Finite Group Databasebeta

← #27273375

Subgroups — #27273375

Order 13568.

462 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 87 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#20281087C211yesmaximal, characteristic
2#20276279C212yesmaximal, complemented
3#20276248C211yesmaximal, characteristic
4#20272539C212yesmaximal, complemented
5C53×#1270C211yesmaximal, characteristic
6C2×#13133194—22no—
7C2×#13133148—21no—
8C2×#13133138—21no—
9#16810546—22no—
10#16810543—21no—
11#16810521C2212yes—
12#16810503—21no—
13#16810493C2211yescharacteristic
14#16810386—22no—
15#16810386—22no—
16#16810378C2211yescharacteristic
17C53×#390C2211yescharacteristic
18C53×#336C2212yes—
19C22×Dic106—21no—
20C23×Dic53—21no—
21C22×#88322—21no—
22#13133194—44no—
23#13133194—44no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53D412yes—
28C2×#11719919—22no—
29C2×#11719919—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—21no—
32#13133148—42no—
33#13133138—42no—
34#13133124—21no—
35#13133124—22no—
36C22×D4×C53D411yescharacteristic
37C22×C8×C53—22no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C42×C53C2311yescharacteristic
41C24×C53—21no—
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53Q1612yes—
46C22×C4×C53—21no—
47C22×C4×C53C2×D412yes—
48C22×C4×C53C2×D411yescharacteristic
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—44no—
52C22×Dic53—41no—
53C22×Dic53—81no—
54C2×Dic106—44no—
55C2×Dic106—81no—
56C2×C8×C53—42no—
57C2×C8×C53—42no—
58C2×C8×C53—42no—
59C4⋊C4×C53—41no—
60C22⋊C4×C53—42no—
61C22⋊C4×C53—42no—
62C42×C53—41no—
63C22⋊Dic53—44no—
64C22⋊Dic53—81no—
65C4⋊Dic53—22no—
66C4⋊Dic53—42no—
67C4⋊Dic53—41no—
68C4⋊Dic53—41no—
69C4⋊Dic53—44no—
70C4⋊Dic53SD1614yes—
71#11719919—44no—
72#11719919—42no—
73#11719919—42no—
74C2×#88322—22no—
75C2×#88322—41no—
76C2×#88322—41no—
77C23×C53—22no—
78C23×C53—42no—
79C23×C53—41no—
80C23×C53—42no—
81C23×C53—41no—
82C23×C53#12011yescharacteristic
83D4×C53—42no—
84D4×C53—81no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53C2×SD1612yes—
92C2×C4×C53C2×Q1611yescharacteristic
93C2×C4×C53AGL(1,ℤ/8)12yes—
94C2×C4×C53#13711yescharacteristic
95C2×Dic53—24no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—84no—
102C2×Dic53—81no—
103C2×Dic53—82no—
104Dic106—84no—
105C8×C53—82no—
106#88322—42no—
107#7826—531nomaximal, Sylow
108C22×C53—42no—
109C22×C53—44no—
110C22×C53—82no—
111C22×C53—81no—
112C22×C53—81no—
113C22×C53#45012yes—
114C22×C53#45111yescharacteristic
115C22×C53#45311yescharacteristic
116C22×C53#45412yes—
117C22×C53#45811yescharacteristic
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—81no—
123Dic53—44no—
124Dic53—82no—
125Dic53—161no—
126#2683—531no—
127#2100—532no—
128#2094—531no—
129#1271—532no—
130#1270D5311yescharacteristic
131C2×C53—82no—
132C2×C53#161411yescharacteristic
133C2×C53#161612yes—
134C2×C53#162911yescharacteristic
135C2×C53#163712yes—
136C2×C53#165711yescharacteristic
137C2×C22⋊Q8—1062no—
138C2×#106—1061no—
139C2×#102—1061no—
140#396—1062no—
141#395—1061no—
142#390C2×D5311yescharacteristic
143#389—532no—
144#386—1061no—
145#384—531no—
146#340—1062no—
147#340—1062no—
148#336—531no—
149#336C2×D5312yes—
150C53#782611yescharacteristic, complemented, Sylow
151C22×Q8—1061no—
152C22×D4D21211yescharacteristic
153C23×C4—1061no—
154C22×C8—22no—
155C22×C8—1061no—
156C22⋊Q8—2124no—
157C22⋊Q8—2124no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—532no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4—1062no—
164C2×C4⋊C4D21211yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—1061no—
167C2×C22⋊C4—1062no—
168C2×C42C22×D5311yescharacteristic
169#106—2122no—
170#102—2122no—
171#95—1061no—
172#95—1062no—
173C24—21no—
174C2×Q8—4241no—
175C2×Q8—2124no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4Dic21212yes—
180C22×C4—21no—
181C22×C4—1062no—
182C22×C4—1062no—
183C22×C4—4241no—
184C22×C4—2124no—
185C22×C4—2121no—
186C22×C4C2×D21211yescharacteristic
187C22×C4D4×D5312yes—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—1062no—
192C2×C8—2121no—
193C2×C8—2121no—
194C4⋊C4—41no—
195C4⋊C4—534no—
196C4⋊C4—1062no—
197C4⋊C4—2121no—
198C4⋊C4—2121no—
199C4⋊C4—2124no—
200C4⋊C4—2122no—
201C4⋊C4—2124no—
202C4⋊C4—2122no—
203C4⋊C4—2122no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—4241no—
207C22⋊C4—2124no—
208C42—41no—
209C23—22no—
210C23—42no—
211C23—41no—
212C23—42no—
213C23—41no—
214C23#1313318711yescharacteristic
215Q8—4244no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—1064no—
225C2×C4—4241no—
226C2×C4—4242no—
227C2×C4—4244no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4SD16×D5312yes—
234C2×C4C2×Dic21211yescharacteristic
235C2×C4#1313324112yes—
236C2×C4#1313325011yescharacteristic
237C8—82no—
238C8—2122no—
239C22—42no—
240C22—44no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1681077512yes—
245C22#1681078111yescharacteristic
246C22#1681079311yescharacteristic
247C22#1681079712yes—
248C22#1681081411yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—81no—
254C4—8481no—
255C4—4242no—
256C4—2124no—
257C2—82no—
258C2#2027374211yescharacteristic
259C2#2027375312yes—
260C2#2027383211yescharacteristic
261C2#2027388312yes—
262C2#2027401411yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1270C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#1270D53
hypocenterC53#7826
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #20276279C2semidirect
4. #20272539C2semidirect
150. C53#7826semidirect

Maximal subgroups

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

subgroupindex
#202810872
#202762792
#202762482
#202725392
C53×#12702
#782653