Finite Group Databasebeta

← #26105559

Subgroups — #26105559

Order 12032.

462 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 95 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#20244951C212yesmaximal, complemented
2#20244921C211yesmaximal, characteristic
3#20241211C212yesmaximal, complemented
4#20241209C211yesmaximal, characteristic
5C47×#2683C211yesmaximal, characteristic
6C2×#13093955—21no—
7C2×#13093945—21no—
8#15688993C2211yescharacteristic
9#15688885—22no—
10#15688885—22no—
11#15688878C2212yes—
12#15688878C2211yescharacteristic
13C2×C22⋊Q8×C47—22no—
14C47×#396—22no—
15C47×#395—21no—
16C47×#390C2211yescharacteristic
17C47×#389C2212yes—
18C47×#386—21no—
19C22×C47⋊C8—21no—
20C22×C47⋊C8—22no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47—22no—
23#13093955—42no—
24#13093945—42no—
25C22×Q8×C47—21no—
26C22×D4×C47D411yescharacteristic
27C23×C4×C47—21no—
28C22⋊Q8×C47—44no—
29C22⋊Q8×C47—44no—
30C2×C4⋊C4×C47—22no—
31C2×C4⋊C4×C47—22no—
32C2×C4⋊C4×C47D412yes—
33C2×C4⋊C4×C47D412yes—
34C2×C4⋊C4×C47D411yescharacteristic
35C2×C22⋊C4×C47—22no—
36C2×C22⋊C4×C47—22no—
37C2×C22⋊C4×C47—21no—
38C2×C42×C47C2311yescharacteristic
39#95×C47—21no—
40#95×C47—22no—
41C24×C47—21no—
42C2×Q8×C47—44no—
43C2×Q8×C47—81no—
44C2×D4×C47—21no—
45C2×D4×C47—41no—
46C2×D4×C47—41no—
47C2×D4×C47Q1612yes—
48C22×C4×C47—22no—
49C22×C4×C47—21no—
50C22×C4×C47—22no—
51C22×C4×C47—44no—
52C22×C4×C47—81no—
53C22×C4×C47C2×D412yes—
54C22×C4×C47C2×D411yescharacteristic
55C22×Dic47—41no—
56C4⋊C4×C47—22no—
57C4⋊C4×C47—44no—
58C4⋊C4×C47—42no—
59C4⋊C4×C47—41no—
60C4⋊C4×C47—42no—
61C4⋊C4×C47—42no—
62C4⋊C4×C47SD1614yes—
63C22⋊C4×C47—44no—
64C22⋊C4×C47—42no—
65C22⋊C4×C47—42no—
66C22⋊C4×C47—81no—
67C42×C47—41no—
68C4⋊Dic47—41no—
69C4⋊Dic47—41no—
70C4⋊Dic47—44no—
71C2×C47⋊C8—22no—
72C2×C47⋊C8—41no—
73C2×C47⋊C8—41no—
74C2×C47⋊C8—42no—
75C2×C47⋊C8—42no—
76C2×C47⋊C8—42no—
77C23×C47—22no—
78C23×C47—42no—
79C23×C47—41no—
80C23×C47—42no—
81C23×C47—41no—
82C23×C47#12011yescharacteristic
83Q8×C47—84no—
84D4×C47—42no—
85D4×C47—81no—
86C2×C4×C47—21no—
87C2×C4×C47—22no—
88C2×C4×C47—22no—
89C2×C4×C47—21no—
90C2×C4×C47—24no—
91C2×C4×C47—42no—
92C2×C4×C47—42no—
93C2×C4×C47—42no—
94C2×C4×C47—42no—
95C2×C4×C47—41no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47—84no—
99C2×C4×C47C2×SD1612yes—
100C2×C4×C47C2×Q1611yescharacteristic
101C2×C4×C47AGL(1,ℤ/8)12yes—
102C2×C4×C47#13711yescharacteristic
103C2×Dic47—81no—
104C2×Dic47—82no—
105C47⋊C8—42no—
106C47⋊C8—82no—
107#7826—471nomaximal, Sylow
108C22×C47—42no—
109C22×C47—44no—
110C22×C47—82no—
111C22×C47—81no—
112C22×C47—81no—
113C22×C47#45012yes—
114C22×C47#45111yescharacteristic
115C22×C47#45311yescharacteristic
116C22×C47#45412yes—
117C22×C47#45811yescharacteristic
118C4×C47—21no—
119C4×C47—21no—
120C4×C47—22no—
121C4×C47—22no—
122C4×C47—44no—
123C4×C47—82no—
