Finite Group Databasebeta

← C2×#20272700 #27294233

Subgroups — C2×#20272700 #27294233

Order 13568.

496 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810405C211yesmaximal, characteristic, complemented
2C2×#16810380C211yesmaximal, characteristic, complemented
3#20272700C214yesmaximal, direct product factor
4C2×C53×#469C211yesmaximal, characteristic
5C2×#13133176—21no—
6C2×#13133157—21no—
7#16810405—21no—
8#16810405C2212yescomplemented
9#16810380—21no—
10#16810380C2212yescomplemented
11C22×D8×C53C411yescharacteristic, complemented
12C22×M16×C53C2211yescharacteristic
13C2×#121×C53—21no—
14C53×#469—21no—
15C53×#469—21no—
16C53×#469C412yescomplemented
17C53×#469C2212yes—
18C2×#106×C53C411yescharacteristic, complemented
19C2×#102×C53—21no—
20#13133176—22no—
21#13133176—22no—
22#13133176—41no—
23#13133176—41no—
24C2×C4⋊Dic53—41no—
25C2×C22⋊Dic53—41no—
26C2×C4×Dic53—22no—
27#13133157—22no—
28#13133157—41no—
29C2×#11719908—41no—
30C22×D4×C53—21no—
31C22×D4×C53—41no—
32C23×C4×C53D411yescharacteristic
33C2×D8×C53—21no—
34C2×D8×C53—41no—
35C2×D8×C53—41no—
36C2×D8×C53C2×C412yescomplemented
37C2×M16×C53—41no—
38C2×M16×C53—41no—
39C2×M16×C53C2×C411yescharacteristic
40C2×M16×C53D411yescharacteristic
41C2×M16×C53D411yescharacteristic
42C2×M16×C53C2311yescharacteristic
43C22×C8×C53C2×C411yescharacteristic
44C22×C8×C53D411yescharacteristic
45#121×C53—22no—
46#121×C53—22no—
47#121×C53—41no—
48#121×C53—41no—
49C2×C4⋊C4×C53—21no—
50C2×C22⋊C4×C53—41no—
51#106×C53—21no—
52#106×C53C2×C412yescomplemented
53#102×C53—22no—
54#102×C53—22no—
55C24×C53—41no—
56C24×C53D811yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—42no—
59C2×D4×C53—42no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—84no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—41no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53D811yescharacteristic
74C22×C4×C53SD1611yescharacteristic
75C22×C4×C53SD1611yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—42no—
78D8×C53—42no—
79D8×C53—42no—
80M16×C53—41no—
81M16×C53—41no—
82M16×C53—41no—
83M16×C53—41no—
84C2×C8×C53—21no—
85C2×C8×C53—21no—
86C2×C8×C53—21no—
87C2×C8×C53—21no—
88C2×C8×C53C22⋊C411yescharacteristic
89C2×C8×C53C22⋊C411yescharacteristic
90C2×C8×C53C22×C411yescharacteristic
91C2×C8×C53C2×D411yescharacteristic
92C4⋊C4×C53—22no—
93C4⋊C4×C53—41no—
94C22⋊C4×C53—42no—
95C22⋊C4×C53—42no—
96C22⋊Dic53—42no—
97C22⋊Dic53—42no—
98C4⋊Dic53—42no—
99C4⋊Dic53—42no—
100C4×Dic53—24no—
101C4×Dic53—42no—
102#11719908—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—42no—
106C23×C53—42no—
107C23×C53—82no—
108C23×C53—82no—
109C23×C53—81no—
110C23×C53—81no—
111C23×C53#10211yescharacteristic
112C23×C53#10211yescharacteristic
113C23×C53C2×D811yescharacteristic
114D4×C53—42no—
115D4×C53—42no—
116D4×C53—84no—
117D4×C53—84no—
118D4×C53—81no—
119D4×C53—81no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53—82no—
132C2×C4×C53#9911yescharacteristic
133C2×C4×C53#9911yescharacteristic
134C2×C4×C53#10211yescharacteristic
135C2×C4×C53#10211yescharacteristic
136C2×C4×C53#10411yescharacteristic
137C2×C4×C53#10411yescharacteristic
138C2×C4×C53C2×C22⋊C411yescharacteristic
139C2×C4×C53C2×SD1611yescharacteristic
140C2×Dic53—44no—
141C2×Dic53—84no—
142C8×C53—21no—
143C8×C53—21no—
144C8×C53—21no—
145C8×C53—21no—
146C2×#1339—531nomaximal, Sylow
147C22×C53—22no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—42no—
153C22×C53—81no—
154C22×C53—82no—
155C22×C53—82no—
156C22×C53—82no—
157C22×C53—162no—
158C22×C53—162no—
159C22×C53#32711yescharacteristic
160C22×C53#32711yescharacteristic
161C22×C53C2×#10211yescharacteristic
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—82no—
165C4×C53#32711yescharacteristic
166C4×C53#32711yescharacteristic
167C4×C53C2×#9911yescharacteristic
168C4×C53C2×#10411yescharacteristic
169Dic53—84no—
170C2×#469D5311yescharacteristic
171C2×#349—531no—
172C2×H(3,ℤ/4)—531no—
173#1339—534no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53—41no—
177C2×C53—41no—
178C2×C53—162no—
179C2×C53#133912yescomplemented
180C2×C53C2×#32711yescharacteristic
181C22×D8Dic5311yescharacteristic, complemented
182C22×M16C2×D5311yescharacteristic
183C2×#121—21no—
184C2×#117—1061no—
