Finite Group Databasebeta

← #27271842

Subgroups — #27271842

Order 13568.

302 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 63 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#20275119C211yesmaximal, characteristic
2C2×#16810363C211yesmaximal, characteristic
3C2×C53×#354C211yesmaximal, characteristic
4#16811101C411yescharacteristic, complemented
5#16810628—21no—
6C2×#13133136C411yescharacteristic, complemented
7C2×#13133128—21no—
8#16810383—21no—
9#16810363—21no—
10#16810363C412yescomplemented
11C2×#124×C53C2211yescharacteristic
12C2×#99×C53—21no—
13C53×#354—21no—
14C53×#354C412yescomplemented
15C2×#11719917—21no—
16C22×#88322—41no—
17Q8×Dic53—42no—
18#13133176—21no—
19#13133176—41no—
20#13133138—42no—
21#13133136—41no—
22#13133128—22no—
23#13133128—41no—
24C22×Q8×C53C2×C411yescharacteristic
25C22×D4×C53D411yescharacteristic
26#124×C53—41no—
27#124×C53C2×C411yescharacteristic
28#124×C53C2×C411yescharacteristic
29#124×C53D411yescharacteristic
30#124×C53Q811yescharacteristic
31C2×C22⋊C4×C53—41no—
32C2×C22⋊C4×C53—41no—
33C2×C42×C53D411yescharacteristic
34#99×C53—22no—
35#99×C53—41no—
36C24×C53—21no—
37C2×Q8×C53—21no—
38C2×Q8×C53—41no—
39C2×Q8×C53—41no—
40C2×Q8×C53C4211yescharacteristic
41C2×Q8×C53C4⋊C411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53C22⋊C411yescharacteristic
46C2×D4×C53C4⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—81no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22⋊C4×C53—41no—
52C22⋊C4×C53—41no—
53C22⋊C4×C53—41no—
54C22⋊C4×C53—41no—
55C22⋊C4×C53—42no—
56C22⋊C4×C53—81no—
57C42×C53—21no—
58C42×C53C22⋊C411yescharacteristic
59C42×C53C4⋊C411yescharacteristic
60C22⋊Dic53—41no—
61C4⋊Dic53—22no—
62C4⋊Dic53—42no—
63C4×Dic53—41no—
64C4×Dic53—42no—
65#11719917—41no—
66#11719917—41no—
67C2×#88322—42no—
68C2×#88322—44no—
69C2×#88322—41no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—41no—
73C23×C53—41no—
74C23×C53—82no—
75C23×C53#9911yescharacteristic
76Q8×C53—42no—
77Q8×C53—41no—
78Q8×C53—41no—
79D4×C53—41no—
80D4×C53—41no—
81D4×C53—81no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—42no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—82no—
92C2×C4×C53—162no—
93C2×C4×C53#9511yescharacteristic
94C2×C4×C53#9911yescharacteristic
95C2×Dic53—42no—
96#88322—44no—
97#88322—81no—
98#7489—531nomaximal, Sylow
99C22×C53—22no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—81no—
103C22×C53—81no—
104C22×C53—82no—
105C22×C53—161no—
106C22×C53#34211yescharacteristic
107C22×C53AΣL(1,𝔽16)11yescharacteristic
108C22×C53#35211yescharacteristic
109C4×C53—21no—
110C4×C53—21no—
111C4×C53—41no—
112C4×C53—41no—
113C4×C53—41no—
114C4×C53—41no—
115C4×C53—162no—
116Dic53—82no—
117C2×#354D5311yescharacteristic
118#1888—531no—
119C2×#329—531no—
120C2×C53—21no—
121C2×C53—21no—
122C2×C53—81no—
123C2×C53—81no—
124C2×C53#139011yescharacteristic
125C2×C53#140412yes—
126C2×#124C2×D5311yescharacteristic
127#519—531no—
128#418—1061no—
129C2×#101—531no—
130C2×#99—21no—
131C2×#97—1061no—
132#354—21no—
133#354Dic5312yescomplemented
134#339—1061no—
135#329—532no—
136#329—1061no—
137C53#748911yescharacteristic, complemented, Sylow
138C22×Q8C4×D5311yescharacteristic
139C22×D4C22⋊D5311yescharacteristic
140C2×M16—1061no—
141C22×C8—2121no—
142#124—41no—
143#124Dic10611yescharacteristic
144#124C4×D5311yescharacteristic
145#124C2×Dic5311yescharacteristic
146#124C22⋊D5311yescharacteristic
147C4×Q8—2122no—
148#117—1061no—
149#117—2121no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C42D21211yescharacteristic
153#102—2122no—
154#101—2121no—
155#99—22no—
156#99—41no—
157#97—1062no—
158#97—2121no—
159C24—21no—
160C2×Q8—21no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8C4×Dic5311yescharacteristic
164C2×Q8#1171991911yescharacteristic
165C2×D4—21no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4#1171991911yescharacteristic
169C2×D4C22⋊Dic5311yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—81no—
173C22×C4#1171992111yescharacteristic
174M16—2121no—
175M16—2121no—
176C2×C8—2121no—
177C2×C8—2122no—
178C2×C8—2124no—
179C4⋊C4—1062no—
180C4⋊C4—2122no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C22⋊C4—81no—
187C22⋊C4—2121no—
188C42—21no—
189C42—2121no—
190C42—2122no—
191C42C4⋊Dic5311yescharacteristic
192C42#1171992111yescharacteristic
193C23—21no—
194C23—21no—
195C23—41no—
196C23—41no—
197C23—82no—
198C23#1313313111yescharacteristic
199Q8—41no—
200Q8—41no—
201Q8—42no—
202D4—41no—
203D4—41no—
204D4—81no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—82no—
215C2×C4—162no—
216C2×C4—2122no—
217C2×C4#1313312411yescharacteristic
218C2×C4#1313313111yescharacteristic
219C8—4241no—
220C8—2124no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22—82no—
227C22—161no—
228C22#1681038911yescharacteristic
229C22#1681040911yescharacteristic
230C22#1681041211yescharacteristic
231C4—21no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—162no—
238C4—4242no—
239C2—21no—
240C2—21no—
241C2—81no—
242C2—81no—
243C2#2027283211yescharacteristic
244C2#2027286512yes—
subgroupG/H
centerC22#16810409
commutatorC2×Q8×C53C42
FrattiniC2×#124C2×D53
FittingC2×C53×#354C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC22×D4×C53D4
cosocleC2×#124×C53C22
hypercenterC2×#354D53
hypocenterC53#7489
LCSG ▷ C2×Q8×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
4. #16811101C4semidirect
6. C2×#13133136C4semidirect
10. #16810363C4semidirect
14. C53×#354C4semidirect
133. #354Dic53semidirect
137. C53#7489semidirect

Maximal subgroups

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

subgroupindex
#202751192
C2×#168103632
C2×C53×#3542
#748953