124C4×C47—81no—
125Dic47—161no—
126#2683D4711yescharacteristic
127#2100—472no—
128#2094—471no—
129#1271—472no—
130#1270—471no—
131C2×C47—82no—
132C2×C47#161411yescharacteristic
133C2×C47#161612yes—
134C2×C47#162911yescharacteristic
135C2×C47#163712yes—
136C2×C47#165711yescharacteristic
137C2×C22⋊Q8—22no—
138C2×#106—941no—
139C2×#102—941no—
140#396—22no—
141#395—21no—
142#390C2×D4711yescharacteristic
143#389C2×D4712yes—
144#386—21no—
145#384—471no—
146#340—942no—
147#340—942no—
148#336—471no—
149#336—472no—
150C47#782611yescharacteristic, complemented, Sylow
151C22×Q8—21no—
152C22×D4C22⋊D4711yescharacteristic
153C23×C4—21no—
154C22×C8—941no—
155C22×C8—942no—
156C22⋊Q8—44no—
157C22⋊Q8—44no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—942no—
161C2×C4⋊C4—941no—
162C2×C4⋊C4C22⋊D4712yes—
163C2×C4⋊C4C22⋊D4712yes—
164C2×C4⋊C4C22⋊D4711yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C42C22×D4711yescharacteristic
169#106—1882no—
170#102—1882no—
171#95—21no—
172#95—22no—
173C24—21no—
174C2×Q8—44no—
175C2×Q8—81no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4#1062533712yes—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—44no—
184C22×C4—81no—
185C22×C4—1881no—
186C22×C4C2×C22⋊D4712yes—
187C22×C4C2×C22⋊D4711yescharacteristic
188C2×C8—942no—
189C2×C8—1882no—
190C2×C8—1882no—
191C2×C8—1882no—
192C2×C8—1881no—
193C2×C8—1881no—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—1881no—
201C4⋊C4—1881no—
202C4⋊C4—1884no—
203C4⋊C4#1062533514yes—
204C22⋊C4—44no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C42—41no—
209C23—22no—
210C23—42no—
211C23—41no—
212C23—42no—
213C23—41no—
214C23#1309399311yescharacteristic
215Q8—84no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—24no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—3761no—
232C2×C4—3762no—
233C2×C4C2×#1062533512yes—
234C2×C4C2×#1062533711yescharacteristic
235C2×C4#1309405012yes—
236C2×C4#1309405511yescharacteristic
237C8—3762no—
238C8—1882no—
239C22—44no—
240C22—42no—
241C22—81no—
242C22—82no—
243C22—81no—
244C22#1568927412yes—
245C22#1568927911yescharacteristic
246C22#1568929111yescharacteristic
247C22#1568929612yes—
248C22#1568931311yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—44no—
254C4—81no—
255C4—82no—
256C4—7521no—
257C2—82no—
258C2#2024241311yescharacteristic
259C2#2024242512yes—
260C2#2024250411yescharacteristic
261C2#2024255312yes—
262C2#2024268611yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2683C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2683D47
hypocenterC47#7826
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. #20244951C2semidirect
3. #20241211C2semidirect
150. C47#7826semidirect

Maximal subgroups

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

subgroupindex
#202449512
#202449212
#202412112
#202412092
C47×#26832
#782647