Finite Group Databasebeta

← #27301144

Subgroups — #27301144

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810740C211yesmaximal, characteristic, complemented
2C2×#16810430C211yesmaximal, characteristic, complemented
3C2×#16810430C211yesmaximal, characteristic, complemented
4#20275604C211yesmaximal, characteristic, complemented
5#20272809C211yesmaximal, characteristic, complemented
6#20272792C211yesmaximal, characteristic, complemented
7C2×C53×#508C211yesmaximal, characteristic
8C2×#13133224—21no—
9C22×#11719908C2211yescharacteristic
10#16810740—22no—
11#16810740—22no—
12C2×#13133153C2211yescharacteristic
13C2×#13133153C2211yescharacteristic
14C2×#13133144—21no—
15C2×#13133144—21no—
16#16810590—21no—
17C2×C8×Dic53C2211yescharacteristic
18#16810430—22no—
19#16810430—22no—
20C2×#135×C53C2211yescharacteristic
21C2×#135×C53C2211yescharacteristic
22C53×#508—22no—
23C53×#508—22no—
24C2×Q32×C53C411yescharacteristic, complemented
25C2×SD32×C53C411yescharacteristic, complemented
26C2×SD32×C53C411yescharacteristic, complemented
27C2×D16×C53C411yescharacteristic, complemented
28C22×C16×C53C2211yescharacteristic
29#13133224—24no—
30#13133224—42no—
31C2×#11719917—21no—
32C2×#11719917—21no—
33C2×#11719917—41no—
34C22×#88322—41no—
35C22×#88322D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×#11719908—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40#13133153—21no—
41#13133153—21no—
42#13133153—21no—
43#13133153D412yes—
44#13133144—24no—
45#13133144—42no—
46C22⋊#88322—41no—
47C22⋊#88322—41no—
48C8×Dic53—21no—
49C8×Dic53—21no—
50C2×#41×C53—21no—
51C2×#41×C53—21no—
52#135×C53—21no—
53#135×C53—21no—
54#135×C53—21no—
55#135×C53—21no—
56#135×C53—21no—
57#135×C53—21no—
58#135×C53D412yes—
59#135×C53D412yes—
60C2×Q16×C53C2×C411yescharacteristic
61C2×Q16×C53C2×C411yescharacteristic
62C2×SD16×C53—21no—
63C2×SD16×C53—21no—
64C2×D8×C53C2×C411yescharacteristic
65C2×D8×C53C2×C411yescharacteristic
66C22×C8×C53C2311yescharacteristic
67Q32×C53—22no—
68SD32×C53—22no—
69SD32×C53—22no—
70D16×C53—22no—
71C2×C16×C53—21no—
72C2×C16×C53—21no—
73C2×C16×C53C2×C411yescharacteristic
74C2×C16×C53C2×C411yescharacteristic
75#41×C53—22no—
76#41×C53—22no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80#41×C53—41no—
81#41×C53—41no—
82#41×C53—41no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53C2×D411yescharacteristic
92C22×Dic53—21no—
93Q16×C53—21no—
94Q16×C53—21no—
95Q16×C53—21no—
96Q16×C53C22⋊C412yes—
97SD16×C53—22no—
98SD16×C53—22no—
99SD16×C53—41no—
100SD16×C53—41no—
101D8×C53—21no—
102D8×C53—21no—
103D8×C53—21no—
104D8×C53C22⋊C412yes—
105C2×C8×C53C22×C411yescharacteristic
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53#4111yescharacteristic
110C2×C8×C53#4111yescharacteristic
111C4×Dic53—21no—
112C4×Dic53—21no—
113#11719917—22no—
114#11719917—44no—
115#11719917—41no—
116#11719917—41no—
117#11719917—41no—
118C2×#88322—21no—
119C2×#88322—21no—
120C2×#88322—44no—
121C2×#88322—41no—
122C2×#88322—41no—
123C2×#88322D811yescharacteristic
124C2×#88322D811yescharacteristic
125C16×C53—21no—
126C16×C53—21no—
127#11719908—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53#13511yescharacteristic
131Q8×C53—22no—
132Q8×C53—41no—
133Q8×C53—41no—
134Q8×C53—41no—
135D4×C53—22no—
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—41no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53—81no—
142D4×C53—81no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—81no—
148C2×C4×C53—81no—
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×Dic53—22no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C8×C53C2×C22⋊C411yescharacteristic
159C8×C53C4×D411yescharacteristic
160C8×C53#12111yescharacteristic
161C8×C53#12311yescharacteristic
162#88322—21no—
163#88322—21no—
164#88322—42no—
165#88322—81no—
166#13011—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—42no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53C4×D811yescharacteristic
176C22×C53#44911yescharacteristic
177C22×C53#46311yescharacteristic
178C4×C53—42no—
179C4×C53—81no—
180C4×C53#38611yescharacteristic
181C4×C53#44211yescharacteristic
182C4×C53#44711yescharacteristic
183C4×C53#45911yescharacteristic
184Dic53—42no—
185C2×#508D5311yescharacteristic
186C2×#443—531no—
187C2×#359—531no—
188C2×#359—531no—
189#1974—531no—
190#1381—531no—
191#1374—531no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—82no—
195C2×C53—161no—
196C2×C53#187911yescharacteristic
197C2×C53#217811yescharacteristic
198C2×C53#217911yescharacteristic
