Finite Group Databasebeta

← #26126488

Subgroups — #26126488

Order 12032.

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

normal subgroups only

This group has 552 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
1#20249138C211yesmaximal, characteristic
2#20241702C211yesmaximal, characteristic, complemented
3#20241390C214yesmaximal, complemented
4C47×#1742C211yesmaximal, characteristic, complemented
5C2×#13094021C411yescharacteristic
6C2×#13093998—21no—
7#15689176—21no—
8#15689040—21no—
9#15689040C2212yes—
10#15689032—21no—
11#15689032C412yes—
12#15689025—21no—
13#15688989C411yescharacteristic
14#15688849C2212yes—
15#15688847—21no—
16C22×M16×C47—21no—
17C2×#117×C47C2211yescharacteristic
18C47×#343—21no—
19C47×#343C2212yes—
20C22×D188—21no—
21C22×D188—21no—
22C22×D188C2×C412yes—
23C22×C22⋊D47—22no—
24#13094021—41no—
25#13094021—41no—
26#13093998—42no—
27#13093998—42no—
28C2×C4⋊Dic47—21no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C4⋊(C47⋊C8)—41no—
33C22⋊(C47⋊C8)—22no—
34C4×C47⋊C8—41no—
35C47⋊#95—21no—
36C47⋊#95C2×C412yes—
37C23×C4×C47D411yescharacteristic
38C2×M16×C47—24no—
39C2×M16×C47—22no—
40C2×M16×C47—41no—
41C2×M16×C47—42no—
42C22×C8×C47—41no—
43#117×C47—41no—
44#117×C47—41no—
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C4⋊C4×C47C2311yescharacteristic
47C2×C22⋊C4×C47C2×C411yescharacteristic
48C2×C22⋊C4×C47D411yescharacteristic
49C2×C42×C47C2×C411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C24×C47C22⋊C411yescharacteristic
52C23×D47—22no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—22no—
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47D812yes—
60C22×C4×C47SD1612yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C2×C22⋊D47—44no—
65C2×C22⋊D47—82no—
66C2×C22⋊D47—82no—
67C22×Dic47—22no—
68C2×D188—42no—
69C2×D188—42no—
70C2×D188—41no—
71C2×D188—41no—
72C2×D188—42no—
73C2×D188—42no—
74C2×D188—82no—
75C2×D188—82no—
76M16×C47—24no—
77M16×C47—44no—
78M16×C47—42no—
79C2×C8×C47—42no—
80C2×C8×C47—44no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C22⋊C4×C47—41no—
84C22⋊C4×C47—41no—
85C42×C47—21no—
86C42×C47—21no—
87C42×C47—21no—
88C42×C47—21no—
89(C2×D47)⋊C4—42no—
90(C2×D47)⋊C4—42no—
91(C2×D47)⋊C4—82no—
92(C2×D47)⋊C4—82no—
93C4⋊Dic47—42no—
94C2×C47⋊C8—42no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47#10212yes—
100C23×C47C2×C22⋊C411yescharacteristic
101C22×D47—42no—
102C22×D47—42no—
103C22×D47—42no—
104C22×D47—42no—
105C22×D47—42no—
106C22×D47—42no—
107C22×D47—42no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47#9911yescharacteristic
123C2×C4×C47#9911yescharacteristic
124C2×C4×C47#10212yes—
125C2×C4×C47#10412yes—
126C2×C4×C47C2×D811yescharacteristic
127C2×C4×C47C2×SD1611yescharacteristic
128C22⋊D47—88no—
129C2×Dic47—42no—
130C2×Dic47—42no—
131C2×Dic47—42no—
132D188—84no—
133D188—84no—
134D188—82no—
135D188—82no—
136C8×C47—44no—
137C47⋊C8—82no—
138#11699—471nomaximal, Sylow
139C22×C47—21no—
140C22×C47—21no—
141C22×C47—22no—
142C22×C47—42no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—81no—
146C22×C47C2×#9911yescharacteristic
147C22×C47C2×#10211yescharacteristic
148C22×C47C2×#10411yescharacteristic
149C2×D47—82no—
150C2×D47—82no—
151C2×D47—82no—
152C2×D47—82no—
153C2×D47—82no—
154C2×D47—82no—
155C2×D47—82no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—41no—
159C4×C47—41no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47#32712yes—
163C4×C47#32712yes—
164Dic47—82no—
165#2619—471no—
166#1742D4711yescharacteristic, complemented
167#1458—471no—
168#1345—474no—
169C2×C47—21no—
