Finite Group Databasebeta

← C7×#62817 #13240033

Subgroups — C7×#62817 #13240033

Order 1792.

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
1C7×#3532C2124yesmaximal, complemented
2C7×#3490C2118yesmaximal, complemented
3C7×#3484C2118yesmaximal, complemented
4C22×C7×#120C213yesmaximal, complemented
5C26×C7C2211yescharacteristic, complemented
6C2×C7×H(5,𝔽2)C2216yescomplemented
7C23×C7×D4C2219yescomplemented
8C7×H(3,𝔽4)C2218yescomplemented
9C7×#560C22124yescomplemented
10C7×#551C22172yescomplemented
11C7×#546C221144yescomplemented
12C7×D42C22172yescomplemented
13C7×#543C2216yescomplemented
14C7×#538C22172yescomplemented
15C7×#535C22136yescomplemented
16C7×#534C22118yescomplemented
17C7×#534C22136yescomplemented
18C7×#525C22136yescomplemented
19C2×C7×#120C22136yescomplemented
20C2×C7×#120C22136yescomplemented
21C2×C7×#120C22112yescomplemented
22C7×#518C22118yescomplemented
23C22×C7×C22⋊C4C2219yescomplemented
24#62817C711yesmaximal, characteristic, direct product factor, Sylow
25C25×C7—412no—
26C25×C7C2319yes—
27C25×C7C2316yes—
28C25×C7C2313yescomplemented
29C7×H(5,𝔽2)—424no—
30C2×C7×#41C23136yescomplemented
31C22×C7×D4—436no—
32C22×C7×D4C23118yes—
33C22×C7×D4C23118yescomplemented
34C22×C7×D4C23172yescomplemented
35C22×C7×D4C23136yescomplemented
36C23×C4×C7C2319yescomplemented
37C7×#127C23124yescomplemented
38C7×#126C23148yescomplemented
39C7×#124C23172yescomplemented
40C7×#123C23172yescomplemented
41C7×#123C23172yescomplemented
42C7×C22⋊Q8C23172yescomplemented
43C7×#121C231144yescomplemented
44C7×#121C23172yescomplemented
45C7×#120C23172yes—
46C7×#120C23124yescomplemented
47C7×#120C23172yescomplemented
48C7×#120C23124yescomplemented
49C7×#120C23172yescomplemented
50C7×#120C23124yescomplemented
51C4×C7×D4C231144yescomplemented
52C7×#117C23136yescomplemented
53C2×C7×C22⋊C4C23118yes—
54C2×C7×C22⋊C4C23118yescomplemented
55C2×C7×C22⋊C4C23172yescomplemented
56C2×C7×C22⋊C4C23136yescomplemented
57#3532C2×C7124yescomplemented
58#3490C2×C7118yescomplemented
59#3484C2×C7118yescomplemented
60C22×#120C2×C713yescomplemented
61C24×C7—236no—
62C24×C7—436no—
63C24×C7—436no—
64C24×C7—418no—
65C24×C7—464no—
66C24×C7C2413yes—
67C24×C7C24118yes—
68C24×C7C2419yes—
69C24×C7C24118yes—
70C24×C7C2412yes—
71C24×C7C2413yes—
72C24×C7C2414yescomplemented
73C7×#41—472no—
74C2×C7×Q8C24112yescomplemented
75C2×C7×D4—272no—
76C2×C7×D4—436no—
77C2×C7×D4—4144no—
78C2×C7×D4—472no—
79C2×C7×D4C24172yes—
80C2×C7×D4C24172yes—
81C2×C7×D4C24136yescomplemented
82C22×C4×C7—418no—
83C22×C4×C7C24118yes—
84C22×C4×C7C24136yes—
85C22×C4×C7C24136yescomplemented
86C7×C4⋊C4C24172yescomplemented
87C7×C22⋊C4C24172yes—
88C7×C22⋊C4C24172yes—
89C7×C22⋊C4C24172yescomplemented
90C42×C7C24124yescomplemented
91C26C22×C711yescharacteristic, complemented
92C2×H(5,𝔽2)C22×C716yescomplemented
93C23×D4C22×C719yescomplemented
94H(3,𝔽4)C22×C718yescomplemented
95#560C22×C7124yescomplemented
96#551C22×C7172yescomplemented
97#546C22×C71144yescomplemented
98D42C22×C7172yescomplemented
99#543C22×C716yescomplemented
100#538C22×C7172yescomplemented
101#535C22×C7136yescomplemented
102#534C22×C7118yescomplemented
103#534C22×C7136yescomplemented
104#525C22×C7136yescomplemented
105C2×#120C22×C7136yescomplemented
