Finite Group Databasebeta

← #26110931

Subgroups — #26110931

Order 12032.

722 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 139 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#20252412C211yesmaximal, characteristic, complemented
2C4×#13094020C211yesmaximal, characteristic
3#20241780C212yesmaximal, complemented
4#20241220C212yesmaximal, complemented
5C47×#1460C211yesmaximal, characteristic, complemented
6C2×#13094020C411yescharacteristic
7C2×C4×D188C2211yescharacteristic
8C2×C4×C22⋊D47—22no—
9D4×C47⋊C8—24no—
10#15689176C2212yes—
11C2×C22⋊(C47⋊C8)—22no—
12#15689026C411yescharacteristic
13#15689024C411yescharacteristic
14#15689023—22no—
15#15688992C411yescharacteristic
16C42×Dic47C2211yescharacteristic
17#15688852—24no—
18C2×C4×D4×C47C2211yescharacteristic
19C47×#431C2212yes—
20C47×#323—22no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—22no—
23C22×C4×D47—21no—
24C22×C47⋊C8—24no—
25#13094020—22no—
26#13094020C814yescomplemented
27C4×D188—22no—
28C4×D188—22no—
29C4×C22⋊D47—48no—
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—22no—
32C2×C22⋊Dic47—22no—
33C2×(C2×D47)⋊C4—22no—
34C2×(C2×D47)⋊C4—21no—
35C2×C4×Dic47—21no—
36C2×C4×Dic47—21no—
37C2×C4×Dic47C2×C411yescharacteristic
38C2×C4×Dic47C2×C411yescharacteristic
39C4⋊(C47⋊C8)D414yescomplemented
40C22⋊(C47⋊C8)—24no—
41C22⋊(C47⋊C8)—22no—
42C4×C47⋊C8—22no—
43C22×D4×C47C2×C411yescharacteristic
44C23×C4×C47D412yes—
45C4×D4×C47—22no—
46C4×D4×C47—22no—
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C22⋊C4×C47D412yes—
49C2×C42×C47C2311yescharacteristic
50C4⋊C8×C47—22no—
51C22⋊C8×C47—22no—
52C4×C8×C47—22no—
53C24×C47C22⋊C412yes—
54C23×D47—21no—
55C2×D4×C47—22no—
56C2×D4×C47—22no—
57C2×D4×C47C2×C812yes—
58C2×D4×C47M1612yes—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—44no—
62C22×C4×C47C22⋊C412yes—
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C2×C22⋊D47—28no—
67C2×C22⋊D47—44no—
68C2×C22⋊D47—44no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C22×Dic47—21no—
72C2×D188—22no—
73C2×D188—22no—
74C2×D188—21no—
75C2×D188C2×C812yescomplemented
76C2×C4×D47—44no—
77C2×C4×D47—41no—
78C2×C4×D47—41no—
79C2×C8×C47—42no—
80C4⋊C4×C47—22no—
81C22⋊C4×C47—24no—
82C42×C47C2×D412yes—
83C42×C47#4112yes—
84C22⋊Dic47—44no—
85(C2×D47)⋊C4—44no—
86(C2×D47)⋊C4—42no—
87C4⋊Dic47—22no—
88Dic47⋊C4—44no—
89C4×Dic47—24no—
90C4×Dic47—21no—
91C4×Dic47—22no—
92C4×Dic47—42no—
93C4×Dic47C2×C812yescomplemented
94C2×C47⋊C8—24no—
95C2×C47⋊C8—84no—
96C23×C47—22no—
97C23×C47—22no—
98C23×C47—42no—
99C23×C47—42no—
100C23×C47C22⋊C814yes—
101C23×C47C2×C22⋊C411yescharacteristic
102C22×D47—22no—
103C22×D47—42no—
104C22×D47—42no—
105C22×D47—41no—
106C22×D47—41no—
107D4×C47—24no—
108D4×C47—24no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—41no—
112C2×C4×C47—44no—
113C2×C4×C47—42no—
114C2×C4×C47—82no—
115C2×C4×C47C4×D412yes—
116C2×C4×C47#12011yescharacteristic
117C2×C4×C47#12112yes—
118C2×C4×C47#12311yescharacteristic
119C2×C4×C47C22×C811yescharacteristic
120C2×C4×C47C2×M1611yescharacteristic
121C2×C4×C47#13112yes—
122C22⋊D47—416no—
123C2×Dic47—22no—
124C2×Dic47—22no—
125C2×Dic47—22no—
126C2×Dic47—22no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—42no—
130C2×Dic47—41no—
131D188—24no—
132D188—42no—
133C4×D47—82no—
134C4×D47—82no—
135C8×C47—82no—
136C47⋊C8—44no—
137#8959—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—24no—
141C22×C47—44no—
142C22×C47—42no—
143C22×C47—42no—
144C22×C47—82no—
145C22×C47#38611yescharacteristic
146C22×C47C2×C22⋊C811yescharacteristic
147C22×C47#40811yescharacteristic
148C2×D47—44no—
149C2×D47—42no—
150C2×D47—82no—
151C2×D47—81no—
152C2×D47—81no—
153C4×C47—22no—
154C4×C47—21no—
155C4×C47—42no—
156C4×C47C8×D412yes—
157C4×C47#43512yes—
158C4×C47#45312yes—
159Dic47—24no—
160Dic47—42no—
161Dic47—81no—
162#2972—471no—
163C4×#127—471no—
164#1478—472no—
165#1460D4711yescharacteristic, complemented
166#1276—472no—
167C2×C47—21no—
