Finite Group Databasebeta

← #44537364

Subgroups — #44537364

Order 31424.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#21463327C211yesmaximal, characteristic, complemented
2C2×D4×D491C211yesmaximal, characteristic, complemented
3#30628323C211yesmaximal, characteristic, complemented
4#30628322C211yesmaximal, characteristic, complemented
5#30628318C211yesmaximal, characteristic, complemented
6#30628315C211yesmaximal, characteristic, complemented
7#30628310C211yesmaximal, characteristic, complemented
8#30628309C211yesmaximal, characteristic, complemented
9#30628305C211yesmaximal, characteristic, complemented
10#30628305C211yesmaximal, characteristic, complemented
11C4×C22⋊D491C211yesmaximal, characteristic, complemented
12(C22⋊D491)⋊C4C211yesmaximal, characteristic, complemented
13C2×(C2×D491)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D491C211yesmaximal, characteristic, complemented
15#121×C491C211yesmaximal, characteristic, complemented
16C23×D491C2211yescharacteristic
17C23×D491C2211yescharacteristic
18#21463327—24no—
19C2×Dic982C2211yescharacteristic, complemented
20C2×D1964C2211yescharacteristic
21C2×C22⋊D491C2211yescharacteristic
22C2×C22⋊D491C2211yescharacteristic
23C2×C22⋊D491C2211yescharacteristic
24C2×C22⋊D491C2211yescharacteristic, complemented
25C2×C22⋊D491C2211yescharacteristic, complemented
26D4×D491—24no—
27C22×Dic491C2211yescharacteristic, complemented
28C22×Dic491C2211yescharacteristic, complemented
29C2×C4×D491C2211yescharacteristic
30C2×C4×D491C2211yescharacteristic, complemented
31C2×C4×D491C2211yescharacteristic, complemented
32Dic491⋊C4C2211yescharacteristic, complemented
33Dic491⋊C4C2211yescharacteristic, complemented
34Dic491⋊C4C2211yescharacteristic, complemented
35C22⋊Dic491C2211yescharacteristic, complemented
36C22⋊Dic491C2211yescharacteristic, complemented
37C22⋊Dic491C2211yescharacteristic, complemented
38(C2×D491)⋊C4C2211yescharacteristic
39(C2×D491)⋊C4C2211yescharacteristic
40(C2×D491)⋊C4C2211yescharacteristic
41(C2×D491)⋊C4C2211yescharacteristic, complemented
42(C2×D491)⋊C4C2211yescharacteristic, complemented
43(C2×D491)⋊C4C2211yescharacteristic, complemented
44(C2×D491)⋊C4C2211yescharacteristic, complemented
45C4×Dic491C2211yescharacteristic, complemented
46C2×D4×C491C2211yescharacteristic
47C2×D4×C491C2211yescharacteristic, complemented
48C2×D4×C491C2211yescharacteristic, complemented
49C22×C4×C491C2211yescharacteristic, complemented
50C4⋊C4×C491C2211yescharacteristic, complemented
51C22⋊C4×C491C2211yescharacteristic, complemented
52C22⋊C4×C491C2211yescharacteristic, complemented
53C22×D491—22no—
54C22×D491—22no—
55C22×D491—22no—
56C22×D491—22no—
57C22×D491—41no—
58C22×D491—41no—
59C22×D491C2311yescharacteristic
60C22×D491C2311yescharacteristic
61C22×D491C2311yescharacteristic
62Dic982—22no—
63D1964—22no—
64C22⋊D491—22no—
65C22⋊D491—22no—
