Finite Group Databasebeta

← #27301515

Subgroups — #27301515

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287818C211yesmaximal, characteristic, complemented
2C2×#16810433C211yesmaximal, characteristic, complemented
3#20276443C211yesmaximal, characteristic, complemented
4#20274774C211yesmaximal, characteristic, complemented
5#20272826C211yesmaximal, characteristic, complemented
6#20272815C211yesmaximal, characteristic, complemented
7C53×#3067C211yesmaximal, characteristic
8#16811040—22no—
9#16811040—22no—
10C2×#13133157—22no—
11C2×#13133157C2211yescharacteristic
12C2×#13133157C2211yescharacteristic
13C2×#13133153C2211yescharacteristic
14C2×#13133149C2211yescharacteristic
15#16810624—21no—
16#16810433—22no—
17#16810430—22no—
18#16810391—21no—
19C53×#575—21no—
20C2×#135×C53C2211yescharacteristic
21C2×#131×C53C2211yescharacteristic
22C53×#473C412yescomplemented
23C53×#472C412yescomplemented
24C53×#471—22no—
25C2×#108×C53C2211yescharacteristic
26C53×#413—21no—
27C2×#11719917—21no—
28C2×C4⋊Dic53—21no—
29#13133157—24no—
30#13133157—21no—
31#13133157—21no—
32#13133157—21no—
33#13133157D412yes—
34C2×#11719908—24no—
35C2×#11719908—21no—
36C2×#11719908D412yes—
37#13133153—21no—
38#13133153—21no—
39#13133149—21no—
40#13133149—21no—
41#13133141—41no—
42#13133138—41no—
43C2×#41×C53—21no—
44C2×#41×C53D411yescharacteristic
45#137×C53—22no—
46#137×C53—41no—
47AGL(1,ℤ/8)×C53—22no—
48AGL(1,ℤ/8)×C53—41no—
49#135×C53—21no—
50#135×C53—21no—
51#135×C53—21no—
52#135×C53—21no—
53#135×C53—42no—
54C2×Q16×C53C2×C411yescharacteristic
55C2×SD16×C53—21no—
56C2×D8×C53C2×C411yescharacteristic
57#131×C53—21no—
58#131×C53—21no—
59#131×C53C2×C412yes—
60#131×C53D412yes—
61C2×M16×C53—21no—
62C2×M16×C53—21no—
63C2×M16×C53D411yescharacteristic
64C22×C8×C53—21no—
65C22×C8×C53C2311yescharacteristic
66#108×C53—21no—
67#108×C53C2×C412yes—
68#101×C53—22no—
69#100×C53—22no—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53—41no—
73#41×C53—41no—
74#41×C53—41no—
75#41×C53—41no—
76#41×C53C22⋊C412yes—
77#41×C53SD1612yes—
78C2×Q8×C53—21no—
79C2×Q8×C53D811yescharacteristic
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—41no—
83C2×D4×C53D811yescharacteristic
84C22×C4×C53—21no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87C22×Dic53—41no—
88Q16×C53—21no—
89Q16×C53—21no—
90Q16×C53—42no—
91SD16×C53—42no—
92SD16×C53—42no—
93SD16×C53—41no—
94SD16×C53—41no—
95D8×C53—21no—
96D8×C53—21no—
97D8×C53—42no—
98M16×C53—21no—
99M16×C53—22no—
100M16×C53—22no—
101M16×C53—22no—
102M16×C53—41no—
103M16×C53C22⋊C412yes—
104C2×C8×C53—21no—
105C2×C8×C53—22no—
106C2×C8×C53—22no—
107C2×C8×C53—41no—
108C2×C8×C53C22×C411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53#4111yescharacteristic
113C2×C8×C53#4111yescharacteristic
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116#11719917—41no—
117#11719917—41no—
118C2×#88322—41no—
119#11719908—22no—
120#11719908—22no—
121C23×C53—21no—
122C23×C53—41no—
123C23×C53#13511yescharacteristic
124Q8×C53—21no—
125Q8×C53—41no—
126Q8×C53—41no—
127Q8×C53#10212yes—
128D4×C53—22no—
129D4×C53—22no—
130D4×C53—21no—
131D4×C53—41no—
132D4×C53—41no—
133D4×C53—81no—
134D4×C53—81no—
135D4×C53#10212yes—
136C2×C4×C53—21no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—41no—
140C2×C4×C53—81no—
141C2×C4×C53—81no—
142C2×C4×C53C2×C22⋊C411yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53C2×D811yescharacteristic
146C2×C4×C53C2×SD1611yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×Dic53—81no—
149C2×Dic53—161no—
150C8×C53—22no—
151C8×C53—42no—
152C8×C53C4×D411yescharacteristic
153C8×C53C4×D411yescharacteristic
154C8×C53#12011yescharacteristic
155C8×C53#12311yescharacteristic
156#88322—81no—
157#13070—531nomaximal, Sylow
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53—161no—
163C22×C53#41611yescharacteristic
164C22×C53#46511yescharacteristic
165C22×C53#46611yescharacteristic
166C4×C53—22no—
167C4×C53—81no—
168C4×C53#38611yescharacteristic
169C4×C53C2×#10211yescharacteristic
170C4×C53#46511yescharacteristic
171C4×C53#46611yescharacteristic
172Dic53—161no—
173#3407—531no—
174#3067D5311yescharacteristic
175C2×#360—531no—
176#2143—531no—
177#1815—531no—
