Finite Group Databasebeta

← #27283124

Subgroups — #27283124

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283111C211yesmaximal, characteristic
2#20274866C211yesmaximal, characteristic
3C4⋊C8×Dic53C211yesmaximal, characteristic
4C2×#16810370C211yesmaximal, characteristic
5C2×#16810369C211yesmaximal, characteristic
6C2×#16810368C211yesmaximal, characteristic
7C2×C53×#332C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×#13133215—22no—
10C2×C4⋊#88322C2211yescharacteristic
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133146—22no—
13C2×C8×Dic53—21no—
14#16810492—21no—
15#16810492—21no—
16#16810492C414yes—
17C42×Dic53C2211yescharacteristic
18#16810377—21no—
19#16810370—22no—
20#16810369—22no—
21#16810368—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C53×#332—22no—
26C22×Dic106—21no—
27C22×#88322—21no—
28C22×#88322—21no—
29#13133217—21no—
30#13133217—21no—
31#13133217—21no—
32#13133217—21no—
33#13133215—44no—
34#13133215—44no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C412yes—
37C2×#11719919—24no—
38C2×#11719919—24no—
39C2×C4×Dic53C2×C412yes—
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D412yes—
42C2×C4×Dic53D411yescharacteristic
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322D412yes—
47#13133146—44no—
48C8×Dic53—42no—
49C22×Q8×C53—21no—
50C22×C8×C53—21no—
51C22×C8×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×C53—21no—
62C4⋊C8×C53D412yes—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C2×Q8×C53—42no—
66C2×Q8×C53—42no—
67C22×C4×C53—22no—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—22no—
72C22×Dic53C22⋊C412yes—
73C22×Dic53C22⋊C412yes—
74C2×Dic106—41no—
75C2×Dic106—41no—
76C2×Dic106—42no—
77C2×Dic106—42no—
78C2×C8×C53—22no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C2×C8×C53—42no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C42×C53C2×D411yescharacteristic
88C42×C53C2×D411yescharacteristic
89C42×C53#4111yescharacteristic
90C42×C53#4111yescharacteristic
91C4⋊Dic53—22no—
92C4⋊Dic53—22no—
93C4⋊Dic53—41no—
94C4⋊Dic53—41no—
95#11719919—48no—
96#11719919—48no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C4×Dic53—24no—
100C4×Dic53D814yes—
101C4×Dic53SD1614yes—
102C4×Dic53SD1612yes—
103C4×Dic53Q1612yes—
104C2×#88322—22no—
105C2×#88322—41no—
106C2×#88322—41no—
107C2×#88322—41no—
108C2×#88322—42no—
109C23×C53C2×C22⋊C411yescharacteristic
110Q8×C53—82no—
111Q8×C53—81no—
112Q8×C53—81no—
113C2×C4×C53—22no—
114C2×C4×C53—42no—
115C2×C4×C53—44no—
116C2×C4×C53C4×D411yescharacteristic
117C2×C4×C53C4×D411yescharacteristic
118C2×C4×C53#12111yescharacteristic
119C2×C4×C53#12111yescharacteristic
120C2×C4×C53#12411yescharacteristic
121C2×C4×C53#12711yescharacteristic
122C2×C4×C53C2×D812yes—
123C2×C4×C53C2×SD1611yescharacteristic
124C2×C4×C53C2×SD1612yes—
125C2×C4×C53C2×Q1611yescharacteristic
126C2×C4×C53#13711yescharacteristic
127C2×C4×C53#13711yescharacteristic
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—42no—
131C2×Dic53—44no—
132C2×Dic53#10114yes—
133C2×Dic53#10214yes—
134C2×Dic53#10214yes—
135C2×Dic53#10314yes—
136Dic106—82no—
137Dic106—81no—
138Dic106—81no—
139C8×C53—42no—
140C8×C53—81no—
141#88322—42no—
142#88322—81no—
143#9692—531nomaximal, Sylow
144C22×C53#39011yescharacteristic
145C22×C53C2×#10112yes—
146C22×C53C2×#10212yes—
147C22×C53C2×#10311yescharacteristic
148C22×C53#41911yescharacteristic
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—82no—
152C4×C53#46111yescharacteristic
153C4×C53#46111yescharacteristic
154C4×C53#46211yescharacteristic
155C4×C53#46211yescharacteristic
156C4×C53#48612yes—
157C4×C53#49211yescharacteristic
158C4×C53#49311yescharacteristic
159Dic53—24no—
160Dic53—82no—
161Dic53#33218yescomplemented
162#2868—531no—
163#1842—531no—
164C4×C4⋊C8—531no—
165C2×#332—531no—
166C2×#332—531no—
167C2×#332—531no—
168C2×#332D5311yescharacteristic
