Finite Group Databasebeta

← #27281454

Subgroups — #27281454

Order 13568.

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

normal subgroups only

This group has 558 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#20279711C211yesmaximal, characteristic
3#20274704C211yesmaximal, characteristic
4#20273387C211yesmaximal, characteristic, complemented
5#20272617C211yesmaximal, characteristic, complemented
6#20272616C211yesmaximal, characteristic, complemented
7C53×#1546C211yesmaximal, characteristic, complemented
8C2×#13133206—21no—
9C2×C4×D212C2211yescharacteristic
10C8×D212—22no—
11#16810688—22no—
12#16810676C2211yescharacteristic
13#16810647C2211yescharacteristic
14C2×#13133133—21no—
15C22⋊C8×D53—22no—
16#16810549C414yes—
17#16810530—21no—
18#16810480—22no—
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#13133206—42no—
31#13133206—42no—
32C4×D212—21no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53D411yescharacteristic
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#13133124—24no—
49#13133124C2×C412yes—
50C22×Q8×C53—21no—
51C2×M16×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2×C412yes—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53D412yes—
62C22⋊C8×C53—21no—
63C4×C8×C53—21no—
64C23×D53C22⋊C412yes—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C22×C4×C53—22no—
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—22no—
76C22×Dic53C22⋊C412yes—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212D812yes—
82C2×D212SD1612yes—
83C2×C4×D53—22no—
84C2×C4×D53—22no—
85C2×C4×D53—44no—
86M16×C53—41no—
87M16×C53—41no—
88C2×C8×C53—22no—
89C2×C8×C53—41no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C42×C53C2×D411yescharacteristic
95C42×C53C2×D411yescharacteristic
96C42×C53#4111yescharacteristic
97C42×C53#4111yescharacteristic
98#11719921—22no—
99#11719921—22no—
100#11719921—48no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53—42no—
104C4×Dic53—41no—
105C4×Dic53—41no—
106#11719917—41no—
107#11719917—41no—
108C2×#88322—22no—
109C2×#88322—41no—
110C8×D53—84no—
111C23×C53C2×C22⋊C411yescharacteristic
112C22×D53—22no—
113C22×D53—22no—
114C22×D53—42no—
115C22×D53—42no—
116C22×D53#10214yes—
117Q8×C53—81no—
118Q8×C53—81no—
119Q8×C53—81no—
120Q8×C53—81no—
121C2×C4×C53—22no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53C4×D411yescharacteristic
127C2×C4×C53C4×D411yescharacteristic
128C2×C4×C53#12011yescharacteristic
129C2×C4×C53#12111yescharacteristic
130C2×C4×C53#12111yescharacteristic
131C2×C4×C53#12311yescharacteristic
132C2×C4×C53C2×D811yescharacteristic
133C2×C4×C53C2×SD1611yescharacteristic
134C2×C4×C53#13511yescharacteristic
135C2×C4×C53#13511yescharacteristic
136C2×Dic53—22no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141D212—22no—
142D212—22no—
143D212—22no—
144D212—22no—
145C4×D53—44no—
146C4×D53—82no—
147C8×C53—42no—
148C8×C53—81no—
149#88322—42no—
150#88322—81no—
151#9393—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53#38611yescharacteristic
155C22×C53C2×#10211yescharacteristic
156C22×C53#41611yescharacteristic
157C2×D53—24no—
158C2×D53—42no—
159C2×D53—42no—
160C2×D53—44no—
161C2×D53—82no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—82no—
166C4×C53#45512yes—
167C4×C53#46511yescharacteristic
168C4×C53#46511yescharacteristic
169C4×C53#46611yescharacteristic
170C4×C53#46611yescharacteristic
171Dic53—42no—
172Dic53—82no—
173#2972—531no—
174#2547—531no—
175#1800—531no—
176#1546—531no—
177#1546D5311yescharacteristic, complemented
178#1303—531no—
179#1303—531no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53#188511yescharacteristic
183C2×C53#189311yescharacteristic
184C2×C53#191511yescharacteristic
185D53—44no—
186C2×C4⋊Q8C2×D5311yescharacteristic
187C2×#124—1061no—
