Finite Group Databasebeta

← C31×#62817 #21519576

Subgroups — C31×#62817 #21519576

Order 7936.

9494 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3532C2124yesmaximal, complemented
2C31×#3490C2118yesmaximal, complemented
3C31×#3484C2118yesmaximal, complemented
4C22×C31×#120C213yesmaximal, complemented
5C26×C31C2211yescharacteristic, complemented
6C2×C31×H(5,𝔽2)C2216yescomplemented
7C23×D4×C31C2219yescomplemented
8C31×H(3,𝔽4)C2218yescomplemented
9C31×#560C22124yescomplemented
10C31×#551C22172yescomplemented
11C31×#546C221144yescomplemented
12D42×C31C22172yescomplemented
13C31×#543C2216yescomplemented
14C31×#538C22172yescomplemented
15C31×#535C22136yescomplemented
16C31×#534C22118yescomplemented
17C31×#534C22136yescomplemented
18C31×#525C22136yescomplemented
19C2×C31×#120C22136yescomplemented
20C2×C31×#120C22136yescomplemented
21C2×C31×#120C22112yescomplemented
22C31×#518C22118yescomplemented
23C22×C22⋊C4×C31C2219yescomplemented
24C25×C31—412no—
25C25×C31C2319yes—
26C25×C31C2316yes—
27C25×C31C2313yescomplemented
28C31×H(5,𝔽2)—424no—
29C2×#41×C31C23136yescomplemented
30C22×D4×C31—436no—
31C22×D4×C31C23118yes—
32C22×D4×C31C23118yescomplemented
33C22×D4×C31C23172yescomplemented
34C22×D4×C31C23136yescomplemented
35C23×C4×C31C2319yescomplemented
36C31×#127C23124yescomplemented
37C31×#126C23148yescomplemented
38C31×#124C23172yescomplemented
39C31×#123C23172yescomplemented
40C31×#123C23172yescomplemented
41C31×C22⋊Q8C23172yescomplemented
42C31×#121C231144yescomplemented
43C31×#121C23172yescomplemented
44C31×#120C23172yes—
45C31×#120C23124yescomplemented
46C31×#120C23172yescomplemented
47C31×#120C23124yescomplemented
48C31×#120C23172yescomplemented
49C31×#120C23124yescomplemented
50C4×D4×C31C231144yescomplemented
51C31×#117C23136yescomplemented
52C2×C22⋊C4×C31C23118yes—
53C2×C22⋊C4×C31C23118yescomplemented
54C2×C22⋊C4×C31C23172yescomplemented
55C2×C22⋊C4×C31C23136yescomplemented
56C24×C31—236no—
57C24×C31—436no—
58C24×C31—436no—
59C24×C31—418no—
60C24×C31—464no—
61C24×C31C2413yes—
62C24×C31C24118yes—
63C24×C31C2419yes—
64C24×C31C24118yes—
65C24×C31C2412yes—
66C24×C31C2413yes—
67C24×C31C2414yescomplemented
68#41×C31—472no—
69C2×Q8×C31C24112yescomplemented
70C2×D4×C31—272no—
71C2×D4×C31—436no—
72C2×D4×C31—4144no—
73C2×D4×C31—472no—
74C2×D4×C31C24172yes—
75C2×D4×C31C24172yes—
76C2×D4×C31C24136yescomplemented
77C22×C4×C31—418no—
78C22×C4×C31C24118yes—
79C22×C4×C31C24136yes—
80C22×C4×C31C24136yescomplemented
81C4⋊C4×C31C24172yescomplemented
82C22⋊C4×C31C24172yes—
83C22⋊C4×C31C24172yes—
84C22⋊C4×C31C24172yescomplemented
85C42×C31C24124yescomplemented
86#62817C3111yesmaximal, characteristic, direct product factor, Sylow
87C23×C31—218no—
88C23×C31—272no—
89C23×C31—218no—
90C23×C31—236no—
91C23×C31—46no—
92C23×C31—418no—
93C23×C31—418no—
94C23×C31—46no—
95C23×C31—436no—
96C23×C31—412no—
97C23×C31—4144no—
98C23×C31—496no—
99C23×C31—824no—
100C23×C31H(5,𝔽2)112yescomplemented
101C23×C31C2516yes—
102C23×C31C25112yes—
103C23×C31C2519yes—
104Q8×C31—412no—
105D4×C31—2144no—
106D4×C31—472no—
107D4×C31—436no—
108C2×C4×C31—236no—
109C2×C4×C31—436no—
110C2×C4×C31—436no—
111C2×C4×C31C25136yes—
112#3532C2×C31124yescomplemented
113#3490C2×C31118yescomplemented
114#3484C2×C31118yescomplemented
115C22×#120C2×C3113yescomplemented
116C22×C31—224no—
117C22×C31—218no—
118C22×C31—236no—