178#1388—531no—
179#1383—531no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—42no—
184C2×C53—161no—
185C2×C53#189311yescharacteristic
186#575—21no—
187C2×#135C2×D5311yescharacteristic
188C2×#131C2×D5311yescharacteristic
189#504—1062no—
190#504—1062no—
191C2×M32—531no—
192C2×M32—531no—
193C2×M32—1062no—
194#473Dic5312yescomplemented
195#472Dic5312yescomplemented
196#471—22no—
197C2×#108—531no—
198C2×#108C2×D5311yescharacteristic
199C2×#106—531no—
200#417—1061no—
201#413—21no—
202#360—1062no—
203#359—1062no—
204#343—1061no—
205C53#1307011yescharacteristic, complemented, Sylow
206C2×#41—21no—
207C2×#41C22⋊D5311yescharacteristic
208#137—22no—
209#137—41no—
210AGL(1,ℤ/8)—22no—
211AGL(1,ℤ/8)—41no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—42no—
217C2×Q16C2×Dic5311yescharacteristic
218C2×SD16—21no—
219C2×D8C2×Dic5311yescharacteristic
220#131—21no—
221#131—21no—
222#131C2×Dic5312yes—
223#131C22⋊D5312yes—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—1061no—
227C2×M16C22⋊D5311yescharacteristic
228C22×C8—21no—
229C22×C8C22×D5311yescharacteristic
230C2×C4⋊C4—1061no—
231M32—532no—
232M32—1064no—
233M32—1061no—
234M32—1061no—
235M32—1061no—
236C2×C16—532no—
237C2×C16—1064no—
238C2×C16—1061no—
239#108—21no—
240#108—1061no—
241#108—1061no—
242#108C2×Dic5312yes—
243#106—1061no—
244#106—1061no—
245#103—2121no—
246#102—2121no—
247#101—22no—
248#100—22no—
249#41—21no—
250#41—21no—
251#41—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255#41#1171992312yes—
256#41C22⋊Dic5312yes—
257C2×Q8—21no—
258C2×Q8#1171992211yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4#1171992211yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—2121no—
266C22×C4C2×C22⋊D5311yescharacteristic
267Q16—21no—
268Q16—21no—
269Q16—42no—
270SD16—42no—
271SD16—42no—
272SD16—41no—
273SD16—41no—
274D8—21no—
275D8—21no—
276D8—42no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—22no—
281M16—41no—
282M16—2121no—
283M16—2121no—
284M16C22⋊Dic5312yes—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—2121no—
290C2×C8D4×D5311yescharacteristic
291C2×C8D4×D5311yescharacteristic
292C2×C8#1171994711yescharacteristic
293C2×C8#1171994711yescharacteristic
294C2×C8C22×Dic5311yescharacteristic
295C2×C8C2×C22⋊D5311yescharacteristic
296C4⋊C4—2121no—
297C4⋊C4—2121no—
298C16—1062no—
299C16—1062no—
300C23—21no—
301C23—41no—
302C23#1313323711yescharacteristic
303Q8—21no—
304Q8—41no—
305Q8—41no—
306Q8#1313313812yes—
307D4—22no—
308D4—22no—
309D4—21no—
310D4—41no—
311D4—41no—
312D4—81no—
313D4—81no—
314D4#1313313812yes—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—8481no—
322C2×C4—4241no—
323C2×C4C2×C22⋊Dic5311yescharacteristic
324C2×C4#1313318911yescharacteristic
325C2×C4#1313318911yescharacteristic
326C2×C4C2×#1171992211yescharacteristic
327C2×C4C2×#1171992311yescharacteristic
328C2×C4#1313323711yescharacteristic
329C8—22no—
330C8—42no—
331C8—4241no—
332C8D4×Dic5311yescharacteristic
333C8D4×Dic5311yescharacteristic
334C8#1313318511yescharacteristic
335C8#1313319811yescharacteristic
336C22—22no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22—161no—
341C22#1681061911yescharacteristic
342C22#1681085011yescharacteristic
343C22#1681085711yescharacteristic
344C4—22no—
345C4—81no—
346C4—8481no—
347C4#1681050311yescharacteristic
348C4C2×#1313313811yescharacteristic
349C4#1681085011yescharacteristic
350C4#1681085711yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—161no—
356C2#2027514411yescharacteristic
subgroupG/H
centerC2#20275144
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3067C2
socleC2×C53#1893
cosocleC22×C8×C53C23
hypercenter#3067D53
hypocenterC53#13070
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #20287818C2semidirect
2. C2×#16810433C2semidirect
3. #20276443C2semidirect
4. #20274774C2semidirect
5. #20272826C2semidirect
6. #20272815C2semidirect
22. C53×#473C4semidirect
23. C53×#472C4semidirect
194. #473Dic53semidirect
195. #472Dic53semidirect
205. C53#13070semidirect

Maximal subgroups

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

subgroupindex
#202878182
C2×#168104332
#202764432
#202747742
#202728262
#202728152
C53×#30672
#1307053