Finite Group Databasebeta

← #27351846

Subgroups — #27351846

Order 13568.

938 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 355 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#20288525C211yesmaximal, characteristic
2#20277636C211yesmaximal, characteristic
3#20273280C216yesmaximal
4#20273215C216yesmaximal
5C53×#2982C211yesmaximal, characteristic
6Q8×Dic106C414yescomplemented
7#16811352C414yescomplemented
8#16811349—24no—
9#16811254—23no—
10#16811180C2216yes—
11#16811101—21no—
12C2×Q8×Dic53—21no—
13#16810522—23no—
14#16810489C2213yes—
15C4×C4⋊Dic53C2213yes—
16#16810361C2214yes—
17#16810361C22112yes—
18C2×C4×Q8×C53C2211yescharacteristic
19C8×Q8×C53—24no—
20C53×#431C2216yes—
21#13133217C2×C416yes—
22#13133217C2×C416yes—
23#13133216—212no—
24#13133209—26no—
25#13133208—212no—
26#13133196—26no—
27#13133196C2314yes—
28Q8×Dic53—24no—
29Q8×Dic53—24no—
30Q8×Dic53—44no—
31C4×Dic106—26no—
32C4×Dic106C2×C4112yes—
33#13133176—26no—
34C2×C4⋊Dic53—23no—
35C2×C4⋊Dic53—23no—
36C2×C4×Dic53—23no—
37#13133124—23no—
38#13133124C2311yescharacteristic
39#13133124C2313yes—
40C22×Q8×C53D411yescharacteristic
41C4×Q8×C53C2×C414yes—
42C4×Q8×C53D414yes—
43C2×C4⋊C4×C53D413yes—
44C2×C42×C53C2313yes—
45C4⋊C8×C53—26no—
46C22⋊C8×C53C2314yes—
47C4×C8×C53—26no—
48C2×Q8×C53C22⋊C414yes—
49C2×Q8×C53SD1614yes—
50C2×Q8×C53Q1614yes—
51C22×C4×C53C2×D413yes—
52C22×C4×C53C2×D413yes—
53C22×C4×C53C2411yescharacteristic
54C22×Dic53—21no—
55C22×Dic53—23no—
56C2×Dic106—26no—
57C2×Dic106C22×C414yes—
58C2×C8×C53—24no—
59C4⋊C4×C53C22⋊C4112yes—
60C42×C53C22×C416yes—
61C42×C53C2×D416yes—
62C22⋊Dic53—24no—
63C4⋊Dic53—26no—
64C4⋊Dic53—26no—
65C4⋊Dic53—26no—
66C4⋊Dic53—46no—
67C4⋊Dic53—46no—
68C4⋊Dic53C22×C414yes—
69#11719919—212no—
70#11719919—24no—
71C4×Dic53—212no—
72C4×Dic53—46no—
73C23×C53C22×D411yescharacteristic
74Q8×C53#103116yes—
75C2×C4×C53—43no—
76C2×C4×C53—41no—
77C2×C4×C53C2×C22⋊C416yes—
78C2×C4×C53C2×C22⋊C416yes—
79C2×C4×C53C2×SD1616yes—
80C2×C4×C53C2×Q1616yes—
81C2×C4×C53C23×C411yescharacteristic
82C2×C4×C53C22×D411yescharacteristic
83C2×Dic53—22no—
84C2×Dic53—24no—
85C2×Dic53—24no—
86C2×Dic53—41no—
87C2×Dic53—41no—
88C2×Dic53—43no—
89C2×Dic53—43no—
90C2×Dic53—43no—
91Dic106—44no—
92Dic106—86no—
93C8×C53—44no—
94#17619—531nomaximal, Sylow
95C22×C53—21no—
96C22×C53—21no—
97C22×C53C22×C22⋊C411yescharacteristic
98C22×C53C22×SD1611yescharacteristic
99C22×C53C22×Q1611yescharacteristic
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—26no—
103C4×C53C2×#103112yes—
104Dic53—42no—
105Dic53—44no—
106Dic53—44no—
107Dic53—83no—
108#3502—531no—
109#2982D5311yescharacteristic
110#2351—531no—
111#1518—536no—
112#1503—536no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#288411yescharacteristic
