Finite Group Databasebeta

← #27271835

Subgroups — #27271835

Order 13568.

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

normal subgroups only

This group has 286 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#20275099C211yesmaximal, characteristic
3C53×#1525C211yesmaximal, characteristic
4#16811390C411yescharacteristic, complemented
5#16810640—21no—
6#16810597—21no—
7#16810597C411yescharacteristic, complemented
8#16810418—21no—
9#16810416C412yescomplemented
10#16810383—21no—
11C53×#532C2211yescharacteristic
12C4×M16×C53—21no—
13C53×#329—21no—
14C53×#329C412yescomplemented
15#13133224—42no—
16C2×#11719917—21no—
17C2×#11719917—41no—
18#13133176—21no—
19#13133176—41no—
20#13133144—42no—
21#13133132—22no—
22#13133132—41no—
23#13133132—41no—
24C2×#41×C53C2×C411yescharacteristic
25C2×#41×C53D411yescharacteristic
26C2×M16×C53—41no—
27C4⋊Q8×C53Q811yescharacteristic
28#127×C53C2×C411yescharacteristic
29#124×C53C2×C411yescharacteristic
30#124×C53D411yescharacteristic
31#121×C53—41no—
32C4×D4×C53—41no—
33C2×C42×C53D411yescharacteristic
34#97×C53—22no—
35C4×C8×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×C53C4211yescharacteristic
46C2×D4×C53C22⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C411yescharacteristic
51M16×C53—44no—
52C2×C8×C53—42no—
53C4⋊C4×C53—41no—
54C22⋊C4×C53—41no—
55C22⋊C4×C53—41no—
56C42×C53—21no—
57C42×C53C22⋊C411yescharacteristic
58C42×C53C4⋊C411yescharacteristic
59C22⋊Dic53—42no—
60C22⋊Dic53—41no—
61C4⋊Dic53—41no—
62C4⋊Dic53—81no—
63C4×Dic53—41no—
64C4×Dic53—81no—
65#11719917—22no—
66#11719917—42no—
67#11719917—41no—
68C2×#88322—41no—
69C2×#88322—42no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53#9911yescharacteristic
73Q8×C53—41no—
74Q8×C53—41no—
75D4×C53—41no—
76D4×C53—41no—
77D4×C53—42no—
78D4×C53—82no—
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—81no—
94C2×Dic53—82no—
95C8×C53—44no—
96#88322—42no—
97#7486—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—22no—
106C4×C53—21no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53AΣL(1,𝔽16)11yescharacteristic
111C4×C53#35211yescharacteristic
112Dic53—161no—
113Dic53—162no—
114#2126—531no—
115#1884—531no—
116#1525D5311yescharacteristic
117C2×C53—21no—
118C2×C53—41no—
119C2×C53—81no—
120C2×C53—81no—
121C2×C53#139011yescharacteristic
122#568—531no—
123#532C2×D5311yescharacteristic
124#421—1061no—
125#410—531no—
126#410—1061no—
127C4×M16—21no—
128#354—1061no—
129#353—532no—
130#339—1061no—
131#329—21no—
132#329Dic5312yescomplemented
133C53#748611yescharacteristic, complemented, Sylow
134C2×#41C4×D5311yescharacteristic
135C2×#41D21211yescharacteristic
136#131—2122no—
137C2×M16—41no—
138C2×M16—1061no—
139C2×M16—2121no—
140C4⋊Q8Dic10611yescharacteristic
141#127C4×D5311yescharacteristic
142#124C2×Dic5311yescharacteristic
143#124C22⋊D5311yescharacteristic
144#121—41no—
145C4×D4—41no—
146#117—1061no—
147#117—2121no—
148C2×C42C22⋊D5311yescharacteristic
149#104—2122no—
150#99—1062no—
151#99—2121no—
152#99—2121no—
153#97—22no—
154C4×C8—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8#1171991911yescharacteristic
160C2×Q8C4⋊Dic5311yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4C4×Dic5311yescharacteristic
165C2×D4#1171992111yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4#1171992111yescharacteristic
170M16—44no—
171M16—1062no—
172M16—2122no—
173M16—2121no—
174C2×C8—42no—
175C2×C8—2121no—
176C2×C8—2122no—
177C4⋊C4—41no—
178C4⋊C4—4241no—
179C4⋊C4—2121no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—2122no—
183C22⋊C4—2121no—
184C42—21no—
185C42—4241no—
186C42—2121no—
187C42#1171991911yescharacteristic
188C42C22⋊Dic5311yescharacteristic
189C23—21no—
190C23—21no—
191C23#1313313111yescharacteristic
192Q8—41no—
193Q8—41no—
194D4—41no—
195D4—41no—
196D4—42no—
197D4—82no—
198D4—81no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—4241no—
211C2×C4—4242no—
212C2×C4#1313312411yescharacteristic
213C2×C4#1313313111yescharacteristic
214C8—44no—
215C8—2122no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22#1681038911yescharacteristic
223C4—22no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—8481no—
229C4—8482no—
230C4#1681041111yescharacteristic
231C4#1681041411yescharacteristic
232C2—21no—
233C2—41no—
234C2—81no—
235C2—81no—
236C2#2027283411yescharacteristic
subgroupG/H
centerC4#16810411
commutatorC2×D4×C53C42
Frattini#532C2×D53
FittingC53×#1525C2
socleC2×C53#1390
prime radical, cosocleC53×#532C22
hypercenter#1525D53
hypocenterC53#7486
LCSG ▷ C2×D4×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1525

Complements

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

NH ≅ G/Ntype
4. #16811390C4semidirect
7. #16810597C4semidirect
9. #16810416C4semidirect
14. C53×#329C4semidirect
132. #329Dic53semidirect
133. C53#7486semidirect

Maximal subgroups

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

subgroupindex
#202763802
#202750992
C53×#15252
#748653