Finite Group Databasebeta

← #26114150

Subgroups — #26114150

Order 12032.

522 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 111 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#20252473C211yesmaximal, characteristic, complemented
2#20248293C211yesmaximal, characteristic
3#20241812C212yesmaximal
4#20241248C212yesmaximal, complemented
5C47×#1801C211yesmaximal, characteristic, complemented
6C2×#13094000—21no—
7C2×C4×D188C2211yescharacteristic
8#15689254—24no—
9#15689180—22no—
10#15689176C2212yes—
11#15689053C412yes—
12#15689040C412yes—
13#15689014—22no—
14#15688989C2211yescharacteristic
15#15688980—21no—
16#15688850—24no—
17C2×C4×Q8×C47C2211yescharacteristic
18C47×#423C2212yes—
19C2×#101×C47—21no—
20C47×#343—21no—
21C22×D188C2×C411yescharacteristic
22C22×C4×D47—21no—
23#13094000—44no—
24C4×D188—22no—
25C4×D188—22no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×Dic47⋊C4—22no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—21no—
32C4⋊(C47⋊C8)—24no—
33C4⋊(C47⋊C8)—22no—
34C22⋊(C47⋊C8)—22no—
35#13093935—24no—
36C4×C47⋊C8D414yes—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95C2×C412yes—
40C22×Q8×C47D411yescharacteristic
41C2×M16×C47—22no—
42C4×Q8×C47—22no—
43C4×Q8×C47—22no—
44C2×C4⋊C4×C47C2×C412yes—
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C42×C47D412yes—
47C2×C42×C47C2311yescharacteristic
48C4⋊C8×C47—22no—
49C4⋊C8×C47—22no—
50#101×C47—42no—
51C23×D47—21no—
52C2×Q8×C47—22no—
53C2×Q8×C47—22no—
54C2×Q8×C47—21no—
55C2×Q8×C47—21no—
56C22×C4×C47C22⋊C412yes—
57C22×C4×C47C22⋊C412yes—
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×Dic47—22no—
62C22×Dic47—21no—
63C2×D188—22no—
64C2×D188—42no—
65C2×C4×D47—42no—
66C2×C4×D47—42no—
67C2×C4×D47—41no—
68C2×C4×D47—41no—
69M16×C47—42no—
70M16×C47—42no—
71C2×C8×C47—42no—
72C4⋊C4×C47—24no—
73C4⋊C4×C47—22no—
74C42×C47—22no—
75C42×C47—22no—
76C42×C47C2×D412yes—
77C42×C47#4112yes—
78(C2×D47)⋊C4—44no—
79(C2×D47)⋊C4—44no—
80(C2×D47)⋊C4—41no—
81(C2×D47)⋊C4—41no—
82C4⋊Dic47—22no—
83C4⋊Dic47—42no—
84Dic47⋊C4—44no—
85C2×C47⋊C8—24no—
86C23×C47C2×C22⋊C411yescharacteristic
87C22×D47—22no—
88C22×D47—41no—
89C22×D47—41no—
90C22×D47—41no—
91C22×D47—41no—
92C22×D47—41no—
93C22×D47—41no—
94Q8×C47—42no—
95Q8×C47—42no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—42no—
103C2×C4×C47C4×D412yes—
104C2×C4×C47#12011yescharacteristic
105C2×C4×C47#12112yes—
106C2×C4×C47#12311yescharacteristic
107C2×C4×C47AGL(1,ℤ/8)12yes—
108C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
109C2×C4×C47#13711yescharacteristic
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—41no—
113C2×Dic47—41no—
114C2×Dic47—41no—
115C2×Dic47—42no—
116D188—24no—
117D188—42no—
118C4×D47—82no—
119C4×D47—82no—
120C8×C47—82no—
121C47⋊C8—44no—
122#9490—471nomaximal, Sylow
123C22×C47—21no—
124C22×C47—21no—
125C22×C47#38611yescharacteristic
126C22×C47#41711yescharacteristic
127C22×C47#41811yescharacteristic
128C2×D47—42no—
129C2×D47—42no—
130C2×D47—42no—
131C2×D47—81no—
132C2×D47—81no—
133C2×D47—81no—
134C2×D47—81no—
135C4×C47—22no—
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—42no—
139C4×C47—42no—
140C4×C47#46812yes—
141C4×C47#46912yes—
142Dic47—82no—
143Dic47—81no—
144#2983—471no—
145#2536—471no—
146#1801D4711yescharacteristic, complemented
147#1486—472no—
148#1286—472no—