106C2×#120C22×C7136yescomplemented
107C2×#120C22×C7112yescomplemented
108#518C22×C7118yescomplemented
109C22×C22⋊C4C22×C719yescomplemented
110C23×C7—218no—
111C23×C7—272no—
112C23×C7—218no—
113C23×C7—236no—
114C23×C7—46no—
115C23×C7—418no—
116C23×C7—418no—
117C23×C7—46no—
118C23×C7—436no—
119C23×C7—412no—
120C23×C7—4144no—
121C23×C7—496no—
122C23×C7—824no—
123C23×C7H(5,𝔽2)112yescomplemented
124C23×C7C2516yes—
125C23×C7C25112yes—
126C23×C7C2519yes—
127C7×Q8—412no—
128C7×D4—2144no—
129C7×D4—472no—
130C7×D4—436no—
131C2×C4×C7—236no—
132C2×C4×C7—436no—
133C2×C4×C7—436no—
134C2×C4×C7C25136yes—
135C25—412no—
136C25C23×C719yes—
137C25C23×C716yes—
138C25C23×C713yescomplemented
139H(5,𝔽2)—424no—
140C2×#41C23×C7136yescomplemented
141C22×D4—436no—
142C22×D4C23×C7118yes—
143C22×D4C23×C7118yescomplemented
144C22×D4C23×C7172yescomplemented
145C22×D4C23×C7136yescomplemented
146C23×C4C23×C719yescomplemented
147#127C23×C7124yescomplemented
148#126C23×C7148yescomplemented
149#124C23×C7172yescomplemented
150#123C23×C7172yescomplemented
151#123C23×C7172yescomplemented
152C22⋊Q8C23×C7172yescomplemented
153#121C23×C71144yescomplemented
154#121C23×C7172yescomplemented
155#120C23×C7172yes—
156#120C23×C7124yescomplemented
157#120C23×C7172yescomplemented
158#120C23×C7124yescomplemented
159#120C23×C7172yescomplemented
160#120C23×C7124yescomplemented
161C4×D4C23×C71144yescomplemented
162#117C23×C7136yescomplemented
163C2×C22⋊C4C23×C7118yes—
164C2×C22⋊C4C23×C7118yescomplemented
165C2×C22⋊C4C23×C7172yescomplemented
166C2×C22⋊C4C23×C7136yescomplemented
167C22×C7—224no—
168C22×C7—218no—
169C22×C7—236no—
170C22×C7—218no—
171C22×C7—412no—
172C22×C7—472no—
173C22×C7—436no—
174C22×C7—48no—
175C22×C7—836no—
176C22×C7—164no—
177C22×C7C2×H(5,𝔽2)118yes—
178C22×C7C2611yescharacteristic
179C4×C7—236no—
180C24—236no—
181C24—436no—
182C24—436no—
183C24—418no—
184C24—464no—
185C24C24×C713yes—
186C24C24×C7118yes—
187C24C24×C719yes—
188C24C24×C7118yes—
189C24C24×C712yes—
190C24C24×C713yes—
191C24C24×C714yescomplemented
192#41—472no—
193C2×Q8C24×C7112yescomplemented
194C2×D4—272no—
195C2×D4—436no—
196C2×D4—4144no—
197C2×D4—472no—
198C2×D4C24×C7172yes—
199C2×D4C24×C7172yes—
200C2×D4C24×C7136yescomplemented
201C22×C4—418no—
202C22×C4C24×C7118yes—
203C22×C4C24×C7136yes—
204C22×C4C24×C7136yescomplemented
205C4⋊C4C24×C7172yescomplemented
206C22⋊C4C24×C7172yes—
207C22⋊C4C24×C7172yes—
208C22⋊C4C24×C7172yescomplemented
209C42C24×C7124yescomplemented
210C2×C7—218no—
211C2×C7—46no—
212C2×C7—412no—
213C2×C7C22×H(5,𝔽2)13yes—
214C23—218no—
215C23—272no—
216C23—218no—
217C23—236no—
218C23—46no—
219C23—418no—
220C23—418no—
221C23—46no—
222C23—436no—
223C23—412no—
224C23—4144no—
225C23—496no—
226C23—824no—
227C23C7×H(5,𝔽2)112yescomplemented
228C23C25×C716yes—
229C23C25×C7112yes—
230C23C25×C719yes—
231Q8—412no—
232D4—2144no—
233D4—472no—
234D4—436no—
235C2×C4—236no—
236C2×C4—436no—
237C2×C4—436no—
238C2×C4C25×C7136yes—
239C7#6281711yescharacteristic, direct product factor, Sylow
240C22—224no—
241C22—218no—
242C22—236no—
243C22—218no—
244C22—412no—
245C22—472no—
246C22—436no—
247C22—48no—
248C22—836no—
249C22—164no—
250C22C2×C7×H(5,𝔽2)118yes—
251C22C26×C711yescharacteristic
252C4—236no—
253C2—218no—
254C2—46no—
255C2—412no—
