Finite Group Databasebeta

← #27301221

Subgroups — #27301221

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283591C211yesmaximal, characteristic, complemented
2C2×#16810430C211yesmaximal, characteristic, complemented
3C2×#16810424C211yesmaximal, characteristic, complemented
4#20275626C211yesmaximal, characteristic, complemented
5#20272824C211yesmaximal, characteristic, complemented
6#20272805C211yesmaximal, characteristic, complemented
7C2×C53×#509C211yesmaximal, characteristic
8#16811106—21no—
9C22×#11719908C2211yescharacteristic
10#16810724—22no—
11Q16×Dic53—21no—
12D8×Dic53—21no—
13C2×#13133153C2211yescharacteristic
14C2×#13133151C2211yescharacteristic
15#16810628—21no—
16#16810624—21no—
17C2×#13133134—21no—
18#16810578C2211yescharacteristic
19#16810430—22no—
20#16810424—22no—
21C2×#135×C53C2211yescharacteristic
22C22×D8×C53C2211yescharacteristic
23C53×#509—22no—
24C53×#509C414yescomplemented
25C2×SD32×C53—21no—
26C2×D16×C53—21no—
27C2×M32×C53C2211yescharacteristic
28C2×#11719917—21no—
29C2×#11719917D411yescharacteristic
30Q8×Dic53—22no—
31D4×Dic53—22no—
32D4×Dic53—42no—
33#13133176—41no—
34#13133176D411yescharacteristic
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—41no—
37C2×#11719908—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40#13133153—21no—
41#13133153—21no—
42#13133151—21no—
43#13133151D412yes—
44#13133141—22no—
45#13133138—22no—
46#13133138—41no—
47#13133138—41no—
48#13133134—42no—
49#13133127—21no—
50C8×Dic53—21no—
51C2×#41×C53—21no—
52C22×D4×C53—21no—
53#135×C53—21no—
54#135×C53—21no—
55#135×C53C2×C412yes—
56#135×C53D412yes—
57C2×Q16×C53D411yescharacteristic
58C2×SD16×C53—21no—
59C2×D8×C53—21no—
60C2×D8×C53—21no—
61C2×D8×C53—21no—
62C2×D8×C53—21no—
63C2×D8×C53—21no—
64C2×D8×C53C2×C412yescomplemented
65C2×D8×C53D411yescharacteristic
66C22×C8×C53C2311yescharacteristic
67SD32×C53—24no—
68D16×C53—24no—
69M32×C53—21no—
70M32×C53C2×C412yescomplemented
71C2×C16×C53—21no—
72C24×C53—41no—
73#41×C53—22no—
74#41×C53—22no—
75#41×C53—41no—
76#41×C53—41no—
77C2×Q8×C53—21no—
78C2×D4×C53—22no—
79C2×D4×C53—21no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53C2×D411yescharacteristic
88C22×Dic53—42no—
89C22×Dic53—41no—
90Q16×C53—21no—
91Q16×C53C22⋊C412yes—
92SD16×C53—22no—
93SD16×C53—22no—
94D8×C53—22no—
95D8×C53—22no—
96D8×C53—21no—
97D8×C53—41no—
98D8×C53—41no—
99D8×C53C22⋊C412yes—
100C2×C8×C53C22×C411yescharacteristic
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53#4111yescharacteristic
105C2×C8×C53#4111yescharacteristic
106C22⋊Dic53—21no—
107C22⋊Dic53—42no—
108C4⋊Dic53—22no—
109C4⋊Dic53—42no—
110C4⋊Dic53—41no—
111C4⋊Dic53D811yescharacteristic
112C4⋊Dic53D811yescharacteristic
113C4×Dic53—21no—
114C4×Dic53—42no—
115#11719917—21no—
116#11719917—21no—
117#11719917—41no—
118#11719917—41no—
119C2×#88322—21no—
120C2×#88322—41no—
121C16×C53—22no—
122#11719908—41no—
123C23×C53—41no—
124C23×C53—42no—
125C23×C53—81no—
126C23×C53—81no—
127C23×C53—161no—
128C23×C53—161no—
129C23×C53AGL(1,ℤ/8)11yescharacteristic
130Q8×C53—22no—
131Q8×C53—41no—
132D4×C53—22no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—42no—
137D4×C53—41no—
138D4×C53—81no—
139D4×C53—81no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—42no—
152C2×Dic53—162no—
153C2×Dic53—161no—
154C8×C53C2×C22⋊C411yescharacteristic
155C8×C53C4×D411yescharacteristic
156C8×C53#12111yescharacteristic
157C8×C53#12311yescharacteristic
158#88322—41no—
159#88322—81no—
160#13022—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—42no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53—82no—
167C22×C53—162no—
168C22×C53—162no—
169C22×C53—161no—
170C22×C53—161no—
171C22×C53#44211yescharacteristic
172C22×C53#44711yescharacteristic
173C22×C53#46311yescharacteristic
174C4×C53—42no—
175C4×C53#38611yescharacteristic
176C4×C53C4×D811yescharacteristic
177C4×C53#44911yescharacteristic
178C4×C53#45911yescharacteristic
179Dic53—41no—
180Dic53—41no—
181Dic53—82no—
182C2×#509D5311yescharacteristic
183#2951—531no—
184C2×#359—531no—
185C2×#357—531no—
186#1978—531no—
187#1387—531no—
188#1379—531no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—82no—
192C2×C53—162no—
193C2×C53#187911yescharacteristic
194C2×C53#219211yescharacteristic
195C2×C53#219311yescharacteristic