119C22×C31—218no—
120C22×C31—412no—
121C22×C31—472no—
122C22×C31—436no—
123C22×C31—48no—
124C22×C31—836no—
125C22×C31—164no—
126C22×C31C2×H(5,𝔽2)118yes—
127C22×C31C2611yescharacteristic
128C4×C31—236no—
129C26C22×C3111yescharacteristic, complemented
130C2×H(5,𝔽2)C22×C3116yescomplemented
131C23×D4C22×C3119yescomplemented
132H(3,𝔽4)C22×C3118yescomplemented
133#560C22×C31124yescomplemented
134#551C22×C31172yescomplemented
135#546C22×C311144yescomplemented
136D42C22×C31172yescomplemented
137#543C22×C3116yescomplemented
138#538C22×C31172yescomplemented
139#535C22×C31136yescomplemented
140#534C22×C31118yescomplemented
141#534C22×C31136yescomplemented
142#525C22×C31136yescomplemented
143C2×#120C22×C31136yescomplemented
144C2×#120C22×C31136yescomplemented
145C2×#120C22×C31112yescomplemented
146#518C22×C31118yescomplemented
147C22×C22⋊C4C22×C3119yescomplemented
148C2×C31—218no—
149C2×C31—46no—
150C2×C31—412no—
151C2×C31C22×H(5,𝔽2)13yes—
152C25—412no—
153C25C23×C3119yes—
154C25C23×C3116yes—
155C25C23×C3113yescomplemented
156H(5,𝔽2)—424no—
157C2×#41C23×C31136yescomplemented
158C22×D4—436no—
159C22×D4C23×C31118yes—
160C22×D4C23×C31118yescomplemented
161C22×D4C23×C31172yescomplemented
162C22×D4C23×C31136yescomplemented
163C23×C4C23×C3119yescomplemented
164#127C23×C31124yescomplemented
165#126C23×C31148yescomplemented
166#124C23×C31172yescomplemented
167#123C23×C31172yescomplemented
168#123C23×C31172yescomplemented
169C22⋊Q8C23×C31172yescomplemented
170#121C23×C311144yescomplemented
171#121C23×C31172yescomplemented
172#120C23×C31172yes—
173#120C23×C31124yescomplemented
174#120C23×C31172yescomplemented
175#120C23×C31124yescomplemented
176#120C23×C31172yescomplemented
177#120C23×C31124yescomplemented
178C4×D4C23×C311144yescomplemented
179#117C23×C31136yescomplemented
180C2×C22⋊C4C23×C31118yes—
181C2×C22⋊C4C23×C31118yescomplemented
182C2×C22⋊C4C23×C31172yescomplemented
183C2×C22⋊C4C23×C31136yescomplemented
184C31#6281711yescharacteristic, direct product factor, Sylow
185C24—236no—
186C24—436no—
187C24—436no—
188C24—418no—
189C24—464no—
190C24C24×C3113yes—
191C24C24×C31118yes—
192C24C24×C3119yes—
193C24C24×C31118yes—
194C24C24×C3112yes—
195C24C24×C3113yes—
196C24C24×C3114yescomplemented
197#41—472no—
198C2×Q8C24×C31112yescomplemented
199C2×D4—272no—
200C2×D4—436no—
201C2×D4—4144no—
202C2×D4—472no—
203C2×D4C24×C31172yes—
204C2×D4C24×C31172yes—
205C2×D4C24×C31136yescomplemented
206C22×C4—418no—
207C22×C4C24×C31118yes—
208C22×C4C24×C31136yes—
209C22×C4C24×C31136yescomplemented
210C4⋊C4C24×C31172yescomplemented
211C22⋊C4C24×C31172yes—
212C22⋊C4C24×C31172yes—
213C22⋊C4C24×C31172yescomplemented
214C42C24×C31124yescomplemented
215C23—218no—
216C23—272no—
217C23—218no—
218C23—236no—
219C23—46no—
220C23—418no—
221C23—418no—
222C23—46no—
223C23—436no—
224C23—412no—
225C23—4144no—
226C23—496no—
227C23—824no—
228C23C31×H(5,𝔽2)112yescomplemented
229C23C25×C3116yes—
230C23C25×C31112yes—
231C23C25×C3119yes—
232Q8—412no—
233D4—2144no—
234D4—472no—
235D4—436no—
236C2×C4—236no—
237C2×C4—436no—
238C2×C4—436no—
239C2×C4C25×C31136yes—
240C22—224no—
241C22—218no—
242C22—236no—
243C22—218no—
244C22—412no—
245C22—472no—
246C22—436no—
247C22—48no—
248C22—836no—
249C22—164no—
250C22C2×C31×H(5,𝔽2)118yes—
251C22C26×C3111yescharacteristic
252C4—236no—
253C2—218no—
254C2—46no—
255C2—412no—