256C2C22×C7×H(5,𝔽2)13yes—
subgroupG/H
center, socleC22×C7C26
commutator, Frattini, cosocleC22C26×C7
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3532C2semidirect
2. C7×#3490C2semidirect
3. C7×#3484C2semidirect
4. C22×C7×#120C2semidirect
5. C26×C7C22semidirect
6. C2×C7×H(5,𝔽2)C22semidirect
7. C23×C7×D4C22semidirect
8. C7×H(3,𝔽4)C22semidirect
9. C7×#560C22semidirect
10. C7×#551C22semidirect
11. C7×#546C22semidirect
12. C7×D42C22semidirect
13. C7×#543C22semidirect
14. C7×#538C22semidirect
15. C7×#535C22semidirect
16. C7×#534C22semidirect
17. C7×#534C22semidirect
18. C7×#525C22semidirect
19. C2×C7×#120C22semidirect
20. C2×C7×#120C22semidirect
21. C2×C7×#120C22semidirect
22. C7×#518C22semidirect
23. C22×C7×C22⋊C4C22semidirect
24. #62817C7direct (N × H)
28. C25×C7C23semidirect
30. C2×C7×#41C23semidirect
33. C22×C7×D4C23semidirect
34. C22×C7×D4C23semidirect
35. C22×C7×D4C23semidirect
36. C23×C4×C7C23semidirect
37. C7×#127C23semidirect
38. C7×#126C23semidirect
39. C7×#124C23semidirect
40. C7×#123C23semidirect
41. C7×#123C23semidirect
42. C7×C22⋊Q8C23semidirect
43. C7×#121C23semidirect
44. C7×#121C23semidirect
46. C7×#120C23semidirect
47. C7×#120C23semidirect
48. C7×#120C23semidirect
49. C7×#120C23semidirect
50. C7×#120C23semidirect
51. C4×C7×D4C23semidirect
52. C7×#117C23semidirect
54. C2×C7×C22⋊C4C23semidirect
55. C2×C7×C22⋊C4C23semidirect
56. C2×C7×C22⋊C4C23semidirect
57. #3532C2×C7semidirect
58. #3490C2×C7semidirect
59. #3484C2×C7semidirect
60. C22×#120C2×C7semidirect
72. C24×C7C24semidirect
74. C2×C7×Q8C24semidirect
81. C2×C7×D4C24semidirect
85. C22×C4×C7C24semidirect
86. C7×C4⋊C4C24semidirect
89. C7×C22⋊C4C24semidirect
90. C42×C7C24semidirect
91. C26C22×C7semidirect
92. C2×H(5,𝔽2)C22×C7semidirect
93. C23×D4C22×C7semidirect
94. H(3,𝔽4)C22×C7semidirect
95. #560C22×C7semidirect
96. #551C22×C7semidirect
97. #546C22×C7semidirect
98. D42C22×C7semidirect
99. #543C22×C7semidirect
100. #538C22×C7semidirect
101. #535C22×C7semidirect
102. #534C22×C7semidirect
103. #534C22×C7semidirect
104. #525C22×C7semidirect
105. C2×#120C22×C7semidirect
106. C2×#120C22×C7semidirect
107. C2×#120C22×C7semidirect
108. #518C22×C7semidirect
109. C22×C22⋊C4C22×C7semidirect
123. C23×C7H(5,𝔽2)semidirect
138. C25C23×C7semidirect
140. C2×#41C23×C7semidirect
143. C22×D4C23×C7semidirect
144. C22×D4C23×C7semidirect
145. C22×D4C23×C7semidirect
146. C23×C4C23×C7semidirect
147. #127C23×C7semidirect
148. #126C23×C7semidirect
149. #124C23×C7semidirect
150. #123C23×C7semidirect
151. #123C23×C7semidirect
152. C22⋊Q8C23×C7semidirect
153. #121C23×C7semidirect
154. #121C23×C7semidirect
156. #120C23×C7semidirect
157. #120C23×C7semidirect
158. #120C23×C7semidirect
159. #120C23×C7semidirect
160. #120C23×C7semidirect
161. C4×D4C23×C7semidirect
162. #117C23×C7semidirect
164. C2×C22⋊C4C23×C7semidirect
165. C2×C22⋊C4C23×C7semidirect
166. C2×C22⋊C4C23×C7semidirect
191. C24C24×C7semidirect
193. C2×Q8C24×C7semidirect
200. C2×D4C24×C7semidirect
204. C22×C4C24×C7semidirect
205. C4⋊C4C24×C7semidirect
208. C22⋊C4C24×C7semidirect
209. C42C24×C7semidirect
227. C23C7×H(5,𝔽2)semidirect
239. C7#62817direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35322
C7×#34902
C7×#34842
C22×C7×#1202
#628177