Finite Group Databasebeta

← C73×#60070 #32969474

Subgroups — C73×#60070 #32969474

Order 18688.

4892 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#585×C73C211yesmaximal, characteristic
2C73×#3428C212yesmaximal, complemented
3C73×#2884C214yesmaximal, complemented
4C73×#2883C2112yesmaximal, complemented
5C2×#410×C73C2112yesmaximal, complemented
6#585×C73—28no—
7#585×C73—28no—
8C2×2−1+4×C73C414yescomplemented
9C2×2−1+4×C73C2212yes—
10C2×H(5,𝔽2)×C73C414yescomplemented
11C2×H(5,𝔽2)×C73C2216yes—
12C22×#41×C73C418yescomplemented
13C22×#41×C73C2216yes—
14C22×#41×C73C2211yescharacteristic
15#520×C73—28no—
16#519×C73C2214yescomplemented
17#518×C73C22112yescomplemented
18C4×#41×C73—28no—
19C2×#117×C73C22112yescomplemented
20#410×C73C22116yescomplemented
21#410×C73C22148yescomplemented
22C2×#99×C73C22148yescomplemented
232−1+4×C73—232no—
242−1+4×C73—216no—
25H(5,𝔽2)×C73—232no—
26H(5,𝔽2)×C73—248no—
27C2×#41×C73—232no—
28C2×#41×C73—232no—
29C2×#41×C73—224no—
30C2×#41×C73—224no—
31C2×#41×C73C2×C414yes—
32C2×#41×C73C2×C414yes—
33C2×#41×C73C2×C4112yescomplemented
34C2×#41×C73C2×C418yescomplemented
35C2×#41×C73C2×C4124yescomplemented
36C2×#41×C73C2×C4112yescomplemented
37C2×#41×C73D418yes—
38C2×#41×C73D418yes—
39C2×#41×C73C2314yes—
40C2×#41×C73C23112yes—
41C22×Q8×C73C2×C418yescomplemented
42C22×Q8×C73C2×C416yescomplemented
43C22×Q8×C73C2311yescharacteristic
44C22×D4×C73C2×C4124yescomplemented
45C22×D4×C73C2×C416yescomplemented
46C22×D4×C73C23112yes—
47C22×D4×C73C2313yes—
48C23×C4×C73C2×C4112yescomplemented
49C23×C4×C73C2313yes—
50C4×Q8×C73—216no—
51C4×D4×C73—248no—
52#117×C73—212no—
53#117×C73—212no—
54#117×C73C2318yes—
55#117×C73C23124yes—
56C2×C4⋊C4×C73—212no—
57C2×C22⋊C4×C73C23124yes—
58C2×C42×C73—212no—
59#99×C73C23164yes—
60C24×C73C22×C4112yes—
61C24×C73C2413yes—
62#41×C73—232no—
63#41×C73—296no—
64#41×C73—224no—
65#41×C73—224no—
66#41×C73—248no—
67#41×C73—28no—
68#41×C73—28no—
69#41×C73—248no—
70#41×C73C22⋊C4116yes—
71#41×C73C22⋊C4116yes—
72#41×C73C22⋊C4132yes—
73C2×Q8×C73—232no—
74C2×Q8×C73—224no—
75C2×Q8×C73C22×C4112yes—
76C2×Q8×C73C22×C414yes—
77C2×Q8×C73C22×C414yes—
78C2×Q8×C73C2×D418yes—
79C2×D4×C73—248no—
80C2×D4×C73—296no—
81C2×D4×C73—224no—
82C2×D4×C73C22×C4112yes—
83C2×D4×C73C22×C414yes—
84C2×D4×C73C22×C4124yes—
85C2×D4×C73C22×C4112yes—
86C2×D4×C73C2×D4124yes—
87C2×D4×C73C2418yes—
88C22×C4×C73—224no—
89C22×C4×C73—224no—
90C22×C4×C73—224no—
91C22×C4×C73C22×C418yes—
92C22×C4×C73C22×C4124yes—
93C22×C4×C73C22×C4112yes—
94C22×C4×C73C22×C416yes—
95C22×C4×C73C22×C416yes—
96C22×C4×C73C2×D4112yes—
97C22×C4×C73C2×D4112yes—
98C22×C4×C73C2411yescharacteristic
99C22×C4×C73C2413yes—
100C4⋊C4×C73—28no—
101C4⋊C4×C73—224no—
102C22⋊C4×C73—224no—
103C22⋊C4×C73C24116yes—
104C42×C73—28no—
105C42×C73—224no—
106C23×C73—224no—
107C23×C73—83no—
108C23×C73—812no—
109C23×C73C23×C418yes—
110C23×C73C23×C416yes—
111C23×C73C22×D4112yes—
