Finite Group Databasebeta

← #27278102

Subgroups — #27278102

Order 13568.

460 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 107 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#20275013C211yesmaximal, characteristic
2#20274919C212yesmaximal, complemented
3#20273033C211yesmaximal, characteristic
4#20272602C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic
6C22×#11719917C2211yescharacteristic
7C2×#13133152—22no—
8C2×C22⋊#88322—21no—
9C2×C22⋊#88322C2211yescharacteristic
10C2×C22⋊#88322C2212yes—
11#16810387—22no—
12#16810378—21no—
13#16810349C412yes—
14#16810347C412yes—
15C2×C4×D4×C53C2211yescharacteristic
16C53×#435—22no—
17C53×#435—22no—
18C53×#432—21no—
19C53×#407C2212yes—
20C2×C22⋊C8×C53—21no—
21C2×#11719917—22no—
22C2×#11719917—42no—
23C22×#88322—21no—
24C22×#88322—22no—
25C22×#88322—21no—
26#13133152—44no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29C22⋊#88322—42no—
30C22⋊#88322C2×C414yes—
31C22×D4×C53Q811yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53C2311yescharacteristic
34C2×M16×C53D412yes—
35C2×M16×C53D412yes—
36C22×C8×C53—21no—
37C22×C8×C53—21no—
38C4×D4×C53—22no—
39C4×D4×C53—22no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53Q811yescharacteristic
43C2×C42×C53D411yescharacteristic
44C4⋊C8×C53—21no—
45C4⋊C8×C53—21no—
46C22⋊C8×C53—22no—
47C22⋊C8×C53—22no—
48C22⋊C8×C53—22no—
49#97×C53—22no—
50C24×C53C4⋊C411yescharacteristic
51C24×C53C2×Q811yescharacteristic
52C2×D4×C53—22no—
53C2×D4×C53—22no—
54C2×D4×C53—41no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53C4⋊C411yescharacteristic
62C22×C4×C53C4⋊C411yescharacteristic
63C22×C4×C53C4⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D412yes—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53#4112yes—
68M16×C53—42no—
69M16×C53—42no—
70C2×C8×C53—22no—
71C2×C8×C53—22no—
72C2×C8×C53—81no—
73C2×C8×C53—81no—
74C4⋊C4×C53—22no—
75C22⋊C4×C53—22no—
76C22⋊C4×C53—22no—
77C42×C53—21no—
78C42×C53—21no—
79#11719917—24no—
80#11719917—42no—
81C2×#88322—22no—
82C2×#88322—24no—
83C2×#88322—42no—
84C2×#88322—42no—
85C2×#88322—41no—
86C2×#88322—41no—
87C2×#88322—41no—
88C2×#88322—41no—
89C2×#88322—41no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53#10812yescomplemented
99C23×C53C2×C4⋊C411yescharacteristic
100C23×C53C22⋊Q812yes—
101D4×C53—44no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—81no—
118C2×C4×C53C4×D412yes—
119C2×C4×C53#12011yescharacteristic
120C2×C4×C53#12311yescharacteristic
121C8×C53—42no—
122C8×C53—42no—
123#88322—44no—
124#88322—42no—
125#88322—81no—
126#8859—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—22no—
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—42no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—81no—
137C22×C53—81no—
138C22×C53—81no—
139C22×C53#38511yescharacteristic
140C22×C53C2×#10811yescharacteristic
141C22×C53#43011yescharacteristic
142C4×C53—22no—
143C4×C53—21no—
144C4×C53—41no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53B(4,𝔽2)11yescharacteristic
148C4×C53#45811yescharacteristic
149#2973D5311yescharacteristic
150#1870—531no—
151#1855—532no—
152#1459—531no—
153#1298—532no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53—42no—
157C2×C53—42no—
158C2×C53#185511yescharacteristic
159C2×C53#185811yescharacteristic
160C2×C53#186711yescharacteristic
161C22×M16—531no—
162C2×C4×D4C2×D5311yescharacteristic
163#435—22no—
164#435—22no—
165#432—21no—
166C2×#108—1062no—
167#407C2×D5312yes—
168C2×C22⋊C8—21no—
169C2×C22⋊C8—532no—
170C2×C22⋊C8—531no—
171C2×C22⋊C8—1061no—
172#341—1062no—
173#336—1061no—
174#324—532no—
175#323—532no—
176C53#885911yescharacteristic, complemented, Sylow
177C22×D4Dic10611yescharacteristic
178C23×C4C22⋊D5311yescharacteristic
179C23×C4C22×D5311yescharacteristic
180C2×M16—1062no—
181C2×M16—2122no—
182C2×M16C22⋊D5312yes—
183C2×M16C22⋊D5312yes—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—1061no—
187C22×C8—1062no—
188C22×C8—1061no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4C4×D5311yescharacteristic
192C2×C22⋊C4Dic10611yescharacteristic
193C2×C22⋊C4C4×D5311yescharacteristic
194C2×C42C22⋊D5311yescharacteristic
195#108—2124no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—534no—
202C22⋊C8—1062no—
203C22⋊C8—1062no—
204C22⋊C8—2122no—
205#97—22no—
206C24#1171991911yescharacteristic
207C24C2×Dic10611yescharacteristic
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4#1171991911yescharacteristic
218C22×C4#1171991911yescharacteristic
219C22×C4#1171991911yescharacteristic
220C22×C4C2×C4×D5311yescharacteristic
221C22×C4#1171994512yes—
222C22×C4C2×C22⋊D5312yes—
223C22×C4C2×C22⋊D5311yescharacteristic
224M16—42no—
225M16—42no—
226M16—1064no—
227M16—2122no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—81no—
231C2×C8—81no—
232C2×C8—1062no—
233C2×C8—1064no—
234C2×C8—2121no—
235C2×C8—2121no—
236C2×C8—2121no—
237C2×C8—2122no—
238C2×C8—2121no—
239C2×C8—2122no—
240C2×C8—2121no—
241C4⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C42—21no—
245C42—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23#1313315212yescomplemented
255C23C2×#1171991911yescharacteristic
256C23#1313319612yes—
257D4—44no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4C4×C22⋊D5312yes—
275C2×C4#1313318611yescharacteristic
276C2×C4#1313320011yescharacteristic
277C8—42no—
278C8—42no—
279C8—4241no—
280C8—2124no—
281C8—2122no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22#1681049611yescharacteristic
295C22C2×#1313315211yescharacteristic
296C22#1681067111yescharacteristic
297C4—21no—
298C4—22no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4#1681081011yescharacteristic
303C4#1681081311yescharacteristic
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2#2027492011yescharacteristic
309C2#2027493511yescharacteristic
310C2#2027499611yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2973C2
socleC22×C53#385
prime radicalC22×D4×C53Q8
cosocleC23×C4×C53C23
hypercenter#2973D53
hypocenterC53#8859
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20274919C2semidirect
4. #20272602C2semidirect
98. C23×C53#108semidirect
176. C53#8859semidirect
254. C23#13133152semidirect

Maximal subgroups

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

subgroupindex
#202750132
#202749192
#202730332
#202726022
C53×#29732
#885953