Finite Group Databasebeta

← #27271863

Subgroups — #27271863

Order 13568.

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

normal subgroups only

This group has 274 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#20275099C211yesmaximal, characteristic
2#20273450C211yesmaximal, characteristic
3C53×#2132C211yesmaximal, characteristic
4#16811390C411yescharacteristic
5#16810647—21no—
6#16810640—21no—
7#16810597C411yescharacteristic
8#16810383—21no—
9#16810372—21no—
10#16810367C412yescomplemented
11C53×#532C2211yescharacteristic
12C53×#413—21no—
13C53×#356—21no—
14C53×#355C412yescomplemented
15#13133224—42no—
16C2×#11719917—21no—
17C2×#11719917—41no—
18C2×#11719917—41no—
19#13133176—21no—
20C4⋊#88322—22no—
21C4⋊#88322—41no—
22#13133144—42no—
23#13133132—41no—
24C2×#41×C53C2×C411yescharacteristic
25C2×#41×C53D411yescharacteristic
26C2×M16×C53—41no—
27C4⋊Q8×C53D411yescharacteristic
28#127×C53C2×C411yescharacteristic
29#124×C53C2×C411yescharacteristic
30#124×C53Q811yescharacteristic
31#121×C53—41no—
32C4×D4×C53—41no—
33C2×C42×C53D411yescharacteristic
34#101×C53—22no—
35#100×C53—41no—
36#41×C53—41no—
37#41×C53—41no—
38#41×C53—41no—
39#41×C53—41no—
40C2×Q8×C53C22⋊C411yescharacteristic
41C2×Q8×C53C4⋊C411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53C4211yescharacteristic
46C2×D4×C53C4⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C411yescharacteristic
51M16×C53—42no—
52M16×C53—81no—
53C2×C8×C53—81no—
54C4⋊C4×C53—41no—
55C22⋊C4×C53—41no—
56C22⋊C4×C53—41no—
57C42×C53—21no—
58C42×C53C22⋊C411yescharacteristic
59C42×C53C4⋊C411yescharacteristic
60C22⋊Dic53—41no—
61C4⋊Dic53—41no—
62C4×Dic53—41no—
63#11719917—22no—
64#11719917—41no—
65#11719917—42no—
66#11719917—82no—
67C2×#88322—41no—
68C2×#88322—42no—
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#9511yescharacteristic
94C2×C4×C53#9911yescharacteristic
95C2×Dic53—81no—
96C8×C53—82no—
97#88322—42no—
98#88322—82no—
99#7496—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×C53AΣL(1,𝔽16)11yescharacteristic
114C4×C53#35211yescharacteristic
115Dic53—161no—
116#2132D5311yescharacteristic
117#1884—531no—
118#1560—531no—
119C2×C53—21no—
120C2×C53—41no—
121C2×C53—81no—
122C2×C53—81no—
123C2×C53#139011yescharacteristic
124#568—531no—
125#532C2×D5311yescharacteristic
126#423—1061no—
127#421—1061no—
128#413—21no—
129#410—531no—
130#356—21no—
131#355Dic5312yescomplemented
132#339—1061no—
133#333—1061no—
134#331—532no—
135C53#749611yescharacteristic, complemented, Sylow
136C2×#41C4×D5311yescharacteristic
137C2×#41C22⋊D5311yescharacteristic
138#131—2122no—
139C2×M16—41no—
140C2×M16—1061no—
141C2×M16—2121no—
142C2×M16—2121no—
143C4⋊Q8C22⋊D5311yescharacteristic
144#127C4×D5311yescharacteristic
145#124Dic10611yescharacteristic
146#124C2×Dic5311yescharacteristic
147#121—41no—
148C4×D4—41no—
149#117—1061no—
150C2×C42D21211yescharacteristic
151C4⋊C8—1062no—
152C4⋊C8—2121no—
153#104—2122no—
154#101—22no—
155#100—41no—
156#99—2121no—
157#41—41no—
158#41—41no—
159#41—41no—
160#41—41no—
161C2×Q8#1171991911yescharacteristic
162C2×Q8C22⋊Dic5311yescharacteristic
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—41no—
166C2×D4C4×Dic5311yescharacteristic
167C2×D4#1171991911yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4#1171992111yescharacteristic
172M16—42no—
173M16—81no—
174M16—1062no—
175M16—4242no—
176M16—2121no—
177M16—2122no—
178C2×C8—81no—
179C2×C8—2122no—
180C2×C8—2121no—
181C2×C8—2122no—
182C4⋊C4—41no—
183C4⋊C4—2121no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—2121no—
187C42—21no—
188C42—2121no—
189C42C4⋊Dic5311yescharacteristic
190C42#1171992111yescharacteristic
191C23—21no—
192C23—21no—
193C23#1313313111yescharacteristic
194Q8—41no—
195Q8—41no—
196D4—41no—
197D4—41no—
198D4—42no—
199D4—81no—
200D4—81no—
201D4—81no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—4241no—
215C2×C4#1313312411yescharacteristic
216C2×C4#1313313111yescharacteristic
217C8—82no—
218C8—4242no—
219C8—2122no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22#1681038911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—8481no—
234C4#1681040911yescharacteristic
235C4#1681041211yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—81no—
240C2#2027283211yescharacteristic
subgroupG/H
centerC2#20272832
commutatorC2×D4×C53C42
Frattini#532C2×D53
FittingC53×#2132C2
socleC2×C53#1390
prime radical, cosocleC53×#532C22
hypercenter#2132D53
hypocenterC53#7496
LCSG ▷ C2×D4×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2132

Complements

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

NH ≅ G/Ntype
10. #16810367C4semidirect
14. C53×#355C4semidirect
131. #355Dic53semidirect
135. C53#7496semidirect

Maximal subgroups

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

subgroupindex
#202750992
#202734502
C53×#21322
#749653