112C23×C73C2511yescharacteristic
113Q8×C73—224no—
114Q8×C73—28no—
115Q8×C73C2×C22⋊C4116yes—
116D4×C73—248no—
117D4×C73—216no—
118D4×C73—224no—
119D4×C73—28no—
120D4×C73C2×C22⋊C4148yes—
121C2×C4×C73—224no—
122C2×C4×C73—28no—
123C2×C4×C73—248no—
124C2×C4×C73—28no—
125C2×C4×C73—216no—
126C2×C4×C73—824no—
127C2×C4×C73C2×C22⋊C4112yes—
128C2×C4×C73C2×C22⋊C4124yes—
129C2×C4×C73C2×C22⋊C4112yes—
130C2×C4×C73C23×C418yes—
131C2×C4×C73C23×C416yes—
132C2×C4×C73C23×C411yescharacteristic
133C2×C4×C73C23×C411yescharacteristic
134C2×C4×C73C22×D4112yes—
135C2×C4×C73C22×D412yes—
136C2×C4×C73C22×D412yes—
137C22×C73—28no—
138C22×C73—412no—
139C22×C73—46no—
140C22×C73—41no—
141C22×C73—48no—
142C22×C73—824no—
143C22×C73C22×C22⋊C4112yes—
144C22×C73C24×C411yescharacteristic
145C22×C73C23×D412yes—
146C4×C73—28no—
147C4×C73—416no—
148C4×C73C22×C22⋊C411yescharacteristic
149C4×C73C22×C22⋊C4112yes—
150C4×C73C22×C22⋊C412yes—
151C4×C73C22×C22⋊C411yescharacteristic
152#60070C7311yesmaximal, characteristic, direct product factor, Sylow
153C2×C73—22no—
154C2×C73—21no—
155C2×C73—212no—
156C2×C73—48no—
157C2×C73C23×C22⋊C411yescharacteristic
158C2×#585C2×C7311yescharacteristic
159#3428C2×C7312yescomplemented
160#2884C2×C7314yescomplemented
161#2883C2×C73112yescomplemented
162C2×#410C2×C73112yescomplemented
163C73#6007011yescharacteristic, direct product factor, Sylow
164#585—28no—
165#585—28no—
166C2×2−1+4C4×C7314yescomplemented
167C2×2−1+4C22×C7312yes—
168C2×H(5,𝔽2)C4×C7314yescomplemented
169C2×H(5,𝔽2)C22×C7316yes—
170C22×#41C4×C7318yescomplemented
171C22×#41C22×C7316yes—
172C22×#41C22×C7311yescharacteristic
173#520—28no—
174#519C22×C7314yescomplemented
175#518C22×C73112yescomplemented
176C4×#41—28no—
177C2×#117C22×C73112yescomplemented
178#410C22×C73116yescomplemented
179#410C22×C73148yescomplemented
180C2×#99C22×C73148yescomplemented
1812−1+4—232no—
1822−1+4—216no—
183H(5,𝔽2)—232no—
184H(5,𝔽2)—248no—
185C2×#41—232no—
186C2×#41—232no—
187C2×#41—224no—
188C2×#41—224no—
189C2×#41C2×C4×C7314yes—
190C2×#41C2×C4×C7314yes—
191C2×#41C2×C4×C73112yescomplemented
192C2×#41C2×C4×C7318yescomplemented
193C2×#41C2×C4×C73124yescomplemented
194C2×#41C2×C4×C73112yescomplemented
195C2×#41D4×C7318yes—
196C2×#41D4×C7318yes—
197C2×#41C23×C7314yes—
198C2×#41C23×C73112yes—
199C22×Q8C2×C4×C7318yescomplemented
200C22×Q8C2×C4×C7316yescomplemented
201C22×Q8C23×C7311yescharacteristic
202C22×D4C2×C4×C73124yescomplemented
203C22×D4C2×C4×C7316yescomplemented
204C22×D4C23×C73112yes—
205C22×D4C23×C7313yes—
206C23×C4C2×C4×C73112yescomplemented
207C23×C4C23×C7313yes—
208C4×Q8—216no—
209C4×D4—248no—
210#117—212no—
211#117—212no—
212#117C23×C7318yes—
213#117C23×C73124yes—
214C2×C4⋊C4—212no—
215C2×C22⋊C4C23×C73124yes—
216C2×C42—212no—
217#99C23×C73164yes—
218C24C22×C4×C73112yes—
219C24C24×C7313yes—
220#41—232no—
221#41—296no—
222#41—224no—
223#41—224no—
224#41—248no—
225#41—28no—
226#41—28no—
