Finite Group Databasebeta

← C2×#20289755 #28310225

Subgroups — C2×#20289755 #28310225

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133292C211yesmaximal, characteristic
2C22×#13133271C211yesmaximal, characteristic
3#20289755C214yesmaximal, direct product factor
4C22×#13133156C211yesmaximal, characteristic
5C22×D53⋊C8C411yescharacteristic
6C22×C4×#6157C411yescharacteristic
7C2×#13133292—22no—
8C2×#13133292—22no—
9C2×#13133292C412yescomplemented
10C2×#13133292C2212yes—
11C22×#11719910C412yescomplemented
12C2×#13133271—22no—
13C2×#13133271—22no—
14C2×#13133271C412yes—
15C2×#13133271C2212yes—
16C22×C8×D53C2211yescharacteristic
17C2×#13133156—22no—
18C2×#13133156—22no—
19C2×#13133156C412yescomplemented
20C2×#13133156C2212yes—
21C22×#11719908C411yescharacteristic, complemented
22C22×C16×C53C411yescharacteristic, complemented
23C23×#6157—21no—
24C2×D53⋊C8—22no—
25C2×D53⋊C8—21no—
26C2×D53⋊C8C814yes—
27C2×D53⋊C8C2×C412yes—
28C22×#88324—21no—
29C2×C4×#6157—22no—
30C2×C4×#6157—21no—
31C2×C4×#6157C814yes—
32C2×C4×#6157C2×C412yes—
33#13133292—24no—
34#13133292—22no—
35#13133292—22no—
36C2×#11719910—24no—
37C2×#11719910C2×C414yescomplemented
38#13133271—24no—
39#13133271—22no—
40#13133271—22no—
41C22×C4×D53C2×C411yescharacteristic
42C22×#88322C2×C411yescharacteristic
43C2×C8×D53C2×C411yescharacteristic
44C2×C8×D53C2×C411yescharacteristic
45C2×C8×D53C2×C411yescharacteristic
46C2×C8×D53D412yes—
47C2×C8×D53D412yes—
48C2×C8×D53D412yes—
49C2×C8×D53Q812yes—
50C2×C8×D53C2311yescharacteristic
51#13133156—24no—
52#13133156—22no—
53#13133156—22no—
54C2×#11719908—22no—
55C2×#11719908C2×C412yescomplemented
56C22×C8×C53C2×C411yescharacteristic
57C2×C16×C53—22no—
58C2×C16×C53C2×C412yescomplemented
59C23×D53M1611yescharacteristic
60C22×#6157—24no—
61C22×#6157—22no—
62C22×#6157—42no—
63C22×#6157—42no—
64C22×C4×C53C4211yescharacteristic
65C22×Dic53M1611yescharacteristic
66C2×C4×D53C4211yescharacteristic
67C2×C4×D53C22⋊C412yes—
68C2×C4×D53C4⋊C412yes—
69C2×C4×D53C2×C812yes—
70C2×C4×D53C2×C812yes—
71C2×C4×D53M1611yescharacteristic
72C2×C4×D53M1611yescharacteristic
73C2×C4×D53C22×C411yescharacteristic
74C2×#88324—24no—
75C2×#88324—22no—
76C4×#6157—24no—
77C4×#6157—22no—
78C4×#6157C2×C814yes—
79D53⋊C8—24no—
80D53⋊C8—22no—
81D53⋊C8C2×C814yes—
82C2×C8×C53C4211yescharacteristic
83C2×C8×C53C22⋊C412yes—
84C2×C8×C53C4⋊C412yes—
85C2×C8×C53C22×C411yescharacteristic
86C2×#88322C4211yescharacteristic
87C2×#88322C22⋊C412yes—
88C2×#88322C4⋊C412yes—
89C2×#88322C22×C411yescharacteristic
90C8×D53C22⋊C412yes—
91C8×D53C22⋊C412yes—
92C8×D53C22⋊C412yes—
93C8×D53C4⋊C412yes—
94C8×D53C4⋊C412yes—
95C8×D53C4⋊C412yes—
96C8×D53C2×D411yescharacteristic
97C8×D53C2×D411yescharacteristic
98C8×D53C2×D411yescharacteristic
99C8×D53C2×Q811yescharacteristic
100#11719910—24no—
101C16×C53—22no—
102#11719908—22no—
103C23×C53#9711yescharacteristic
