Finite Group Databasebeta

← C2×#41×#88322 #28248526

Subgroups — C2×#41×#88322 #28248526

Order 13568.

2566 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#88322C2116yesmaximal, direct product factor
2C2×Q8×#88322C212yesmaximal, complemented
3C2×D4×#88322C216yesmaximal, complemented
4C2×#16810676C213yesmaximal, complemented
5C22×C4×#88322C213yesmaximal, complemented
6C2×C4×#41×C53C211yesmaximal, characteristic
7C23×#88322C2216yes—
8Q8×#88322C22116yescomplemented
9D4×#88322C22148yescomplemented
10#16810676C22124yescomplemented
11C2×C4⋊#88322C2216yes—
12C2×C22⋊#88322C2216yes—
13C2×C4×#88322C2213yes—
14C2×C4×#88322C2211yescharacteristic
15C2×C4×#88322C2216yes—
16C2×C4×#88322C22124yescomplemented
17C22×#41×C53C411yescharacteristic
18C4×#41×C53C418yes—
19C4×#41×C53C2218yes—
20C2×C4×Q8×C53C411yescharacteristic
21C2×C4×Q8×C53C2211yescharacteristic
22C2×C4×D4×C53C413yes—
23C2×C4×D4×C53C2213yes—
24C2×#117×C53C413yes—
25C22×C42×C53C2213yes—
26C22×#88322—224no—
27C22×#88322C2316yes—
28C22×#88322C23124yes—
29C22×#88322C2312yes—
30C4⋊#88322C23124yes—
31C22⋊#88322C23124yes—
32C4×#88322C2314yes—
33C4×#88322C23112yes—
34C4×#88322C23124yes—
35C2×#41×C53C8116yescomplemented
36C2×#41×C53C2×C418yes—
37C22×Q8×C53C2×C411yescharacteristic
38C22×D4×C53C2×C413yes—
39C23×C4×C53C2×C413yes—
40C23×C4×C53C2×C413yes—
41C23×C4×C53C2313yes—
42C4×Q8×C53C2×C418yes—
43C4×Q8×C53C2×C414yes—
44C4×Q8×C53C2314yes—
45C4×D4×C53C2×C4124yes—
46C4×D4×C53C2×C4112yes—
47C4×D4×C53C23112yes—
48#117×C53C2×C4124yes—
49C2×C4⋊C4×C53C2×C413yes—
50C2×C4⋊C4×C53C2×C413yes—
51C2×C22⋊C4×C53C2×C416yes—
52C2×C42×C53C2×C413yes—
53C2×C42×C53C2×C413yes—
54C2×C42×C53C2×C4112yes—
55C2×C42×C53C2313yes—
56C2×C42×C53C23112yes—
57C2×C42×C53C2311yescharacteristic
58C24×C53C22×C413yes—
59#41×C53C2×C8164yescomplemented
60C2×Q8×C53C2×C818yescomplemented
61C2×Q8×C53C22×C414yes—
62C2×D4×C53C2×C8124yescomplemented
63C2×D4×C53C22×C4112yes—
64C22×C4×C53—212no—
65C22×C4×C53—212no—
66C22×C4×C53C2×C8124yescomplemented
67C22×C4×C53C22×C413yes—
68C22×C4×C53C22×C416yes—
69C22×C4×C53C22×C413yes—
70C22×C4×C53C22×C4112yes—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C22×C413yes—
73C22×C4×C53C22×C4112yes—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2416yes—
76C22×C4×C53C2411yescharacteristic
77C4⋊C4×C53C22×C4112yes—
78C4⋊C4×C53C22×C4112yes—
79C22⋊C4×C53C22×C4124yes—
80C42×C53C22×C412yes—
81C42×C53C22×C4112yes—
82C42×C53C22×C416yes—
83C42×C53C22×C4112yes—
84C42×C53C2412yes—
85C42×C53C2416yes—
86C2×#88322—212no—
87C2×#88322—248no—
88C2×#88322#4118yesdirect product factor
89C2×#88322C24112yes—
90C2×#88322C2414yes—
91C23×C53—26no—
92C23×C53—26no—
93C23×C53C22×C8112yes—
94C23×C53C23×C416yes—
95C23×C53C23×C411yescharacteristic
96Q8×C53C22×C8116yescomplemented
97D4×C53C22×C8148yescomplemented
98C2×C4×C53—26no—
99C2×C4×C53—26no—
