Finite Group Databasebeta

← #27387051

Subgroups — #27387051

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic212C211yesmaximal, characteristic
2#20277189C211yesmaximal, characteristic
3C4×#13133182C211yesmaximal, characteristic
4#20275348C212yesmaximal
5#20275285C212yesmaximal
6#20275057C212yesmaximal
7#20274980C212yesmaximal
8C4×#13133140C211yesmaximal, characteristic
9C4×#13133140C211yesmaximal, characteristic
10#20274460C211yesmaximal, characteristic
11C4×#106×C53C211yesmaximal, characteristic
12C22×Dic212C2211yescharacteristic
13C2×C4×Dic106C2211yescharacteristic
14C2×C4×Dic106C2211yescharacteristic
15C2×#13133182C2212yes—
16C4×Dic212C418yescomplemented
17C2×#13133151C2211yescharacteristic
18C2×#13133142C2211yescharacteristic
19C2×#13133142C2211yescharacteristic
20C2×#13133140C2212yes—
21C2×#13133140C2212yes—
22C2×C4×#88322C2211yescharacteristic
23#16810528C2212yes—
24#16810519C2212yes—
25#16810505—22no—
26C4⋊C4×Dic53C2212yes—
27C4×#11719919—21no—
28C4×#11719919—21no—
29#16810474—22no—
30#16810469—21no—
31C42×Dic53—21no—
32C42×Dic53C2211yescharacteristic
33#16810385C2212yes—
34#16810385C2212yes—
35#16810377C2212yes—
36#16810377C2212yes—
37C2×#106×C53C2212yes—
38C2×C4×C8×C53C2211yescharacteristic
39C4×C4⋊C4×C53C2211yescharacteristic
40C4×C4⋊C4×C53C2211yescharacteristic
41C53×#340C2212yes—
42C22×Dic106C2311yescharacteristic
43C22×Dic106C2311yescharacteristic
44C2×Dic212—24no—
45C2×Dic212C2×C414yes—
46C22×#88322C2312yes—
47C4×Dic106—22no—
48C4×Dic106—22no—
49C4×Dic106C2×C414yes—
50C4×Dic106C2×C414yes—
51#13133182—24no—
52#13133182—24no—
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×#11719919—21no—
56C2×#11719919—21no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×C4×Dic53—21no—
60C2×C4×Dic53—21no—
61C2×C4×Dic53—22no—
62C2×C4×Dic53—22no—
63C2×C4×Dic53D412yes—
64C2×C4×Dic53D411yescharacteristic
65C2×C4×Dic53D411yescharacteristic
66C2×C4×Dic53C2312yes—
67#13133151C2×C414yes—
68#13133142C2×C414yes—
69#13133142C2×C414yes—
70#13133140—24no—
71#13133140—24no—
72C4×#88322—22no—
73#13133124—22no—
74#13133124—22no—
75#13133124—22no—
76#13133124—22no—
77C22×C8×C53C2311yescharacteristic
78C22×C8×C53C2311yescharacteristic
79C2×C4⋊C4×C53C2312yes—
80C2×C4⋊C4×C53C2312yes—
81C2×C42×C53—21no—
82C2×C42×C53—21no—
83C2×C42×C53C2311yescharacteristic
84#106×C53—24no—
85C4×C8×C53C2×C414yes—
86#95×C53—21no—
87#95×C53—21no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2411yescharacteristic
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53C2×D412yes—
98C22×Dic53C2×D412yes—
99C2×Dic106—22no—
100C2×Dic106—21no—
101C2×Dic106—22no—
102C2×Dic106—22no—
103C2×Dic106—21no—
104C2×Dic106C22×C412yes—
105C2×Dic106C22×C412yes—
106C2×Dic106#4114yes—
107C2×C8×C53—22no—
108C2×C8×C53—22no—
109C2×C8×C53C22×C412yes—
110C2×C8×C53C22×C412yes—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53#4114yes—
115C42×C53—42no—
116C42×C53—42no—
117C42×C53C22×C412yes—
118C42×C53#4112yes—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121C4⋊Dic53C22×C412yes—
