Finite Group Databasebeta

← #27281464

Subgroups — #27281464

Order 13568.

594 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 117 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#20283738C211yesmaximal, characteristic
2#20279669C211yesmaximal, characteristic
3#20274704C211yesmaximal, characteristic
4#20273400C211yesmaximal, characteristic, complemented
5#20272617C211yesmaximal, characteristic, complemented
6#20272616C211yesmaximal, characteristic, complemented
7C53×#1549C211yesmaximal, characteristic, complemented
8C2×#13133193—22no—
9C2×C4×D212C2211yescharacteristic
10C8×D212—22no—
11#16810688—22no—
12#16810676C2211yescharacteristic
13#16810647C2211yescharacteristic
14C2×#13133133—21no—
15C22⋊C8×D53—22no—
16#16810558C414yes—
17#16810524—21no—
18#16810492—21no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810391—21no—
21#16810369—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C53×#431C2211yescharacteristic
24C53×#423C2211yescharacteristic
25C53×#332—22no—
26C22×D212D411yescharacteristic
27C22×C4×D53D412yes—
28C2×#11719917—21no—
29C2×C8×D53—24no—
30#13133193—48no—
31C4×D212—21no—
32C4×D212—21no—
33C4×D212—21no—
34C4×D212—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—24no—
38C2×#11719921—24no—
39C2×#11719921C2×C412yes—
40C2×C4×Dic53—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322D412yes—
44#13133133—42no—
45C22⋊#88322—21no—
46#13133129—24no—
47C4×#88322—21no—
48#13133124C2×C412yes—
49C22×Q8×C53—21no—
50C2×M16×C53—21no—
51C4⋊Q8×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53C2×C412yes—
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53D412yes—
61C22⋊C8×C53—21no—
62C4×C8×C53—21no—
63C23×D53C22⋊C412yes—
64C2×Q8×C53—44no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C22×C4×C53—22no—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—22no—
72C22×Dic53C22⋊C412yes—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212SD1612yes—
78C2×D212Q1612yes—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53—44no—
82M16×C53—41no—
83M16×C53—41no—
84C2×C8×C53—22no—
85C2×C8×C53—41no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C42×C53C2×D411yescharacteristic
91C42×C53C2×D411yescharacteristic
92C42×C53#4111yescharacteristic
93C42×C53#4111yescharacteristic
94#11719921—22no—
95#11719921—22no—
96#11719921—48no—
97C4⋊Dic53—24no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100#11719919—48no—
101C4×Dic53—41no—
102C4×Dic53—41no—
103#11719917—41no—
104#11719917—41no—
105C2×#88322—22no—
106C2×#88322—41no—
107C8×D53—84no—
108C23×C53C2×C22⋊C411yescharacteristic
109C22×D53—22no—
110C22×D53—22no—
111C22×D53—42no—
112C22×D53—42no—
113C22×D53#10314yes—
114Q8×C53—82no—
115Q8×C53—82no—
116C2×C4×C53—22no—
117C2×C4×C53—41no—
118C2×C4×C53—44no—
119C2×C4×C53—42no—
120C2×C4×C53C4×D411yescharacteristic
121C2×C4×C53C4×D411yescharacteristic
122C2×C4×C53#12011yescharacteristic
123C2×C4×C53#12111yescharacteristic
124C2×C4×C53#12111yescharacteristic
125C2×C4×C53#12311yescharacteristic
126C2×C4×C53C2×SD1611yescharacteristic
127C2×C4×C53C2×Q1611yescharacteristic
128C2×C4×C53#13511yescharacteristic
129C2×C4×C53#13511yescharacteristic
130C2×Dic53—24no—
131C2×Dic53—22no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135D212—22no—
136D212—22no—
137D212—22no—
138D212—22no—
139C4×D53—44no—
140C4×D53—82no—
141C8×C53—42no—
142C8×C53—81no—
143#88322—42no—
144#88322—81no—
145#9395—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53#38611yescharacteristic
149C22×C53C2×#10311yescharacteristic
150C22×C53#41611yescharacteristic
151C2×D53—24no—
152C2×D53—42no—
153C2×D53—42no—
154C2×D53—44no—
155C2×D53—82no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—82no—
160C4×C53#45312yes—
161C4×C53#46511yescharacteristic
162C4×C53#46511yescharacteristic
163C4×C53#46711yescharacteristic
164C4×C53#46711yescharacteristic
165Dic53—44no—
166Dic53—42no—
167#2972—531no—
168#2542—531no—
169#1800—531no—
170#1549—531no—
171#1549D5311yescharacteristic, complemented
172#1303—531no—
