Finite Group Databasebeta

← #26120776

Subgroups — #26120776

Order 12032.

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

normal subgroups only

This group has 406 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#20251660C211yesmaximal, characteristic
2#20243557C211yesmaximal, characteristic
3#20243556C212yesmaximal
4#20241270C212yesmaximal
5C47×#1295C211yesmaximal, characteristic
6C2×#13093958—22no—
7C2×#13093957—21no—
8C2×#13093956—22no—
9C2×#13093955—21no—
10C2×#13093935C2211yescharacteristic
11#15689065—22no—
12#15689062—21no—
13#15689044—22no—
14#15689043—21no—
15#15688993C2211yescharacteristic
16#15688989C2212yes—
17C2×#97×C47C2212yes—
18C47×#382C2211yescharacteristic
19C22×Dic94—21no—
20C22×C47⋊C8—21no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47—22no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47D412yes—
25C2×C4⋊Dic47D412yes—
26C2×Dic47⋊C4—22no—
27#13093958—44no—
28#13093957—42no—
29#13093956—44no—
30#13093955—42no—
31#13093935—21no—
32#13093935Q812yescomplemented
33C47⋊#95—21no—
34C47⋊#95—22no—
35C47⋊#95—21no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C22×C8×C47—22no—
39C2×C4⋊C4×C47D411yescharacteristic
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C42×C47C2311yescharacteristic
42#97×C47—22no—
43#97×C47Q814yescomplemented
44#95×C47—22no—
45C22×C4×C47—21no—
46C22×C4×C47—21no—
47C22×C4×C47C2×D411yescharacteristic
48C22×C4×C47C2×D412yes—
49C22×Dic47—22no—
50C22×Dic47—22no—
51C22×Dic47—41no—
52C2×Dic94—82no—
53C2×Dic94—81no—
54C2×C8×C47—24no—
55C2×C8×C47—24no—
56C2×C8×C47—24no—
57C4⋊C4×C47—41no—
58C4⋊C4×C47—41no—
59C42×C47C2×Q812yes—
60C42×C47C2×Q811yescharacteristic
61C42×C47#4111yescharacteristic
62C4⋊Dic47—42no—
63C4⋊Dic47—42no—
64C4⋊Dic47—41no—
65C4⋊Dic47—42no—
66C4⋊Dic47—41no—
67C4⋊Dic47—42no—
68C4⋊Dic47—81no—
69Dic47⋊C4—82no—
70C2×C47⋊C8—22no—
71C2×C47⋊C8—22no—
72C2×C47⋊C8—22no—
73C23×C47#12011yescharacteristic
74C2×C4×C47—21no—
75C2×C4×C47—22no—
76C2×C4×C47—21no—
77C2×C4×C47—22no—
78C2×C4×C47—42no—
79C2×C4×C47—41no—
80C2×C4×C47—41no—
81C2×C4×C47—42no—
82C2×C4×C47C22⋊Q812yes—
83C2×C4×C47C22⋊Q811yescharacteristic
84C2×C4×C47C4⋊Q812yes—
85C2×C4×C47C4⋊Q811yescharacteristic
86C2×Dic47—42no—
87C2×Dic47—42no—
88C2×Dic47—42no—
89C2×Dic47—42no—
90C2×Dic47—42no—
91C2×Dic47—42no—
92C2×Dic47—82no—
93C2×Dic47—81no—
94Dic94—84no—
95C8×C47—28no—
96C47⋊C8—24no—
97#10583—471nomaximal, Sylow
98C22×C47#39511yescharacteristic
99C22×C47#45312yes—
100C22×C47#45311yescharacteristic
101C22×C47#45612yes—
102C22×C47#45611yescharacteristic
103C4×C47—21no—
104C4×C47—22no—
105C4×C47—22no—
106C4×C47—21no—
107C4×C47—81no—
108C4×C47—81no—
109Dic47—82no—
110Dic47—82no—
111Dic47—161no—
112#2844—471no—
113#1847—472no—
114#1847—471no—
115#1295—472no—
116#1295D4711yescharacteristic
117C2×C47#165511yescharacteristic
118C2×C47#166011yescharacteristic
119C2×C47#166012yes—
120C2×C47#206012yes—
121C2×C47#206011yescharacteristic
122C2×#107—942no—
123C2×#107—941no—
124C2×#106—942no—
125C2×#106—941no—
126C2×#97—471no—
127C2×#97C2×D4712yes—
128#400—942no—
129#400—941no—
130#395—942no—
131#395—941no—
132#384—471no—
133#382—472no—
134#382C2×D4711yescharacteristic
135C47#1058311yescharacteristic, complemented, Sylow
136C22×Q8—941no—
137C22×C8—22no—
138C22×C8—941no—
139C2×C4⋊C4—472no—
140C2×C4⋊C4—472no—
141C2×C4⋊C4—942no—
142C2×C4⋊C4—941no—
143C2×C4⋊C4—941no—
144C2×C4⋊C4—942no—
145C2×C4⋊C4D18811yescharacteristic
146C2×C4⋊C4D18811yescharacteristic
147C2×C42C22×D4711yescharacteristic
148#107—1882no—
149#107—1884no—
150#106—1882no—
151#106—1884no—
152#97—22no—
153#97—472no—
154#97—941no—
155#97Dic9414yescomplemented
156#95—22no—
157#95—942no—
158#95—941no—
159#95—941no—
160#95—942no—
161#95—942no—
162C2×Q8—3762no—
163C2×Q8—3761no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—942no—
167C22×C4—942no—
168C22×C4—1881no—
169C22×C4C2×D18811yescharacteristic
170C22×C4D4×D4712yes—
171C2×C8—24no—
172C2×C8—24no—
173C2×C8—24no—
174C2×C8—942no—
175C2×C8—942no—
176C2×C8—942no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—3762no—
180C4⋊C4—3761no—
181C4⋊C4—1882no—
182C4⋊C4—1881no—
183C4⋊C4—1882no—
184C4⋊C4—1881no—
185C4⋊C4—1882no—
186C4⋊C4—1882no—
187C42C2×Dic9412yes—
188C42#1062535211yescharacteristic
189C42Q8×D4711yescharacteristic
190C23#1309399411yescharacteristic
191Q8—3764no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—3762no—
201C2×C4—3761no—
202C2×C4—1882no—
203C2×C4—1882no—
204C2×C4—1882no—
205C2×C4—1882no—
206C2×C4—1882no—
207C2×C4—1882no—
208C2×C4#1309400112yes—
209C2×C4#1309400411yescharacteristic
210C2×C4#1309402312yes—
211C2×C4#1309402411yescharacteristic
212C8—28no—
213C8—944no—
214C22#1568904411yescharacteristic
215C22#1568929212yes—
216C22#1568929311yescharacteristic
217C22#1568930712yes—
218C22#1568930811yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—81no—
224C4—81no—
225C4—3762no—
226C4—3762no—
227C4—7521no—
228C2#2024267811yescharacteristic
229C2#2024270011yescharacteristic
230C2#2024270112yes—
231C2#2024473212yes—
232C2#2024473411yescharacteristic
subgroupG/H
centerC23#13093994
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1295C2
socle, prime radicalC23×C47#120
hypercenter#1295D47
hypocenterC47#10583
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
32. #13093935Q8semidirect
43. #97×C47Q8semidirect
135. C47#10583semidirect
155. #97Dic94semidirect

Maximal subgroups

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

subgroupindex
#202516602
#202435572
#202435562
#202412702
C47×#12952
#1058347