Finite Group Databasebeta

← #27300964

Subgroups — #27300964

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287818C211yesmaximal, characteristic, complemented
2#20276343C211yesmaximal, characteristic, complemented
3C2×#16810407C211yesmaximal, characteristic, complemented
4#20276329C211yesmaximal, characteristic, complemented
5#20272804C211yesmaximal, characteristic, complemented
6#20272801C211yesmaximal, characteristic, complemented
7C53×#2955C211yesmaximal, characteristic
8#16811040—22no—
9#16811040—22no—
10C2×#13133157—22no—
11C2×#13133157C2211yescharacteristic
12C2×#13133157C2211yescharacteristic
13C2×#13133157C2211yescharacteristic
14C22×#11719908C2211yescharacteristic
15#16810407—22no—
16#16810407—22no—
17C22⋊#11719908—22no—
18C53×#575C411yescharacteristic
19C2×#131×C53C2211yescharacteristic
20C2×#131×C53C2211yescharacteristic
21C53×#444—22no—
22C53×#443—22no—
23C53×#430C411yescharacteristic
24C53×#421C411yescharacteristic
25C53×#421C411yescharacteristic
26C53×#405C2211yescharacteristic
27#13133157—24no—
28#13133157—21no—
29#13133157—21no—
30#13133157—21no—
31#13133157—21no—
32#13133157—21no—
33#13133157D412yes—
34C2×#11719908—24no—
35C2×#11719908—21no—
36C2×#11719908—21no—
37C2×#11719908—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40C2×#11719908D412yes—
41C2×#41×C53C2×C411yescharacteristic
42C2×#41×C53C2×C411yescharacteristic
43#137×C53—21no—
44#137×C53C812yes—
45AGL(1,ℤ/8)×C53—21no—
46AGL(1,ℤ/8)×C53C812yes—
47#135×C53—22no—
48#131×C53—21no—
49#131×C53—21no—
50#131×C53—21no—
51#131×C53—21no—
52#131×C53—21no—
53#131×C53—21no—
54#131×C53D412yes—
55#131×C53D412yes—
56C2×M16×C53—21no—
57C2×M16×C53—21no—
58C2×M16×C53C2×C411yescharacteristic
59C2×M16×C53C2×C411yescharacteristic
60C2×M16×C53C2×C411yescharacteristic
61C22×C8×C53—21no—
62C22×C8×C53—21no—
63C22×C8×C53C2311yescharacteristic
64#117×C53C2×C411yescharacteristic
65#108×C53—22no—
66#104×C53—22no—
67#104×C53—22no—
68#97×C53—21no—
69C4×C8×C53—21no—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53—21no—
73#41×C53—21no—
74#41×C53—21no—
75#41×C53—21no—
76#41×C53C22⋊C412yes—
77#41×C53C2×C812yes—
78C2×Q8×C53C2×C811yescharacteristic
79C2×Q8×C53M1611yescharacteristic
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53C2×C811yescharacteristic
83C2×D4×C53M1611yescharacteristic
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53C22×C411yescharacteristic
87Q16×C53—22no—
88SD16×C53—22no—
89SD16×C53—22no—
90D8×C53—22no—
91M16×C53—21no—
92M16×C53—21no—
93M16×C53—22no—
94M16×C53—22no—
95M16×C53—21no—
96M16×C53—21no—
97M16×C53—41no—
98M16×C53—41no—
99M16×C53C22⋊C412yes—
100M16×C53C2×C812yes—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53—22no—
104C2×C8×C53—22no—
105C2×C8×C53—21no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53#4111yescharacteristic
113C2×C8×C53#4111yescharacteristic
114C4⋊C4×C53—21no—
115C22⋊C4×C53—21no—
116C42×C53—21no—
117#11719908—22no—
118#11719908—22no—
119#11719908—41no—
120#11719908—41no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53#13111yescharacteristic
124Q8×C53—21no—
125Q8×C53—21no—
126Q8×C53—21no—
127Q8×C53C22⋊C812yes—
128D4×C53—22no—
129D4×C53—22no—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53—21no—
133D4×C53—41no—
134D4×C53—41no—
135D4×C53C22⋊C812yes—
136C2×C4×C53—21no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—21no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53C22×C811yescharacteristic
147C2×C4×C53C2×M1611yescharacteristic
148C2×C4×C53#13111yescharacteristic
149C8×C53—22no—
150C8×C53—22no—
151C8×C53—41no—
152C8×C53#12011yescharacteristic
153C8×C53#12111yescharacteristic
154C8×C53#12111yescharacteristic
155C8×C53#12311yescharacteristic
156#12979—531nomaximal, Sylow
157C22×C53—22no—
158C22×C53—41no—
159C22×C53—42no—
160C22×C53—41no—
161C22×C53—81no—
162C22×C53#40811yescharacteristic
163C22×C53C8×D411yescharacteristic
164C22×C53#43511yescharacteristic
165C4×C53—22no—
166C4×C53—41no—
167C4×C53—81no—
168C4×C53#38611yescharacteristic
169C4×C53C2×C22⋊C811yescharacteristic
170C4×C53C8×D411yescharacteristic
171C4×C53#43511yescharacteristic
172#3407—531no—
173#2955D5311yescharacteristic
174#2117—531no—
175C2×#350—531no—
176#2113—531no—
177#1379—531no—