169C2×C53C2×#33212yes—
170C2×C53#154612yes—
171C2×C53#195711yescharacteristic
172C2×C53#196611yescharacteristic
173C2×C53#198111yescharacteristic
174C2×C4⋊Q8—531no—
175C2×C4⋊Q8—1062no—
176C2×C4⋊Q8C2×D5311yescharacteristic
177C2×C4⋊C8—531no—
178C2×C4⋊C8—531no—
179C2×C4⋊C8—1062no—
180C2×C4⋊C8C2×D5311yescharacteristic
181C2×C4⋊C8C2×D5311yescharacteristic
182C2×C4×C8—1061no—
183#384—534no—
184#384—1061no—
185#384—1061no—
186C43—531no—
187#336—1061no—
188#332—22no—
189#332—1062no—
190#332—1062no—
191#332—1062no—
192C53#969211yescharacteristic, complemented, Sylow
193C22×Q8—21no—
194C22×Q8—1061no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—1061no—
198C22×C8—1061no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—1061no—
204C4⋊Q8—1061no—
205C4⋊Q8—1061no—
206C4⋊Q8—1061no—
207C4⋊Q8—2124no—
208C4⋊Q8—2124no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—532no—
211C2×C4⋊C4—1064no—
212C2×C4⋊C4—1064no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4C4×D5312yes—
215C2×C42—532no—
216C2×C42—531no—
217C2×C42—532no—
218C2×C42—531no—
219C2×C42C22×D5311yescharacteristic
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—532no—
224C4⋊C8—1061no—
225C4⋊C8—1061no—
226C4⋊C8—1061no—
227C4⋊C8—2124no—
228C4⋊C8C22⋊D5312yes—
229C4×C8—2122no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—2121no—
235C2×Q8—2121no—
236C2×Q8—2122no—
237C2×Q8—2122no—
238C22×C4—22no—
239C22×C4—532no—
240C22×C4—532no—
241C22×C4—1062no—
242C22×C4C2×C4×D5311yescharacteristic
243C22×C4D4×D5311yescharacteristic
244C22×C4D4×D5311yescharacteristic
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—1062no—
251C2×C8—2121no—
252C2×C8—2122no—
253C2×C8—2121no—
254C2×C8—2121no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—1062no—
260C4⋊C4—1062no—
261C4⋊C4—2128no—
262C4⋊C4—2128no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C42—534no—
266C42—532no—
267C42—534no—
268C42—532no—
269C42—1064no—
270C42—1061no—
271C42—1061no—
272C42#1171994511yescharacteristic
273C42D4×D5311yescharacteristic
274C42#1171994711yescharacteristic
275C42C2×C22⋊D5311yescharacteristic
276C23C22⋊C4×D5311yescharacteristic
277Q8—82no—
278Q8—81no—
279Q8—81no—
280Q8—4242no—
281Q8—4241no—
282Q8—4241no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—534no—
287C2×C4—534no—
288C2×C4—534no—
289C2×C4—534no—
290C2×C4—1062no—
291C2×C4—1062no—
292C2×C4—2122no—
293C2×C4—2124no—
294C2×C4#1313317711yescharacteristic
295C2×C4C4×C22⋊D5311yescharacteristic
296C2×C4#1313318911yescharacteristic
297C2×C4#1313319011yescharacteristic
298C2×C4#1313320311yescharacteristic
299C2×C4#1313321311yescharacteristic
300C2×C4D8×D5312yes—
301C2×C4SD16×D5311yescharacteristic
302C2×C4SD16×D5312yes—
303C2×C4Q16×D5311yescharacteristic
304C2×C4#1313324611yescharacteristic
305C2×C4#1313324911yescharacteristic
306C8—42no—
307C8—81no—
308C8—4241no—
309C8—2122no—
310C22#1681052311yescharacteristic
311C22#101×D5312yes—
312C22#102×D5312yes—
313C22#103×D5311yescharacteristic
314C22#1681063011yescharacteristic
315C4—21no—
316C4—21no—
317C4—82no—
318C4—538no—
319C4—1064no—
320C4—4242no—
321C4#1681082611yescharacteristic
322C4#1681082711yescharacteristic
323C4#1681083111yescharacteristic
324C4#1681083211yescharacteristic
325C4#1681096812yes—
326C4#1681099411yescharacteristic
327C4#1681099811yescharacteristic
328C2#332×D5312yes—
329C2#2027338512yes—
330C2#2027551911yescharacteristic
331C2#2027555711yescharacteristic
332C2#2027563611yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#332C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×#332D53
hypocenterC53#9692
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
161. Dic53#332semidirect
192. C53#9692semidirect

Maximal subgroups

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

subgroupindex
#202831112
#202748662
C4⋊C8×Dic532
C2×#168103702
C2×#168103692
C2×#168103682
C2×C53×#3322
#969253