Finite Group Databasebeta

← #26119902

Subgroups — #26119902

Order 12032.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15689404C211yesmaximal, characteristic
2#20243377C211yesmaximal, characteristic
3#20242072C211yesmaximal, characteristic, complemented
4#20242070C211yesmaximal, characteristic, complemented
5#20241778C211yesmaximal, characteristic
6#20241285C211yesmaximal, characteristic, complemented
7C47×#1570C211yesmaximal, characteristic, complemented
8C2×#13094024C2211yescharacteristic
9C2×C4×D188C2211yescharacteristic
10#15689404—22no—
11#15689404—22no—
12#15689176C2211yescharacteristic
13C2×#13093958—21no—
14#15689147C2211yescharacteristic
15C2×#13093946—21no—
16C2×#13093940—21no—
17#15688891—21no—
18#15688871—22no—
19#15688870—22no—
20#15688850—22no—
21C47×#431C2211yescharacteristic
22C47×#423C2211yescharacteristic
23C2×C4⋊C8×C47C2211yescharacteristic
24C47×#334—22no—
25C22×Dic94—21no—
26C22×D188D411yescharacteristic
27C22×C4×D47—21no—
28C2×#10625329—21no—
29#13094024—21no—
30#13094024—21no—
31#13094024—21no—
32#13094024D412yes—
33C4×D188—21no—
34C4×D188—21no—
35C4×D188—21no—
36C4×D188—21no—
37C2×C4⋊Dic47—21no—
38C2×C4⋊Dic47—21no—
39C2×C4⋊Dic47D411yescharacteristic
40C2×(C2×D47)⋊C4—21no—
41#13093958—24no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45#13093946—24no—
46#13093940—42no—
47C22⋊(C47⋊C8)—21no—
48C4×C47⋊C8—21no—
49C2×M16×C47—21no—
50C22×C8×C47—21no—
51C2×C42×C47C2311yescharacteristic
52C4⋊C8×C47—21no—
53C4⋊C8×C47—21no—
54C4⋊C8×C47—21no—
55C4⋊C8×C47D412yes—
56C4⋊C8×C47D412yes—
57C22⋊C8×C47—21no—
58C4×C8×C47—21no—
59C23×D47—21no—
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47#4111yescharacteristic
62C22×C4×C47#4111yescharacteristic
63C22×Dic47—21no—
64C22×Dic47—41no—
65C2×D188—21no—
66C2×D188—21no—
67C2×D188—21no—
68C2×D188—21no—
69C2×D188—21no—
70C2×D188SD1612yes—
71C2×C4×D47—41no—
72C2×C4×D47—41no—
73C2×C4×D47—82no—
74C2×Dic94—22no—
75C2×Dic94—41no—
76C2×Dic94—82no—
77M16×C47—41no—
78M16×C47—41no—
79C2×C8×C47—22no—
80C2×C8×C47—22no—
81C2×C8×C47—41no—
82C2×C8×C47—41no—
83C2×C8×C47—41no—
84C42×C47C2×D411yescharacteristic
85C42×C47C2×D411yescharacteristic
86C42×C47C2×D411yescharacteristic
87C42×C47#4111yescharacteristic
88(C2×D47)⋊C4—41no—
89(C2×D47)⋊C4—41no—
90C4⋊Dic47—22no—
91C4⋊Dic47—21no—
92C4⋊Dic47—22no—
93C4⋊Dic47—21no—
94C4⋊Dic47—41no—
95C4⋊Dic47—41no—
96#10625329—41no—
97#10625329—41no—
98C2×C47⋊C8—41no—
99C2×C47⋊C8—41no—
100C23×C47#12311yescharacteristic
101C22×D47—22no—
102C22×D47—41no—
103C22×D47—41no—
104C22×D47—81no—
105C22×D47—81no—
106C2×C4×C47—22no—
107C2×C4×C47—41no—
108C2×C4×C47#12011yescharacteristic
109C2×C4×C47#12111yescharacteristic
110C2×C4×C47#12111yescharacteristic
111C2×C4×C47#12111yescharacteristic
112C2×C4×C47#12311yescharacteristic
113C2×C4×C47#12411yescharacteristic
114C2×C4×C47C2×SD1611yescharacteristic
115C2×C4×C47#13511yescharacteristic
116C2×C4×C47#13711yescharacteristic
117C2×C4×C47#13711yescharacteristic
118C2×Dic47—42no—
119C2×Dic47—41no—
120C2×Dic47—82no—
121C2×Dic47—81no—
122D188—22no—
123D188—22no—
124D188—41no—
125D188—41no—
126C4×D47—82no—
127C4×D47—82no—
128Dic94—82no—
129Dic94—82no—
130C8×C47—42no—
131C8×C47—42no—
132C8×C47—81no—
133C47⋊C8—81no—
134C47⋊C8—81no—
135#10448—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47#39411yescharacteristic
139C22×C47#48311yescharacteristic
140C22×C47#48511yescharacteristic
141C2×D47—42no—
142C2×D47—82no—
143C2×D47—81no—
144C2×D47—81no—
145C2×D47—82no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47#44811yescharacteristic
150C4×C47#45211yescharacteristic
151C4×C47#46511yescharacteristic
152C4×C47#47011yescharacteristic
153C4×C47#47212yes—
154Dic47—82no—
155Dic47—81no—
156C2×#474—471no—
157#1800—471no—
158#1570D4711yescharacteristic, complemented
159#1549—471no—
160#1548—471no—
161#1478—471no—