178#1378—531no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—42no—
183C2×C53—81no—
184C2×C53#186911yescharacteristic
185#575Dic5311yescharacteristic
186C2×#131C2×D5311yescharacteristic
187C2×#131C2×D5311yescharacteristic
188#504—1062no—
189#504—1062no—
190C2×M32—531no—
191C2×M32—531no—
192C2×M32—531no—
193C2×M32—1062no—
194C22×C16—531no—
195#444—22no—
196#443—22no—
197#430Dic5311yescharacteristic
198#421Dic5311yescharacteristic
199#421Dic5311yescharacteristic
200#405C2×D5311yescharacteristic
201#350—1062no—
202#350—1062no—
203C22⋊C16—1062no—
204C53#1297911yescharacteristic, complemented, Sylow
205C2×#41C2×Dic5311yescharacteristic
206C2×#41C2×Dic5311yescharacteristic
207#137—21no—
208#137#8832212yes—
209AGL(1,ℤ/8)—21no—
210AGL(1,ℤ/8)#8832212yes—
211#135—22no—
212#131—21no—
213#131—21no—
214#131—21no—
215#131—21no—
216#131—21no—
217#131—21no—
218#131C22⋊D5312yes—
219#131C22⋊D5312yes—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16C2×Dic5311yescharacteristic
223C2×M16C2×Dic5311yescharacteristic
224C2×M16C2×Dic5311yescharacteristic
225C22×C8—21no—
226C22×C8—21no—
227C22×C8C22×D5311yescharacteristic
228#117C2×Dic5311yescharacteristic
229M32—532no—
230M32—1064no—
231M32—1061no—
232M32—1061no—
233M32—1061no—
234M32—1061no—
235M32—1061no—
236C2×C16—532no—
237C2×C16—1064no—
238C2×C16—1061no—
239C2×C16—1061no—
240C2×C16—1061no—
241C2×C16—1061no—
242C2×C16—1061no—
243#108—22no—
244#104—22no—
245#104—22no—
246#97—21no—
247C4×C8—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251#41—21no—
252#41—21no—
253#41—21no—
254#41C2×#8832212yes—
255#41C22⋊Dic5312yes—
256C2×Q8C2×#8832211yescharacteristic
257C2×Q8#1171991711yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4C2×#8832211yescharacteristic
261C2×D4#1171991711yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4C22×Dic5311yescharacteristic
265Q16—22no—
266SD16—22no—
267SD16—22no—
268D8—22no—
269M16—21no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—21no—
274M16—21no—
275M16—41no—
276M16—41no—
277M16C2×#8832212yes—
278M16C22⋊Dic5312yes—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8D4×D5311yescharacteristic
287C2×C8D4×D5311yescharacteristic
288C2×C8#1171994711yescharacteristic
289C2×C8#1171994711yescharacteristic
290C2×C8C2×C22⋊D5311yescharacteristic
291C2×C8C2×C22⋊D5311yescharacteristic
292C4⋊C4—21no—
293C22⋊C4—21no—
294C42—21no—
295C16—1062no—
296C16—1062no—
297C16—2121no—
298C16—2121no—
299C23—21no—
300C23—21no—
301C23#1313322411yescharacteristic
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8C22⋊#8832212yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—41no—
313D4C22⋊#8832212yes—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4C2×C22⋊Dic5311yescharacteristic
322C2×C4D4×Dic5311yescharacteristic
323C2×C4D4×Dic5311yescharacteristic
324C2×C4C22×#8832211yescharacteristic
325C2×C4C2×#1171991711yescharacteristic
326C2×C4#1313322411yescharacteristic
327C8—22no—
328C8—22no—
329C8—41no—
330C8#1313318511yescharacteristic
331C8#1313318911yescharacteristic
332C8#1313318911yescharacteristic
333C8#1313319811yescharacteristic
334C22—22no—
335C22—41no—
336C22—42no—
337C22—41no—
338C22—81no—
339C22#1681059011yescharacteristic
340C22D4×#8832211yescharacteristic
341C22#1681069511yescharacteristic
342C4—22no—
343C4—41no—
344C4—81no—
345C4#1681050311yescharacteristic
346C4C2×C22⋊#8832211yescharacteristic
347C4D4×#8832211yescharacteristic
348C4#1681069511yescharacteristic
349C2—21no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—81no—
354C2#2027500811yescharacteristic
subgroupG/H
centerC8#13133185
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#2955C2
socleC2×C53#1869
prime radicalC53×#575C4
cosocleC22×C8×C53C23
hypercenter#2955D53
hypocenterC53#12979
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2955

Complements

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

NH ≅ G/Ntype
1. #20287818C2semidirect
2. #20276343C2semidirect
3. C2×#16810407C2semidirect
4. #20276329C2semidirect
5. #20272804C2semidirect
6. #20272801C2semidirect
204. C53#12979semidirect

Maximal subgroups

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

subgroupindex
#202878182
#202763432
C2×#168104072
#202763292
#202728042
#202728012
C53×#29552
#1297953