Finite Group Databasebeta

← #27271887

Subgroups — #27271887

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#20276380C211yesmaximal, characteristic
2#20275548C211yesmaximal, characteristic
3C53×#2126C211yesmaximal, characteristic
4#16810647C411yescharacteristic, complemented
5C2×#13133144—21no—
6C2×#13133144—21no—
7#16810597—21no—
8#16810469C411yescharacteristic, complemented
9#16810418C412yescomplemented
10#16810416—21no—
11C53×#532C2211yescharacteristic
12C53×#410—21no—
13C53×#354C412yescomplemented
14C53×#353—21no—
15C2×#11719917—21no—
16#13133176—41no—
17C2×C4×Dic53—21no—
18C4⋊#88322—41no—
19#13133144—42no—
20#13133144—42no—
21#13133132—22no—
22#13133132—41no—
23#13133124—41no—
24C2×#41×C53D411yescharacteristic
25C2×#41×C53D411yescharacteristic
26C4⋊Q8×C53Q811yescharacteristic
27#127×C53D411yescharacteristic
28#124×C53C2×C411yescharacteristic
29#124×C53C2×C411yescharacteristic
30#121×C53—41no—
31C4×D4×C53—41no—
32#117×C53—41no—
33C2×C42×C53C2×C411yescharacteristic
34#99×C53—22no—
35#99×C53—41no—
36#41×C53—41no—
37#41×C53—41no—
38#41×C53—41no—
39#41×C53—41no—
40C2×Q8×C53C4⋊C411yescharacteristic
41C2×Q8×C53C4⋊C411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53C22⋊C411yescharacteristic
46C2×D4×C53C22⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×Dic53—41no—
52C4⋊C4×C53—41no—
53C4⋊C4×C53—81no—
54C22⋊C4×C53—41no—
55C22⋊C4×C53—41no—
56C22⋊C4×C53—42no—
57C42×C53—21no—
58C42×C53—81no—
59C42×C53C4211yescharacteristic
60C42×C53C4⋊C411yescharacteristic
61C22⋊Dic53—42no—
62C4⋊Dic53—81no—
63C4×Dic53—22no—
64C4×Dic53—41no—
65C4×Dic53—81no—
66#11719917—41no—
67#11719917—41no—
68C2×#88322—41no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53#9911yescharacteristic
72Q8×C53—41no—
73Q8×C53—41no—
74D4×C53—41no—
75D4×C53—41no—
76D4×C53—81no—
77D4×C53—81no—
78D4×C53—82no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—81no—
90C2×C4×C53—82no—
91C2×C4×C53#9511yescharacteristic
92C2×C4×C53#9911yescharacteristic
93C2×Dic53—42no—
94C2×Dic53—82no—
95C2×Dic53—82no—
96#88322—81no—
97#7506—531nomaximal, Sylow
98C22×C53—21no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—81no—
103C22×C53—81no—
104C22×C53#34211yescharacteristic
105C4×C53—21no—
106C4×C53—41no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53—162no—
111C4×C53#35311yescharacteristic
112C4×C53#35411yescharacteristic
113Dic53—82no—
114Dic53—162no—
115#2126—531no—
116#2126D5311yescharacteristic
117#1963—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#410—21no—
128#410—1061no—
129#376—531no—
130#354—532no—
131#354Dic5312yescomplemented
132#353—21no—
133#353—1061no—
134C53#750611yescharacteristic, complemented, Sylow
135C2×#41D21211yescharacteristic
136C2×#41C22⋊D5311yescharacteristic
137C2×M16—1061no—
138C4⋊Q8Dic10611yescharacteristic
139#127C22⋊D5311yescharacteristic
140#124C4×D5311yescharacteristic
141#124C2×Dic5311yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—41no—
145#117—2121no—
146C2×C42—1061no—
147C2×C42C4×D5311yescharacteristic
148C4⋊C8—2121no—
149#104—2122no—
150#104—2122no—
151#99—22no—
152#99—41no—
153#99—1062no—
154#99—2121no—
155#95—2121no—
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160C2×Q8#1171991911yescharacteristic
161C2×Q8C4⋊Dic5311yescharacteristic
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—41no—
165C2×D4#1171992111yescharacteristic
166C2×D4C22⋊Dic5311yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2121no—
171C22×C4#1171992111yescharacteristic
172M16—2121no—
173M16—2121no—
174C2×C8—2121no—
175C4⋊C4—41no—
176C4⋊C4—81no—
177C4⋊C4—4241no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—42no—
181C22⋊C4—2122no—
182C42—21no—
183C42—81no—
184C42—1062no—
185C42—4241no—
186C42—2121no—
187C42C4×Dic5311yescharacteristic
188C42#1171991911yescharacteristic
189C23—21no—
190C23—21no—
191C23#1313313111yescharacteristic
192Q8—41no—
193Q8—41no—
194D4—41no—
195D4—41no—
196D4—81no—
197D4—81no—
198D4—82no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—82no—
211C2×C4—4242no—
212C2×C4—4242no—
213C2×C4—2122no—
214C2×C4#1313312411yescharacteristic
215C2×C4#1313313111yescharacteristic
216C8—4241no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—81no—
222C22—81no—
223C22#1681038911yescharacteristic
224C4—21no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—41no—
229C4—162no—
230C4—8482no—
231C4—4242no—
232C4#1681041511yescharacteristic
233C4#1681041711yescharacteristic
234C2—21no—
235C2—41no—
236C2—81no—
237C2—81no—
238C2#2027283911yescharacteristic
subgroupG/H
centerC4#16810415
commutatorC42×C53C42
Frattini#532C2×D53
FittingC53×#2126C2
socleC2×C53#1393
prime radical, cosocleC53×#532C22
hypercenter#2126D53
hypocenterC53#7506
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2126

Complements

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

NH ≅ G/Ntype
4. #16810647C4semidirect
8. #16810469C4semidirect
9. #16810418C4semidirect
13. C53×#354C4semidirect
131. #354Dic53semidirect
134. C53#7506semidirect

Maximal subgroups

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

subgroupindex
#202763802
#202755482
C53×#21262
#750653