Finite Group Databasebeta

← #27271859

Subgroups — #27271859

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#20276380C211yesmaximal, characteristic
2#20275099C211yesmaximal, characteristic
3C53×#1536C211yesmaximal, characteristic
4#16811390C411yescharacteristic, complemented
5#16810640—21no—
6#16810597—21no—
7#16810597C411yescharacteristic, complemented
8#16810418—21no—
9#16810416C412yescomplemented
10#16810383—21no—
11C53×#532C2211yescharacteristic
12C53×#432—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
26C4⋊Q8×C53Q811yescharacteristic
27#127×C53C2×C411yescharacteristic
28#124×C53C2×C411yescharacteristic
29#124×C53D411yescharacteristic
30#121×C53—41no—
31C4×D4×C53—41no—
32C2×C42×C53D411yescharacteristic
33C4⋊C8×C53—41no—
34C22⋊C8×C53—41no—
35#97×C53—22no—
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
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—42no—
59C22⋊Dic53—41no—
60C4⋊Dic53—41no—
61C4⋊Dic53—81no—
62C4×Dic53—41no—
63C4×Dic53—81no—
64#11719917—22no—
65#11719917—41no—
66#11719917—42no—
67C2×#88322—41no—
68C2×#88322—42no—
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#9511yescharacteristic
91C2×C4×C53#9911yescharacteristic
92C2×Dic53—81no—
93C2×Dic53—82no—
94C8×C53—82no—
95#88322—42no—
96#7494—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×C53AΣL(1,𝔽16)11yescharacteristic
110C4×C53#35211yescharacteristic
111Dic53—161no—
112Dic53—162no—
113#2126—531no—
114#1884—531no—
115#1536D5311yescharacteristic
116C2×C53—21no—
117C2×C53—41no—
118C2×C53—81no—
119C2×C53—81no—
120C2×C53#139011yescharacteristic
121#568—531no—
122#532C2×D5311yescharacteristic
123#432—21no—
124#421—1061no—
125#410—531no—
126#410—1061no—
127#354—1061no—
128#353—532no—
129#339—1061no—
130#329—21no—
131#329Dic5312yescomplemented
132C53#749411yescharacteristic, complemented, Sylow
133C2×#41C4×D5311yescharacteristic
134C2×#41D21211yescharacteristic
135#131—2122no—
136C2×M16—1061no—
137C2×M16—2121no—
138C4⋊Q8Dic10611yescharacteristic
139#127C4×D5311yescharacteristic
140#124C2×Dic5311yescharacteristic
141#124C22⋊D5311yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—1061no—
145#117—2121no—
146C2×C42C22⋊D5311yescharacteristic
147C4⋊C8—41no—
148#104—2122no—
149#99—1062no—
150#99—2121no—
151#99—2121no—
152C22⋊C8—41no—
153#97—22no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158C2×Q8#1171991911yescharacteristic
159C2×Q8C4⋊Dic5311yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4C4×Dic5311yescharacteristic
164C2×D4#1171992111yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4#1171992111yescharacteristic
169M16—1062no—
170M16—2121no—
171M16—2122no—
172C2×C8—42no—
173C2×C8—2121no—
174C2×C8—2122no—
175C4⋊C4—41no—
176C4⋊C4—4241no—
177C4⋊C4—2121no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—2122no—
181C22⋊C4—2121no—
182C42—21no—
183C42—4241no—
184C42—2121no—
185C42#1171991911yescharacteristic
186C42C22⋊Dic5311yescharacteristic
187C23—21no—
188C23—21no—
189C23#1313313111yescharacteristic
190Q8—41no—
191Q8—41no—
192D4—41no—
193D4—41no—
194D4—42no—
195D4—81no—
196D4—82no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—4241no—
209C2×C4—4242no—
210C2×C4#1313312411yescharacteristic
211C2×C4#1313313111yescharacteristic
212C8—82no—
213C8—2122no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—81no—
219C22—81no—
220C22#1681038911yescharacteristic
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—8481no—
227C4—8482no—
228C4#1681041111yescharacteristic
229C4#1681041411yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—81no—
234C2#2027283411yescharacteristic
subgroupG/H
centerC2#20272834
commutatorC2×D4×C53C42
Frattini#532C2×D53
FittingC53×#1536C2
socleC2×C53#1390
prime radical, cosocleC53×#532C22
hypercenter#1536D53
hypocenterC53#7494
LCSG ▷ C2×D4×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1536

Complements

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

NH ≅ G/Ntype
4. #16811390C4semidirect
7. #16810597C4semidirect
9. #16810416C4semidirect
14. C53×#329C4semidirect
131. #329Dic53semidirect
132. C53#7494semidirect

Maximal subgroups

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

subgroupindex
#202763802
#202750992
C53×#15362
#749453