Finite Group Databasebeta

← C2×#22596605 #32203864

Subgroups — C2×#22596605 #32203864

Order 18176.

3294 conjugacy classes of subgroups; only the 627 normal subgroups are shown.

normal subgroups only

This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 627 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596605C2116yesmaximal, direct product factor
2C2×#17985935C213yesmaximal, complemented
3C2×#17985923C212yesmaximal, complemented
4C2×#17985921C216yesmaximal, complemented
5C2×#17985854C213yesmaximal
6C2×#399×C71C211yesmaximal, characteristic, complemented
7C25×D71C2211yescharacteristic
8C22×C4⋊Dic71C2213yescomplemented
9C22×Dic71⋊C4C2216yescomplemented
10C22×C22⋊Dic71C2213yes—
11C22×(C2×D71)⋊C4C2216yes—
12C22×(C2×D71)⋊C4C2216yes—
13#17985935C22124yescomplemented
14#17985923C22116yescomplemented
15#17985921C22148yescomplemented
16#17985854C22124yes—
17C2×C71⋊#95C2213yescomplemented
18C22×C4⋊C4×C71C2213yescomplemented
19C22×C22⋊C4×C71C2213yes—
20#399×C71C2218yescomplemented
21C2×#95×C71C2211yescharacteristic, complemented
22C24×D71—46no—
23C24×D71—46no—
24C24×D71Q818yes—
25C24×D71C2312yes—
26C24×D71C2314yes—
27C23×Dic71C2316yes—
28C2×C4⋊Dic71—46no—
29C2×C4⋊Dic71C23112yescomplemented
30C2×Dic71⋊C4—412no—
31C2×Dic71⋊C4C23124yescomplemented
32C2×C22⋊Dic71—46no—
33C2×C22⋊Dic71C23112yes—
34C2×(C2×D71)⋊C4—412no—
35C2×(C2×D71)⋊C4—412no—
36C2×(C2×D71)⋊C4C23124yes—
37C2×(C2×D71)⋊C4C23124yes—
38C71⋊#95C23112yescomplemented
39C25×C71C2311yescharacteristic
40C23×C4×C71C2313yes—
41C23×C4×C71C2313yes—
42C2×C4⋊C4×C71—46no—
43C2×C4⋊C4×C71C23112yescomplemented
44C2×C22⋊C4×C71—46no—
45C2×C22⋊C4×C71C23112yes—
46#95×C71C2314yescomplemented
47C23×D71—212no—
48C23×D71—212no—
49C23×D71—448no—
50C23×D71—412no—
51C23×D71—412no—
52C23×D71—416no—
53C23×D71—448no—
54C23×D71C2×Q818yes—
55C23×D71C2×Q8116yes—
56C23×D71C2414yes—
57C22×Dic71—212no—
58C22×Dic71—212no—
59C22×Dic71—212no—
60C22×Dic71C24112yes—
61C4⋊Dic71—412no—
62Dic71⋊C4—424no—
63C22⋊Dic71—412no—
64(C2×D71)⋊C4—424no—
65(C2×D71)⋊C4—424no—
66C24×C71—26no—
67C24×C71—26no—
68C24×C71C2×Q814yes—
69C24×C71C2412yes—
70C24×C71C2411yescharacteristic
71C22×C4×C71—26no—
72C22×C4×C71—26no—
73C22×C4×C71—212no—
74C22×C4×C71—212no—
75C22×C4×C71C2416yes—
76C22×C4×C71C2416yes—
77C4⋊C4×C71—412no—
78C22⋊C4×C71—412no—
79C23×C71—224no—
80C23×C71—26no—
81C23×C71—26no—
82C23×C71—44no—
83C23×C71—412no—
84C23×C71C22×Q814yes—
85C23×C71C22×Q812yes—
86C23×C71H(5,𝔽2)16yes—
87C23×C71H(5,𝔽2)16yes—
88C23×C71C2511yescharacteristic
89C22×D71—248no—
90C22×D71—212no—
91C22×D71—212no—
92C22×D71—448no—
93C22×D71—4128no—
94C22×D71—416no—
95C22×D71—448no—
96C22×D71—48no—
97C22×D71—424no—
98C22×D71C22×Q818yes—
99C2×C4×C71—26no—
100C2×C4×C71—26no—
101C2×C4×C71—212no—
102C2×C4×C71—212no—
103C2×C4×C71—412no—
104C2×C4×C71—412no—
105C2×Dic71—212no—
106C2×Dic71—212no—
107C2×Dic71—212no—
108C2×Dic71—424no—
109C22×C71—212no—
110C22×C71—416no—
111C22×C71—42no—
112C22×C71—46no—
113C22×C71#543112yes—
114C22×C71#56014yes—
115C22×C71#560112yes—
116C22×C71C23×Q811yescharacteristic
117C22×C71C2×H(5,𝔽2)13yes—
118C22×C71C2×H(5,𝔽2)13yes—
119C2×D71—224no—
120C2×D71—464no—
121C2×D71—432no—
122C2×D71—44no—
123C2×D71—412no—
124C4×C71—46no—
125C4×C71—46no—
126Dic71—412no—
127C2×#2842—711nomaximal, Sylow
128C2×C71—44no—
129C2×C71#284218yescomplemented
130C2×C71C2×#54313yes—
131C2×C71C2×#56011yescharacteristic
132C2×C71C2×#56013yes—
133D71—48no—
134#2842—7116no—
135C2×#399—713no—
136C2×#399D7111yescharacteristic, complemented
137C2×#396—712no—
138C2×#396—716no—
139C2×#379—713no—
140C71C2×#284211yescharacteristic, complemented, Sylow