162#1301—471no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47#203511yescharacteristic
166C2×C47#204511yescharacteristic
167C2×C47#204911yescharacteristic
168D47—82no—
169C2×C4⋊Q8—471no—
170C2×C4×D4—471no—
171#474—942no—
172#474—942no—
173#431—471no—
174#431C2×D4711yescharacteristic
175C2×#107—941no—
176#423—471no—
177#423C2×D4711yescharacteristic
178C2×C4⋊C8C2×D4711yescharacteristic
179C2×#102—941no—
180C2×#100—941no—
181#343—941no—
182#334—22no—
183#333—942no—
184#332—942no—
185D4⋊C8—942no—
186C47#1044811yescharacteristic, complemented, Sylow
187C22×Q8—941no—
188C22×D4—471no—
189C23×C4—941no—
190C2×M16—21no—
191C2×M16—941no—
192C22×C8—21no—
193C4⋊Q8—472no—
194C4⋊Q8—941no—
195C4⋊Q8—941no—
196C4⋊Q8—941no—
197C4×D4—941no—
198C4×D4—941no—
199C4×D4—941no—
200C4×D4—941no—
201C2×C4⋊C4—471no—
202C2×C4⋊C4—941no—
203C2×C4⋊C4—941no—
204C2×C22⋊C4—941no—
205C2×C42C22×D4711yescharacteristic
206#107—944no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—941no—
211C4⋊C8—941no—
212C4⋊C8—941no—
213C4⋊C8D18812yes—
214C4⋊C8C22⋊D4712yes—
215#102—944no—
216#100—1882no—
217C22⋊C8—21no—
218C22⋊C8—941no—
219C4×C8—21no—
220C4×C8—941no—
221C24—941no—
222C2×Q8—942no—
223C2×Q8—3762no—
224C2×Q8—1881no—
225C2×D4—472no—
226C2×D4—941no—
227C2×D4—941no—
228C2×D4—941no—
229C2×D4—941no—
230C2×D4—941no—
231C22×C4—941no—
232C22×C4—3762no—
233C22×C4—1881no—
234C22×C4—1881no—
235C22×C4—1881no—
236C22×C4#1062535011yescharacteristic
237C22×C4D4×D4711yescharacteristic
238C22×C4#1062535411yescharacteristic
239M16—41no—
240M16—41no—
241M16—1881no—
242M16—1881no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—1881no—
249C2×C8—1881no—
250C4⋊C4—942no—
251C4⋊C4—941no—
252C4⋊C4—941no—
253C4⋊C4—942no—
254C4⋊C4—1881no—
255C4⋊C4—1881no—
256C22⋊C4—1881no—
257C22⋊C4—1881no—
258C42C2×D18811yescharacteristic
259C42D4×D4711yescharacteristic
260C42#1062535411yescharacteristic
261C42C2×C22⋊D4711yescharacteristic
262C23—942no—
263C23—3761no—
264C23—3761no—
265C23—1881no—
266C23—1881no—
267C23#1309400911yescharacteristic
268Q8—3762no—
269Q8—3762no—
270D4—942no—
271D4—942no—
272D4—1881no—
273D4—1881no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—3762no—
277C2×C4—3762no—
278C2×C4—3762no—
279C2×C4—3761no—
280C2×C4—1881no—
281C2×C4—1882no—
282C2×C4#1309399211yescharacteristic
283C2×C4#1309399811yescharacteristic
284C2×C4#1309399911yescharacteristic
285C2×C4#1309399911yescharacteristic
286C2×C4#1309400911yescharacteristic
287C2×C4#1309401311yescharacteristic
288C2×C4SD16×D4711yescharacteristic
289C2×C4#1309404511yescharacteristic
290C2×C4#1309405311yescharacteristic
291C2×C4#1309405311yescharacteristic
292C8—42no—
293C8—42no—
294C8—81no—
295C8—3761no—
296C8—3761no—
297C22—21no—
298C22—21no—
299C22—3762no—
300C22—3762no—
301C22—3761no—
302C22—3761no—
303C22—1882no—
304C22#1568904011yescharacteristic
305C22#1568945511yescharacteristic
306C22#1568946711yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—3761no—
311C4—3762no—
312C4#1568926011yescharacteristic
313C4#1568928211yescharacteristic
314C4#1568935311yescharacteristic
315C4#1568938211yescharacteristic
316C4#1568939312yes—
317C2—21no—
318C2—21no—
319C2—3762no—
320C2#2024458611yescharacteristic
321C2#2024464411yescharacteristic
322C2#2024467011yescharacteristic
subgroupG/H
centerC22#15689040
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1570C2
socleC22×C47#394
prime radicalC22×D188D4
hypercenter#1570D47
hypocenterC47#10448
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
3. #20242072C2semidirect
4. #20242070C2semidirect
6. #20241285C2semidirect
7. C47×#1570C2semidirect
158. #1570D47semidirect
186. C47#10448semidirect

Maximal subgroups

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

subgroupindex
C2×#156894042
#202433772
#202420722
#202420702
#202417782
#202412852
C47×#15702
#1044847