100C2×C4×C53—212no—
101C2×C4×C53—224no—
102C2×C4×C53—212no—
103C2×C4×C53C22×C814yes—
104C2×C4×C53C22×C8112yes—
105C2×C4×C53C22×C8148yescomplemented
106C2×C4×C53#13114yes—
107C2×C4×C53C23×C416yes—
108C2×C4×C53C23×C416yes—
109C2×C4×C53C23×C411yescharacteristic
110C2×C4×C53C23×C412yes—
111C2×C4×C53C23×C416yes—
112C2×C4×C53C23×C412yes—
113C2×C4×C53C2×#4114yes—
114C2×C4×C53C2511yescharacteristic
115#88322—212no—
116#88322C2×#4118yesdirect product factor
117C2×C8×#41—531nomaximal, Sylow
118C22×C53—26no—
119C22×C53—224no—
120C22×C53—26no—
121C22×C53C4×#4112yes—
122C22×C53C23×C8112yes—
123C22×C53C23×C812yes—
124C22×C53C2×#13112yes—
125C22×C53C24×C411yescharacteristic
126C4×C53—26no—
127C4×C53—26no—
128C4×C53C4×#4112yes—
129C4×C53C23×C814yes—
130C4×C53C23×C8112yes—
131C4×C53C2×#13114yes—
132C4×C53C22×#4112yes—
133C2×C4×#41D5311yescharacteristic
134C8×#41—5316no—
135C2×C8×Q8—532no—
136C2×C8×D4—536no—
137C2×#431—533no—
138C22×C4×C8—533no—
139C2×C53—212no—
140C2×C53C8×#4114yescomplemented
141C2×C53C2×C4×#4111yescharacteristic
142C2×C53C24×C811yescharacteristic
143C2×C53C22×#13111yescharacteristic
144C22×#41Dic5311yescharacteristic
145C23×C8—536no—
146C4×#41Dic5318yes—
147C4×#41C2×D5318yes—
148C2×C4×Q8Dic5311yescharacteristic
149C2×C4×Q8C2×D5311yescharacteristic
150C2×C4×D4Dic5313yes—
151C2×C4×D4C2×D5313yes—
152C2×#117Dic5313yes—
153C22×C42C2×D5313yes—
154C8×Q8—5316no—
155C8×D4—5348no—
156#431—5324no—
157C2×C4⋊C8—536no—
158C2×C22⋊C8—536no—
159C2×C4×C8—536no—
160C2×C4×C8—5324no—
161C2×C4×C8—531no—
162C2×C4×C8—533no—
163C53C2×C8×#4111yescharacteristic, complemented, Sylow
164C2×#41#88322116yesdirect product factor
165C2×#41C2×Dic5318yes—
166C22×Q8C2×Dic5311yescharacteristic
167C22×D4C2×Dic5313yes—
168C23×C4C2×Dic5313yes—
169C23×C4C2×Dic5313yes—
170C23×C4C22×D5313yes—
171C22×C8—532no—
172C22×C8—536no—
173C22×C8—5324no—
174C22×C8—10624no—
175C4×Q8C2×Dic5318yes—
176C4×Q8C2×Dic5314yes—
177C4×Q8C22×D5314yes—
178C4×D4C2×Dic53124yes—
179C4×D4C2×Dic53112yes—
180C4×D4C22×D53112yes—
181#117C2×Dic53124yes—
182C2×C4⋊C4C2×Dic5313yes—
183C2×C4⋊C4C2×Dic5313yes—
184C2×C22⋊C4C2×Dic5316yes—
185C2×C42C2×Dic5313yes—
186C2×C42C2×Dic5313yes—
187C2×C42C2×Dic53112yes—
188C2×C42C22×D5313yes—
189C2×C42C22×D53112yes—
190C2×C42C22×D5311yescharacteristic
191C4⋊C8—5324no—
192C22⋊C8—5324no—
193C4×C8—534no—
194C4×C8—5312no—
195C4×C8—5324no—
196C24C22×Dic5313yes—
197#41C2×#88322164yesdirect product factor
198C2×Q8C2×#8832218yescomplemented
199C2×Q8C22×Dic5314yes—
200C2×D4C2×#88322124yescomplemented
201C2×D4C22×Dic53112yes—
202C22×C4—212no—
203C22×C4—212no—
204C22×C4C2×#88322124yescomplemented
205C22×C4C22×Dic5313yes—
206C22×C4C22×Dic5316yes—
207C22×C4C22×Dic5313yes—
208C22×C4C22×Dic53112yes—
209C22×C4C22×Dic5311yescharacteristic
210C22×C4C22×Dic5313yes—
211C22×C4C22×Dic53112yes—
212C22×C4C22×Dic5311yescharacteristic
