Finite Group Databasebeta

← #27271881

Subgroups — #27271881

Order 13568.

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

normal subgroups only

This group has 270 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#20276400C211yesmaximal, characteristic
2#20276380C211yesmaximal, characteristic
3C53×#1963C211yesmaximal, characteristic
4#16810607—21no—
5#16810597—21no—
6#16810422C412yescomplemented
7#16810420—21no—
8#16810418—21no—
9#16810416C412yescomplemented
10C53×#532C2211yescharacteristic
11C53×#423C411yescharacteristic, complemented
12C2×#104×C53—21no—
13C2×#104×C53—21no—
14C53×#376C411yescharacteristic, complemented
15C2×#11719917—41no—
16#13133176—41no—
17#13133136—22no—
18#13133134—41no—
19#13133132—22no—
20#13133132—41no—
21C2×#41×C53D411yescharacteristic
22C2×#41×C53D411yescharacteristic
23C2×M16×C53—21no—
24C4⋊Q8×C53C2×C411yescharacteristic
25#127×C53C2×C411yescharacteristic
26#124×C53D411yescharacteristic
27#124×C53Q811yescharacteristic
28#121×C53—41no—
29C4×D4×C53—41no—
30C2×C42×C53—21no—
31C2×C42×C53C2×C411yescharacteristic
32C4⋊C8×C53—41no—
33#104×C53—42no—
34#104×C53—42no—
35#95×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×C53—41no—
51C22×C4×C53C22⋊C411yescharacteristic
52M16×C53—41no—
53M16×C53—41no—
54C2×C8×C53—41no—
55C4⋊C4×C53—41no—
56C22⋊C4×C53—41no—
57C22⋊C4×C53—41no—
58C42×C53—21no—
59C42×C53—22no—
60C42×C53—41no—
61C42×C53C4211yescharacteristic
62C42×C53C4⋊C411yescharacteristic
63C22⋊Dic53—42no—
64C4⋊Dic53—81no—
65C4×Dic53—81no—
66#11719917—42no—
67#11719917—81no—
68C2×#88322—81no—
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—42no—
90C2×C4×C53—81no—
91C2×C4×C53—82no—
92C2×C4×C53#9511yescharacteristic
93C2×C4×C53#9911yescharacteristic
94C2×Dic53—82no—
95C8×C53—81no—
96#88322—82no—
97#7503—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—82no—
111C4×C53#35311yescharacteristic
112C4×C53#35411yescharacteristic
113Dic53—162no—
114#2132—531no—
115#2126—531no—
116#1963D5311yescharacteristic
117C2×C53—21no—
118C2×C53—41no—
119C2×C53—81no—
120C2×C53—81no—
121C2×C53#139311yescharacteristic
122#532C2×D5311yescharacteristic
123#423Dic5311yescharacteristic, complemented
124C2×#104—21no—
125C2×#104—21no—
126#413—1061no—
127#410—1061no—
128#376Dic5311yescharacteristic, complemented
129#356—532no—
130#355—1061no—
131#354—1061no—
132#353—532no—
133C53#750311yescharacteristic, complemented, Sylow
134C2×#41C22⋊D5311yescharacteristic
135C2×#41C22⋊D5311yescharacteristic
136C2×M16—21no—
137C2×M16—2121no—
138C4⋊Q8C4×D5311yescharacteristic
139#127C4×D5311yescharacteristic
140#124Dic10611yescharacteristic
141#124D21211yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—2121no—
145C2×C42—21no—
146C2×C42C2×Dic5311yescharacteristic
147C4⋊C8—41no—
148#104—42no—
149#104—42no—
150#101—1062no—
151#100—2121no—
152#99—1062no—
153#99—2121no—
154#95—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8#1171991911yescharacteristic
160C2×Q8#1171992111yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4#1171991911yescharacteristic
165C2×D4#1171992111yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4C22⋊Dic5311yescharacteristic
171M16—41no—
172M16—41no—
173M16—4241no—
174M16—2122no—
175C2×C8—41no—
176C2×C8—4241no—
177C4⋊C4—41no—
178C4⋊C4—4241no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—2122no—
182C42—21no—
183C42—22no—
184C42—41no—
185C42—4241no—
186C42C4×Dic5311yescharacteristic
187C42C4⋊Dic5311yescharacteristic
188C23—21no—
189C23—21no—
190C23#1313313211yescharacteristic
191Q8—41no—
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—81no—
196D4—81no—
197D4—82no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—81no—
210C2×C4—82no—
211C2×C4—4242no—
212C2×C4#1313312411yescharacteristic
213C2×C4#1313313211yescharacteristic
214C8—81no—
215C8—4242no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22#1681039011yescharacteristic
223C4—21no—
224C4—41no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—82no—
229C4—8482no—
230C4#1681041611yescharacteristic
231C4#1681041811yescharacteristic
232C2—21no—
233C2—41no—
234C2—81no—
235C2—81no—
236C2#2027284011yescharacteristic
subgroupG/H
centerC4#16810418
commutatorC42×C53C42
Frattini#532C2×D53
FittingC53×#1963C2
socleC2×C53#1393
prime radical, cosocleC53×#532C22
hypercenter#1963D53
hypocenterC53#7503
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1963

Complements

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

NH ≅ G/Ntype
6. #16810422C4semidirect
9. #16810416C4semidirect
11. C53×#423C4semidirect
14. C53×#376C4semidirect
123. #423Dic53semidirect
128. #376Dic53semidirect
133. C53#7503semidirect

Maximal subgroups

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

subgroupindex
#202764002
#202763802
C53×#19632
#750353