Finite Group Databasebeta

← #26115758

Subgroups — #26115758

Order 12032.

406 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 113 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×#15689475C211yesmaximal, characteristic
2#20243559C211yesmaximal, characteristic
3#20242069C211yesmaximal, characteristic
4#20242061C211yesmaximal, characteristic
5#20242059C211yesmaximal, characteristic
6#20241681C211yesmaximal, characteristic
7C47×#1549C211yesmaximal, characteristic
8C2×#13094024C2211yescharacteristic
9#15689475—21no—
10#15689475—21no—
11#15689475C414yes—
12#15689176C2211yescharacteristic
13C2×#13093959—22no—
14#15689147C2211yescharacteristic
15C2×#13093947—22no—
16C2×C4×C47⋊C8C2211yescharacteristic
17#15688872—22no—
18#15688870—22no—
19#15688868—22no—
20#15688845—21no—
21#15688844—21no—
22C2×C4⋊Q8×C47C2211yescharacteristic
23C47×#431C2211yescharacteristic
24C47×#423C2211yescharacteristic
25C47×#332—22no—
26C22×Dic94—21no—
27C2×#10625329—21no—
28C22×C47⋊C8D412yes—
29#13094024—21no—
30#13094024—21no—
31#13094024—21no—
32#13094024C2×C412yes—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—21no—
35#13093959—44no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)D412yes—
39#13093947—24no—
40#13093947—24no—
41C22⋊(C47⋊C8)—21no—
42C4×C47⋊C8—21no—
43C4×C47⋊C8C2×C412yes—
44C4×C47⋊C8D411yescharacteristic
45C4×C47⋊C8D411yescharacteristic
46C22×Q8×C47—21no—
47C2×M16×C47—21no—
48C4⋊Q8×C47—21no—
49C4⋊Q8×C47—21no—
50C4⋊Q8×C47—21no—
51C4⋊Q8×C47C2×C412yes—
52C2×C4⋊C4×C47—21no—
53C2×C4⋊C4×C47—21no—
54C2×C42×C47C2311yescharacteristic
55C4⋊C8×C47—21no—
56C4⋊C8×C47—21no—
57C4⋊C8×C47D412yes—
58C22⋊C8×C47—21no—
59C4×C8×C47—21no—
60C2×Q8×C47—22no—
61C2×Q8×C47—41no—
62C2×Q8×C47—82no—
63C22×C4×C47—41no—
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47#4111yescharacteristic
66C22×C4×C47#4111yescharacteristic
67C22×Dic47—41no—
68C2×Dic94—22no—
69C2×Dic94—42no—
70C2×Dic94—42no—
71C2×Dic94—41no—
72M16×C47—42no—
73C2×C8×C47—22no—
74C2×C8×C47—41no—
75C4⋊C4×C47—22no—
76C4⋊C4×C47—22no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C42×C47C22×C411yescharacteristic
80C42×C47C2×D411yescharacteristic
81C42×C47C2×D411yescharacteristic
82C42×C47C2×D411yescharacteristic
83C4⋊Dic47—22no—
84C4⋊Dic47—22no—
85C4⋊Dic47—41no—
86C4⋊Dic47—41no—
87#10625329—42no—
88C2×C47⋊C8—22no—
89C2×C47⋊C8—22no—
90C2×C47⋊C8—22no—
91C2×C47⋊C8—41no—
92C2×C47⋊C8C22⋊C412yes—
93C2×C47⋊C8C22⋊C412yes—
94C23×C47#12411yescharacteristic
95Q8×C47—82no—
96Q8×C47—82no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—42no—
100C2×C4×C47—82no—
101C2×C4×C47C2×C22⋊C411yescharacteristic
102C2×C4×C47C4×D411yescharacteristic
103C2×C4×C47C4×D411yescharacteristic
104C2×C4×C47#12111yescharacteristic
105C2×C4×C47#12111yescharacteristic
106C2×C4×C47#12711yescharacteristic
107C2×C4×C47AGL(1,ℤ/8)12yes—
108C2×C4×C47#13712yes—
109C2×Dic47—42no—
110C2×Dic47—82no—
111Dic94—44no—
112Dic94—81no—
113Dic94—81no—
114C8×C47—42no—
115C8×C47—81no—
116C47⋊C8—22no—
117C47⋊C8—22no—
118C47⋊C8—42no—
119C47⋊C8—81no—
120#9775—471nomaximal, Sylow
121C22×C47—21no—
122C22×C47—21no—
123C22×C47#39011yescharacteristic
124C22×C47#48912yes—
125C4×C47—21no—
126C4×C47—21no—
127C4×C47—82no—
128C4×C47#41712yes—
129C4×C47#47112yes—
130C4×C47#47312yes—
131C4×C47#49611yescharacteristic
132C4×C47#49711yescharacteristic
