Finite Group Databasebeta

← #27542628

Subgroups — #27542628

Order 13568.

2550 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811180C211yesmaximal, characteristic
2#20277282C214yesmaximal, complemented
3#20277270C214yesmaximal, complemented
4#20277243C218yesmaximal
5#20277197C212yesmaximal
6#20277189C214yesmaximal, complemented
7C4×#13133182C214yesmaximal, complemented
8#117×Dic53C212yesmaximal
9C2×C42×Dic53C211yesmaximal, characteristic
10C53×#2285C211yesmaximal, characteristic
11#16811180C4116yescomplemented
12C2×#13133217C2212yes—
13C2×#13133209C2212yes—
14C2×#13133196C2214yes—
15C2×C4×Dic106C2214yes—
16C2×#13133182C2218yes—
17C2×#13133176C2212yes—
18C22×C4×Dic53C2211yescharacteristic
19C22×C4×Dic53C2214yes—
20C22×C4×Dic53C2211yescharacteristic
21#16810528C2218yes—
22#16810519C2218yes—
23#16810519C2218yes—
24#16810505C22116yes—
25#16810495C2218yes—
26C4⋊C4×Dic53C2218yes—
27C4⋊C4×Dic53C2218yes—
28C4×#11719919C2218yes—
29#16810474C22116yes—
30C22⋊C4×Dic53C2218yes—
31#16810469C2214yes—
32C42×Dic53C2212yes—
33C42×Dic53C2212yes—
34C42×Dic53C2212yes—
35C42×Dic53C2212yes—
36C42×Dic53C2214yes—
37C2×#117×C53C2212yes—
38C22×C42×C53C2211yescharacteristic
39C53×#384C2212yes—
40C53×#382C2212yes—
41C53×#380C2214yes—
42C4×C4⋊C4×C53C2214yes—
43C22×Dic106C2314yes—
44C23×Dic53C2312yes—
45C23×Dic53C2312yes—
46#13133217C2×C4116yescomplemented
47#13133209C2×C4116yescomplemented
48#13133196C2×C4132yes—
49C4×Dic106C2×C4132yescomplemented
50#13133182—232no—
51#13133176C2×C4116yes—
52C2×C4⋊Dic53C2314yes—
53C2×C4⋊Dic53C2314yes—
54C2×#11719919C23116yes—
55C2×#11719919C2318yes—
56C2×C22⋊Dic53C2314yes—
57C2×C4×Dic53—216no—
58C2×C4×Dic53—24no—
59C2×C4×Dic53Q818yescomplemented
60C2×C4×Dic53C2314yes—
61C2×C4×Dic53C2312yes—
62C2×C4×Dic53C2314yes—
63C2×C4×Dic53C2314yes—
64C2×C4×Dic53C2312yes—
65C2×C4×Dic53C23116yes—
66C2×C4×Dic53C2314yes—
67C2×C4×Dic53C2311yescharacteristic
68C2×C4×Dic53C2311yescharacteristic
69C2×C4×Dic53C2312yes—
70C2×C4×Dic53C2314yes—
71#13133124C23116yes—
72#13133124C23116yes—
73C23×C4×C53C2312yes—
74C23×C4×C53C2311yescharacteristic
75#117×C53—28no—
76C2×C4⋊C4×C53C2314yes—
77C2×C4⋊C4×C53C2318yes—
78C2×C22⋊C4×C53C2314yes—
79C2×C42×C53C2×C418yes—
80C2×C42×C53C2314yes—
81C2×C42×C53C2312yes—
82C2×C42×C53C2311yescharacteristic
83C2×C42×C53C2311yescharacteristic
84#95×C53C2318yes—
85C24×C53C2411yescharacteristic
86C22×C4×C53—28no—
87C22×C4×C53C22×C418yes—
88C22×C4×C53C22×C414yes—
89C22×C4×C53C2×Q814yes—
90C22×C4×C53#4114yes—
91C22×C4×C53C2418yes—
92C22×C4×C53C2412yes—
93C22×C4×C53C2411yescharacteristic
94C22×C4×C53C2411yescharacteristic
95C22×C4×C53C2412yes—
96C22×Dic53—24no—
97C22×Dic53—28no—
98C22×Dic53—28no—
99C22×Dic53C2×Q818yes—
100C22×Dic53C2418yes—
101C22×Dic53C2412yes—
102C22×Dic53C2414yes—
103C22×Dic53C2412yes—
104C2×Dic106—216no—
105C2×Dic106C22×C4116yes—
106C4⋊C4×C53—28no—
107C4⋊C4×C53—216no—
108C22⋊C4×C53—28no—
109C42×C53—28no—
110C42×C53C22×C418yes—
111C42×C53C22×C414yes—
112C42×C53C22×C414yes—
113C22⋊Dic53C22×C4116yes—