256C2C22×C31×H(5,𝔽2)13yes—
subgroupG/H
center, socleC22×C31C26
commutator, Frattini, cosocleC22C26×C31
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3532C2semidirect
2. C31×#3490C2semidirect
3. C31×#3484C2semidirect
4. C22×C31×#120C2semidirect
5. C26×C31C22semidirect
6. C2×C31×H(5,𝔽2)C22semidirect
7. C23×D4×C31C22semidirect
8. C31×H(3,𝔽4)C22semidirect
9. C31×#560C22semidirect
10. C31×#551C22semidirect
11. C31×#546C22semidirect
12. D42×C31C22semidirect
13. C31×#543C22semidirect
14. C31×#538C22semidirect
15. C31×#535C22semidirect
16. C31×#534C22semidirect
17. C31×#534C22semidirect
18. C31×#525C22semidirect
19. C2×C31×#120C22semidirect
20. C2×C31×#120C22semidirect
21. C2×C31×#120C22semidirect
22. C31×#518C22semidirect
23. C22×C22⋊C4×C31C22semidirect
27. C25×C31C23semidirect
29. C2×#41×C31C23semidirect
32. C22×D4×C31C23semidirect
33. C22×D4×C31C23semidirect
34. C22×D4×C31C23semidirect
35. C23×C4×C31C23semidirect
36. C31×#127C23semidirect
37. C31×#126C23semidirect
38. C31×#124C23semidirect
39. C31×#123C23semidirect
40. C31×#123C23semidirect
41. C31×C22⋊Q8C23semidirect
42. C31×#121C23semidirect
43. C31×#121C23semidirect
45. C31×#120C23semidirect
46. C31×#120C23semidirect
47. C31×#120C23semidirect
48. C31×#120C23semidirect
49. C31×#120C23semidirect
50. C4×D4×C31C23semidirect
51. C31×#117C23semidirect
53. C2×C22⋊C4×C31C23semidirect
54. C2×C22⋊C4×C31C23semidirect
55. C2×C22⋊C4×C31C23semidirect
67. C24×C31C24semidirect
69. C2×Q8×C31C24semidirect
76. C2×D4×C31C24semidirect
80. C22×C4×C31C24semidirect
81. C4⋊C4×C31C24semidirect
84. C22⋊C4×C31C24semidirect
85. C42×C31C24semidirect
86. #62817C31direct (N × H)
100. C23×C31H(5,𝔽2)semidirect
112. #3532C2×C31semidirect
113. #3490C2×C31semidirect
114. #3484C2×C31semidirect
115. C22×#120C2×C31semidirect
129. C26C22×C31semidirect
130. C2×H(5,𝔽2)C22×C31semidirect
131. C23×D4C22×C31semidirect
132. H(3,𝔽4)C22×C31semidirect
133. #560C22×C31semidirect
134. #551C22×C31semidirect
135. #546C22×C31semidirect
136. D42C22×C31semidirect
137. #543C22×C31semidirect
138. #538C22×C31semidirect
139. #535C22×C31semidirect
140. #534C22×C31semidirect
141. #534C22×C31semidirect
142. #525C22×C31semidirect
143. C2×#120C22×C31semidirect
144. C2×#120C22×C31semidirect
145. C2×#120C22×C31semidirect
146. #518C22×C31semidirect
147. C22×C22⋊C4C22×C31semidirect
155. C25C23×C31semidirect
157. C2×#41C23×C31semidirect
160. C22×D4C23×C31semidirect
161. C22×D4C23×C31semidirect
162. C22×D4C23×C31semidirect
163. C23×C4C23×C31semidirect
164. #127C23×C31semidirect
165. #126C23×C31semidirect
166. #124C23×C31semidirect
167. #123C23×C31semidirect
168. #123C23×C31semidirect
169. C22⋊Q8C23×C31semidirect
170. #121C23×C31semidirect
171. #121C23×C31semidirect
173. #120C23×C31semidirect
174. #120C23×C31semidirect
175. #120C23×C31semidirect
176. #120C23×C31semidirect
177. #120C23×C31semidirect
178. C4×D4C23×C31semidirect
179. #117C23×C31semidirect
181. C2×C22⋊C4C23×C31semidirect
182. C2×C22⋊C4C23×C31semidirect
183. C2×C22⋊C4C23×C31semidirect
184. C31#62817direct (N × H)
196. C24C24×C31semidirect
198. C2×Q8C24×C31semidirect
205. C2×D4C24×C31semidirect
209. C22×C4C24×C31semidirect
210. C4⋊C4C24×C31semidirect
213. C22⋊C4C24×C31semidirect
214. C42C24×C31semidirect
228. C23C31×H(5,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C31×#35322
C31×#34902
C31×#34842
C22×C31×#1202
#6281731