Finite Group Databasebeta

← #27273373

Subgroups — #27273373

Order 13568.

462 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 95 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#20276278C212yesmaximal, complemented
2#20276248C211yesmaximal, characteristic
3#20272538C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic
5C53×#2683C211yesmaximal, characteristic
6C2×#13133148—21no—
7C2×#13133138—21no—
8#16810493C2211yescharacteristic
9#16810385—22no—
10#16810385—22no—
11#16810378C2211yescharacteristic
12#16810378C2212yes—
13C2×C22⋊Q8×C53—22no—
14C53×#396—22no—
15C53×#395—21no—
16C53×#390C2211yescharacteristic
17C53×#389C2212yes—
18C53×#386—21no—
19C22×#88322—21no—
20C22×#88322—22no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—22no—
23#13133148—42no—
24#13133138—42no—
25C22×Q8×C53—21no—
26C22×D4×C53D411yescharacteristic
27C23×C4×C53—21no—
28C22⋊Q8×C53—44no—
29C22⋊Q8×C53—44no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53D412yes—
33C2×C4⋊C4×C53D412yes—
34C2×C4⋊C4×C53D411yescharacteristic
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—22no—
38C2×C42×C53C2311yescharacteristic
39#95×C53—21no—
40#95×C53—22no—
41C24×C53—21no—
42C2×Q8×C53—44no—
43C2×Q8×C53—81no—
44C2×D4×C53—21no—
45C2×D4×C53—41no—
46C2×D4×C53—41no—
47C2×D4×C53Q1612yes—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—21no—
51C22×C4×C53—44no—
52C22×C4×C53—81no—
53C22×C4×C53C2×D412yes—
54C22×C4×C53C2×D411yescharacteristic
55C22×Dic53—41no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—44no—
58C4⋊C4×C53—42no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53SD1614yes—
63C22⋊C4×C53—44no—
64C22⋊C4×C53—42no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—81no—
67C42×C53—41no—
68C4⋊Dic53—41no—
69C4⋊Dic53—41no—
70C4⋊Dic53—44no—
71C2×#88322—22no—
72C2×#88322—41no—
73C2×#88322—41no—
74C2×#88322—42no—
75C2×#88322—42no—
76C2×#88322—42no—
77C23×C53—22no—
78C23×C53—42no—
79C23×C53—41no—
80C23×C53—42no—
81C23×C53—41no—
82C23×C53#12011yescharacteristic
83Q8×C53—84no—
84D4×C53—42no—
85D4×C53—81no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—24no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—41no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—84no—
99C2×C4×C53C2×SD1612yes—
100C2×C4×C53C2×Q1611yescharacteristic
101C2×C4×C53AGL(1,ℤ/8)12yes—
102C2×C4×C53#13711yescharacteristic
103C2×Dic53—81no—
104C2×Dic53—82no—
105#88322—42no—
106#88322—82no—
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—44no—
123C4×C53—82no—
124C4×C53—81no—
125Dic53—161no—
126#2683D5311yescharacteristic
127#2100—532no—
128#2094—531no—
129#1271—532no—
130#1270—531no—
131C2×C53—82no—
132C2×C53#161411yescharacteristic
133C2×C53#161612yes—
134C2×C53#162911yescharacteristic
135C2×C53#163712yes—
136C2×C53#165711yescharacteristic
137C2×C22⋊Q8—22no—
138C2×#106—1061no—
139C2×#102—1061no—
140#396—22no—
141#395—21no—
142#390C2×D5311yescharacteristic
143#389C2×D5312yes—
144#386—21no—
145#384—531no—
146#340—1062no—
147#340—1062no—
148#336—531no—
149#336—532no—
150C53#782611yescharacteristic, complemented, Sylow
151C22×Q8—21no—
152C22×D4C22⋊D5311yescharacteristic
153C23×C4—21no—
154C22×C8—1061no—
155C22×C8—1062no—
156C22⋊Q8—44no—
157C22⋊Q8—44no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4—1061no—
162C2×C4⋊C4C22⋊D5312yes—
163C2×C4⋊C4C22⋊D5312yes—
164C2×C4⋊C4C22⋊D5311yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C42C22×D5311yescharacteristic
169#106—2122no—
170#102—2122no—
171#95—21no—
172#95—22no—
173C24—21no—
174C2×Q8—44no—
175C2×Q8—81no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4#1171992512yes—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—44no—
184C22×C4—81no—
185C22×C4—2121no—
186C22×C4C2×C22⋊D5312yes—
187C22×C4C2×C22⋊D5311yescharacteristic
188C2×C8—1062no—
189C2×C8—2122no—
190C2×C8—2122no—
191C2×C8—2122no—
192C2×C8—2121no—
193C2×C8—2121no—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—2121no—
201C4⋊C4—2121no—
202C4⋊C4—2124no—
203C4⋊C4#1171992314yes—
204C22⋊C4—44no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C42—41no—
209C23—22no—
210C23—42no—
211C23—41no—
212C23—42no—
213C23—41no—
214C23#1313318611yescharacteristic
215Q8—84no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—24no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—4241no—
232C2×C4—4242no—
233C2×C4C2×#1171992312yes—
234C2×C4C2×#1171992511yescharacteristic
235C2×C4#1313324312yes—
236C2×C4#1313324811yescharacteristic
237C8—4242no—
238C8—2122no—
239C22—42no—
240C22—44no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1681077412yes—
245C22#1681077911yescharacteristic
246C22#1681079111yescharacteristic
247C22#1681079612yes—
248C22#1681081311yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—44no—
254C4—82no—
255C4—81no—
256C4—8481no—
257C2—82no—
258C2#2027374011yescharacteristic
259C2#2027375212yes—
260C2#2027383111yescharacteristic
261C2#2027388012yes—
262C2#2027401311yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2683C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#2683D53
hypocenterC53#7826
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. #20276278C2semidirect
3. #20272538C2semidirect
150. C53#7826semidirect

Maximal subgroups

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

subgroupindex
#202762782
#202762482
#202725382
#202725362
C53×#26832
#782653