Finite Group Databasebeta

← #26120918

Subgroups — #26120918

Order 12032.

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

normal subgroups only

This group has 306 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
1#20244376C211yesmaximal, characteristic
2#20244164C211yesmaximal, characteristic
3#20242141C211yesmaximal, characteristic
4#20242068C211yesmaximal, characteristic
5C2×#15688871C211yesmaximal, characteristic
6#20241262C211yesmaximal, characteristic
7C47×#1571C211yesmaximal, characteristic
8C2×#13094024C2211yescharacteristic
9C2×#13093960—21no—
10#15689147C2211yescharacteristic
11C2×C4⋊(C47⋊C8)C2211yescharacteristic
12C4×#10625329C2211yescharacteristic
13#15688891—21no—
14#15688891—21no—
15#15688888—21no—
16#15688875—21no—
17#15688873—21no—
18#15688871—22no—
19#15688871—21no—
20#15688871—21no—
21C47×#431C2211yescharacteristic
22C47×#423C2211yescharacteristic
23C2×C4⋊C8×C47C2211yescharacteristic
24C47×#335—22no—
25C22×Dic94Q812yes—
26C2×#10625329—21no—
27C2×#10625329D411yescharacteristic
28C2×#10625329D411yescharacteristic
29C22×C47⋊C8—21no—
30#13094024—21no—
31#13094024—21no—
32#13094024—21no—
33#13094024—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47—21no—
36#13093960—42no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41#13093935—21no—
42C4×C47⋊C8—21no—
43C2×M16×C47—21no—
44C22×C8×C47—21no—
45C2×C42×C47C2311yescharacteristic
46C4⋊C8×C47—21no—
47C4⋊C8×C47—21no—
48C4⋊C8×C47—21no—
49C4⋊C8×C47D411yescharacteristic
50C4⋊C8×C47D411yescharacteristic
51C4⋊C8×C47D412yes—
52C22⋊C8×C47—21no—
53C4×C8×C47—21no—
54C22×C4×C47C2×D411yescharacteristic
55C22×C4×C47C2×Q811yescharacteristic
56C22×C4×C47#4111yescharacteristic
57C22×Dic47—22no—
58C2×Dic94—22no—
59C2×Dic94—22no—
60C2×Dic94—82no—
61M16×C47—41no—
62M16×C47—41no—
63C2×C8×C47—22no—
64C2×C8×C47—21no—
65C2×C8×C47—21no—
66C2×C8×C47—41no—
67C2×C8×C47—41no—
68C2×C8×C47—41no—
69C42×C47C2×D411yescharacteristic
70C42×C47C2×D411yescharacteristic
71C42×C47#4111yescharacteristic
72C42×C47#4111yescharacteristic
73C4⋊Dic47—41no—
74C4⋊Dic47—41no—
75C4⋊Dic47—41no—
76C4⋊Dic47—41no—
77#10625329—21no—
78#10625329—21no—
79#10625329—21no—
80#10625329—41no—
81#10625329—41no—
82#10625329Q1612yes—
83C2×C47⋊C8—21no—
84C2×C47⋊C8—21no—
85C2×C47⋊C8—41no—
86C2×C47⋊C8—41no—
87C2×C47⋊C8—81no—
88C23×C47C22⋊Q811yescharacteristic
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—41no—
92C2×C4×C47#12111yescharacteristic
93C2×C4×C47#12111yescharacteristic
94C2×C4×C47#12111yescharacteristic
95C2×C4×C47C22⋊Q811yescharacteristic
96C2×C4×C47C22⋊Q811yescharacteristic
97C2×C4×C47#12611yescharacteristic
98C2×C4×C47C2×Q1611yescharacteristic
99C2×C4×C47#13511yescharacteristic
100C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
101C2×C4×C47#13711yescharacteristic
102C2×Dic47—42no—
103C2×Dic47—82no—
104Dic94—82no—
105Dic94—82no—
106C8×C47—42no—
107C8×C47—41no—
108C8×C47—41no—
109C8×C47—81no—
110C47⋊C8—22no—
111C47⋊C8—41no—
112C47⋊C8—81no—
113C47⋊C8—81no—
114#10605—471nomaximal, Sylow
115C22×C47—21no—
116C22×C47—21no—
117C22×C47#39611yescharacteristic
118C22×C47#47711yescharacteristic
119C22×C47#47911yescharacteristic
120C4×C47—21no—
121C4×C47—21no—
122C4×C47—21no—
123C4×C47#46211yescharacteristic
124C4×C47#46311yescharacteristic
125C4×C47#46711yescharacteristic
126C4×C47#46811yescharacteristic
127C4×C47#47111yescharacteristic
128C4×C47#47311yescharacteristic