170C2×C47—21no—
171C2×C47—42no—
172C2×C47C2×#32711yescharacteristic
173C2×C47#212211yescharacteristic
174C2×C47#213411yescharacteristic
175D47—162no—
176C22×M16—21no—
177C2×#127—471no—
178C2×#121—941no—
179C2×#117C2×D4711yescharacteristic
180#431—941no—
181#394—472no—
182#394—941no—
183#392—472no—
184#392—941no—
185#390—941no—
186#382—471no—
187#343—21no—
188#343C2×D4712yes—
189#324—472no—
190#323—941no—
191C47#1169911yescharacteristic, complemented, Sylow
192C22×D4—472no—
193C22×D4—942no—
194C22×D4—941no—
195C22×D4—941no—
196C23×C4C22⋊D4711yescharacteristic
197C2×M16—24no—
198C2×M16—22no—
199C2×M16—41no—
200C2×M16—42no—
201C22×C8—41no—
202#127—1881no—
203#127—1881no—
204#121—1882no—
205#121—1882no—
206#117—41no—
207#117—41no—
208C2×C4⋊C4—941no—
209C2×C4⋊C4C22⋊D4711yescharacteristic
210C2×C4⋊C4C22×D4711yescharacteristic
211C2×C22⋊C4—942no—
212C2×C22⋊C4—942no—
213C2×C22⋊C4—942no—
214C2×C22⋊C4C4×D4711yescharacteristic
215C2×C22⋊C4D18811yescharacteristic
216C2×C42C4×D4711yescharacteristic
217C2×C42D18811yescharacteristic
218C4⋊C8—1881no—
219C22⋊C8—942no—
220C4×C8—1881no—
221#95—472no—
222#95—941no—
223C24—942no—
224C24(C2×D47)⋊C411yescharacteristic
225C2×D4—3762no—
226C2×D4—3762no—
227C2×D4—3762no—
228C2×D4—3762no—
229C2×D4—1882no—
230C2×D4—1882no—
231C2×D4—1884no—
232C2×D4—1881no—
233C2×D4—1881no—
234C2×D4—1882no—
235C2×D4—1882no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—942no—
240C22×C4(C2×D47)⋊C411yescharacteristic
241C22×C4(C2×D47)⋊C411yescharacteristic
242C22×C4(C2×D47)⋊C411yescharacteristic
243C22×C4#1062533412yes—
244C22×C4#1062533612yes—
245C22×C4C2×C4×D4711yescharacteristic
246C22×C4C2×D18811yescharacteristic
247C22×C4C2×C22⋊D4711yescharacteristic
248M16—24no—
249M16—44no—
250M16—42no—
251C2×C8—42no—
252C2×C8—44no—
253C2×C8—1882no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—1882no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—3762no—
260C22⋊C4—3762no—
261C22⋊C4—1882no—
262C22⋊C4—1882no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—1882no—
272C23—1882no—
273C23—1882no—
274C23—1882no—
275C23—1882no—
276C23—1882no—
277C23—1882no—
278C23#1309394412yes—
279C23C2×(C2×D47)⋊C411yescharacteristic
280D4—3764no—
281D4—3764no—
282D4—3768no—
283D4—3762no—
284D4—3762no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4—1882no—
302C2×C4#1309393811yescharacteristic
303C2×C4#1309393811yescharacteristic
304C2×C4#1309394412yes—
305C2×C4#1309395212yes—
306C2×C4C2×#1062533411yescharacteristic
307C2×C4C2×#1062533611yescharacteristic
308C8—44no—
309C8—3762no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—3762no—
318C22—3762no—
319C22—3762no—
320C22—3762no—
321C22—3762no—
322C22—3762no—
323C22—3762no—
324C22C2×#1309393811yescharacteristic
325C22C2×#1309394411yescharacteristic
326C22C2×#1309395211yescharacteristic
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—3762no—
334C4#1568885612yes—
335C4#1568885612yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—7522no—
340C2C2×#1568885611yescharacteristic
341C2#2024503911yescharacteristic
342C2#2024507911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1742C2
socleC22×C47C2×#99
prime radical#20249138C2
cosocleC2×C4⋊C4×C47C23
hypercenter#1742D47
hypocenterC47#11699
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. #20241702C2semidirect
3. #20241390C2semidirect
4. C47×#1742C2semidirect
166. #1742D47semidirect
191. C47#11699semidirect

Maximal subgroups

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

subgroupindex
#202491382
#202417022
#202413902
C47×#17422
#1169947