Finite Group Databasebeta

← #26117244

Subgroups — #26117244

Order 12032.

636 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 103 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#20253834C211yesmaximal, characteristic, complemented
2#20253388C212yesmaximal, complemented
3#20241786C212yesmaximal, complemented
4#20241219C211yesmaximal, characteristic, complemented
5C47×#1955C211yesmaximal, characteristic, complemented
6C2×#13094046—22no—
7#15689673C2212yes—
8#15689517—22no—
9#15689512—22no—
10#15689436—22no—
11#15689403—22no—
12#15689318—22no—
13#15689257—22no—
14C4×#10625334—24no—
15#15689176C2212yes—
16#15688850—24no—
17C2×C4×D4×C47C2211yescharacteristic
18C2×C4⋊C8×C47C2211yescharacteristic
19C2×C22⋊C8×C47—22no—
20C2×#97×C47C2211yescharacteristic
21C2×#10625350—22no—
22#13094046—44no—
23#13094046—44no—
24C2×Dic188—22no—
25C2×#10625325—24no—
26C2×D376—22no—
27C2×#10625334—24no—
28#13094024D412yes—
29#13094021D412yes—
30#13094012—22no—
31#13093998—22no—
32#13093998—22no—
33C4×D188—22no—
34C4×D188—22no—
35#13093957—24no—
36C4⋊(C47⋊C8)—22no—
37#13093945—24no—
38#13093944—24no—
39C22⋊(C47⋊C8)—22no—
40C4×C47⋊C8D414yes—
41C22×D4×C47D411yescharacteristic
42C23×C4×C47—21no—
43C22×C8×C47—21no—
44C22×C8×C47D412yes—
45C4×D4×C47—22no—
46C4×D4×C47—22no—
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C22⋊C4×C47—21no—
49C2×C42×C47C2311yescharacteristic
50C4⋊C8×C47—22no—
51C22⋊C8×C47—44no—
52#97×C47—22no—
53C24×C47—21no—
54C2×D4×C47—22no—
55C2×D4×C47—42no—
56C22×C4×C47—21no—
57C22×C4×C47—41no—
58C22×C4×C47—41no—
59C22×C4×C47—42no—
60C22×C4×C47—42no—
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47#4111yescharacteristic
64C2×C22⋊D47—42no—
65#10625350—44no—
66#10625350—44no—
67C2×D188—22no—
68C2×D188—22no—
69C2×D188—22no—
70C2×C4×D47—42no—
71C2×Dic94—22no—
72C2×C8×C47—22no—
73C2×C8×C47—22no—
74C2×C8×C47—41no—
75C2×C8×C47—42no—
76C2×C8×C47D814yescomplemented
77C4⋊C4×C47—22no—
78C22⋊C4×C47—42no—
79C42×C47C2×D412yes—
80C42×C47C2×D412yes—
81#10625334—48no—
82(C2×D47)⋊C4—42no—
83(C2×D47)⋊C4—42no—
84C4⋊Dic47—22no—
85C4⋊Dic47—22no—
86C2×C47⋊C8—24no—
87Dic188—44no—
88D376—44no—
89#10625325—48no—
90C23×C47—22no—
91C23×C47—41no—
92C23×C47—42no—
93C23×C47—41no—
94C23×C47—42no—
95C23×C47#12111yescharacteristic
96C22×D47—42no—
97D4×C47—24no—
98D4×C47—42no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—42no—
103C2×C4×C47—82no—
104C2×C4×C47—81no—
105C2×C4×C47—81no—
106C2×C4×C47#12011yescharacteristic
107C2×C4×C47#12012yes—
108C2×C4×C47#12112yes—
109C2×C4×C47#12711yescharacteristic
110C2×C4×C47C2×D812yes—
111C2×C4×C47AGL(1,ℤ/8)12yes—
112C22⋊D47—84no—
113C2×Dic47—42no—
114D188—44no—
115D188—44no—
116D188—44no—
117C4×D47—82no—
118C4×D47—82no—
119Dic94—44no—
120C8×C47—24no—
121C8×C47—81no—
122C47⋊C8—44no—
123#10032—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—44no—
127C22×C47—42no—
128C22×C47—82no—
129C22×C47—81no—
130C22×C47—81no—
131C22×C47#39211yescharacteristic
132C22×C47#45912yes—
133C2×D47—82no—
134C2×D47—82no—
135C2×D47—82no—
136C4×C47—22no—
137C4×C47—22no—
138C4×C47—81no—
139C4×C47#44712yes—
140C4×C47#49311yescharacteristic
141C4×C47#49611yescharacteristic
142Dic47—82no—
143#3150—471no—
144#3100—472no—
145#1955D4711yescharacteristic, complemented
146#1480—472no—
147#1275—471no—
148C2×C47—22no—
