Finite Group Databasebeta

← #27271822

Subgroups — #27271822

Order 13568.

278 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 59 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#20276380C211yesmaximal, characteristic
2#20273450C211yesmaximal, characteristic
3C53×#1884C211yesmaximal, characteristic
4#16810647—21no—
5#16810597—21no—
6#16810418C412yescomplemented
7#16810416—21no—
8#16810372C412yescomplemented
9#16810367—21no—
10C53×#568C411yescharacteristic, complemented
11C53×#532C2211yescharacteristic
12C53×#421—21no—
13C53×#410C411yescharacteristic, complemented
14C53×#339—21no—
15C2×#11719917—41no—
16#13133176—41no—
17C4⋊#88322—22no—
18C4⋊#88322—41no—
19#13133132—22no—
20#13133132—41no—
21C2×#41×C53C2×C411yescharacteristic
22C2×#41×C53D411yescharacteristic
23#131×C53—42no—
24C2×M16×C53—21no—
25C2×M16×C53—41no—
26C4⋊Q8×C53C2×C411yescharacteristic
27#127×C53D411yescharacteristic
28#124×C53C2×C411yescharacteristic
29#124×C53Q811yescharacteristic
30#121×C53—41no—
31C4×D4×C53—41no—
32#117×C53—21no—
33C2×C42×C53D411yescharacteristic
34#104×C53—42no—
35#99×C53—41no—
36#41×C53—41no—
37#41×C53—41no—
38#41×C53—41no—
39#41×C53—41no—
40C2×Q8×C53C4211yescharacteristic
41C2×Q8×C53C4⋊C411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53C22⋊C411yescharacteristic
46C2×D4×C53C4⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C411yescharacteristic
51M16×C53—22no—
52M16×C53—42no—
53M16×C53—41no—
54C2×C8×C53—41no—
55C2×C8×C53—42no—
56C4⋊C4×C53—41no—
57C4⋊C4×C53—41no—
58C22⋊C4×C53—41no—
59C22⋊C4×C53—41no—
60C22⋊C4×C53—41no—
61C42×C53—21no—
62C42×C53—41no—
63C42×C53C22⋊C411yescharacteristic
64C42×C53C4⋊C411yescharacteristic
65C22⋊Dic53—42no—
66C4⋊Dic53—81no—
67C4×Dic53—81no—
68#11719917—82no—
69C2×#88322—42no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53#9911yescharacteristic
73Q8×C53—41no—
74Q8×C53—41no—
75D4×C53—41no—
76D4×C53—41no—
77D4×C53—42no—
78D4×C53—81no—
79D4×C53—81no—
80D4×C53—81no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—81no—
94C2×C4×C53#9511yescharacteristic
95C2×C4×C53#9911yescharacteristic
96C2×Dic53—82no—
97C8×C53—42no—
98#88322—82no—
99#7482—531nomaximal, Sylow
100C22×C53—21no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—81no—
105C22×C53—81no—
106C22×C53#34211yescharacteristic
107C4×C53—21no—
108C4×C53—21no—
109C4×C53—21no—
110C4×C53—41no—
111C4×C53—41no—
112C4×C53—41no—
113C4×C53—161no—
114C4×C53AΣL(1,𝔽16)11yescharacteristic
115C4×C53#35211yescharacteristic
116Dic53—162no—
117#2126—531no—
118#1884D5311yescharacteristic
119#1560—531no—
120C2×C53—21no—
121C2×C53—41no—
122C2×C53—81no—
123C2×C53—81no—
124C2×C53#139011yescharacteristic
125#568Dic5311yescharacteristic, complemented
126#532C2×D5311yescharacteristic
127#423—1061no—
128#421—21no—
129#410—1061no—
130#410Dic5311yescharacteristic, complemented
131#354—532no—
132#353—1061no—
133#339—21no—
134#333—532no—
135#331—1061no—
136C53#748211yescharacteristic, complemented, Sylow
137C2×#41C2×Dic5311yescharacteristic
138C2×#41C22⋊D5311yescharacteristic
139#131—42no—
140C2×M16—21no—
141C2×M16—41no—
142C2×M16—2121no—
143C4⋊Q8C4×D5311yescharacteristic
144#127D21211yescharacteristic
145#124Dic10611yescharacteristic
146#124C4×D5311yescharacteristic
147#121—41no—
148C4×D4—41no—
149#117—21no—
150#117—2121no—
151C2×C42C22⋊D5311yescharacteristic
152C4⋊C8—1062no—
153C4⋊C8—2121no—
154#104—42no—
155#99—41no—
156#99—1062no—
157#99—2121no—
158#41—41no—
159#41—41no—
160#41—41no—
161#41—41no—
162C2×Q8C4×Dic5311yescharacteristic
163C2×Q8#1171991911yescharacteristic
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—41no—
167C2×D4C4⋊Dic5311yescharacteristic
168C2×D4#1171992111yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4C22⋊Dic5311yescharacteristic
173M16—22no—
174M16—42no—
175M16—41no—
176M16—4242no—
177C2×C8—41no—
178C2×C8—42no—
179C2×C8—2122no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—4241no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—2122no—
187C42—21no—
188C42—41no—
189C42—4241no—
190C42#1171991911yescharacteristic
191C42#1171992111yescharacteristic
192C23—21no—
193C23—21no—
194C23#1313313211yescharacteristic
195Q8—41no—
196Q8—41no—
197D4—41no—
198D4—41no—
199D4—42no—
200D4—81no—
201D4—81no—
202D4—81no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—81no—
216C2×C4—4242no—
217C2×C4#1313312411yescharacteristic
218C2×C4#1313313211yescharacteristic
219C8—42no—
220C8—4242no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—81no—
227C22#1681039011yescharacteristic
228C4—21no—
229C4—21no—
230C4—21no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—161no—
235C4—8482no—
236C4#1681041011yescharacteristic
237C4#1681041311yescharacteristic
238C2—21no—
239C2—41no—
240C2—81no—
241C2—81no—
242C2#2027283311yescharacteristic
subgroupG/H
centerC2#20272833
commutatorC2×Q8×C53C42
Frattini#532C2×D53
FittingC53×#1884C2
socleC2×C53#1390
prime radical, cosocleC53×#532C22
hypercenter#1884D53
hypocenterC53#7482
LCSG ▷ C2×Q8×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1884

Complements

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

NH ≅ G/Ntype
6. #16810418C4semidirect
8. #16810372C4semidirect
10. C53×#568C4semidirect
13. C53×#410C4semidirect
125. #568Dic53semidirect
130. #410Dic53semidirect
136. C53#7482semidirect

Maximal subgroups

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

subgroupindex
#202763802
#202734502
C53×#18842
#748253