Finite Group Databasebeta

← #28314008

Subgroups — #28314008

Order 13568.

1162 conjugacy classes of subgroups; only the 344 normal subgroups are shown.

normal subgroups only

This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 344 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#20291608C212yesmaximal, complemented
2#20289959C216yesmaximal, complemented
3#20289948C216yesmaximal
4C2×#16811099C211yesmaximal, characteristic
5Q8×#88324—28no—
6#16811639C22112yes—
7#16811469C22112yes—
8#16811467C2214yes—
9C22×Q8×D53C411yescharacteristic
10#16811099—28no—
11C2×Q8×Dic53C2211yescharacteristic
12C2×C4×Dic106C413yes—
13C2×#13133182C2213yes—
14C2×#13133175C413yes—
15C2×#13133174C2213yes—
16C2×C4×Q8×C53C411yescharacteristic
17C4⋊#88324—212no—
18C22⋊#88324C2318yes—
19C4×#88324—212no—
20C22×Dic106C2×C413yes—
21C2×Q8×D53—24no—
22C2×Q8×D53—24no—
23C2×Q8×D53C818yescomplemented
24C22×C4×D53C2×C413yes—
25Q8×Dic53D418yes—
26C4×Dic106—212no—
27#13133182—212no—
28#13133175—212no—
29#13133174—212no—
30C2×C4⋊Dic53C2×C413yes—
31C2×#11719919C2×C413yes—
32C2×#11719919C2313yes—
33C2×#11719921C2×C413yes—
34C2×#11719921C2311yescharacteristic
35C2×C4×Dic53C2×C416yes—
36C2×C4×Dic53C2313yes—
37C22×Q8×C53C2×C411yescharacteristic
38C4×Q8×C53—24no—
39C2×C4⋊C4×C53C2×C413yes—
40C2×C42×C53C2×C413yes—
41C23×D53C22×C411yescharacteristic
42C2×Q8×C53C22⋊C414yes—
43C2×Q8×C53C2×C814yes—
44C2×Q8×C53M1614yes—
45C22×C4×C53C22×C411yescharacteristic
46C22×C4×C53C22×C413yes—
47C22×C4×C53C22×C413yes—
48C22×Dic53C22×C413yes—
49C22×Dic53C22×C413yes—
50C22×Dic53C2411yescharacteristic
51Q8×D53—232no—
52C2×C4×D53—26no—
53C2×C4×D53—26no—
54C2×C4×D53C2×C8112yes—
55C2×Dic106—26no—
56C2×Dic106—26no—
57C2×Dic106C2×C8112yes—
58C2×#88324—28no—
59C4⋊C4×C53—26no—
60C42×C53—26no—
61#11719921—26no—
62#11719921—22no—
63C4⋊Dic53C22⋊C4112yes—
64#11719919—26no—
65#11719919—26no—
66C4×Dic53—212no—
67C4×Dic53C2×D4112yes—
68C23×C53C23×C411yescharacteristic
69C22×D53—21no—
70C22×D53—21no—
71C22×D53—41no—
72C22×D53—41no—
73C22×D53C22×C812yes—
74Q8×C53C22⋊C8116yes—
75C2×C4×C53—23no—
76C2×C4×C53—21no—
77C2×C4×C53—43no—
78C2×C4×C53—41no—
79C2×C4×C53C2×C22⋊C416yes—
80C2×C4×C53C22×C816yes—
81C2×C4×C53C2×M1616yes—
82C2×Dic53—23no—
83C2×Dic53—23no—
84C2×Dic53—43no—
85C2×Dic53—41no—
86C2×Dic53C2×C22⋊C416yes—
87C2×Dic53C22×C816yes—
88C2×Dic53C22×D412yes—
89C4×D53—224no—
90Dic106—224no—
91#88324—48no—
92#17352—531nomaximal, Sylow
93C22×C53—21no—
94C22×C53—21no—
95C22×C53C22×C22⋊C411yescharacteristic
96C22×C53C23×C811yescharacteristic
97C22×C53C22×M1611yescharacteristic
98C2×D53—22no—
99C2×D53—42no—
100C2×D53—41no—
101C2×D53—41no—
102C2×D53—42no—
103C4×C53—43no—
104C4×C53—41no—
105C4×C53C2×C22⋊C8112yes—
106Dic53—26no—
107Dic53—22no—
108Dic53—26no—
109C2×#519—531no—
110#2982—532no—
111#1461—536no—
