Finite Group Databasebeta

← #26111222

Subgroups — #26111222

Order 12032.

458 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 115 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×#15689493C211yesmaximal, characteristic
2#20244360C212yesmaximal
3C2×#15688865C211yesmaximal, characteristic
4#20241269C212yesmaximal
5C47×#2382C211yesmaximal, characteristic
6C2×#13094024C2211yescharacteristic
7#15689493—22no—
8#15689493C414yescomplemented
9C2×#13093947—22no—
10C2×#13093935C2212yes—
11C2×#13093935C2211yescharacteristic
12#15688885—22no—
13#15688885—22no—
14#15688865—22no—
15C2×#125×C47C2211yescharacteristic
16C47×#389—22no—
17C47×#387—22no—
18C47×#387—22no—
19C47×#382C2212yes—
20C22×Dic94—21no—
21C22×C47⋊C8—21no—
22C22×C47⋊C8—22no—
23#13094024—22no—
24#13094024—21no—
25#13094024C2×C412yes—
26C2×C4⋊Dic47—22no—
27#13093947—28no—
28#13093935—21no—
29#13093935—22no—
30#13093935—22no—
31#13093935C2×C412yes—
32#125×C47—22no—
33#125×C47C2×C412yes—
34#125×C47D412yes—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—22no—
37C2×C4⋊C4×C47D412yes—
38C2×C4⋊C4×C47D412yes—
39C2×C4⋊C4×C47D412yes—
40C2×C42×C47—22no—
41C2×C42×C47C2311yescharacteristic
42#95×C47—22no—
43#95×C47—22no—
44#95×C47—22no—
45#95×C47—22no—
46C22×C4×C47—22no—
47C22×C4×C47—22no—
48C22×C4×C47—22no—
49C22×C4×C47C2×D411yescharacteristic
50C22×C4×C47C2×D412yes—
51C22×Dic47—41no—
52C2×Dic94—22no—
53C2×Dic94—42no—
54C2×Dic94—42no—
55C2×Dic94—41no—
56C4⋊C4×C47—24no—
57C4⋊C4×C47—24no—
58C4⋊C4×C47—22no—
59C4⋊C4×C47—42no—
60C4⋊C4×C47—42no—
61C4⋊C4×C47—82no—
62C4⋊C4×C47C22⋊C414yes—
63C42×C47—82no—
64C42×C47C22×C411yescharacteristic
65C42×C47C2×D411yescharacteristic
66C42×C47#4112yes—
67C4⋊Dic47—24no—
68C4⋊Dic47—42no—
69C2×C47⋊C8—24no—
70C2×C47⋊C8—22no—
71C2×C47⋊C8—41no—
72C2×C47⋊C8—41no—
73C2×C47⋊C8—42no—
74C2×C47⋊C8—42no—
75C23×C47#12011yescharacteristic
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—24no—
81C2×C4×C47—42no—
82C2×C4×C47—42no—
83C2×C4×C47—42no—
84C2×C4×C47—42no—
85C2×C4×C47—42no—
86C2×C4×C47—42no—
87C2×C4×C47—42no—
88C2×C4×C47—42no—
89C2×C4×C47C2×C22⋊C411yescharacteristic
90C2×C4×C47C4×D412yes—
91C2×C4×C47#12112yes—
92C2×C4×C47#12311yescharacteristic
93C2×Dic47—42no—
94C2×Dic47—81no—
95C2×Dic47—81no—
96Dic94—44no—
97Dic94—82no—
98C47⋊C8—44no—
99C47⋊C8—42no—
100#9004—471nomaximal, Sylow
101C22×C47#38611yescharacteristic
102C22×C47#45411yescharacteristic
103C22×C47#45411yescharacteristic
104C22×C47#45512yes—
105C22×C47#45612yes—
106C4×C47—22no—
107C4×C47—22no—
108C4×C47—21no—
109C4×C47—21no—
110C4×C47—44no—
111C4×C47—82no—
112C4×C47—82no—
113Dic47—82no—
114C2×#491—471no—
115#2382D4711yescharacteristic
116#1993—472no—
117C2×#330—471no—
118#1295—472no—
119C2×C47#166211yescharacteristic
120C2×C47#166311yescharacteristic
121C2×C47#166412yes—
122C2×C47#188212yes—
123C2×C47#188811yescharacteristic
124C2×C4⋊Q8—471no—
125C2×#125C2×D4711yescharacteristic
126#491—474no—
127#491—942no—
128C2×#103—942no—
129C2×#97—471no—
130C2×#97—472no—
131#389—22no—
132#387—22no—
133#387—22no—
134#382C2×D4712yes—
135#340—942no—
136#340—942no—
137#330—942no—
138C47#900411yescharacteristic, complemented, Sylow
139C22×Q8—941no—
140C22×C8—942no—
141C22×C8—941no—
142C4⋊Q8—472no—
143C4⋊Q8—941no—
144C4⋊Q8—942no—
145#125—22no—
146#125C4×D4712yes—
147#125D18812yes—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—942no—
151C2×C4⋊C4C22⋊D4712yes—
152C2×C4⋊C4C22⋊D4712yes—
153C2×C4⋊C4C22⋊D4712yes—
154C2×C42—22no—
155C2×C42C22×D4711yescharacteristic
156#103—948no—
157#97—472no—
158#97—941no—
159#97—942no—
160#97—942no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165C2×Q8—942no—
166C2×Q8—1882no—
167C2×Q8—1882no—
168C2×Q8—1881no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—1881no—
173C22×C4C2×C22⋊D4711yescharacteristic
174C22×C4C2×C22⋊D4712yes—
175C2×C8—944no—
176C2×C8—942no—
177C2×C8—1881no—
178C2×C8—1881no—
179C2×C8—1882no—
180C2×C8—1882no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—82no—
187C4⋊C4—944no—
188C4⋊C4—1882no—
189C4⋊C4(C2×D47)⋊C414yes—
190C42—82no—
191C42C2×C4×D4711yescharacteristic
192C42C2×D18811yescharacteristic
193C42#1062535012yes—
194C23#1309399311yescharacteristic
195Q8—3762no—
196Q8—1884no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—3761no—
211C2×C4—3761no—
212C2×C4—1882no—
213C2×C4C2×(C2×D47)⋊C411yescharacteristic
214C2×C4C4×C22⋊D4712yes—
215C2×C4#1309399812yes—
216C2×C4#1309400711yescharacteristic
217C8—1884no—
218C8—1882no—
219C22#1568900211yescharacteristic
220C22#1568929611yescharacteristic
221C22#1568929611yescharacteristic
222C22#1568930112yes—
223C22#1568930612yes—
224C4—22no—
225C4—22no—
226C4—21no—
227C4—21no—
228C4—44no—
229C4—82no—
230C4—82no—
231C4—3762no—
232C2#2024271111yescharacteristic
233C2#2024271611yescharacteristic
234C2#2024272212yes—
235C2#2024375712yes—
236C2#2024379111yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2382C2
socle, prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#2382D47
hypocenterC47#9004
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2382

Complements

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

NH ≅ G/Ntype
8. #15689493C4semidirect
138. C47#9004semidirect

Maximal subgroups

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

subgroupindex
C2×#156894932
#202443602
C2×#156888652
#202412692
C47×#23822
#900447