168C2×C47—21no—
169C2×C47—44no—
170C2×C47#186911yescharacteristic
171C2×C47#188411yescharacteristic
172C2×C47#188711yescharacteristic
173D47—82no—
174C2×#127—471no—
175C2×C4×D4—471no—
176C2×C4×D4—942no—
177C2×C4×D4C2×D4711yescharacteristic
178C8×D4—944no—
179#431—472no—
180#431C2×D4712yes—
181C2×C22⋊C8—942no—
182#390—471no—
183#390—471no—
184#390—942no—
185#384—471no—
186C43—471no—
187D4⋊C8—944no—
188#323—22no—
189C47#895911yescharacteristic, complemented, Sylow
190C22×D4—471no—
191C22×D4—942no—
192C22×D4C4×D4711yescharacteristic
193C23×C4—941no—
194C23×C4D18812yes—
195C22×C8—944no—
196#127—474no—
197#127—942no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—942no—
201C4×D4—942no—
202C4×D4—1888no—
203C2×C4⋊C4—471no—
204C2×C4⋊C4—942no—
205C2×C4⋊C4C4×D4711yescharacteristic
206C2×C22⋊C4—942no—
207C2×C22⋊C4—942no—
208C2×C22⋊C4—941no—
209C2×C22⋊C4C22⋊D4712yes—
210C2×C42—471no—
211C2×C42—471no—
212C2×C42—941no—
213C2×C42—941no—
214C2×C42C22×D4711yescharacteristic
215C4⋊C8—22no—
216C4⋊C8—474no—
217C22⋊C8—22no—
218C22⋊C8—944no—
219C22⋊C8—942no—
220C4×C8—22no—
221C4×C8—942no—
222C24—941no—
223C24(C2×D47)⋊C412yes—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—472no—
227C2×D4—948no—
228C2×D4—942no—
229C2×D4—941no—
230C2×D4—942no—
231C2×D4—1884no—
232C2×D4—1884no—
233C2×D4C8×D4712yes—
234C2×D4#1062532412yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—44no—
238C22×C4—941no—
239C22×C4—941no—
240C22×C4—941no—
241C22×C4—1884no—
242C22×C4—1881no—
243C22×C4—1881no—
244C22×C4(C2×D47)⋊C412yes—
245C22×C4C2×C4×D4711yescharacteristic
246C22×C4C2×D18811yescharacteristic
247C22×C4C2×C22⋊D4711yescharacteristic
248C2×C8—42no—
249C2×C8—944no—
250C2×C8—3764no—
251C4⋊C4—22no—
252C4⋊C4—942no—
253C4⋊C4—1884no—
254C22⋊C4—24no—
255C22⋊C4—1884no—
256C22⋊C4—1884no—
257C22⋊C4—1882no—
258C42—472no—
259C42—944no—
260C42—941no—
261C42—942no—
262C42—1882no—
263C42D4×D4712yes—
264C42#1062535212yes—
265C23—22no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—942no—
270C23—1882no—
271C23—1882no—
272C23—1881no—
273C23—1881no—
274C23(C2×D47)⋊C814yes—
275C23C2×(C2×D47)⋊C411yescharacteristic
276D4—24no—
277D4—24no—
278D4—944no—
279D4—18816no—
280D4—1882no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—82no—
287C2×C4—942no—
288C2×C4—942no—
289C2×C4—942no—
290C2×C4—942no—
291C2×C4—3762no—
292C2×C4—3762no—
293C2×C4—1882no—
294C2×C4—1881no—
295C2×C4—1881no—
296C2×C4—1881no—
297C2×C4(C22⋊D47)⋊C412yes—
298C2×C4#1309399411yescharacteristic
299C2×C4#1309399612yes—
300C2×C4#1309400811yescharacteristic
301C2×C4C2×C8×D4711yescharacteristic
302C2×C4C2×#1062532411yescharacteristic
303C2×C4#1309403212yes—
304C8—82no—
305C8—1884no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—44no—
310C22—42no—
311C22—42no—
312C22—82no—
313C22—3762no—
314C22—3761no—
315C22—3761no—
316C22—1884no—
317C22—1882no—
318C22#1568900011yescharacteristic
319C22C2×(C2×D47)⋊C811yescharacteristic
320C22#1568909111yescharacteristic
321C4—22no—
322C4—21no—
323C4—42no—
324C4—944no—
325C4—3761no—
326C4—1882no—
327C4(C22⋊D47)⋊C812yes—
328C4#1568919112yes—
329C4#1568929312yes—
330C2—21no—
331C2—21no—
332C2—44no—
333C2—3762no—
334C2#2024367811yescharacteristic
335C2#2024376911yescharacteristic
336C2#2024378411yescharacteristic
subgroupG/H
centerC2×C4#13093994
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1460C2
socleC22×C47#386
prime radicalC2×#13094020C4
cosocleC2×C42×C47C23
hypercenter#1460D47
hypocenterC47#8959
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
1. #20252412C2semidirect
3. #20241780C2semidirect
4. #20241220C2semidirect
5. C47×#1460C2semidirect
26. #13094020C8semidirect
39. C4⋊(C47⋊C8)D4semidirect
75. C2×D188C2×C8semidirect
93. C4×Dic47C2×C8semidirect
165. #1460D47semidirect
189. C47#8959semidirect

Maximal subgroups

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

subgroupindex
#202524122
C4×#130940202
#202417802
#202412202
C47×#14602
#895947