Finite Group Databasebeta

← #27300916

Subgroups — #27300916

Order 13568.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#16810420C211yesmaximal, characteristic
2#20275119C211yesmaximal, characteristic
3#20274929C212yesmaximal, complemented
4#20272842C212yesmaximal, complemented
5C53×#2379C211yesmaximal, characteristic
6#16811101C2211yescharacteristic
7#16810663—22no—
8#16810628—21no—
9C2×#13133136C2211yescharacteristic
10C2×#13133136C2212yes—
11#16810420C414yescomplemented
12#16810383—21no—
13#16810383—22no—
14C2×#124×C53C2211yescharacteristic
15C53×#391C2212yes—
16C53×#388—22no—
17C53×#386—21no—
18C53×#385—22no—
19C53×#382—21no—
20C2×#11719917—22no—
21C2×#11719917—21no—
22Q8×Dic53—41no—
23Q8×Dic53—41no—
24#13133176—21no—
25#13133176—22no—
26#13133150—42no—
27#13133148—42no—
28#13133138—41no—
29#13133138—41no—
30#13133136—22no—
31#13133136—22no—
32#13133136C2×C414yescomplemented
33C22×Q8×C53C2311yescharacteristic
34C22×D4×C53D411yescharacteristic
35C23×C4×C53—21no—
36#124×C53—22no—
37#124×C53C2×C412yes—
38#124×C53Q812yes—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53D412yes—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—22no—
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D412yes—
47#95×C53—22no—
48#95×C53—22no—
49#95×C53—22no—
50C24×C53—21no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53C22×C411yescharacteristic
54C2×Q8×C53C2×Q811yescharacteristic
55C2×Q8×C53#4112yes—
56C2×D4×C53—21no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C2×D4×C53C4⋊C412yes—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—82no—
64C22×C4×C53—81no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D412yes—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—81no—
69C4⋊C4×C53—81no—
70C22⋊C4×C53—24no—
71C22⋊C4×C53—24no—
72C22⋊C4×C53—82no—
73C22⋊C4×C53—81no—
74C42×C53—21no—
75C42×C53—42no—
76C42×C53C4⋊C412yes—
77C22⋊Dic53—41no—
78C4⋊Dic53—21no—
79C4⋊Dic53—21no—
80C4⋊Dic53—22no—
81C4⋊Dic53—22no—
82C4×Dic53—41no—
83C4×Dic53—42no—
84#11719917—24no—
85#11719917—42no—
86#11719917—41no—
87#11719917—41no—
88C2×#88322—42no—
89C2×#88322—41no—
90C23×C53—22no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53#12011yescharacteristic
96Q8×C53—41no—
97Q8×C53—41no—
98Q8×C53—42no—
99D4×C53—42no—
100D4×C53—81no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—84no—
115C2×C4×C53C2×C4⋊C411yescharacteristic
116C2×C4×C53C4×D412yes—
117C2×C4×C53C22⋊Q812yes—
118C2×C4×C53#12311yescharacteristic
119C2×Dic53—41no—
120C2×Dic53—41no—
121#88322—44no—
122#88322—81no—
123#12970—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—44no—
127C22×C53—42no—
128C22×C53—82no—
129C22×C53—81no—
130C22×C53—81no—
131C22×C53#38511yescharacteristic
132C22×C53B(4,𝔽2)11yescharacteristic
133C22×C53#45811yescharacteristic
134C4×C53—22no—
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—22no—
138C4×C53—42no—
139C4×C53—82no—
140C4×C53—82no—
141Dic53—81no—
142Dic53—81no—
143#2379D5311yescharacteristic
144C2×#355—531no—
145#1888—531no—
146#1857—532no—
147#1394—532no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—82no—
151C2×C53#186711yescharacteristic
152C2×C53#220211yescharacteristic
153C2×C53#220311yescharacteristic
154C2×#124C2×D5311yescharacteristic
155#519—531no—
156#428—1062no—
157#418—1061no—
158C2×#101—532no—
159C2×#101—531no—
160#391C2×D5312yes—
161#388—22no—
162#386—21no—
163#385—22no—
164#382—21no—
165#355—534no—
166#339—1062no—
167#339—1061no—
168C53#1297011yescharacteristic, complemented, Sylow
169C22×Q8C22×D5311yescharacteristic
170C22×D4C22⋊D5311yescharacteristic
171C23×C4—21no—
172C2×M16—1062no—
173C2×M16—1061no—
174#124—22no—
175#124Dic10612yes—
176#124C4×D5312yes—
177C4×Q8—2121no—
178C4×Q8—2121no—
179#117—1061no—
180#117—1062no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C22⋊D5312yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C42C22⋊D5311yescharacteristic
188C2×C42C22⋊D5312yes—
189#107—2122no—
190#106—2122no—
191#102—2121no—
192#102—2121no—
193#101—534no—
194#101—1062no—
195#101—1062no—
196#95—22no—
197#95—22no—
198#95—22no—
199C24—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C2×Dic10611yescharacteristic
203C2×Q8C2×C4×D5311yescharacteristic
204C2×Q8#1171994512yes—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4#1171991912yes—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—82no—
213C22×C4—81no—
214C22×C4C2×C22⋊D5311yescharacteristic
215C22×C4C2×C22⋊D5312yes—
216M16—1064no—
217M16—2121no—
218M16—2121no—
219M16—2122no—
220C2×C8—2121no—
221C2×C8—2122no—
222C4⋊C4—42no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—1061no—
226C4⋊C4—1061no—
227C4⋊C4—1062no—
228C4⋊C4—1062no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—82no—
232C22⋊C4—81no—
233C22⋊C4—2121no—
234C42—21no—
235C42—42no—
236C42—2122no—
237C42—2121no—
238C42#1171991912yes—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23#1313318611yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—42no—
248D4—42no—
249D4—81no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—84no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4C2×#1171991911yescharacteristic
267C2×C4C4×C22⋊D5312yes—
268C2×C4#1313319612yes—
269C2×C4#1313320011yescharacteristic
270C8—4241no—
271C8—2124no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22#1681049611yescharacteristic
280C22#1681081011yescharacteristic
281C22#1681081311yescharacteristic
282C4—21no—
283C4—22no—
284C4—21no—
285C4—22no—
286C4—42no—
287C4—82no—
288C4—82no—
289C4—4241no—
290C4—4241no—
291C2—21no—
292C2—21no—
293C2—82no—
294C2#2027499611yescharacteristic
295C2#2027673211yescharacteristic
296C2#2027673611yescharacteristic
subgroupG/H
centerC22#16810810
commutatorC2×Q8×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#2379C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
cosocleC22×Q8×C53C23
hypercenter#2379D53
hypocenterC53#12970
LCSG ▷ C2×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2379

Complements

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

NH ≅ G/Ntype
3. #20274929C2semidirect
4. #20272842C2semidirect
11. #16810420C4semidirect
32. #13133136C2×C4semidirect
168. C53#12970semidirect

Maximal subgroups

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

subgroupindex
C2×#168104202
#202751192
#202749292
#202728422
C53×#23792
#1297053