141C26—711no—
142C22×C4⋊C4—716no—
143C22×C4⋊C4—713no—
144C22×C4⋊C4C2×D7113yescomplemented
145C22×C22⋊C4—713no—
146C22×C22⋊C4—716no—
147C22×C22⋊C4—716no—
148C22×C22⋊C4C2×D7113yes—
149#399—7124no—
150#399C2×D7118yescomplemented
151#396—7148no—
152#396—7116no—
153#379—7124no—
154C2×#95—713no—
155C2×#95C2×D7111yescharacteristic, complemented
156C25—2846no—
157C25—2846no—
158C25—718no—
159C25—714no—
160C25—712no—
161C25C22×D7111yescharacteristic
162C23×C4—716no—
163C23×C4C22×D7113yes—
164C23×C4C22×D7113yes—
165C2×C4⋊C4—46no—
166C2×C4⋊C4—28412no—
167C2×C4⋊C4—2846no—
168C2×C4⋊C4—7124no—
169C2×C4⋊C4—7112no—
170C2×C4⋊C4C22×D71112yescomplemented
171C2×C22⋊C4—46no—
172C2×C22⋊C4—28412no—
173C2×C22⋊C4—28412no—
174C2×C22⋊C4—2846no—
175C2×C22⋊C4—7112no—
176C2×C22⋊C4—7124no—
177C2×C22⋊C4—7124no—
178C2×C22⋊C4C22×D71112yes—
179#95—7112no—
180#95C22×D7114yescomplemented
181C24—26no—
182C24—26no—
183C24—28412no—
184C24—28412no—
185C24—28448no—
186C24—28416no—
187C24—28448no—
188C24—7116no—
189C24—718no—
190C24—714no—
191C24—14212no—
192C24—14212no—
193C24Q8×D7114yes—
194C24C23×D7112yes—
195C24C23×D7111yescharacteristic
196C22×C4—26no—
197C22×C4—26no—
198C22×C4—212no—
199C22×C4—212no—
200C22×C4—7112no—
201C22×C4—14212no—
202C22×C4—14212no—
203C22×C4—14212no—
204C22×C4C23×D7116yes—
205C22×C4C23×D7116yes—
206C4⋊C4—412no—
207C4⋊C4—28424no—
208C4⋊C4—28412no—
209C22⋊C4—412no—
210C22⋊C4—28424no—
211C22⋊C4—28424no—
212C22⋊C4—28412no—
213C23—26no—
214C23—26no—
215C23—224no—
216C23—44no—
217C23—412no—
218C23—28448no—
219C23—28416no—
220C23—28448no—
221C23—284128no—
222C23—2848no—
223C23—28424no—
224C23—718no—
225C23—14212no—
226C23—14212no—
227C23—14248no—
228C23C2×Q8×D7114yes—
229C23C2×Q8×D7112yes—
230C23#1455617316yes—
231C23#1455617416yes—
232C23C24×D7111yescharacteristic
233C2×C4—26no—
234C2×C4—26no—
235C2×C4—212no—
236C2×C4—212no—
237C2×C4—412no—
238C2×C4—412no—
239C2×C4—28424no—
240C2×C4—14212no—
241C2×C4—14212no—
242C2×C4—14212no—
243C22—212no—
244C22—42no—
245C22—46no—
246C22—416no—
247C22—28464no—
248C22—2844no—
249C22—28412no—
250C22—28432no—
251C22—14224no—
252C22#17986624112yes—
253C22#1798673614yes—
254C22#17986739112yes—
255C22C22×Q8×D7111yescharacteristic
256C22C2×#1455617313yes—
257C22C2×#1455617413yes—
258C4—46no—
259C4—46no—
260C4—28412no—
261C2—44no—
262C2—2848no—
263C2#2259660518yesdirect product factor
264C2C2×#1798662413yes—
265C2C2×#1798673611yescharacteristic
266C2C2×#1798673913yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#399×C71C2
socleC24×C71C24
prime radicalC25×D71C22
hypercenterC2×#399D71
hypocenterC71C2×#2842
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. #22596605C2direct (N × H)
2. C2×#17985935C2semidirect
3. C2×#17985923C2semidirect
4. C2×#17985921C2semidirect
6. C2×#399×C71C2semidirect
8. C22×C4⋊Dic71C22semidirect
9. C22×Dic71⋊C4C22semidirect
13. #17985935C22semidirect
14. #17985923C22semidirect
15. #17985921C22semidirect
17. C2×C71⋊#95C22semidirect
18. C22×C4⋊C4×C71C22semidirect
20. #399×C71C22semidirect
21. C2×#95×C71C22semidirect
29. C2×C4⋊Dic71C23semidirect
31. C2×Dic71⋊C4C23semidirect
38. C71⋊#95C23semidirect
43. C2×C4⋊C4×C71C23semidirect
46. #95×C71C23semidirect
129. C2×C71#2842semidirect
136. C2×#399D71semidirect
140. C71C2×#2842semidirect
144. C22×C4⋊C4C2×D71semidirect
150. #399C2×D71semidirect
155. C2×#95C2×D71semidirect
170. C2×C4⋊C4C22×D71semidirect
180. #95C22×D71semidirect
263. C2#22596605direct (N × H)

Maximal subgroups

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

subgroupindex
#225966052
C2×#179859352
C2×#179859232
C2×#179859212
C2×#179858542
C2×#399×C712
C2×#284271