114C4⋊Dic53C22×C4116yes—
115C4⋊Dic53C22×C4116yes—
116#11719919—232no—
117#11719919C22×C4132yes—
118C4×Dic53—28no—
119C4×Dic53—24no—
120C4×Dic53—24no—
121C4×Dic53—28no—
122C4×Dic53—28no—
123C4×Dic53—232no—
124C4×Dic53C22×C4116yes—
125C4×Dic53C2×Q814yes—
126C4×Dic53C2×Q818yes—
127C4×Dic53C2×Q814yes—
128C4×Dic53#41116yes—
129C23×C53—24no—
130C23×C53C23×C412yes—
131C23×C53C22×Q812yes—
132C23×C53C2×#4112yes—
133C23×C53C2511yescharacteristic
134C2×C4×C53—24no—
135C2×C4×C53—216no—
136C2×C4×C53—216no—
137C2×C4×C53—28no—
138C2×C4×C53C4×Q8116yescomplemented
139C2×C4×C53C23×C414yes—
140C2×C4×C53C23×C412yes—
141C2×C4×C53C23×C412yes—
142C2×C4×C53C23×C414yes—
143C2×C4×C53C22×Q812yes—
144C2×C4×C53C22×Q812yes—
145C2×C4×C53C2×#4118yes—
146C2×C4×C53C2×#4112yes—
147C2×C4×C53C2×#4112yes—
148C2×Dic53—216no—
149C2×Dic53—216no—
150C2×Dic53—24no—
151C2×Dic53—24no—
152C2×Dic53—24no—
153C2×Dic53—216no—
154C2×Dic53C23×C4116yes—
155C2×Dic53C22×Q814yes—
156C2×Dic53C22×Q814yes—
157C2×Dic53C2×#4118yes—
158C2×Dic53C2×#4118yes—
159Dic106—232no—
160C4×#533—531nomaximal, Sylow
161C22×C53—24no—
162C22×C53—24no—
163C22×C53—24no—
164C22×C53C2×C4×Q814yes—
165C22×C53C24×C411yescharacteristic
166C22×C53C23×Q811yescharacteristic
167C22×C53C22×#4114yes—
168C22×C53C22×#4111yescharacteristic
169C4×C53—24no—
170C4×C53—48no—
171C4×C53C2×C4×Q814yes—
172C4×C53C2×C4×Q814yes—
173C4×C53C4×#4118yes—
174Dic53—216no—
175Dic53—24no—
176Dic53—28no—
177Dic53#53318yescomplemented
178C2×#533—531no—
179C4×C4⋊Q8—534no—
180C4×#125—534no—
181C4×C22⋊Q8—538no—
182#2296—532no—
183#2295—534no—
184#2285D5311yescharacteristic
185C42×Q8—534no—
186C4×#117—532no—
187C2×C43—531no—
188C2×C53—24no—
189C2×C53C22×C4×Q811yescharacteristic
190C2×C53C2×C4×#4112yes—
191C2×C53C2×#53312yes—
192C2×C53#344412yes—
193#533—5316no—
194C2×C4⋊Q8—532no—
195C2×#125—532no—
196C2×C22⋊Q8—534no—
197C2×C4×Q8—538no—
198C2×C4×Q8—534no—
199C2×#117—532no—
200C2×#117C2×D5312yes—
201C22×C42—531no—
202C22×C42—531no—
203C22×C42—534no—
204C22×C42C2×D5311yescharacteristic
205#391—538no—
206#389—538no—
207#389—538no—
208#387—5316no—
209#385—538no—
210#384C2×D5312yes—
211#382C2×D5312yes—
212#380C2×D5314yes—
213C4×C4⋊C4—538no—
214C4×C4⋊C4—538no—
215C4×C4⋊C4—538no—
216C4×C4⋊C4—5316no—
217C4×C4⋊C4C2×D5314yes—
218C4×C22⋊C4—538no—
219#376—534no—
220C43—532no—
221C43—532no—
222C43—532no—
223C43—532no—
224C43—534no—
225C53C4×#53311yescharacteristic, complemented, Sylow
226C22×Q8—534no—
227C23×C4—532no—
228C23×C4—532no—
229C23×C4C22×D5312yes—
230C23×C4C22×D5311yescharacteristic
231C4⋊Q8—5316no—
232#125—5316no—
233C22⋊Q8—5332no—
234C4×Q8—5332no—
235C4×Q8—10632no—
236#117—28no—
237#117—5316no—
238C2×C4⋊C4—534no—
239C2×C4⋊C4—538no—
240C2×C4⋊C4—534no—
241C2×C4⋊C4—5316no—
242C2×C4⋊C4C22×D5314yes—
243C2×C4⋊C4C22×D5318yes—
244C2×C22⋊C4—534no—
245C2×C22⋊C4C22×D5314yes—
246C2×C42—538no—
247C2×C42—534no—
248C2×C42—532no—
249C2×C42—534no—
250C2×C42—531no—
251C2×C42—531no—
252C2×C42—5316no—
253C2×C42—534no—
254C2×C42—534no—
255C2×C42—532no—
256C2×C42—534no—