173#1303—531no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53#188711yescharacteristic
177C2×C53#189211yescharacteristic
178C2×C53#191511yescharacteristic
179D53—44no—
180C2×C4⋊Q8C2×D5311yescharacteristic
181C2×C22⋊Q8—1062no—
182C2×C4×D4—531no—
183C8×D4—1062no—
184C8×D4—1062no—
185#431—531no—
186#431C2×D5311yescharacteristic
187#423—531no—
188#423C2×D5311yescharacteristic
189C2×#100—1061no—
190C2×C22⋊C8—1062no—
191#399—534no—
192#390—1061no—
193#384—1061no—
194C4×C4⋊C4—531no—
195#343—1061no—
196#332—22no—
197#332—1062no—
198C53#939511yescharacteristic, complemented, Sylow
199C22×Q8—21no—
200C22×D4—531no—
201C23×C4—532no—
202C2×M16—21no—
203C2×M16—1061no—
204C22×C8—1064no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C22⋊Q8—2128no—
210C4×D4—1061no—
211C4×D4—1061no—
212C4×D4—1061no—
213C4×D4—1061no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1064no—
218C2×C4⋊C4C4×D5312yes—
219C2×C22⋊C4—532no—
220C2×C22⋊C4—1064no—
221C2×C42—1061no—
222C2×C42C22×D5311yescharacteristic
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—532no—
226C4⋊C8—1061no—
227C4⋊C8—1061no—
228C4⋊C8C22⋊D5312yes—
229#100—2122no—
230C22⋊C8—21no—
231C22⋊C8—1064no—
232C22⋊C8—1061no—
233C4×C8—21no—
234C4×C8—1061no—
235#95—532no—
236C24—532no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×D4—532no—
241C2×D4—532no—
242C2×D4—1061no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4—1061no—
246C22×C4—22no—
247C22×C4—532no—
248C22×C4—1062no—
249C22×C4—1062no—
250C22×C4—1062no—
251C22×C4—2124no—
252C22×C4C2×C4×D5311yescharacteristic
253C22×C4D4×D5311yescharacteristic
254C22×C4D4×D5311yescharacteristic
255M16—41no—
256M16—41no—
257M16—2121no—
258M16—2121no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—1062no—
262C2×C8—4244no—
263C2×C8—2121no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—1064no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—2128no—
272C22⋊C4—1062no—
273C22⋊C4—1062no—
274C22⋊C4—2128no—
275C42—2121no—
276C42—2121no—
277C42#1171994511yescharacteristic
278C42D4×D5311yescharacteristic
279C42#1171994711yescharacteristic
280C42C2×C22⋊D5311yescharacteristic
281C23—534no—
282C23—1062no—
283C23—1062no—
284C23—2122no—
285C23—2122no—
286C23C22⋊C4×D5311yescharacteristic
287Q8—82no—
288Q8—82no—
289D4—1062no—
290D4—1062no—
291D4—1062no—
292D4—1062no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—1064no—
298C2×C4—1062no—
299C2×C4—4242no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4—2124no—
303C2×C4—2122no—
304C2×C4#1313317711yescharacteristic
305C2×C4C4×C22⋊D5311yescharacteristic
306C2×C4#1313318511yescharacteristic
307C2×C4#1313318811yescharacteristic
308C2×C4#1313319011yescharacteristic
309C2×C4#1313319911yescharacteristic
310C2×C4SD16×D5311yescharacteristic
311C2×C4Q16×D5311yescharacteristic
312C2×C4#1313323811yescharacteristic
313C2×C4#1313324011yescharacteristic
314C8—42no—
315C8—81no—
316C8—4241no—
317C8—2122no—
318C22—21no—
319C22—21no—
320C22—1064no—
321C22—4242no—
322C22—2124no—
323C22—2122no—
324C22—2122no—
325C22#1681049911yescharacteristic
326C22#103×D5311yescharacteristic
327C22#1681061811yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—82no—
332C4—2124no—
333C4—2122no—
334C4#1681078912yes—
335C4#1681084911yescharacteristic
336C4#1681085511yescharacteristic
337C4#1681086111yescharacteristic
338C4#1681086311yescharacteristic
339C2—21no—
340C2—21no—
341C2—2124no—
342C2#2027511011yescharacteristic
343C2#2027513511yescharacteristic
344C2#2027526411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1549C2
socleC22×C53#386
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1549D53
hypocenterC53#9395
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
4. #20273400C2semidirect
5. #20272617C2semidirect
6. #20272616C2semidirect
7. C53×#1549C2semidirect
171. #1549D53semidirect
198. C53#9395semidirect

Maximal subgroups

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

subgroupindex
#202837382
#202796692
#202747042
#202734002
#202726172
#202726162
C53×#15492
#939553