188C2×C4×D4—531no—
189C8×D4—1062no—
190C8×D4—1062no—
191#431—531no—
192#431C2×D5311yescharacteristic
193#423—531no—
194#423C2×D5311yescharacteristic
195C2×#100—1061no—
196C2×C22⋊C8—1062no—
197#397—534no—
198#391—1061no—
199#380—1062no—
200C4×C4⋊C4—531no—
201#343—1061no—
202#332—22no—
203#332—1062no—
204C53#939311yescharacteristic, complemented, Sylow
205C22×Q8—21no—
206C22×D4—531no—
207C23×C4—532no—
208C2×M16—21no—
209C2×M16—1061no—
210C22×C8—1064no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215#124—2122no—
216#124—2122no—
217C4×D4—1061no—
218C4×D4—1061no—
219C4×D4—1061no—
220C4×D4—1061no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4C4×D5312yes—
225C2×C22⋊C4—532no—
226C2×C22⋊C4—1064no—
227C2×C42—1061no—
228C2×C42C22×D5311yescharacteristic
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—532no—
232C4⋊C8—1061no—
233C4⋊C8—1061no—
234C4⋊C8C22⋊D5312yes—
235#100—2122no—
236C22⋊C8—21no—
237C22⋊C8—1064no—
238C22⋊C8—1061no—
239C4×C8—21no—
240C4×C8—1061no—
241#95—532no—
242#95—1064no—
243C24—532no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—532no—
251C2×D4—532no—
252C2×D4—1061no—
253C2×D4—1061no—
254C2×D4—1061no—
255C2×D4—1061no—
256C22×C4—22no—
257C22×C4—532no—
258C22×C4—1062no—
259C22×C4—1062no—
260C22×C4—1062no—
261C22×C4—2124no—
262C22×C4C2×C4×D5311yescharacteristic
263C22×C4D4×D5311yescharacteristic
264C22×C4D4×D5311yescharacteristic
265M16—41no—
266M16—41no—
267M16—2121no—
268M16—2121no—
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—1062no—
272C2×C8—4244no—
273C2×C8—2121no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—2122no—
281C22⋊C4—1062no—
282C22⋊C4—1062no—
283C22⋊C4—2128no—
284C42—2121no—
285C42—2121no—
286C42#1171994511yescharacteristic
287C42D4×D5311yescharacteristic
288C42#1171994711yescharacteristic
289C42C2×C22⋊D5311yescharacteristic
290C23—534no—
291C23—1062no—
292C23—1062no—
293C23—2122no—
294C23—2122no—
295C23C22⋊C4×D5311yescharacteristic
296Q8—81no—
297Q8—81no—
298Q8—81no—
299Q8—81no—
300D4—1062no—
301D4—1062no—
302D4—1062no—
303D4—1062no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—1062no—
310C2×C4—4242no—
311C2×C4—2122no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4—2124no—
315C2×C4—2122no—
316C2×C4#1313317711yescharacteristic
317C2×C4C4×C22⋊D5311yescharacteristic
318C2×C4#1313318511yescharacteristic
319C2×C4#1313318811yescharacteristic
320C2×C4#1313319011yescharacteristic
321C2×C4#1313319911yescharacteristic
322C2×C4D8×D5311yescharacteristic
323C2×C4SD16×D5311yescharacteristic
324C2×C4#1313323611yescharacteristic
325C2×C4#1313323811yescharacteristic
326C8—42no—
327C8—81no—
328C8—4241no—
329C8—2122no—
330C22—21no—
331C22—21no—
332C22—1064no—
333C22—4242no—
334C22—2124no—
335C22—2122no—
336C22—2122no—
337C22#1681049911yescharacteristic
338C22#102×D5311yescharacteristic
339C22#1681061611yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—82no—
344C4—4242no—
345C4—2122no—
346C4#1681079912yes—
347C4#1681084911yescharacteristic
348C4#1681085111yescharacteristic
349C4#1681085611yescharacteristic
350C4#1681085811yescharacteristic
351C2—21no—
352C2—21no—
353C2—2124no—
354C2#2027510011yescharacteristic
355C2#2027514011yescharacteristic
356C2#2027526411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1546C2
socleC22×C53#386
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1546D53
hypocenterC53#9393
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
4. #20273387C2semidirect
5. #20272617C2semidirect
6. #20272616C2semidirect
7. C53×#1546C2semidirect
177. #1546D53semidirect
204. C53#9393semidirect

Maximal subgroups

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

subgroupindex
#202837382
#202797112
#202747042
#202733872
#202726172
#202726162
C53×#15462
#939353