Finite Group Databasebeta

← #27271868

Subgroups — #27271868

Order 13568.

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

normal subgroups only

This group has 290 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#20275544C211yesmaximal, characteristic
3C53×#2132C211yesmaximal, characteristic
4#16810647C411yescharacteristic, complemented
5C2×#13133144—21no—
6C2×#13133144—21no—
7#16810597—21no—
8C42×Dic53C411yescharacteristic, complemented
9#16810418—21no—
10#16810416C412yescomplemented
11C53×#532C2211yescharacteristic
12C53×#413—21no—
13C53×#356C412yescomplemented
14C53×#355—21no—
15C2×#11719917—21no—
16#13133176—41no—
17C2×C4×Dic53—21no—
18C2×C4×Dic53—41no—
19C4⋊#88322—41no—
20#13133144—42no—
21#13133144—42no—
22#13133132—22no—
23#13133132—41no—
24C2×#41×C53D411yescharacteristic
25C2×#41×C53D411yescharacteristic
26C2×M16×C53—41no—
27C4⋊Q8×C53C2×C411yescharacteristic
28#127×C53C2×C411yescharacteristic
29#124×C53D411yescharacteristic
30#124×C53Q811yescharacteristic
31#121×C53—41no—
32C4×D4×C53—41no—
33C2×C42×C53C2×C411yescharacteristic
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×C53C22⋊C411yescharacteristic
46C2×D4×C53C4⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×Dic53—41no—
52M16×C53—42no—
53M16×C53—81no—
54C2×C8×C53—81no—
55C4⋊C4×C53—41no—
56C22⋊C4×C53—41no—
57C22⋊C4×C53—41no—
58C42×C53—21no—
59C42×C53C4211yescharacteristic
60C42×C53C4⋊C411yescharacteristic
61C22⋊Dic53—42no—
62C4⋊Dic53—81no—
63C4×Dic53—22no—
64C4×Dic53—41no—
65C4×Dic53—44no—
66C4×Dic53—81no—
67#11719917—41no—
68#11719917—41no—
69C2×#88322—41no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53#9911yescharacteristic
73Q8×C53—41no—
74Q8×C53—41no—
75D4×C53—41no—
76D4×C53—41no—
77D4×C53—82no—
78D4×C53—81no—
79D4×C53—81no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53#9511yescharacteristic
92C2×C4×C53#9911yescharacteristic
93C2×Dic53—42no—
94C2×Dic53—44no—
95C2×Dic53—82no—
96C8×C53—82no—
97#88322—81no—
98#7497—531nomaximal, Sylow
99C22×C53—21no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—81no—
104C22×C53—81no—
105C22×C53#34211yescharacteristic
106C4×C53—22no—
107C4×C53—21no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53—41no—
111C4×C53#35311yescharacteristic
112C4×C53#35411yescharacteristic
113Dic53—44no—
114Dic53—162no—
115#2132D5311yescharacteristic
116#2126—531no—
117#1962—531no—
118C2×C53—21no—
119C2×C53—41no—
120C2×C53—81no—
121C2×C53—81no—
122C2×C53#139311yescharacteristic
123#532C2×D5311yescharacteristic
124#423—531no—
125C2×#104—1061no—
126C2×#104—1061no—
127#413—21no—
128#410—1061no—
129C43—531no—
130#356Dic5312yescomplemented
131#355—21no—
132#354—1061no—
133#353—532no—
134C53#749711yescharacteristic, complemented, Sylow
135C2×#41D21211yescharacteristic
136C2×#41C22⋊D5311yescharacteristic
137C2×M16—41no—
138C2×M16—1061no—
139C4⋊Q8C2×Dic5311yescharacteristic
140#127C4×D5311yescharacteristic
141#124Dic10611yescharacteristic
142#124C22⋊D5311yescharacteristic
143#121—41no—
144C4×D4—41no—
145#117—2121no—
146C2×C42—1061no—
147C2×C42—2121no—
148C2×C42C4×D5311yescharacteristic
149C4⋊C8—2121no—
150#104—2122no—
151#104—2122no—
152#101—22no—
153#100—41no—
154#99—1062no—
155#99—2121no—
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160C2×Q8C4⋊Dic5311yescharacteristic
161C2×Q8C22⋊Dic5311yescharacteristic
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—41no—
165C2×D4#1171991911yescharacteristic
166C2×D4#1171992111yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2121no—
171C22×C4#1171992111yescharacteristic
172M16—42no—
173M16—81no—
174M16—2121no—
175M16—2121no—
176C2×C8—81no—
177C2×C8—2121no—
178C4⋊C4—41no—
179C4⋊C4—4241no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—2122no—
183C42—21no—
184C42—1062no—
185C42—4241no—
186C42—2121no—
187C42—2124no—
188C42C4×Dic5311yescharacteristic
189C42#1171991911yescharacteristic
190C23—21no—
191C23—21no—
192C23#1313313111yescharacteristic
193Q8—41no—
194Q8—41no—
195D4—41no—
196D4—41no—
197D4—82no—
198D4—81no—
199D4—81no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—4242no—
212C2×C4—2122no—
213C2×C4—2124no—
214C2×C4#1313312411yescharacteristic
215C2×C4#1313313111yescharacteristic
216C8—82no—
217C8—4241no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22#1681038911yescharacteristic
225C4—22no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—8482no—
231C4—2124no—
232C4#1681041511yescharacteristic
233C4#1681041711yescharacteristic
234C2—21no—
235C2—41no—
236C2—81no—
237C2—81no—
238C2#2027283911yescharacteristic
subgroupG/H
centerC4#16810415
commutatorC42×C53C42
Frattini#532C2×D53
FittingC53×#2132C2
socleC2×C53#1393
prime radical, cosocleC53×#532C22
hypercenter#2132D53
hypocenterC53#7497
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2132

Complements

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

NH ≅ G/Ntype
4. #16810647C4semidirect
8. C42×Dic53C4semidirect
10. #16810416C4semidirect
13. C53×#356C4semidirect
130. #356Dic53semidirect
134. C53#7497semidirect

Maximal subgroups

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

subgroupindex
#202763802
#202755442
C53×#21322
#749753