Finite Group Databasebeta

← #26108820

Subgroups — #26108820

Order 12032.

656 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 131 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#20242114C212yesmaximal, complemented
2#20242015C212yesmaximal, complemented
3#20241722C212yesmaximal
4C2×C47×#541C211yesmaximal, characteristic
5#15689180C2214yes—
6#15688872—24no—
7#15688865—24no—
8#15688849—22no—
9#15688847—22no—
10C47×#541—24no—
11C47×#541—24no—
12C2×C4⋊Q8×C47C2211yescharacteristic
13C2×#126×C47—22no—
14C2×#125×C47—22no—
15C2×#125×C47C2211yescharacteristic
16C2×C22⋊Q8×C47C412yes—
17C2×C4×Q8×C47C412yes—
18C2×#117×C47C2211yescharacteristic
19C4⋊(C47⋊C8)—24no—
20C22⋊(C47⋊C8)—24no—
21#13093935—24no—
22C22×Q8×C47C2×C412yes—
23C23×C4×C47C2×C411yescharacteristic
24C4⋊Q8×C47—22no—
25C4⋊Q8×C47D414yes—
26#126×C47—216no—
27#125×C47—216no—
28#125×C47—22no—
29#125×C47D414yes—
30C22⋊Q8×C47—24no—
31C22⋊Q8×C47—24no—
32C4×Q8×C47—24no—
33C4×Q8×C47—24no—
34#117×C47—22no—
35#117×C47—22no—
36C2×C4⋊C4×C47—22no—
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47C2×C412yes—
41C2×C4⋊C4×C47D412yes—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47D412yes—
44C2×C42×C47—22no—
45C2×C42×C47C2×C411yescharacteristic
46C2×C42×C47C2311yescharacteristic
47C24×C47C22⋊C411yescharacteristic
48C2×Q8×C47—22no—
49C2×Q8×C47—44no—
50C2×Q8×C47C22⋊C414yes—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—42no—
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C412yes—
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D412yes—
60C4⋊C4×C47—28no—
61C4⋊C4×C47—28no—
62C4⋊C4×C47—22no—
63C4⋊C4×C47—28no—
64C4⋊C4×C47—28no—
65C4⋊C4×C47—24no—
66C4⋊C4×C47C22⋊C414yes—
67C22⋊C4×C47—28no—
68C22⋊C4×C47—24no—
69C42×C47—28no—
70C42×C47—21no—
71C42×C47—21no—
72C42×C47C2×D412yes—
73C42×C47C2×D412yes—
74C2×C47⋊C8—44no—
75C23×C47—21no—
76C23×C47—22no—
77C23×C47—42no—
78C23×C47C2×C22⋊C411yescharacteristic
79Q8×C47—44no—
80Q8×C47—44no—
81C2×C4×C47—22no—
82C2×C4×C47—22no—
83C2×C4×C47—24no—
84C2×C4×C47—24no—
85C2×C4×C47—21no—
86C2×C4×C47—22no—
87C2×C4×C47—42no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47—44no—
91C2×C4×C47—44no—
92C2×C4×C47—42no—
93C2×C4×C47C2×C22⋊C411yescharacteristic
94C2×C4×C47C2×C22⋊C411yescharacteristic
95C2×C4×C47#12014yes—
96C47⋊C8—84no—
97#8527—471nomaximal, Sylow
98C22×C47—22no—
99C22×C47—42no—
100C22×C47—41no—
101C22×C47—82no—
102C22×C47#37911yescharacteristic
103C22×C47#41312yes—
104C4×C47—24no—
105C4×C47—21no—
106C4×C47—21no—
107C4×C47—42no—
108C4×C47—44no—
109C4×C47—41no—
110C4×C47—44no—
111C2×#541D4711yescharacteristic
112#1558—472no—
113#1534—472no—
114#1464—472no—
115C2×C47—22no—
116C2×C47—81no—
117C2×C47#178511yescharacteristic
118C2×C47#179612yes—
119#541—24no—
120#541—24no—
121C2×C4⋊Q8C2×D4711yescharacteristic
122C2×#126—22no—
123C2×#125—22no—
124C2×#125C2×D4711yescharacteristic
125C2×C22⋊Q8Dic4712yes—
126C2×C4×Q8Dic4712yes—
127C2×#117C2×D4711yescharacteristic
128#432—474no—
129#333—944no—
130#330—944no—
131#324—942no—
132#323—942no—
133C47#852711yescharacteristic, complemented, Sylow
134C22×Q8C2×Dic4712yes—
135C23×C4C2×Dic4711yescharacteristic
136C4⋊Q8—22no—
137C4⋊Q8C22⋊D4714yes—
138#126—216no—
139#125—216no—
140#125—22no—
141#125C22⋊D4714yes—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146#117—22no—
147#117—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C2×Dic4712yes—
153C2×C4⋊C4C22⋊D4712yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4C22⋊D4712yes—
156C2×C42—22no—
157C2×C42C2×Dic4711yescharacteristic
158C2×C42C22×D4711yescharacteristic
159C4⋊C8—944no—
160C22⋊C8—944no—
161#97—944no—
162C24C22⋊Dic4711yescharacteristic
163C2×Q8—22no—
164C2×Q8—44no—
165C2×Q8C22⋊Dic4714yes—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4C22⋊Dic4711yescharacteristic
172C22×C4C22⋊Dic4712yes—
173C22×C4C22×Dic4711yescharacteristic
174C22×C4C2×C22⋊D4712yes—
175C2×C8—1884no—
176C4⋊C4—28no—
177C4⋊C4—28no—
178C4⋊C4—22no—
179C4⋊C4—28no—
180C4⋊C4—28no—
181C4⋊C4—24no—
182C4⋊C4C22⋊Dic4714yes—
183C22⋊C4—28no—
184C22⋊C4—24no—
185C42—28no—
186C42—21no—
187C42—21no—
188C42C2×C22⋊D4712yes—
189C42C2×C22⋊D4712yes—
190C23—21no—
191C23—22no—
192C23—42no—
193C23C2×C22⋊Dic4711yescharacteristic
194Q8—44no—
195Q8—44no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4C2×C22⋊Dic4711yescharacteristic
209C2×C4C2×C22⋊Dic4711yescharacteristic
210C2×C4#1309399314yes—
211C8—3764no—
212C22—22no—
213C22—41no—
214C22—42no—
215C22—82no—
216C22#1568897911yescharacteristic
217C22#1568910712yes—
218C4—21no—
219C4—21no—
220C4—24no—
221C4—41no—
222C4—44no—
223C4—44no—
224C4—42no—
225C2—22no—
226C2—81no—
227C2#2024331611yescharacteristic
228C2#2024336312yes—
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#541C2
socleC22×C47#379
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenterC2×#541D47
hypocenterC47#8527
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. #20242114C2semidirect
2. #20242015C2semidirect
133. C47#8527semidirect

Maximal subgroups

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

subgroupindex
#202421142
#202420152
#202417222
C2×C47×#5412
#852747