133Dic47—82no—
134C2×#487—471no—
135#1848—471no—
136#1549D4711yescharacteristic
137#1548—471no—
138#1546—471no—
139#1546—471no—
140#1452—471no—
141C2×C47—22no—
142C2×C47#195911yescharacteristic
143C2×C47#197711yescharacteristic
144C2×C47#197911yescharacteristic
145C2×C4⋊Q8—471no—
146C2×C4⋊Q8C2×D4711yescharacteristic
147#487—474no—
148#487—941no—
149#487—941no—
150#431—471no—
151#431C2×D4711yescharacteristic
152C2×#108—942no—
153#423—471no—
154#423C2×D4711yescharacteristic
155C2×#103—942no—
156C2×C4×C8—471no—
157#333—942no—
158#332—22no—
159#332—942no—
160#332—942no—
161#322—941no—
162#322—941no—
163C47#977511yescharacteristic, complemented, Sylow
164C22×Q8—21no—
165C22×Q8—941no—
166C2×M16—21no—
167C2×M16—941no—
168C22×C8—472no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—472no—
173C4⋊Q8—941no—
174C4⋊Q8—941no—
175C4⋊Q8—941no—
176C4⋊Q8C4×D4712yes—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—941no—
180C2×C4⋊C4—941no—
181C2×C42C22×D4711yescharacteristic
182#108—1884no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—472no—
186C4⋊C8—941no—
187C4⋊C8—941no—
188C4⋊C8C22⋊D4712yes—
189#103—944no—
190#103—944no—
191C22⋊C8—21no—
192C22⋊C8—941no—
193C4×C8—21no—
194C4×C8—472no—
195C4×C8—471no—
196C4×C8—471no—
197C4×C8—941no—
198C2×Q8—22no—
199C2×Q8—41no—
200C2×Q8—82no—
201C2×Q8—942no—
202C2×Q8—1882no—
203C2×Q8—1882no—
204C2×Q8—1881no—
205C22×C4—41no—
206C22×C4—1881no—
207C22×C4#1062535011yescharacteristic
208C22×C4D4×D4711yescharacteristic
209C22×C4#1062535211yescharacteristic
210M16—42no—
211M16—1882no—
212C2×C8—22no—
213C2×C8—41no—
214C2×C8—472no—
215C2×C8—472no—
216C2×C8—942no—
217C2×C8—942no—
218C2×C8—942no—
219C2×C8—1881no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—942no—
225C4⋊C4—942no—
226C4⋊C4—1881no—
227C4⋊C4—1881no—
228C42C2×C4×D4711yescharacteristic
229C42D4×D4711yescharacteristic
230C42D4×D4711yescharacteristic
231C42C2×C22⋊D4711yescharacteristic
232C23#1309401011yescharacteristic
233Q8—82no—
234Q8—82no—
235Q8—3761no—
236Q8—3761no—
237Q8—1884no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—42no—
241C2×C4—82no—
242C2×C4—3762no—
243C2×C4—1882no—
244C2×C4C22⋊C4×D4711yescharacteristic
245C2×C4(C22⋊D47)⋊C411yescharacteristic
246C2×C4C4×C22⋊D4711yescharacteristic
247C2×C4#1309399611yescharacteristic
248C2×C4#1309399711yescharacteristic
249C2×C4#1309402011yescharacteristic
250C2×C4#1309405112yes—
251C2×C4#1309405612yes—
252C8—42no—
253C8—81no—
254C8—942no—
255C8—942no—
256C8—3761no—
257C8—1882no—
258C22—21no—
259C22—21no—
260C22#1568902311yescharacteristic
261C22#1568948412yes—
262C4—21no—
263C4—21no—
264C4—82no—
265C4—3762no—
266C4#1568912312yes—
267C4#1568938412yes—
268C4#1568939712yes—
269C4#1568950911yescharacteristic
270C4#1568951611yescharacteristic
271C2—22no—
272C2#2024420111yescharacteristic
273C2#2024429411yescharacteristic
274C2#2024430011yescharacteristic
subgroupG/H
centerC22#15689023
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1549C2
socleC22×C47#390
prime radicalC23×C47#124
cosocleC2×C42×C47C23
hypercenter#1549D47
hypocenterC47#9775
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
163. C47#9775semidirect

Maximal subgroups

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

subgroupindex
C2×#156894752
#202435592
#202420692
#202420612
#202420592
#202416812
C47×#15492
#977547