Finite Group Databasebeta

← #27271828

Subgroups — #27271828

Order 13568.

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

normal subgroups only

This group has 266 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#20273350C211yesmaximal, characteristic
3C53×#1884C211yesmaximal, characteristic
4#16810680—21no—
5#16810607—21no—
6#16810422C412yescomplemented
7#16810420—21no—
8#16810363—21no—
9#16810363C412yescomplemented
10C53×#568C411yescharacteristic
11C53×#532C2211yescharacteristic
12C53×#421—21no—
13C53×#410C411yescharacteristic
14C53×#339—21no—
15C2×#11719917—41no—
16C4⋊#88322—41no—
17#13133136—22no—
18#13133134—41no—
19C22⋊#88322—41no—
20#13133128—22no—
21C2×#41×C53C2×C411yescharacteristic
22C2×#41×C53D411yescharacteristic
23#131×C53—42no—
24C2×M16×C53—21no—
25C2×M16×C53—41no—
26C4⋊Q8×C53C2×C411yescharacteristic
27#127×C53Q811yescharacteristic
28#124×C53C2×C411yescharacteristic
29#124×C53D411yescharacteristic
30#121×C53—41no—
31C4×D4×C53—41no—
32#117×C53—21no—
33C2×C42×C53D411yescharacteristic
34#104×C53—42no—
35#99×C53—41no—
36#41×C53—41no—
37#41×C53—41no—
38#41×C53—41no—
39#41×C53—41no—
40C2×Q8×C53C4211yescharacteristic
41C2×Q8×C53C22⋊C411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53C4⋊C411yescharacteristic
46C2×D4×C53C4⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C411yescharacteristic
51M16×C53—22no—
52M16×C53—42no—
53M16×C53—41no—
54C2×C8×C53—41no—
55C2×C8×C53—42no—
56C4⋊C4×C53—41no—
57C4⋊C4×C53—41no—
58C22⋊C4×C53—41no—
59C22⋊C4×C53—41no—
60C22⋊C4×C53—41no—
61C42×C53—21no—
62C42×C53—41no—
63C42×C53C22⋊C411yescharacteristic
64C42×C53C4⋊C411yescharacteristic
65#11719917—42no—
66#11719917—81no—
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—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—81no—
91C2×C4×C53#9511yescharacteristic
92C2×C4×C53#9911yescharacteristic
93C8×C53—42no—
94#88322—82no—
95#88322—82no—
96#7484—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—21no—
105C4×C53—41no—
106C4×C53—41no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C53—161no—
110C4×C53AΣL(1,𝔽16)11yescharacteristic
111C4×C53#35211yescharacteristic
112#2132—531no—
113#1884D5311yescharacteristic
114#1536—531no—
115C2×C53—21no—
116C2×C53—41no—
117C2×C53—81no—
118C2×C53—81no—
119C2×C53#139011yescharacteristic
120#568Dic5311yescharacteristic
121#532C2×D5311yescharacteristic
122#432—1061no—
123#421—21no—
124#413—1061no—
125#410Dic5311yescharacteristic
126#356—532no—
127#355—1061no—
128#339—21no—
129#329—532no—
130#329—1061no—
131C53#748411yescharacteristic, complemented, Sylow
132C2×#41C2×Dic5311yescharacteristic
133C2×#41C22⋊D5311yescharacteristic
134#131—42no—
135C2×M16—21no—
136C2×M16—41no—
137C2×M16—2121no—
138C4⋊Q8C4×D5311yescharacteristic
139#127Dic10611yescharacteristic
140#124C4×D5311yescharacteristic
141#124D21211yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—21no—
145C2×C42C22⋊D5311yescharacteristic
146C4⋊C8—2121no—
147#104—42no—
148#101—1062no—
149#100—2121no—
150#99—41no—
151C22⋊C8—2121no—
152#97—1062no—
153#41—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157C2×Q8C4×Dic5311yescharacteristic
158C2×Q8#1171992111yescharacteristic
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4#1171991911yescharacteristic
163C2×D4C4⋊Dic5311yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4C22⋊Dic5311yescharacteristic
168M16—22no—
169M16—42no—
170M16—41no—
171M16—4241no—
172M16—2122no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—4241no—
176C2×C8—2122no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C42—21no—
183C42—41no—
184C42#1171991911yescharacteristic
185C42#1171992111yescharacteristic
186C23—21no—
187C23—21no—
188C23#1313313211yescharacteristic
189Q8—41no—
190Q8—41no—
191D4—41no—
192D4—41no—
193D4—42no—
194D4—81no—
195D4—82no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4#1313312411yescharacteristic
209C2×C4#1313313211yescharacteristic
210C8—42no—
211C8—4242no—
212C8—4242no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22—81no—
219C22#1681039011yescharacteristic
220C4—21no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—161no—
226C4#1681041011yescharacteristic
227C4#1681041311yescharacteristic
228C2—21no—
229C2—41no—
230C2—81no—
231C2—81no—
232C2#2027283311yescharacteristic
subgroupG/H
centerC2#20272833
commutatorC2×Q8×C53C42
Frattini#532C2×D53
FittingC53×#1884C2
socleC2×C53#1390
prime radical, cosocleC53×#532C22
hypercenter#1884D53
hypocenterC53#7484
LCSG ▷ C2×Q8×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1884

Complements

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

NH ≅ G/Ntype
6. #16810422C4semidirect
9. #16810363C4semidirect
131. C53#7484semidirect

Maximal subgroups

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

subgroupindex
#202764002
#202733502
C53×#18842
#748453