66C22⋊D491—22no—
67C22⋊D491—41no—
68C22⋊D491D414yescomplemented
69C2×Dic491—22no—
70C2×Dic491—22no—
71C2×Dic491C2311yescharacteristic
72C2×Dic491C2311yescharacteristic
73C2×Dic491C2311yescharacteristic
74C2×Dic491C2311yescharacteristic
75C2×Dic491C2311yescharacteristic
76C4×D491—22no—
77C4×D491—22no—
78C4×D491—41no—
79C23×C491C2311yescharacteristic
80C23×C491C2311yescharacteristic
81C23×C491C2311yescharacteristic
82D4×C491—22no—
83D4×C491—22no—
84D4×C491—41no—
85C2×C4×C491—41no—
86C2×C4×C491C2311yescharacteristic
87C2×C4×C491C2311yescharacteristic
88C2×C4×C491C2311yescharacteristic
89C2×C4×C491C2311yescharacteristic
90C22×C491—21no—
91C22×C491—21no—
92C22×C491—21no—
93C22×C491—21no—
94C22×C491—21no—
95C22×C491—21no—
96C22×C491—21no—
97C22×C491—21no—
98C22×C491C2×D412yes—
99C22×C491C2411yescharacteristic
100C2×D491—21no—
101C2×D491—21no—
102C2×D491—21no—
103C2×D491—21no—
104C2×D491—21no—
105C2×D491—21no—
106C2×D491—21no—
107C2×D491—44no—
108C2×D491—44no—
109C2×D491C2×D412yes—
110C2×D491#4112yescomplemented
111C4×C491—21no—
112C4×C491—21no—
113C4×C491—21no—
114C4×C491—21no—
115Dic491—21no—
116Dic491—21no—
117Dic491—21no—
118Dic491—21no—
119Dic491C2×D412yes—
120C2×C491—22no—
121C2×C491—41no—
122C2×C491—41no—
123C2×C491C22×D411yescharacteristic
124C2×C491C2×#4111yescharacteristic
125C2×C491H(5,𝔽2)11yescharacteristic
126D491—22no—
127D491—22no—
128D491—41no—
129C491#54611yescharacteristic, complemented, Sylow
130#546—4911nomaximal, Sylow
131C2×#41—4911no—
132C22×D4—4911no—
133#124—4911no—
134#123—4911no—
135#123—4911no—
136C22⋊Q8—4911no—
137#121—4911no—
138#121—4911no—
139#121D49111yescharacteristic, complemented
140#120—4911no—
141#120—4911no—
142C4×D4—4911no—
143C4×D4—4911no—
144C2×C22⋊C4—4911no—
145C2×C22⋊C4—4911no—
146C24—4911no—
147C24—4911no—
148#41—9824no—
149C2×Q8—4911no—
150C2×D4—9824no—
151C2×D4—4911no—
152C2×D4—4911no—
153C2×D4—4911no—
154C2×D4—4911no—
155C2×D4—4911no—
156C2×D4—4911no—
157C2×D4C2×D49111yescharacteristic
158C2×D4C2×D49111yescharacteristic, complemented
159C2×D4C2×D49111yescharacteristic, complemented
160C22×C4—4911no—
161C22×C4—4911no—
162C22×C4—4911no—
163C22×C4—4911no—
164C22×C4—4911no—
165C22×C4C2×D49111yescharacteristic, complemented
166C4⋊C4—4911no—
167C4⋊C4—4911no—
168C4⋊C4—4911no—
169C4⋊C4C2×D49111yescharacteristic, complemented
170C22⋊C4—4911no—
171C22⋊C4—4911no—
172C22⋊C4—4911no—
173C22⋊C4—4911no—
174C22⋊C4—4911no—
175C22⋊C4—4911no—
176C22⋊C4—4911no—
177C22⋊C4—4911no—
178C22⋊C4—4911no—
179C22⋊C4—4911no—
180C22⋊C4C2×D49111yescharacteristic, complemented
181C22⋊C4C2×D49111yescharacteristic, complemented
182C42—4911no—
183C23—19641no—
184C23—19641no—
185C23—9822no—
186C23—9822no—
187C23—9822no—
188C23—9822no—
189C23—4911no—
190C23—4911no—
191C23—4911no—
192C23C22×D49111yescharacteristic
193C23C22×D49111yescharacteristic
194C23C22×D49111yescharacteristic
