Finite Group Databasebeta

← #27271841

Subgroups — #27271841

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#20275099C211yesmaximal, characteristic
3C53×#1543C211yesmaximal, characteristic
4#16811390C411yescharacteristic
5#16810640—21no—
6#16810607—21no—
7#16810597C411yescharacteristic
8#16810422—21no—
9#16810420C412yescomplemented
10#16810383—21no—
11C53×#532C2211yescharacteristic
12C53×#431—21no—
13C53×#333—21no—
14C53×#331C412yescomplemented
15#13133224—42no—
16C2×#11719917—21no—
17C2×#11719917—41no—
18C2×#11719917—41no—
19#13133176—21no—
20#13133144—42no—
21#13133136—22no—
22#13133134—41no—
23#13133132—41no—
24C2×#41×C53C2×C411yescharacteristic
25C2×#41×C53D411yescharacteristic
26C4⋊Q8×C53D411yescharacteristic
27#127×C53C2×C411yescharacteristic
28#124×C53C2×C411yescharacteristic
29#124×C53Q811yescharacteristic
30#121×C53—41no—
31C4×D4×C53—41no—
32C2×C42×C53D411yescharacteristic
33C4⋊C8×C53—22no—
34C22⋊C8×C53—41no—
35C4×C8×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
51C2×C8×C53—42no—
52C4⋊C4×C53—41no—
53C22⋊C4×C53—41no—
54C22⋊C4×C53—41no—
55C42×C53—21no—
56C42×C53C22⋊C411yescharacteristic
57C42×C53C4⋊C411yescharacteristic
58C22⋊Dic53—41no—
59C4⋊Dic53—41no—
60C4×Dic53—41no—
61#11719917—22no—
62#11719917—42no—
63#11719917—41no—
64#11719917—42no—
65#11719917—81no—
66C2×#88322—41no—
67C2×#88322—42no—
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—42no—
77D4×C53—82no—
78D4×C53—81no—
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—42no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53#9511yescharacteristic
91C2×C4×C53#9911yescharacteristic
92C2×Dic53—81no—
93C8×C53—82no—
94#88322—42no—
95#88322—82no—
96#7488—531nomaximal, Sylow
97C22×C53—21no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—81no—
102C22×C53—81no—
103C22×C53#34211yescharacteristic
104C4×C53—22no—
105C4×C53—21no—
106C4×C53—41no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C53AΣL(1,𝔽16)11yescharacteristic
110C4×C53#35211yescharacteristic
111Dic53—161no—
112#2132—531no—
113#1884—531no—
114#1543D5311yescharacteristic
115C2×C53—21no—
116C2×C53—41no—
117C2×C53—81no—
118C2×C53—81no—
119C2×C53#139011yescharacteristic
120#568—531no—
121#532C2×D5311yescharacteristic
122#431—21no—
123#421—1061no—
124#413—1061no—
125#410—531no—
126#356—1061no—
127#355—532no—
128#339—1061no—
129#333—21no—
130#331Dic5312yescomplemented
131C53#748811yescharacteristic, complemented, Sylow
132C2×#41C4×D5311yescharacteristic
133C2×#41D21211yescharacteristic
134#131—2122no—
135C2×M16—1061no—
136C2×M16—2121no—
137C2×M16—2121no—
138C4⋊Q8C22⋊D5311yescharacteristic
139#127C2×Dic5311yescharacteristic
140#124Dic10611yescharacteristic
141#124C4×D5311yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—1061no—
145C2×C42C22⋊D5311yescharacteristic
146C4⋊C8—22no—
147#104—2122no—
148#101—1062no—
149#100—2121no—
150#99—2121no—
151C22⋊C8—41no—
152C4×C8—41no—
153#41—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157C2×Q8#1171991911yescharacteristic
158C2×Q8#1171992111yescharacteristic
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4C4×Dic5311yescharacteristic
163C2×D4C4⋊Dic5311yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4#1171992111yescharacteristic
168M16—1062no—
169M16—4241no—
170M16—2122no—
171M16—2122no—
172M16—2121no—
173C2×C8—42no—
174C2×C8—4241no—
175C2×C8—2121no—
176C2×C8—2122no—
177C4⋊C4—41no—
178C4⋊C4—2121no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—2121no—
182C42—21no—
183C42—2121no—
184C42#1171991911yescharacteristic
185C42C22⋊Dic5311yescharacteristic
186C23—21no—
187C23—21no—
188C23#1313313111yescharacteristic
189Q8—41no—
190Q8—41no—
191D4—41no—
192D4—41no—
193D4—42no—
194D4—82no—
195D4—81no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—4241no—
208C2×C4#1313312411yescharacteristic
209C2×C4#1313313111yescharacteristic
210C8—82no—
211C8—4242no—
212C8—2122no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22—81no—
219C22#1681038911yescharacteristic
220C4—22no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—8481no—
226C4#1681041111yescharacteristic
227C4#1681041411yescharacteristic
228C2—21no—
229C2—41no—
230C2—81no—
231C2—81no—
232C2#2027283411yescharacteristic
subgroupG/H
centerC4#16810411
commutatorC2×D4×C53C42
Frattini#532C2×D53
FittingC53×#1543C2
socleC2×C53#1390
prime radical, cosocleC53×#532C22
hypercenter#1543D53
hypocenterC53#7488
LCSG ▷ C2×D4×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1543

Complements

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

NH ≅ G/Ntype
9. #16810420C4semidirect
14. C53×#331C4semidirect
130. #331Dic53semidirect
131. C53#7488semidirect

Maximal subgroups

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

subgroupindex
#202764002
#202750992
C53×#15432
#748853