149C2×C47—82no—
150C2×C47#201011yescharacteristic
151C2×C47#201111yescharacteristic
152C2×C47#201711yescharacteristic
153D47—162no—
154C2×#135—942no—
155#532—472no—
156C2×C4×D4C2×D4711yescharacteristic
157#497—942no—
158#496—942no—
159#480—942no—
160#474—942no—
161#459—942no—
162#447—942no—
163C4×D8—944no—
164#431—472no—
165C2×C4⋊C8C2×D4711yescharacteristic
166C2×C22⋊C8—22no—
167C2×#97C2×D4711yescharacteristic
168D4⋊C8—944no—
169C47#1003211yescharacteristic, complemented, Sylow
170C2×#41—942no—
171C22×D4D18811yescharacteristic
172C23×C4—21no—
173#135—1884no—
174#135—1884no—
175C2×Q16—942no—
176C2×SD16—944no—
177C2×D8—942no—
178C2×D8—944no—
179C22×C8—21no—
180C22×C8C22⋊D4712yes—
181C4⋊Q8—472no—
182#127—472no—
183#124—942no—
184#121—942no—
185#121—942no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—942no—
189C4×D4—942no—
190C2×C4⋊C4D18811yescharacteristic
191C2×C22⋊C4—21no—
192C2×C42C22×D4711yescharacteristic
193#107—944no—
194C4⋊C8—22no—
195C4⋊C8—942no—
196#102—944no—
197#102—944no—
198C22⋊C8—44no—
199C22⋊C8—942no—
200#97—22no—
201C4×C8—474no—
202C24—21no—
203#41—1884no—
204#41—1884no—
205C2×Q8—942no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—942no—
209C2×D4—942no—
210C2×D4—942no—
211C2×D4—1882no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—1882no—
218C22×C4C2×D18811yescharacteristic
219C22×C4#1062535011yescharacteristic
220C22×C4C2×C22⋊D4711yescharacteristic
221Q16—1884no—
222SD16—1888no—
223D8—1884no—
224D8—1888no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—41no—
228C2×C8—42no—
229C2×C8—944no—
230C2×C8#1062533414yescomplemented
231C4⋊C4—22no—
232C4⋊C4—942no—
233C4⋊C4—942no—
234C22⋊C4—42no—
235C22⋊C4—1882no—
236C22⋊C4—1882no—
237C42D4×D4712yes—
238C42D4×D4712yes—
239C23—22no—
240C23—41no—
241C23—42no—
242C23—41no—
243C23—42no—
244C23—1882no—
245C23#1309399811yescharacteristic
246Q8—1884no—
247D4—24no—
248D4—42no—
249D4—3764no—
250D4—1884no—
251D4—1884no—
252D4—1884no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—3762no—
261C2×C4—3762no—
262C2×C4—1882no—
263C2×C4#1309399211yescharacteristic
264C2×C4#1309399412yes—
265C2×C4#1309399712yes—
266C2×C4#1309402011yescharacteristic
267C2×C4C2×#1062533412yes—
268C2×C4#1309404912yes—
269C8—24no—
270C8—81no—
271C8—1884no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22—3762no—
280C22—3762no—
281C22—3762no—
282C22#1568903211yescharacteristic
283C22#1568931812yes—
284C4—22no—
285C4—22no—
286C4—81no—
287C4—3762no—
288C4#1568925612yes—
289C4#1568949911yescharacteristic
290C4#1568951311yescharacteristic
291C2—22no—
292C2—82no—
293C2—7522no—
294C2#2024445511yescharacteristic
295C2#2024445811yescharacteristic
296C2#2024449211yescharacteristic
subgroupG/H
centerC2×C4#13094020
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1955C2
socleC22×C47#392
hypercenter#1955D47
hypocenterC47#10032
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. #20253834C2semidirect
2. #20253388C2semidirect
3. #20241786C2semidirect
4. #20241219C2semidirect
5. C47×#1955C2semidirect
76. C2×C8×C47D8semidirect
145. #1955D47semidirect
169. C47#10032semidirect
230. C2×C8#10625334semidirect

Maximal subgroups

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

subgroupindex
#202538342
#202533882
#202417862
#202412192
C47×#19552
#1003247