195Q8—9822no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—19641no—
200D4—9822no—
201D4—9822no—
202D4—9822no—
203D4—9822no—
204D4—4914no—
205C2×C4—41no—
206C2×C4—19641no—
207C2×C4—9822no—
208C2×C4—9822no—
209C2×C4—9822no—
210C2×C4—9822no—
211C2×C4—4911no—
212C2×C4—4911no—
213C2×C4—4911no—
214C2×C4—4911no—
215C2×C4—4911no—
216C2×C4C22×D49111yescharacteristic
217C2×C4C22×D49111yescharacteristic
218C2×C4C22×D49111yescharacteristic
219C2×C4C22×D49111yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—19644no—
229C22—19644no—
230C22—9821no—
231C22—9821no—
232C22—9821no—
233C22—9821no—
234C22—9821no—
235C22—9821no—
236C22—9821no—
237C22—4912no—
238C22—4912no—
239C22D4×D49112yes—
240C22C23×D49111yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—9821no—
246C4—9821no—
247C4—9821no—
248C4—9821no—
249C4—4912no—
250C2—22no—
251C2—41no—
252C2—41no—
253C2—19641no—
254C2—9822no—
255C2—9822no—
256C2C2×D4×D49111yescharacteristic
257C2#41×D49111yescharacteristic
258C2#3062838211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D491
commutator, socle, cosocleC22×C491C24
Fitting#121×C491C2
hypercenter#121D491
hypocenterC491#546
LCSG ▷ C22×C491 ▷ C491
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#21463327C2semidirect
2. C2×D4×D491C2semidirect
3. #30628323C2semidirect
4. #30628322C2semidirect
5. #30628318C2semidirect
6. #30628315C2semidirect
7. #30628310C2semidirect
8. #30628309C2semidirect
9. #30628305C2semidirect
10. #30628305C2semidirect
11. C4×C22⋊D491C2semidirect
12. (C22⋊D491)⋊C4C2semidirect
13. C2×(C2×D491)⋊C4C2semidirect
14. C22⋊C4×D491C2semidirect
15. #121×C491C2semidirect
19. C2×Dic982C22semidirect
24. C2×C22⋊D491C22semidirect
25. C2×C22⋊D491C22semidirect
27. C22×Dic491C22semidirect
28. C22×Dic491C22semidirect
30. C2×C4×D491C22semidirect
31. C2×C4×D491C22semidirect
32. Dic491⋊C4C22semidirect
33. Dic491⋊C4C22semidirect
34. Dic491⋊C4C22semidirect
35. C22⋊Dic491C22semidirect
36. C22⋊Dic491C22semidirect
37. C22⋊Dic491C22semidirect
41. (C2×D491)⋊C4C22semidirect
42. (C2×D491)⋊C4C22semidirect
43. (C2×D491)⋊C4C22semidirect
44. (C2×D491)⋊C4C22semidirect
45. C4×Dic491C22semidirect
47. C2×D4×C491C22semidirect
48. C2×D4×C491C22semidirect
49. C22×C4×C491C22semidirect
50. C4⋊C4×C491C22semidirect
51. C22⋊C4×C491C22semidirect
52. C22⋊C4×C491C22semidirect
68. C22⋊D491D4semidirect
110. C2×D491#41semidirect
129. C491#546semidirect
139. #121D491semidirect
158. C2×D4C2×D491semidirect
159. C2×D4C2×D491semidirect
165. C22×C4C2×D491semidirect
169. C4⋊C4C2×D491semidirect
180. C22⋊C4C2×D491semidirect
181. C22⋊C4C2×D491semidirect

Maximal subgroups

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

subgroupindex
C2×#214633272
C2×D4×D4912
#306283232
#306283222
#306283182
#306283152
#306283102
#306283092
#306283052
C4×C22⋊D4912
(C22⋊D491)⋊C42
C2×(C2×D491)⋊C42
C22⋊C4×D4912
#121×C4912
#546491