104C22×D53—22no—
105C22×D53—22no—
106C22×D53#9711yescharacteristic
107C22×D53C22⋊C812yes—
108C22×D53C4⋊C812yes—
109C22×D53C2×M1611yescharacteristic
110C2×#6157—24no—
111C2×#6157—42no—
112C2×#6157—42no—
113C2×#6157—48no—
114C2×C4×C53#9512yes—
115C2×C4×C53C4×C812yes—
116C2×C4×C53#9711yescharacteristic
117C2×C4×C53C2×C4211yescharacteristic
118C2×Dic53#9711yescharacteristic
119C2×Dic53C22⋊C812yes—
120C2×Dic53C4⋊C812yes—
121C2×Dic53C2×M1611yescharacteristic
122C4×D53#9512yes—
123C4×D53C4×C812yes—
124C4×D53C22⋊C812yes—
125C4×D53C22⋊C812yes—
126C4×D53C4⋊C812yes—
127C4×D53C4⋊C812yes—
128C4×D53C2×C22⋊C411yescharacteristic
129C4×D53C2×C4⋊C411yescharacteristic
130C4×D53C22×C811yescharacteristic
131C4×D53C22×C811yescharacteristic
132#88324—24no—
133C8×C53#9512yes—
134C8×C53C2×C22⋊C411yescharacteristic
135C8×C53C2×C4⋊C411yescharacteristic
136#88322#9512yes—
137#88322C2×C22⋊C411yescharacteristic
138#88322C2×C4⋊C411yescharacteristic
139C2×#1375—531nomaximal, Sylow
140C22×C53#33612yes—
141C22×C53#34712yes—
142C22×C53#34712yes—
143C22×C53C2×#9711yescharacteristic
144C2×D53—21no—
145C2×D53—21no—
146C2×D53—21no—
147C2×D53—21no—
148C2×D53—24no—
149C2×D53—24no—
150C2×D53#33612yes—
151C2×D53C2×C22⋊C811yescharacteristic
152C2×D53C2×C4⋊C811yescharacteristic
153#6157—44no—
154C4×C53#33612yes—
155C4×C53C2×#9511yescharacteristic
156C4×C53C2×C4×C811yescharacteristic
157Dic53#33612yes—
158Dic53C2×C22⋊C811yescharacteristic
159Dic53C2×C4⋊C811yescharacteristic
160C22×M32—531no—
161C22×M32—531no—
162C22×#97—531no—
163#1375—534no—
164C2×C53#137514yescomplemented
165C2×C53C2×#33611yescharacteristic
166C2×C53C2×#34711yescharacteristic
167C2×C53C2×#34711yescharacteristic
168D53—21no—
169D53—22no—
170D53—21no—
171C23×C8—531no—
172C23×C8—531no—
173C22×C42—531no—
174C2×M32—532no—
175C2×M32—532no—
176C2×M32—532no—
177C2×M32—532no—
178C2×M32—1062no—
179C2×M32—1062no—
180C2×M32—1062no—
181C2×M32—1062no—
182C22×C16—532no—
183C22×C16—531no—
184C22×C16#615711yescharacteristic, complemented
185C2×#97—532no—
186C2×#97—532no—
187C2×#97—1062no—
188C2×#97—1062no—
189C53C2×#137511yescharacteristic, complemented, Sylow
190C23×C4—531no—
191C23×C4—1061no—
192C22×C8—531no—
193C22×C8—531no—
194C22×C8—531no—
195C22×C8—532no—
196C22×C8—532no—
197C22×C8—532no—
198C22×C8—532no—
199C22×C8—531no—
200C22×C8—531no—
201C22×C8—532no—
202C22×C8—534no—
203C22×C8—1062no—
204C22×C8—1061no—
205C22×C8—1061no—
206C22×C8C2×#615711yescharacteristic
207C2×C42—532no—
208C2×C42—534no—
209C2×C42—1062no—
210C2×C42—1061no—
211M32—1062no—
212M32—1062no—
213M32—1064no—
214M32—1064no—
215M32—1062no—
216M32—1062no—
217C2×C16—22no—
218C2×C16—534no—
219C2×C16—532no—
220C2×C16—1064no—
221C2×C16—1062no—
222C2×C16C2×#615712yescomplemented
223#97—1064no—
224#97—1062no—
225#97—1062no—
226C24—531no—
227C22×C4—531no—
228C22×C4—532no—
229C22×C4—532no—
230C22×C4—532no—
