Finite Group Databasebeta

← #27279182

Subgroups — #27279182

Order 13568.

528 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283279C211yesmaximal, characteristic
2#20273144C212yesmaximal, complemented
3#20273029C211yesmaximal, characteristic
4#20272618C212yesmaximal, complemented
5C53×#2396C211yesmaximal, characteristic
6C22×#11719917—21no—
7#16810676C2212yes—
8#16810650—24no—
9#16810647—21no—
10#16810647C2212yes—
11Q8⋊#88322—24no—
12#16810349—21no—
13#16810347—21no—
14C2×C4×Q8×C53C2211yescharacteristic
15C2×#117×C53C2211yescharacteristic
16C53×#400C411yescharacteristic
17C53×#399C411yescharacteristic
18C53×#398—21no—
19C53×#396—21no—
20C53×#395C411yescharacteristic
21C53×#393C411yescharacteristic
22C4×C4⋊C4×C53—21no—
23C4×C4⋊C4×C53C2211yescharacteristic
24C4×C22⋊C4×C53—21no—
25C2×#11719917—22no—
26C2×#11719917—22no—
27C2×#11719917—42no—
28C2×#11719917—42no—
29C22×#88322—22no—
30C4⋊#88322—24no—
31C4⋊#88322—22no—
32C4⋊#88322—22no—
33C4⋊#88322D414yes—
34C22⋊#88322—22no—
35C4×#88322—22no—
36C22×Q8×C53C2×C411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C4×Q8×C53—22no—
39C4×Q8×C53—22no—
40#117×C53—22no—
41#117×C53—22no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C42×C53—21no—
54C2×C42×C53—21no—
55C2×C42×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53C2×C411yescharacteristic
62#95×C53C2×C411yescharacteristic
63C24×C53C22⋊C411yescharacteristic
64C2×Q8×C53—22no—
65C2×Q8×C53—42no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53SD1612yes—
78C22×C4×C53Q1612yes—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C4⋊C4×C53—22no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—42no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—22no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—41no—
96C42×C53—41no—
97C42×C53—41no—
98C42×C53—41no—
99C42×C53C2×D412yes—
100C42×C53#4112yes—
101#11719917—44no—
102#11719917—42no—
103#11719917—42no—
104C2×#88322—24no—
105C2×#88322—44no—
106C2×#88322—42no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53#10312yes—
112C23×C53C2×C22⋊C411yescharacteristic
113Q8×C53—24no—
114Q8×C53—42no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53#10312yes—
140C2×C4×C53C4×D412yes—
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53#12112yes—
143C2×C4×C53#12311yescharacteristic
144C2×C4×C53C2×SD1611yescharacteristic
145C2×C4×C53C2×Q1611yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53#13711yescharacteristic
148#88322—44no—
149#88322—82no—
150#9033—531nomaximal, Sylow
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—22no—
154C22×C53—42no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—81no—
158C22×C53#38611yescharacteristic
159C22×C53C2×#10311yescharacteristic
160C22×C53#41711yescharacteristic
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—22no—
164C4×C53—42no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—42no—
168C4×C53—81no—
169C4×C53—81no—
170C4×C53#45011yescharacteristic
171C4×C53#45111yescharacteristic
172C4×C53#48311yescharacteristic
173C4×C53#48411yescharacteristic
174#2905—531no—
175#2396D5311yescharacteristic
176#1487—532no—
177#1458—531no—
178#1303—532no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—42no—
182C2×C53#187211yescharacteristic
183C2×C53#188411yescharacteristic
184C2×C53#189111yescharacteristic
185C22×M16—1061no—
186C2×C4×Q8C2×D5311yescharacteristic
187C2×#117C2×D5311yescharacteristic
188#431—532no—
189#424—1064no—
190#423—532no—
191#423—1061no—
192#400Dic5311yescharacteristic
193#399Dic5311yescharacteristic
194#398—21no—
195#396—21no—
196#395Dic5311yescharacteristic
197#393Dic5311yescharacteristic
198C4×C4⋊C4—21no—
199C4×C4⋊C4C2×D5311yescharacteristic
200C4×C22⋊C4—21no—
201Q8⋊C8—1064no—
202#324—1061no—
203#323—1061no—
204C53#903311yescharacteristic, complemented, Sylow
205C22×Q8C2×Dic5311yescharacteristic
206C23×C4C22⋊D5311yescharacteristic
207C2×M16—1062no—
208C2×M16—1062no—
209C2×M16—2122no—
210C2×M16—2122no—
211C22×C8—1062no—
212C4×Q8—22no—
213C4×Q8—22no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×Dic5311yescharacteristic
221C2×C4⋊C4C2×Dic5311yescharacteristic
222C2×C4⋊C4C22⋊D5311yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C2×Dic5311yescharacteristic
226C2×C22⋊C4C22⋊D5311yescharacteristic
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C22⋊D5311yescharacteristic
230C2×C42C22×D5311yescharacteristic
231C4⋊C8—534no—
232C4⋊C8—1064no—
233C4⋊C8—1062no—
234C4⋊C8—1062no—
235C22⋊C8—1062no—
236C4×C8—1062no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95C2×Dic5311yescharacteristic
242#95C2×Dic5311yescharacteristic
243C24C22⋊Dic5311yescharacteristic
244C2×Q8—22no—
245C2×Q8—42no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4#1171992412yes—
255C22×C4#1171992512yes—
256C22×C4C22⋊Dic5311yescharacteristic
257C22×C4C22⋊Dic5311yescharacteristic
258C22×C4C22⋊Dic5311yescharacteristic
259C22×C4C22×Dic5311yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262M16—2124no—
263M16—2122no—
264M16—2122no—
265C2×C8—1064no—
266C2×C8—2122no—
267C2×C8—2124no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C42—21no—
280C42—21no—
281C42—41no—
282C42—41no—
283C42—41no—
284C42—41no—
285C42D4×D5312yes—
286C42#1171994712yes—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23#1313314112yes—
292C23C2×C22⋊Dic5311yescharacteristic
293Q8—24no—
294Q8—42no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4#1313314112yes—
320C2×C4D4×Dic5312yes—
321C2×C4#1313318511yescharacteristic
322C2×C4#1313318912yes—
323C2×C4#1313319811yescharacteristic
324C2×C4C2×#1171992411yescharacteristic
325C2×C4C2×#1171992511yescharacteristic
326C2×C4#1313324211yescharacteristic
327C2×C4#1313324711yescharacteristic
328C8—4242no—
329C8—2124no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22#1681050311yescharacteristic
338C22C2×#1313314111yescharacteristic
339C22#1681062411yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—42no—
344C4—41no—
345C4—42no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4#1681077311yescharacteristic
350C4#1681077811yescharacteristic
351C4#1681095111yescharacteristic
352C4#1681095611yescharacteristic
353C2—21no—
354C2—21no—
355C2—42no—
356C2#2027502311yescharacteristic
357C2#2027509911yescharacteristic
358C2#2027513411yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2396C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2396D53
hypocenterC53#9033
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2396

Complements

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

NH ≅ G/Ntype
2. #20273144C2semidirect
4. #20272618C2semidirect
204. C53#9033semidirect

Maximal subgroups

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

subgroupindex
#202832792
#202731442
#202730292
#202726182
C53×#23962
#903353