196C2×#135C2×D5311yescharacteristic
197C22×D8C2×D5311yescharacteristic
198#520—1061no—
199#509—22no—
200#509Dic5314yescomplemented
201C2×SD32—21no—
202C2×D16—21no—
203C2×M32C2×D5311yescharacteristic
204C22×C16—531no—
205#440—1062no—
206C4×Q16—1061no—
207C4×D8—1061no—
208C2×#108—531no—
209C2×#107—531no—
210#418—1061no—
211#417—1061no—
212C2×#100—1061no—
213#405—531no—
214#359—1062no—
215#357—1062no—
216C53#1302211yescharacteristic, complemented, Sylow
217C2×#41—21no—
218C22×D4—21no—
219#135—21no—
220#135—21no—
221#135C2×Dic5312yes—
222#135C22⋊D5312yes—
223C2×Q16C22⋊D5311yescharacteristic
224C2×SD16—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—21no—
229C2×D8—21no—
230C2×D8C2×Dic5312yescomplemented
231C2×D8C22⋊D5311yescharacteristic
232C2×M16—531no—
233C2×M16—1061no—
234C22×C8C22×D5311yescharacteristic
235C4×Q8—1062no—
236C4×D4—1062no—
237C4×D4—2122no—
238#117—531no—
239#117—2121no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—2121no—
242SD32—24no—
243D16—24no—
244M32—21no—
245M32C2×Dic5312yescomplemented
246C2×C16—21no—
247C2×C16—1061no—
248C2×C16—1061no—
249C2×C16—1061no—
250#108—1061no—
251#108—1061no—
252#107—532no—
253#107—1061no—
254#103—1062no—
255#102—1062no—
256#102—2121no—
257#102—2121no—
258#100—2122no—
259#97—1061no—
260C4×C8—1061no—
261C24—41no—
262#41—22no—
263#41—22no—
264#41—41no—
265#41—41no—
266C2×Q8—21no—
267C2×D4—22no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C22×C4—41no—
276C22×C4—2122no—
277C22×C4—2121no—
278C22×C4D4×D5311yescharacteristic
279Q16—21no—
280Q16C22⋊Dic5312yes—
281SD16—22no—
282SD16—22no—
283D8—22no—
284D8—22no—
285D8—21no—
286D8—41no—
287D8—41no—
288D8C22⋊Dic5312yes—
289M16—1061no—
290M16—1061no—
291M16—2121no—
292M16—2121no—
293C2×C8—1061no—
294C2×C8—2121no—
295C2×C8D4×D5311yescharacteristic
296C2×C8#1171994711yescharacteristic
297C2×C8#1171994711yescharacteristic
298C2×C8C22×Dic5311yescharacteristic
299C2×C8C2×C22⋊D5311yescharacteristic
300C2×C8C2×C22⋊D5311yescharacteristic
301C4⋊C4—531no—
302C4⋊C4—531no—
303C4⋊C4—1062no—
304C4⋊C4—2122no—
305C4⋊C4—2121no—
306C22⋊C4—1061no—
307C22⋊C4—2122no—
308C42—1061no—
309C42—2122no—
310C16—22no—
311C16—2121no—
312C23—41no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—161no—
317C23—161no—
318C23#1313324111yescharacteristic
319Q8—22no—
320Q8—41no—
321D4—22no—
322D4—42no—
323D4—42no—
324D4—42no—
325D4—41no—
326D4—42no—
327D4—81no—
328D4—81no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—8481no—
333C2×C4—8482no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—2122no—
337C2×C4D4×Dic5311yescharacteristic
338C2×C4#1313318511yescharacteristic
339C2×C4#1313318911yescharacteristic
340C2×C4D8×D5311yescharacteristic
341C2×C4#1313323611yescharacteristic
342C2×C4#1313324111yescharacteristic
343C8—4241no—
344C8—2121no—
345C8C2×C22⋊Dic5311yescharacteristic
346C8D4×Dic5311yescharacteristic
347C8#1313318911yescharacteristic
348C8#1313319811yescharacteristic
349C22—21no—
350C22—21no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22—82no—
355C22—162no—
356C22—162no—
357C22—161no—
358C22—161no—
359C22#1681073511yescharacteristic
360C22#1681075511yescharacteristic
361C22#1681083611yescharacteristic
362C4—42no—
363C4—4242no—
364C4—2121no—
365C4—2121no—
366C4#1681050311yescharacteristic
367C4D8×Dic5311yescharacteristic
368C4#1681076511yescharacteristic
369C4#1681081611yescharacteristic
370C2—21no—
371C2—21no—
372C2—82no—
373C2—162no—
374C2#2027506511yescharacteristic
375C2#2027666911yescharacteristic
376C2#2027667611yescharacteristic
subgroupG/H
centerC22#16810755
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC2×C53×#509C2
socleC22×C53#447
cosocleC22×C8×C53C23
hypercenterC2×#509D53
hypocenterC53#13022
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. #20283591C2semidirect
2. C2×#16810430C2semidirect
3. C2×#16810424C2semidirect
4. #20275626C2semidirect
5. #20272824C2semidirect
6. #20272805C2semidirect
24. C53×#509C4semidirect
64. C2×D8×C53C2×C4semidirect
70. M32×C53C2×C4semidirect
200. #509Dic53semidirect
216. C53#13022semidirect
230. C2×D8C2×Dic53semidirect
245. M32C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202835912
C2×#168104302
C2×#168104242
#202756262
#202728242
#202728052
C2×C53×#5092
#1302253