231C22×C4—532no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—1062no—
237C22×C4—1064no—
238C22×C4—2122no—
239C22×C4—2122no—
240C22×C4C4×#615711yescharacteristic
241C2×C8—532no—
242C2×C8—532no—
243C2×C8—531no—
244C2×C8—531no—
245C2×C8—532no—
246C2×C8—532no—
247C2×C8—532no—
248C2×C8—532no—
249C2×C8—531no—
250C2×C8—531no—
251C2×C8—532no—
252C2×C8—531no—
253C2×C8—532no—
254C2×C8—531no—
255C2×C8—534no—
256C2×C8—1062no—
257C2×C8—1064no—
258C2×C8—1062no—
259C2×C8—1064no—
260C2×C8C4×#615711yescharacteristic
261C2×C8C4⋊#615712yes—
262C2×C8C22⋊#615712yes—
263C2×C8C22×#615711yescharacteristic
264C42—534no—
265C42—1062no—
266C42—1064no—
267C16—22no—
268C16—1064no—
269C16—1062no—
270C23—531no—
271C23—531no—
272C23—532no—
273C23—532no—
274C23—1062no—
275C23—1062no—
276C23#1313327111yescharacteristic
277C2×C4—531no—
278C2×C4—531no—
279C2×C4—532no—
280C2×C4—532no—
281C2×C4—532no—
282C2×C4—531no—
283C2×C4—531no—
284C2×C4—532no—
285C2×C4—531no—
286C2×C4—531no—
287C2×C4—532no—
288C2×C4—532no—
289C2×C4—532no—
290C2×C4—532no—
291C2×C4—1064no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4—2128no—
295C2×C4(C2×#6157)⋊C412yes—
296C2×C4C8×#615712yes—
297C2×C4#1313327111yescharacteristic
298C2×C4C2×C4×#615711yescharacteristic
299C8—532no—
300C8—531no—
301C8—531no—
302C8—1064no—
303C8(C2×#6157)⋊C412yes—
304C8C2×C22⋊#615711yescharacteristic
305C8C2×C4⋊#615711yescharacteristic
306C22—531no—
307C22—532no—
308C22—531no—
309C22—1061no—
310C22—1061no—
311C22—1061no—
312C22—1061no—
313C22—1064no—
314C22—1064no—
315C22(C2×#6157)⋊C812yes—
316C22#1681151312yes—
317C22#1681151412yes—
318C22C2×#1313327111yescharacteristic
319C4—531no—
320C4—532no—
321C4—531no—
322C4—2124no—
323C4(C2×#6157)⋊C812yes—
324C4C2×(C2×#6157)⋊C411yescharacteristic
325C4C2×C8×#615711yescharacteristic
326C2—1061no—
327C2—1062no—
328C2—1061no—
329C2#2028975514yesdirect product factor
330C2C2×(C2×#6157)⋊C811yescharacteristic
331C2C2×#1681151311yescharacteristic
332C2C2×#1681151411yescharacteristic
subgroupG/H
centerC22×C4C4×#6157
commutatorC4×C53C2×C4×C8
FrattiniC2×C8C22×#6157
FittingC22×C16×C53C4
socleC23×C53#97
prime radicalC23×D53M16
cosocleC2×C8×D53C23
hypercenterC22×C16#6157
hypocenterC53C2×#1375
LCSG ▷ C4×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C16

Complements

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

NH ≅ G/Ntype
3. #20289755C2direct (N × H)
9. C2×#13133292C4semidirect
11. C22×#11719910C4semidirect
19. C2×#13133156C4semidirect
21. C22×#11719908C4semidirect
22. C22×C16×C53C4semidirect
37. C2×#11719910C2×C4semidirect
55. C2×#11719908C2×C4semidirect
58. C2×C16×C53C2×C4semidirect
164. C2×C53#1375semidirect
184. C22×C16#6157semidirect
189. C53C2×#1375semidirect
222. C2×C16C2×#6157semidirect
329. C2#20289755direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332922
C22×#131332712
#202897552
C22×#131331562
C2×#137553