112#1458—536no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53C22×C22⋊C811yescharacteristic
116C2×C53#288411yescharacteristic
117C2×C53#288911yescharacteristic
118D53—42no—
119C23×Q8—531no—
120#519—1068no—
121C2×C4×Q8—531no—
122C2×C4×Q8—533no—
123C2×C4×Q8—533no—
124C2×C4×Q8#615711yescharacteristic
125C2×#117—533no—
126C2×#117—533no—
127C8×Q8—1068no—
128#431—5312no—
129#324—5312no—
130#323—534no—
131C53#1735211yescharacteristic, complemented, Sylow
132C22×Q8—538no—
133C22×Q8—533no—
134C22×Q8—1064no—
135C22×Q8—1064no—
136C22×Q8C2×#615711yescharacteristic
137C23×C4—533no—
138C4×Q8—24no—
139C4×Q8—538no—
140C4×Q8—10612no—
141C4×Q8—10612no—
142#117—10612no—
143#117—10612no—
144C2×C4⋊C4—533no—
145C2×C4⋊C4—533no—
146C2×C4⋊C4—533no—
147C2×C4⋊C4C2×#615713yes—
148C2×C22⋊C4—531no—
149C2×C22⋊C4—533no—
150C2×C42—533no—
151C2×C42—536no—
152C2×C42C2×#615713yes—
153C4⋊C8—10612no—
154C22⋊C8—538no—
155C4×C8—10612no—
156C24—531no—
157C2×Q8—5312no—
158C2×Q8—10632no—
159C2×Q8—1066no—
160C2×Q8—1066no—
161C2×Q8D53⋊C814yes—
162C2×Q8#1171993614yes—
163C2×Q8C22⋊#615714yes—
164C22×C4—533no—
165C22×C4—533no—
166C22×C4—531no—
167C22×C4—5312no—
168C22×C4—1066no—
169C22×C4—1066no—
170C22×C4C22×#615713yes—
171C22×C4C22×#615711yescharacteristic
172C22×C4C22×#615713yes—
173C2×C8—1068no—
174C4⋊C4—26no—
175C4⋊C4—5312no—
176C4⋊C4—1066no—
177C4⋊C4—1066no—
178C22⋊C4—1062no—
179C22⋊C4—1066no—
180C42—26no—
181C42—5312no—
182C42—10612no—
183C23—532no—
184C23—1061no—
185C23—1061no—
186C23—2121no—
187C23—2121no—
188C23C23×#615711yescharacteristic
189Q8—10624no—
190Q8#13133274116yes—
191C2×C4—21no—
192C2×C4—23no—
193C2×C4—41no—
194C2×C4—43no—
195C2×C4—536no—
196C2×C4—532no—
197C2×C4—536no—
198C2×C4—1063no—
199C2×C4—1063no—
200C2×C4—10624no—
201C2×C4—2123no—
202C2×C4—2121no—
203C2×C4C2×C22⋊#615716yes—
204C2×C4C2×D53⋊C816yes—
205C2×C4C2×#1171993616yes—
206C8—2128no—
207C22—21no—
208C22—21no—
209C22—1062no—
210C22—2122no—
211C22—2122no—
212C22—2121no—
213C22—2121no—
214C22C22×C22⋊#615711yescharacteristic
215C22C22×D53⋊C811yescharacteristic
216C22C22×#1171993611yescharacteristic
217C4—41no—
218C4—43no—
219C4—1066no—
220C4—1066no—
221C4—1062no—
222C4C2×#13133274112yes—
223C2—21no—
224C2—21no—
225C2—2122no—
226C2C22×#1313327411yescharacteristic
227C2#2029131711yescharacteristic
228C2#2029132811yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC22×C53C23×C8
FrattiniC23C23×#6157
FittingC2×C4×Q8×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C22×C4
cosocleC22×Dic53C24
hypercenterC2×C4×Q8#6157
hypocenterC53#17352
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20291608C2semidirect
2. #20289959C2semidirect
23. C2×Q8×D53C8semidirect
131. C53#17352semidirect

Maximal subgroups

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

subgroupindex
#202916082
#202899592
#202899482
C2×#168110992
#1735253