122C4⋊Dic53C22×C412yes—
123#11719919—28no—
124#11719919—24no—
125#11719919—24no—
126#11719919—44no—
127C4×Dic53—28no—
128C4×Dic53—24no—
129C4×Dic53—24no—
130C4×Dic53—22no—
131C4×Dic53—22no—
132C4×Dic53—22no—
133C4×Dic53—22no—
134C4×Dic53—24no—
135C4×Dic53—44no—
136C2×#88322—22no—
137C2×#88322—22no—
138C2×#88322—22no—
139Dic212—28no—
140C23×C53C22×D411yescharacteristic
141C2×C4×C53—22no—
142C2×C4×C53—24no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53C4×D412yes—
149C2×C4×C53C4×D412yes—
150C2×C4×C53#13712yes—
151C2×C4×C53#13712yes—
152C2×C4×C53C23×C411yescharacteristic
153C2×C4×C53C2×#4112yes—
154C2×C4×C53C2×#4112yes—
155C2×C4×C53C2×#4111yescharacteristic
156C2×Dic53—22no—
157C2×Dic53—24no—
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—22no—
161C2×Dic53—22no—
162C2×Dic53—22no—
163C2×Dic53—22no—
164C2×Dic53—22no—
165C2×Dic53—22no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53#13514yes—
170Dic106—22no—
171Dic106—24no—
172Dic106—22no—
173Dic106—22no—
174Dic106—42no—
175Dic106#11714yes—
176C8×C53—22no—
177C8×C53—22no—
178#88322—42no—
179#21116—531nomaximal, Sylow
180C22×C53C2×C4×D411yescharacteristic
181C22×C53#52512yes—
182C22×C53C2×#13512yes—
183C22×C53C2×#13711yescharacteristic
184C22×C53C2×#13711yescharacteristic
185C4×C53—22no—
186C4×C53—44no—
187C4×C53—82no—
188C4×C53#44114yes—
189C4×C53C2×#11711yescharacteristic
190C4×C53C4×#4111yescharacteristic
191C4×C53#52912yes—
192Dic53—24no—
193Dic53—24no—
194Dic53—42no—
195Dic53—42no—
196Dic53—42no—
197Dic53—42no—
198C2×C4×Q16—531no—
199#2295—531no—
200C42×Q8—531no—
201#1927—532no—
202#1918—532no—
203#1878—532no—
204#1865—532no—
205C4×#106D5311yescharacteristic
206C4×#103—531no—
207C4×#103—531no—
208#1744—531no—
209C2×C53#229011yescharacteristic
210C2×C53C4×#13511yescharacteristic
211C2×C53C2×#44111yescharacteristic
212C2×C53#310712yes—
213C2×C53#310812yes—
214C22×Q16—531no—
215C2×C4×Q8—532no—
216C2×C4×Q8—531no—
217C2×C4×Q8—531no—
218C4×Q16—538no—
219C2×#107—531no—
220C2×#106C2×D5312yes—
221C2×#103—531no—
222C2×#103—531no—
223C2×#103—532no—
224C2×#103—532no—
225C2×C4×C8—531no—
226C2×C4×C8C2×D5311yescharacteristic
227#391—532no—
228#389—532no—
229#387—1062no—
230C4×C4⋊C4—532no—
231C4×C4⋊C4—1062no—
232C4×C4⋊C4—1061no—
233C4×C4⋊C4—1061no—
234C4×C4⋊C4C2×D5311yescharacteristic
235C4×C4⋊C4C2×D5311yescharacteristic
236#376—1061no—
237C43—531no—
238C43—1061no—
239#340—532no—
240#340—532no—
241#340C2×D5312yes—
242#336—532no—
243#336—532no—
244C53#2111611yescharacteristic, complemented, Sylow
245C22×Q8—531no—
246C22×Q8—531no—
247C2×Q16—534no—
248C2×Q16—1064no—
249C22×C8—532no—
250C22×C8C22×D5311yescharacteristic
251C22×C8C22×D5311yescharacteristic
252C4×Q8—534no—
253C4×Q8—534no—
254C4×Q8—1064no—
255C4×Q8—1064no—
256C4×Q8—1062no—
257C4×Q8—1062no—
258C2×C4⋊C4—531no—
259C2×C4⋊C4—531no—
260C2×C4⋊C4—1062no—
261C2×C4⋊C4—1061no—
262C2×C4⋊C4—1061no—
263C2×C4⋊C4—1062no—
264C2×C4⋊C4C22×D5312yes—
265C2×C4⋊C4C22×D5312yes—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—532no—