116C2×C53C22×#10311yescharacteristic
117C2×C53#290111yescharacteristic
118Q82—534no—
119#557—534no—
120#557—1064no—
121#544—1063no—
122#533—536no—
123#519—1061no—
124C2×C4×Q8—1061no—
125C2×C4×Q8C2×D5311yescharacteristic
126C8×Q8—24no—
127#431C2×D5316yes—
128#389—1063no—
129#382—533no—
130C4×C4⋊C4—533no—
131#328—534no—
132#328—5312no—
133C53#1761911yescharacteristic, complemented, Sylow
134C22×Q8D21211yescharacteristic
135C4⋊Q8—536no—
136C4⋊Q8—536no—
137C4⋊Q8—10612no—
138#125—10612no—
139#125—1066no—
140C22⋊Q8—534no—
141C22⋊Q8—1066no—
142C4×Q8—5312no—
143C4×Q8—1064no—
144C4×Q8—1064no—
145C4×Q8—1066no—
146C4×Q8—2124no—
147C4×Q8C4×D5314yes—
148C4×Q8C22⋊D5314yes—
149#117—1066no—
150C2×C4⋊C4—1063no—
151C2×C4⋊C4—1063no—
152C2×C4⋊C4D21213yes—
153C2×C42—1063no—
154C2×C42C22×D5313yes—
155C4⋊C8—26no—
156C22⋊C8C22×D5314yes—
157C4×C8—26no—
158#95—531no—
159#95—533no—
160#95—1063no—
161C2×Q8—534no—
162C2×Q8—1066no—
163C2×Q8#1171991314yes—
164C2×Q8Dic21214yes—
165C2×Q8#1171992114yes—
166C22×C4—1061no—
167C22×C4—1063no—
168C22×C4C2×D21213yes—
169C22×C4C2×D21213yes—
170C22×C4C23×D5311yescharacteristic
171C2×C8—24no—
172C4⋊C4—534no—
173C4⋊C4—1066no—
174C4⋊C4—1066no—
175C4⋊C4—10612no—
176C4⋊C4—1066no—
177C4⋊C4—1064no—
178C4⋊C4—2126no—
179C4⋊C4—2126no—
180C4⋊C4#11719921112yes—
181C22⋊C4—1064no—
182C42—10612no—
183C42—2126no—
184C42C2×C4×D5316yes—
185C42C2×C22⋊D5316yes—
186C23C22×D21211yescharacteristic
187Q8—4246no—
188Q8—2124no—
189Q8#13133142116yes—
190C2×C4—43no—
191C2×C4—41no—
192C2×C4—1062no—
193C2×C4—1064no—
194C2×C4—1064no—
195C2×C4—2123no—
196C2×C4—2123no—
197C2×C4—2123no—
198C2×C4—2121no—
199C2×C4—2121no—
200C2×C4C2×#1171992116yes—
201C2×C4C2×#1171992116yes—
202C2×C4C2×#1171991316yes—
203C2×C4C2×Dic21216yes—
204C2×C4C22×C4×D5311yescharacteristic
205C2×C4C22×C22⋊D5311yescharacteristic
206C8—44no—
207C22—21no—
208C22—21no—
209C22C22×#1171992111yescharacteristic
210C22C22×#1171991311yescharacteristic
211C22C22×Dic21211yescharacteristic
212C4—21no—
213C4—21no—
214C4—26no—
215C4—4243no—
216C4—2122no—
217C4—2124no—
218C4—2124no—
219C4C2×#13133142112yes—
220C2—21no—
221C2—21no—
222C2#2028317811yescharacteristic
223C2C22×#1313314211yescharacteristic
224C2#2028326011yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2982C2
socleC22×C53C22×C22⋊C4
prime radicalC23×C53C22×D4
cosocleC22×C4×C53C24
hypercenter#2982D53
hypocenterC53#17619
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2982

Complements

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

NH ≅ G/Ntype
6. Q8×Dic106C4semidirect
7. #16811352C4semidirect
133. C53#17619semidirect

Maximal subgroups

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

subgroupindex
#202885252
#202776362
#202732802
#202732152
C53×#29822
#1761953