185C2×M32—1061no—
186#469—21no—
187#469—21no—
188#469Dic5312yescomplemented
189#469C2×D5312yes—
190C2×#106Dic5311yescharacteristic, complemented
191C2×#102—21no—
192#349—532no—
193#349—1061no—
194H(3,ℤ/4)—532no—
195H(3,ℤ/4)—1061no—
196C53C2×#133911yescharacteristic, complemented, Sylow
197C22×D4—21no—
198C22×D4—41no—
199C23×C4C22⋊D5311yescharacteristic
200C2×D8—21no—
201C2×D8—41no—
202C2×D8—41no—
203C2×D8C2×Dic5312yescomplemented
204C2×M16—41no—
205C2×M16—41no—
206C2×M16C2×Dic5311yescharacteristic
207C2×M16C22⋊D5311yescharacteristic
208C2×M16C22⋊D5311yescharacteristic
209C2×M16C22×D5311yescharacteristic
210C22×C8C2×Dic5311yescharacteristic
211C22×C8C22⋊D5311yescharacteristic
212#121—22no—
213#121—22no—
214#121—41no—
215#121—41no—
216#117—1062no—
217#117—1062no—
218#117—2121no—
219#117—2121no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—2121no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—2121no—
224C2×C42—1062no—
225M32—1062no—
226M32—2121no—
227C2×C16—2121no—
228#106—21no—
229#106C2×Dic5312yescomplemented
230#102—22no—
231#102—22no—
232C24—41no—
233C24#1171992211yescharacteristic
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—84no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—2122no—
249C22×C4#1171992211yescharacteristic
250C22×C4#1171992311yescharacteristic
251C22×C4#1171992311yescharacteristic
252C22×C4C22⋊Dic5311yescharacteristic
253C22×C4C22⋊Dic5311yescharacteristic
254C22×C4C2×C22⋊D5311yescharacteristic
255D8—42no—
256D8—42no—
257M16—41no—
258M16—41no—
259M16—41no—
260M16—41no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8C22⋊Dic5311yescharacteristic
266C2×C8C22⋊Dic5311yescharacteristic
267C2×C8C22×Dic5311yescharacteristic
268C2×C8C2×C22⋊D5311yescharacteristic
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—2122no—
272C4⋊C4—2122no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—2122no—
276C22⋊C4—2122no—
277C42—1064no—
278C42—2122no—
279C16—2122no—
280C23—21no—
281C23—21no—
282C23—42no—
283C23—42no—
284C23—82no—
285C23—82no—
286C23—81no—
287C23—81no—
288C23#1313313811yescharacteristic
289C23#1313313811yescharacteristic
290C23C2×#1171992211yescharacteristic
291D4—42no—
292D4—42no—
293D4—84no—
294D4—84no—
295D4—81no—
296D4—81no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—4244no—
310C2×C4—2124no—
311C2×C4#1313313211yescharacteristic
312C2×C4#1313313211yescharacteristic
313C2×C4#1313313811yescharacteristic
314C2×C4#1313313811yescharacteristic
315C2×C4#1313314411yescharacteristic
316C2×C4#1313314411yescharacteristic
317C2×C4C2×C22⋊Dic5311yescharacteristic
318C2×C4C2×#1171992311yescharacteristic
319C8—21no—
320C8—21no—
321C8—21no—
322C8—21no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—42no—
329C22—81no—
330C22—82no—
331C22—82no—
332C22—82no—
333C22—162no—
334C22—162no—
335C22#1681035711yescharacteristic
336C22#1681035711yescharacteristic
337C22C2×#1313313811yescharacteristic
338C4—21no—
339C4—21no—
340C4—82no—
341C4—4244no—
342C4#1681035711yescharacteristic
343C4#1681035711yescharacteristic
344C4C2×#1313313211yescharacteristic
345C4C2×#1313314411yescharacteristic
346C2—21no—
347C2—21no—
348C2—41no—
349C2—41no—
350C2—162no—
351C2#2027270012yesdirect product factor
352C2C2×#1681035711yescharacteristic
subgroupG/H
centerC22#16810357
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
Fitting, prime radicalC2×C53×#469C2
socleC22×C53#327
cosocleC2×M16×C53C23
hypercenterC2×#469D53
hypocenterC53C2×#1339
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ C2×#469

Complements

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

NH ≅ G/Ntype
1. C2×#16810405C2semidirect
2. C2×#16810380C2semidirect
3. #20272700C2direct (N × H)
8. #16810405C22semidirect
10. #16810380C22semidirect
11. C22×D8×C53C4semidirect
16. C53×#469C4semidirect
18. C2×#106×C53C4semidirect
36. C2×D8×C53C2×C4semidirect
52. #106×C53C2×C4semidirect
179. C2×C53#1339semidirect
181. C22×D8Dic53semidirect
188. #469Dic53semidirect
190. C2×#106Dic53semidirect
196. C53C2×#1339semidirect
203. C2×D8C2×Dic53semidirect
229. #106C2×Dic53semidirect
351. C2#20272700direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168104052
C2×#168103802
#202727002
C2×C53×#4692
C2×#133953