269C2×C42—531no—
270C2×C42—531no—
271C2×C42—532no—
272C2×C42—1062no—
273C2×C42—1061no—
274C2×C42—1061no—
275C2×C42—1062no—
276C2×C42C22×D5311yescharacteristic
277#107—534no—
278#106—24no—
279#103—534no—
280#103—534no—
281#103—1064no—
282#103—1064no—
283C4×C8—1062no—
284C4×C8C4×D5314yes—
285#95—21no—
286#95—21no—
287#95—1062no—
288#95—1062no—
289#95—1062no—
290#95—1062no—
291C2×Q8—532no—
292C2×Q8—532no—
293C2×Q8—534no—
294C2×Q8—1062no—
295C2×Q8—1061no—
296C2×Q8—1061no—
297C2×Q8—1062no—
298C2×Q8—1062no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—532no—
302C22×C4—532no—
303C22×C4—1061no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4—1061no—
307C22×C4D4×D5311yescharacteristic
308C22×C4D4×D5311yescharacteristic
309C22×C4C23×D5311yescharacteristic
310Q16—1068no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—1062no—
314C2×C8—1062no—
315C2×C8—1062no—
316C2×C8C2×C4×D5312yes—
317C2×C8C2×C4×D5312yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—532no—
322C4⋊C4—532no—
323C4⋊C4—1068no—
324C4⋊C4—1064no—
325C4⋊C4—1064no—
326C4⋊C4—1061no—
327C4⋊C4—1061no—
328C4⋊C4—2124no—
329C4⋊C4#1171994514yes—
330C42—42no—
331C42—42no—
332C42—1064no—
333C42—1068no—
334C42—1064no—
335C42—1064no—
336C42—1062no—
337C42—1062no—
338C42—1062no—
339C42—1062no—
340C42—2124no—
341C42C2×C4×D5312yes—
342C42#1171994912yes—
343C23C2×D4×D5311yescharacteristic
344Q8—534no—
345Q8—1062no—
346Q8—1064no—
347Q8—1062no—
348Q8—1062no—
349Q8—2122no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—534no—
358C2×C4—1062no—
359C2×C4—1062no—
360C2×C4—1064no—
361C2×C4—1062no—
362C2×C4—1062no—
363C2×C4—1062no—
364C2×C4—1062no—
365C2×C4—1062no—
366C2×C4—1062no—
367C2×C4—1062no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4#1313317912yes—
372C2×C4#1313317912yes—
373C2×C4#1313324612yes—
374C2×C4#1313324612yes—
375C2×C4C22×C4×D5311yescharacteristic
376C2×C4#41×D5312yes—
377C2×C4C2×#1171994512yes—
378C2×C4C2×#1171994911yescharacteristic
379C8—22no—
380C8—22no—
381C8—2122no—
382C22C2×#1313317911yescharacteristic
383C22#1681113212yes—
384C22#135×D5312yes—
385C22C2×#1313324611yescharacteristic
386C22C2×#1313324611yescharacteristic
387C4—22no—
388C4—44no—
389C4—82no—
390C4—1064no—
391C4—1064no—
392C4—2122no—
393C4—2122no—
394C4—2122no—
395C4—2122no—
396C4#1681072914yes—
397C4#117×D5311yescharacteristic
398C4C4×#1171994511yescharacteristic
399C4#1681116012yes—
400C2#2027715611yescharacteristic
401C2#2028348611yescharacteristic
402C2C2×#1681072911yescharacteristic
403C2#2028478212yes—
404C2#2028479312yes—
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#106×C53C2
socle, prime radicalC23×C53C22×D4
cosocleC22×C4×C53C24
hypercenterC4×#106D53
hypocenterC53#21116
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
16. C4×Dic212C4semidirect
244. C53#21116semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2122
#202771892
C4×#131331822
#202753482
#202752852
#202750572
#202749802
C4×#131331402
#202744602
C4×#106×C532
#2111653