescharacteristic
165Dic47—44no—
166Dic47—42no—
167#2972—471no—
168#2542—471no—
169#1800—471no—
170#1549—471no—
171#1549D4711yescharacteristic, complemented
172#1303—471no—
173#1303—471no—
174C2×C47—21no—
175C2×C47—21no—
176C2×C47#188711yescharacteristic
177C2×C47#189211yescharacteristic
178C2×C47#191511yescharacteristic
179D47—44no—
180C2×C4⋊Q8C2×D4711yescharacteristic
181C2×C22⋊Q8—942no—
182C2×C4×D4—471no—
183C8×D4—942no—
184C8×D4—942no—
185#431—471no—
186#431C2×D4711yescharacteristic
187#423—471no—
188#423C2×D4711yescharacteristic
189C2×#100—941no—
190C2×C22⋊C8—942no—
191#399—474no—
192#390—941no—
193#384—941no—
194C4×C4⋊C4—471no—
195#343—941no—
196#332—22no—
197#332—942no—
198C47#939511yescharacteristic, complemented, Sylow
199C22×Q8—21no—
200C22×D4—471no—
201C23×C4—472no—
202C2×M16—21no—
203C2×M16—941no—
204C22×C8—944no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C22⋊Q8—1888no—
210C4×D4—941no—
211C4×D4—941no—
212C4×D4—941no—
213C4×D4—941no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—471no—
216C2×C4⋊C4—941no—
217C2×C4⋊C4—944no—
218C2×C4⋊C4C4×D4712yes—
219C2×C22⋊C4—472no—
220C2×C22⋊C4—944no—
221C2×C42—941no—
222C2×C42C22×D4711yescharacteristic
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—472no—
226C4⋊C8—941no—
227C4⋊C8—941no—
228C4⋊C8C22⋊D4712yes—
229#100—1882no—
230C22⋊C8—21no—
231C22⋊C8—944no—
232C22⋊C8—941no—
233C4×C8—21no—
234C4×C8—941no—
235#95—472no—
236C24—472no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×D4—472no—
241C2×D4—472no—
242C2×D4—941no—
243C2×D4—941no—
244C2×D4—941no—
245C2×D4—941no—
246C22×C4—22no—
247C22×C4—472no—
248C22×C4—942no—
249C22×C4—942no—
250C22×C4—942no—
251C22×C4—1884no—
252C22×C4C2×C4×D4711yescharacteristic
253C22×C4D4×D4711yescharacteristic
254C22×C4D4×D4711yescharacteristic
255M16—41no—
256M16—41no—
257M16—1881no—
258M16—1881no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—942no—
262C2×C8—3764no—
263C2×C8—1881no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—944no—
269C4⋊C4—941no—
270C4⋊C4—941no—
271C4⋊C4—1888no—
272C22⋊C4—942no—
273C22⋊C4—942no—
274C22⋊C4—1888no—
275C42—1881no—
276C42—1881no—
277C42#1062535011yescharacteristic
278C42D4×D4711yescharacteristic
279C42#1062535211yescharacteristic
280C42C2×C22⋊D4711yescharacteristic
281C23—474no—
282C23—942no—
283C23—942no—
284C23—1882no—
285C23—1882no—
286C23C22⋊C4×D4711yescharacteristic
287Q8—82no—
288Q8—82no—
289D4—942no—
290D4—942no—
291D4—942no—
292D4—942no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—944no—
298C2×C4—942no—
299C2×C4—3762no—
300C2×C4—1882no—
301C2×C4—1882no—
302C2×C4—1884no—
303C2×C4—1882no—
304C2×C4(C22⋊D47)⋊C411yescharacteristic
305C2×C4C4×C22⋊D4711yescharacteristic
306C2×C4#1309399211yescharacteristic
307C2×C4#1309399511yescharacteristic
308C2×C4#1309399711yescharacteristic
309C2×C4#1309400611yescharacteristic
310C2×C4SD16×D4711yescharacteristic
311C2×C4Q16×D4711yescharacteristic
312C2×C4#1309404511yescharacteristic
313C2×C4#1309404711yescharacteristic
314C8—42no—
315C8—81no—
316C8—3761no—
317C8—1882no—
318C22—21no—
319C22—21no—
320C22—944no—
321C22—3762no—
322C22—1884no—
323C22—1882no—
324C22—1882no—
325C22#1568899911yescharacteristic
326C22#103×D4711yescharacteristic
327C22#1568911811yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—82no—
332C4—1884no—
333C4—1882no—
334C4#1568928912yes—
335C4#1568934911yescharacteristic
336C4#1568935511yescharacteristic
337C4#1568936111yescharacteristic
338C4#1568936311yescharacteristic
339C2—21no—
340C2—21no—
341C2—1884no—
342C2#2024378311yescharacteristic
343C2#2024380811yescharacteristic
344C2#2024393711yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1549C2
socleC22×C47#386
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1549D47
hypocenterC47#9395
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
4. #20242073C2semidirect
5. #20241290C2semidirect
6. #20241289C2semidirect
7. C47×#1549C2semidirect
171. #1549D47semidirect
198. C47#9395semidirect

Maximal subgroups

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

subgroupindex
#202524112
#202483422
#202433772
#202420732
#202412902
#202412892
C47×#15492
#939547