257C2×C42—532no—
258C2×C42—10616no—
259C2×C42—1064no—
260C2×C42C4×D5318yes—
261C2×C42C22×D5314yes—
262C2×C42C22×D5311yescharacteristic
263C2×C42C22×D5311yescharacteristic
264C2×C42C22×D5312yes—
265#95—5316no—
266#95—5316no—
267#95C22×D5318yes—
268C24C23×D5311yescharacteristic
269C2×Q8—5316no—
270C2×Q8—10616no—
271C22×C4—28no—
272C22×C4—532no—
273C22×C4—538no—
274C22×C4—532no—
275C22×C4—534no—
276C22×C4—538no—
277C22×C4—1068no—
278C22×C4—1068no—
279C22×C4—1064no—
280C22×C4C2×C4×D5318yes—
281C22×C4C2×C4×D5314yes—
282C22×C4#1171994714yes—
283C22×C4Q8×D5314yes—
284C22×C4C23×D5318yes—
285C22×C4C23×D5312yes—
286C22×C4C23×D5311yescharacteristic
287C22×C4C23×D5312yes—
288C22×C4C23×D5311yescharacteristic
289C4⋊C4—216no—
290C4⋊C4—28no—
291C4⋊C4—5316no—
292C4⋊C4—5332no—
293C4⋊C4—5316no—
294C4⋊C4—10632no—
295C22⋊C4—28no—
296C22⋊C4—5316no—
297C42—28no—
298C42—5316no—
299C42—534no—
300C42—538no—
301C42—534no—
302C42—5316no—
303C42—1068no—
304C42—1068no—
305C42—10632no—
306C42—1068no—
307C42—1064no—
308C42—1064no—
309C42C2×C4×D5318yes—
310C42C2×C4×D5314yes—
311C42C2×C4×D5314yes—
312C23—24no—
313C23C22×C4×D5312yes—
314C23C2×Q8×D5312yes—
315C23C2×#1171994712yes—
316C23C24×D5311yescharacteristic
317Q8—10632no—
318C2×C4—24no—
319C2×C4—216no—
320C2×C4—216no—
321C2×C4—28no—
322C2×C4—534no—
323C2×C4—538no—
324C2×C4—5316no—
325C2×C4—534no—
326C2×C4—538no—
327C2×C4—10616no—
328C2×C4—10616no—
329C2×C4—1064no—
330C2×C4—1064no—
331C2×C4—10616no—
332C2×C4—1064no—
333C2×C4#13133182116yescomplemented
334C2×C4C22×C4×D5314yes—
335C2×C4C22×C4×D5312yes—
336C2×C4C22×C4×D5312yes—
337C2×C4C22×C4×D5314yes—
338C2×C4C2×Q8×D5312yes—
339C2×C4C2×Q8×D5312yes—
340C2×C4#41×D5318yes—
341C2×C4C2×#1171994712yes—
342C2×C4C2×#1171994712yes—
343C22—24no—
344C22—24no—
345C22—24no—
346C22C2×#1313318214yes—
347C22C23×C4×D5311yescharacteristic
348C22C22×Q8×D5311yescharacteristic
349C22C2×#41×D5314yes—
350C22C22×#1171994711yescharacteristic
351C4—24no—
352C4—48no—
353C4—538no—
354C4—1064no—
355C4—1068no—
356C4—10616no—
357C4C2×#1313318214yes—
358C4C2×#1313318214yes—
359C4#1681109118yes—
360C2—24no—
361C2C22×#1313318211yescharacteristic
362C2C2×#1681109112yes—
363C2#533×D5312yes—
364C2#2028801012yes—
subgroupG/H
centerC22×C4C23×D53
commutatorC22×C53C24×C4
FrattiniC23C24×D53
FittingC53×#2285C2
socle, cosocleC23×C53C25
prime radicalC24×C53C24
hypercenter#2285D53
hypocenterC53C4×#533
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #20277282C2semidirect
3. #20277270C2semidirect
6. #20277189C2semidirect
7. C4×#13133182C2semidirect
11. #16811180C4semidirect
46. #13133217C2×C4semidirect
47. #13133209C2×C4semidirect
49. C4×Dic106C2×C4semidirect
59. C2×C4×Dic53Q8semidirect
138. C2×C4×C53C4×Q8semidirect
177. Dic53#533semidirect
225. C53C4×#533semidirect
333. C2×C4#13133182semidirect

Maximal subgroups

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

subgroupindex
C2×#168111802
#202772822
#202772702
#202772432
#202771972
#202771892
C4×#131331822
#117×Dic532
C2×C42×Dic532
C53×#22852
C4×#53353