213C22×C4C23×D5316yes—
214C22×C4C23×D5311yescharacteristic
215C2×C8—538no—
216C2×C8—534no—
217C2×C8—5312no—
218C2×C8—10648no—
219C2×C8—10612no—
220C4⋊C4C22×Dic53112yes—
221C4⋊C4C22×Dic53112yes—
222C22⋊C4C22×Dic53124yes—
223C42C22×Dic5312yes—
224C42C22×Dic53112yes—
225C42C22×Dic53112yes—
226C42C22×Dic5316yes—
227C42C23×D5312yes—
228C42C23×D5316yes—
229C23—26no—
230C23—26no—
231C23C22×#88322112yes—
232C23C23×Dic5316yes—
233C23C23×Dic5311yescharacteristic
234Q8C22×#88322116yescomplemented
235D4C22×#88322148yescomplemented
236C2×C4—26no—
237C2×C4—26no—
238C2×C4—212no—
239C2×C4—224no—
240C2×C4—212no—
241C2×C4C22×#8832214yes—
242C2×C4C22×#88322112yes—
243C2×C4C22×#88322148yescomplemented
244C2×C4#1313322414yes—
245C2×C4C23×Dic5316yes—
246C2×C4C23×Dic5316yes—
247C2×C4C23×Dic5311yescharacteristic
248C2×C4C23×Dic5312yes—
249C2×C4C23×Dic5316yes—
250C2×C4C23×Dic5312yes—
251C2×C4#41×D5314yes—
252C2×C4C24×D5311yescharacteristic
253C8—538no—
254C8—10612no—
255C22—224no—
256C22—26no—
257C22—26no—
258C22#41×Dic5312yes—
259C22C23×#88322112yes—
260C22C23×#8832212yes—
261C22C2×#1313322412yes—
262C22C24×Dic5311yescharacteristic
263C4—26no—
264C4—26no—
265C4#41×Dic5312yes—
266C4C23×#8832214yes—
267C4C23×#88322112yes—
268C4C2×#1313322414yes—
269C4C2×#41×D5312yes—
270C2—212no—
271C2#41×#8832214yesdirect product factor
272C2C2×#41×Dic5311yescharacteristic
273C2C24×#8832211yescharacteristic
274C2C22×#1313322411yescharacteristic
subgroupG/H
centerC2×C42C22×D53
commutatorC2×C53C24×C8
FrattiniC2×C4C24×D53
FittingC2×C4×#41×C53C2
socleC23×C53C23×C4
prime radicalC22×#41×C53C4
cosocleC2×C4×C53C25
hypercenterC2×C4×#41D53
hypocenterC53C2×C8×#41
LCSG ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C42 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #41×#88322C2direct (N × H)
2. C2×Q8×#88322C2semidirect
3. C2×D4×#88322C2semidirect
4. C2×#16810676C2semidirect
5. C22×C4×#88322C2semidirect
8. Q8×#88322C22semidirect
9. D4×#88322C22semidirect
10. #16810676C22semidirect
16. C2×C4×#88322C22semidirect
35. C2×#41×C53C8semidirect
59. #41×C53C2×C8semidirect
60. C2×Q8×C53C2×C8semidirect
62. C2×D4×C53C2×C8semidirect
66. C22×C4×C53C2×C8semidirect
88. C2×#88322#41direct (N × H)
96. Q8×C53C22×C8semidirect
97. D4×C53C22×C8semidirect
105. C2×C4×C53C22×C8semidirect
116. #88322C2×#41direct (N × H)
140. C2×C53C8×#41semidirect
163. C53C2×C8×#41semidirect
164. C2×#41#88322direct (N × H)
197. #41C2×#88322direct (N × H)
198. C2×Q8C2×#88322semidirect
200. C2×D4C2×#88322semidirect
204. C22×C4C2×#88322semidirect
234. Q8C22×#88322semidirect
235. D4C22×#88322semidirect
243. C2×C4C22×#88322semidirect
271. C2#41×#88322direct (N × H)

Maximal subgroups

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

subgroupindex
#41×#883222
C2×Q8×#883222
C2×D4×#883222
C2×#168106762
C22×C4×#883222
C2×C4×#41×C532
C2×C8×#4153