149C2×C47—21no—
150C2×C47—21no—
151C2×C47#189111yescharacteristic
152C2×C47#189511yescharacteristic
153C2×C47#191211yescharacteristic
154D47—82no—
155C2×C22⋊Q8—941no—
156C2×C4×Q8C2×D4711yescharacteristic
157C2×C4×D4—471no—
158#446—944no—
159#432—942no—
160#431—472no—
161#423C2×D4712yes—
162C2×#101—21no—
163#398—472no—
164#394—472no—
165#388—942no—
166#382—471no—
167#380—941no—
168#343—21no—
169D4⋊C8—944no—
170C47#949011yescharacteristic, complemented, Sylow
171C22×Q8C22⋊D4711yescharacteristic
172C22×D4—471no—
173C23×C4—941no—
174C2×M16—22no—
175C22⋊Q8—1884no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—942no—
179C4×D4—942no—
180C2×C4⋊C4—471no—
181C2×C4⋊C4—942no—
182C2×C4⋊C4—941no—
183C2×C4⋊C4C4×D4712yes—
184C2×C4⋊C4C22⋊D4711yescharacteristic
185C2×C22⋊C4—942no—
186C2×C22⋊C4—942no—
187C2×C22⋊C4—941no—
188C2×C42D18812yes—
189C2×C42C22×D4711yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—944no—
193C4⋊C8—942no—
194#101—42no—
195C22⋊C8—942no—
196#97—944no—
197C4×C8—474no—
198#95—472no—
199#95—942no—
200#95—941no—
201C24—941no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×D4—942no—
207C2×D4—1882no—
208C22×C4—942no—
209C22×C4—941no—
210C22×C4—1882no—
211C22×C4—1882no—
212C22×C4—1881no—
213C22×C4—1881no—
214C22×C4(C2×D47)⋊C412yes—
215C22×C4(C2×D47)⋊C412yes—
216C22×C4C2×C4×D4711yescharacteristic
217C22×C4C2×D18811yescharacteristic
218C22×C4C2×C22⋊D4711yescharacteristic
219M16—42no—
220M16—42no—
221C2×C8—42no—
222C2×C8—944no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—942no—
226C4⋊C4—1884no—
227C4⋊C4—1882no—
228C22⋊C4—1884no—
229C22⋊C4—1884no—
230C22⋊C4—1881no—
231C22⋊C4—1881no—
232C42—22no—
233C42—22no—
234C42D4×D4712yes—
235C42#1062535212yes—
236C23—942no—
237C23—1881no—
238C23—1881no—
239C23—1881no—
240C23—1881no—
241C23—1881no—
242C23—1881no—
243C23C2×(C2×D47)⋊C411yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—944no—
247D4—1882no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—3762no—
256C2×C4—3762no—
257C2×C4—1882no—
258C2×C4—1882no—
259C2×C4—1881no—
260C2×C4—1881no—
261C2×C4—1882no—
262C2×C4—1881no—
263C2×C4(C22⋊D47)⋊C412yes—
264C2×C4#1309399411yescharacteristic
265C2×C4#1309399612yes—
266C2×C4#1309400811yescharacteristic
267C2×C4#1309404912yes—
268C2×C4#1309405011yescharacteristic
269C2×C4#1309405511yescharacteristic
270C8—82no—
271C8—1884no—
272C22—21no—
273C22—21no—
274C22—3761no—
275C22—3761no—
276C22—3761no—
277C22—3761no—
278C22—1882no—
279C22—1882no—
280C22—1882no—
281C22#1568900011yescharacteristic
282C22#1568912211yescharacteristic
283C22#1568912711yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—3762no—
290C4—3761no—
291C4#1568937112yes—
292C4#1568937412yes—
293C2—21no—
294C2—21no—
295C2—3762no—
296C2#2024380411yescharacteristic
297C2#2024382411yescharacteristic
298C2#2024392311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1801C2
socleC22×C47#386
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1801D47
hypocenterC47#9490
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #20252473C2semidirect
4. #20241248C2semidirect
5. C47×#1801C2semidirect
146. #1801D47semidirect
170. C47#9490semidirect

Maximal subgroups

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

subgroupindex
#202524732
#202482932
#202418122
#202412482
C47×#18012
#949047