129Dic47—82no—
130#1997—471no—
131#1950—471no—
132#1571D4711yescharacteristic
133#1565—471no—
134#1548—471no—
135C2×#333—471no—
136#1292—471no—
137C2×C47—21no—
138C2×C47—21no—
139C2×C47#206411yescharacteristic
140C2×C47#206811yescharacteristic
141C2×C47#206911yescharacteristic
142C2×C4⋊Q8—471no—
143#431C2×D4711yescharacteristic
144C2×#108—941no—
145#423—471no—
146#423C2×D4711yescharacteristic
147C2×C4⋊C8—471no—
148C2×C4⋊C8C2×D4711yescharacteristic
149C4×M16—471no—
150#343—941no—
151#343—941no—
152#341—941no—
153#335—22no—
154#335—941no—
155#334—941no—
156#333—941no—
157#333—941no—
158#333—942no—
159C47#1060511yescharacteristic, complemented, Sylow
160C22×Q8—472no—
161C2×M16—21no—
162C2×M16—471no—
163C2×M16—471no—
164C2×M16—941no—
165C22×C8—21no—
166C22×C8—941no—
167C4⋊Q8—941no—
168C4⋊Q8—941no—
169C4⋊Q8—941no—
170C4⋊Q8—941no—
171C2×C4⋊C4—941no—
172C2×C4⋊C4—941no—
173C2×C42C22×D4711yescharacteristic
174#108—1882no—
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—941no—
179C4⋊C8—941no—
180C4⋊C8—941no—
181C4⋊C8—941no—
182C4⋊C8D18811yescharacteristic
183C4⋊C8D18811yescharacteristic
184C4⋊C8C22⋊D4712yes—
185C22⋊C8—21no—
186#97—941no—
187C4×C8—21no—
188C4×C8—941no—
189C2×Q8—942no—
190C2×Q8—942no—
191C2×Q8—3762no—
192C22×C4—942no—
193C22×C4#1062535011yescharacteristic
194C22×C4D4×D4711yescharacteristic
195C22×C4Q8×D4711yescharacteristic
196M16—41no—
197M16—41no—
198M16—472no—
199M16—941no—
200M16—941no—
201M16—941no—
202M16—1881no—
203M16—1881no—
204C2×C8—22no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—941no—
211C2×C8—941no—
212C2×C8—3761no—
213C2×C8—1881no—
214C2×C8—1881no—
215C4⋊C4—1881no—
216C4⋊C4—1881no—
217C4⋊C4—1881no—
218C4⋊C4—1881no—
219C42C2×D18811yescharacteristic
220C42#1062535211yescharacteristic
221C42#1062535411yescharacteristic
222C42C2×C22⋊D4711yescharacteristic
223C23#1309400211yescharacteristic
224Q8—3762no—
225Q8—3762no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—3762no—
230C2×C4—1882no—
231C2×C4#1309399611yescharacteristic
232C2×C4#1309399811yescharacteristic
233C2×C4#1309399911yescharacteristic
234C2×C4#1309400011yescharacteristic
235C2×C4#1309400411yescharacteristic
236C2×C4#1309401911yescharacteristic
237C2×C4Q16×D4711yescharacteristic
238C2×C4#1309404311yescharacteristic
239C2×C4#1309404811yescharacteristic
240C2×C4#1309405611yescharacteristic
241C8—42no—
242C8—41no—
243C8—41no—
244C8—81no—
245C8—942no—
246C8—3761no—
247C8—3761no—
248C8—1881no—
249C22—21no—
250C22—21no—
251C22#1568904511yescharacteristic
252C22#1568941711yescharacteristic
253C22#1568942911yescharacteristic
254C4—21no—
255C4—21no—
256C4—21no—
257C4—3762no—
258C4#1568933111yescharacteristic
259C4#1568933611yescharacteristic
260C4#1568936511yescharacteristic
261C4#1568937011yescharacteristic
262C4#1568938611yescharacteristic
263C4#1568939811yescharacteristic
264C2—21no—
265C2—21no—
266C2#2024475411yescharacteristic
267C2#2024477811yescharacteristic
268C2#2024478511yescharacteristic
subgroupG/H
centerC22#15689045
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1571C2
socleC22×C47#396
prime radicalC23×C47C22⋊Q8
hypercenter#1571D47
hypocenterC47#10605
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
159. C47#10605semidirect

Maximal subgroups

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

subgroupindex
#202443762
#202441642
#202421412
#202420682
C2×#156888712
#202412622
C47×#15712
#1060547