199C2×#135C2×D5311yescharacteristic
200C2×#135C2×D5311yescharacteristic
201C2×#131—1061no—
202#508—22no—
203#508—22no—
204C2×Q32Dic5311yescharacteristic, complemented
205C2×SD32Dic5311yescharacteristic, complemented
206C2×SD32Dic5311yescharacteristic, complemented
207C2×D16Dic5311yescharacteristic, complemented
208C22×C16—531no—
209C22×C16C2×D5311yescharacteristic
210#443—1062no—
211#443—1062no—
212C2×#108—531no—
213C2×#108—531no—
214C2×#104—1061no—
215C2×#104—1061no—
216#408—1061no—
217C2×C4×C8—531no—
218#359—1062no—
219#359—1062no—
220C53#1301111yescharacteristic, complemented, Sylow
221C2×#41—21no—
222C2×#41—21no—
223#135—21no—
224#135—21no—
225#135—21no—
226#135—21no—
227#135—21no—
228#135—21no—
229#135C22⋊D5312yes—
230#135C22⋊D5312yes—
231C2×Q16C2×Dic5311yescharacteristic
232C2×Q16C2×Dic5311yescharacteristic
233C2×SD16—21no—
234C2×SD16—21no—
235C2×D8C2×Dic5311yescharacteristic
236C2×D8C2×Dic5311yescharacteristic
237#131—1064no—
238#131—2122no—
239C2×M16—1061no—
240C2×M16—1061no—
241C2×M16—2121no—
242C22×C8—531no—
243C22×C8—2121no—
244C22×C8C22×D5311yescharacteristic
245C2×C42—531no—
246Q32—22no—
247SD32—22no—
248SD32—22no—
249D16—22no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—1061no—
253C2×C16—1061no—
254C2×C16—1061no—
255C2×C16C2×Dic5311yescharacteristic
256C2×C16C2×Dic5311yescharacteristic
257#108—532no—
258#108—1061no—
259#108—1061no—
260#108—1061no—
261#104—1064no—
262#104—2122no—
263C22⋊C8—2121no—
264C22⋊C8—2121no—
265C4×C8—1061no—
266C4×C8—1061no—
267#41—22no—
268#41—22no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273#41—41no—
274#41—41no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—1061no—
284C22×C4D4×D5311yescharacteristic
285Q16—21no—
286Q16—21no—
287Q16—21no—
288Q16C22⋊Dic5312yes—
289SD16—22no—
290SD16—22no—
291SD16—41no—
292SD16—41no—
293D8—21no—
294D8—21no—
295D8—21no—
296D8C22⋊Dic5312yes—
297M16—1062no—
298M16—2124no—
299M16—2121no—
300M16—2121no—
301M16—2121no—
302C2×C8—531no—
303C2×C8—531no—
304C2×C8—1061no—
305C2×C8—1061no—
306C2×C8—2124no—
307C2×C8—2121no—
308C2×C8—2121no—
309C2×C8D4×D5311yescharacteristic
310C2×C8#1171994711yescharacteristic
311C2×C8#1171994711yescharacteristic
312C2×C8C22×Dic5311yescharacteristic
313C2×C8C2×C22⋊D5311yescharacteristic
314C2×C8C2×C22⋊D5311yescharacteristic
315C42—1061no—
316C42—1061no—
317C16—21no—
318C16—21no—
319C16—2121no—
320C23—41no—
321C23—41no—
322C23#1313323611yescharacteristic
323Q8—22no—
324Q8—41no—
325Q8—41no—
326Q8—41no—
327D4—22no—
328D4—42no—
329D4—42no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—81no—
334D4—81no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—1062no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4D4×Dic5311yescharacteristic
345C2×C4#1313318511yescharacteristic
346C2×C4#1313318911yescharacteristic
347C2×C4D8×D5311yescharacteristic
348C2×C4#1313324111yescharacteristic
349C2×C4#1313324111yescharacteristic
350C8—1061no—
351C8—1061no—
352C8—4241no—
353C8—2122no—
354C8C2×C22⋊Dic5311yescharacteristic
355C8D4×Dic5311yescharacteristic
356C8#1313318911yescharacteristic
357C8#1313319811yescharacteristic
358C22—21no—
359C22—21no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22D8×Dic5311yescharacteristic
367C22#1681076511yescharacteristic
368C22#1681083611yescharacteristic
369C4—42no—
370C4—81no—
371C4—2122no—
372C4#1681050311yescharacteristic
373C4#1681073511yescharacteristic
374C4#1681075511yescharacteristic
375C4#1681081611yescharacteristic
376C2—21no—
377C2—21no—
378C2—82no—
379C2—161no—
380C2#2027506511yescharacteristic
381C2#2027659611yescharacteristic
382C2#2027660111yescharacteristic
subgroupG/H
centerC2×C4D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC2×C53×#508C2
socleC22×C53C4×D8
cosocleC22×C8×C53C23
hypercenterC2×#508D53
hypocenterC53#13011
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#508

Complements

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

NH ≅ G/Ntype
1. C2×#16810740C2semidirect
2. C2×#16810430C2semidirect
3. C2×#16810430C2semidirect
4. #20275604C2semidirect
5. #20272809C2semidirect
6. #20272792C2semidirect
24. C2×Q32×C53C4semidirect
25. C2×SD32×C53C4semidirect
26. C2×SD32×C53C4semidirect
27. C2×D16×C53C4semidirect
204. C2×Q32Dic53semidirect
205. C2×SD32Dic53semidirect
206. C2×SD32Dic53semidirect
207. C2×D16Dic53semidirect
220. C53#13011semidirect

Maximal subgroups

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

subgroupindex
C2×#168107402
C2×#168104302
#202756042
#202728092
#202727922
C2×C53×#5082
#1301153