Finite Group Databasebeta

← #27300900

Subgroups — #27300900

Order 13568.

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

normal subgroups only

This group has 516 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#20277889C211yesmaximal, characteristic
2C2×#16810418C211yesmaximal, characteristic
3#20274996C212yesmaximal
4#20272834C212yesmaximal
5C53×#1885C211yesmaximal, characteristic
6C2×#13133132C2212yes—
7#16810530—21no—
8#16810516—22no—
9#16810503C2212yes—
10#16810498—22no—
11#16810495—22no—
12#16810489—21no—
13#16810418C414yescomplemented
14#16810390—22no—
15C2×#124×C53C2211yescharacteristic
16C53×#518C2211yescharacteristic
17C53×#419—21no—
18C2×#100×C53C2211yescharacteristic
19C53×#339—21no—
20C23×Dic53D412yes—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×#11719919—42no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53D412yes—
28C2×C4×Dic53—21no—
29#13133132—22no—
30#13133132C2×C414yes—
31#13133124—22no—
32#13133124—22no—
33#13133124—22no—
34#13133124—42no—
35C22×Q8×C53D411yescharacteristic
36C22×D4×C53C2311yescharacteristic
37C2×M16×C53—21no—
38#124×C53—22no—
39#124×C53C2×C412yes—
40#124×C53Q812yes—
41C4×D4×C53—41no—
42C4×D4×C53—41no—
43#117×C53—21no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C42×C53D411yescharacteristic
47#103×C53—41no—
48#103×C53—41no—
49#100×C53—22no—
50C24×C53C2×D412yes—
51C2×Q8×C53—21no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53C4⋊C412yescomplemented
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53C22×C411yescharacteristic
58C2×D4×C53C2×Q811yescharacteristic
59C2×D4×C53#4112yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—42no—
68C22×Dic53—42no—
69C22×Dic53—82no—
70M16×C53—41no—
71M16×C53—41no—
72C2×C8×C53—41no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—21no—
75C22⋊C4×C53—24no—
76C22⋊C4×C53—24no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—41no—
79C42×C53—21no—
80C42×C53—41no—
81C42×C53C4⋊C412yescomplemented
82C22⋊Dic53—24no—
83C22⋊Dic53—42no—
84C22⋊Dic53—42no—
85C22⋊Dic53—82no—
86C4⋊Dic53—81no—
87C4⋊Dic53—81no—
88C4⋊Dic53—81no—
89#11719919—84no—
90C4×Dic53—81no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53C4×D412yes—
96C23×C53#12011yescharacteristic
97C23×C53C22⋊Q812yes—
98Q8×C53—42no—
99Q8×C53—81no—
100D4×C53—41no—
101D4×C53—41no—
102D4×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—82no—
114C2×C4×C53—82no—
115C2×C4×C53C2×C4⋊C411yescharacteristic
116C2×C4×C53#12311yescharacteristic
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—44no—
122C2×Dic53—44no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—82no—
126C2×Dic53—82no—
127C2×Dic53—82no—
128C8×C53—81no—
129#12967—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—82no—
139C22×C53#38511yescharacteristic
140C22×C53B(4,𝔽2)11yescharacteristic
141C22×C53#45811yescharacteristic
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—42no—
145C4×C53—42no—
146C4×C53—81no—
147C4×C53—81no—
148Dic53—82no—
149Dic53—84no—
150Dic53—82no—
151#2379—531no—
152C2×#354—531no—
153#1885D5311yescharacteristic
154#1867—532no—
155#1390—532no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—42no—
159C2×C53—42no—
160C2×C53#186711yescharacteristic
161C2×C53#219911yescharacteristic
162C2×C53#220111yescharacteristic
163C2×#124C2×D5311yescharacteristic
164#518C2×D5311yescharacteristic
165#419—21no—
166C2×#100C2×D5311yescharacteristic
167C2×#99—532no—
168#391—1061no—
169#388—1062no—
170#386—532no—
171#385—1062no—
172#385—1062no—
173#382—1061no—
174#354—534no—
175#342—1062no—
176#339—21no—
177C53#1296711yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D5311yescharacteristic
179C22×D4C22×D5311yescharacteristic
180C23×C4—532no—
181C2×M16—21no—
182#124—22no—
183#124Dic10612yes—
184#124C4×D5312yes—
185C4×D4—41no—
186C4×D4—41no—
187#117—21no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—2122no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—532no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C42—1061no—
198C2×C42C22⋊D5311yescharacteristic
199#103—41no—
200#103—41no—
201#100—22no—
202#99—534no—
203#99—1062no—
204#95—1062no—
205#95—1062no—
206#95—1062no—
207#95—2122no—
208C24D4×D5312yes—
209C2×Q8—21no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#1171991912yescomplemented
213C2×D4—41no—
214C2×D4—41no—
215C2×D4C2×Dic10611yescharacteristic
216C2×D4C2×C4×D5311yescharacteristic
217C2×D4#1171994712yes—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—1062no—
223C22×C4—1062no—
224C22×C4—4242no—
225C22×C4—2122no—
226C22×C4—2122no—
227C22×C4C2×C22⋊D5311yescharacteristic
228M16—41no—
229M16—41no—
230C2×C8—41no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—4241no—
234C4⋊C4—4241no—
235C4⋊C4—4241no—
236C4⋊C4—4244no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—42no—
240C22⋊C4—41no—
241C22⋊C4—1064no—
242C22⋊C4—4242no—
243C22⋊C4—2122no—
244C22⋊C4—2122no—
245C42—21no—
246C42—41no—
247C42—4241no—
248C42#1171991912yescomplemented
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23#1313317712yes—
254C23#1313318511yescharacteristic
255C23#1313319412yes—
256Q8—42no—
257Q8—81no—
258D4—41no—
259D4—41no—
260D4—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—4242no—
274C2×C4—4242no—
275C2×C4—4242no—
276C2×C4—2124no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4—2124no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4C2×#1171991911yescharacteristic
285C2×C4#1313319811yescharacteristic
286C8—81no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—82no—
296C22#1681049511yescharacteristic
297C22#1681080911yescharacteristic
298C22#1681081211yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—81no—
304C4—81no—
305C4—4242no—
306C4—4244no—
307C4—4242no—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2027499511yescharacteristic
313C2#2027671511yescharacteristic
314C2#2027672711yescharacteristic
subgroupG/H
centerC22#16810809
commutatorC2×D4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1885C2
socleC22×C53B(4,𝔽2)
prime radical, cosocleC22×D4×C53C23
hypercenter#1885D53
hypocenterC53#12967
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1885

Complements

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

NH ≅ G/Ntype
13. #16810418C4semidirect
54. C2×Q8×C53C4⋊C4semidirect
81. C42×C53C4⋊C4semidirect
177. C53#12967semidirect
212. C2×Q8#11719919semidirect
248. C42#11719919semidirect

Maximal subgroups

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

subgroupindex
#202778892
C2×#168104182
#202749962
#202728342
C53×#18852
#1296753