227#41—248no—
228#41C22⋊C4×C73116yes—
229#41C22⋊C4×C73116yes—
230#41C22⋊C4×C73132yes—
231C2×Q8—232no—
232C2×Q8—224no—
233C2×Q8C22×C4×C73112yes—
234C2×Q8C22×C4×C7314yes—
235C2×Q8C22×C4×C7314yes—
236C2×Q8C2×D4×C7318yes—
237C2×D4—248no—
238C2×D4—296no—
239C2×D4—224no—
240C2×D4C22×C4×C73112yes—
241C2×D4C22×C4×C7314yes—
242C2×D4C22×C4×C73124yes—
243C2×D4C22×C4×C73112yes—
244C2×D4C2×D4×C73124yes—
245C2×D4C24×C7318yes—
246C22×C4—224no—
247C22×C4—224no—
248C22×C4—224no—
249C22×C4C22×C4×C7318yes—
250C22×C4C22×C4×C73124yes—
251C22×C4C22×C4×C73112yes—
252C22×C4C22×C4×C7316yes—
253C22×C4C22×C4×C7316yes—
254C22×C4C2×D4×C73112yes—
255C22×C4C2×D4×C73112yes—
256C22×C4C24×C7311yescharacteristic
257C22×C4C24×C7313yes—
258C4⋊C4—28no—
259C4⋊C4—224no—
260C22⋊C4—224no—
261C22⋊C4C24×C73116yes—
262C42—28no—
263C42—224no—
264C23—224no—
265C23—83no—
266C23—812no—
267C23C23×C4×C7318yes—
268C23C23×C4×C7316yes—
269C23C22×D4×C73112yes—
270C23C25×C7311yescharacteristic
271Q8—224no—
272Q8—28no—
273Q8C2×C22⋊C4×C73116yes—
274D4—248no—
275D4—216no—
276D4—224no—
277D4—28no—
278D4C2×C22⋊C4×C73148yes—
279C2×C4—224no—
280C2×C4—28no—
281C2×C4—28no—
282C2×C4—248no—
283C2×C4—216no—
284C2×C4—824no—
285C2×C4C2×C22⋊C4×C73112yes—
286C2×C4C2×C22⋊C4×C73112yes—
287C2×C4C2×C22⋊C4×C73124yes—
288C2×C4C23×C4×C7318yes—
289C2×C4C23×C4×C7311yescharacteristic
290C2×C4C23×C4×C7316yes—
291C2×C4C23×C4×C7311yescharacteristic
292C2×C4C22×D4×C7312yes—
293C2×C4C22×D4×C73112yes—
294C2×C4C22×D4×C7312yes—
295C22—28no—
296C22—412no—
297C22—46no—
298C22—41no—
299C22—48no—
300C22—824no—
301C22C22×C22⋊C4×C73112yes—
302C22C24×C4×C7311yescharacteristic
303C22C23×D4×C7312yes—
304C4—28no—
305C4—416no—
306C4C22×C22⋊C4×C7311yescharacteristic
307C4C22×C22⋊C4×C7311yescharacteristic
308C4C22×C22⋊C4×C73112yes—
309C4C22×C22⋊C4×C7312yes—
310C2—22no—
311C2—21no—
312C2—212no—
313C2—48no—
314C2C23×C22⋊C4×C7311yescharacteristic
subgroupG/H
centerC4×C73C22×C22⋊C4
commutatorC22C24×C4×C73
Frattini, cosocleC23C25×C73
socleC2×C73C23×C22⋊C4
prime radicalC2×#585×C73C2
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C73 ◁ C22×#41×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#3428C2semidirect
3. C73×#2884C2semidirect
4. C73×#2883C2semidirect
5. C2×#410×C73C2semidirect
8. C2×2−1+4×C73C4semidirect
10. C2×H(5,𝔽2)×C73C4semidirect
12. C22×#41×C73C4semidirect
16. #519×C73C22semidirect
17. #518×C73C22semidirect
19. C2×#117×C73C22semidirect
20. #410×C73C22semidirect
21. #410×C73C22semidirect
22. C2×#99×C73C22semidirect
33. C2×#41×C73C2×C4semidirect
34. C2×#41×C73C2×C4semidirect
35. C2×#41×C73C2×C4semidirect
36. C2×#41×C73C2×C4semidirect
41. C22×Q8×C73C2×C4semidirect
42. C22×Q8×C73C2×C4semidirect
44. C22×D4×C73C2×C4semidirect
45. C22×D4×C73C2×C4semidirect
48. C23×C4×C73C2×C4semidirect
152. #60070C73direct (N × H)
159. #3428C2×C73semidirect
160. #2884C2×C73semidirect
161. #2883C2×C73semidirect
162. C2×#410C2×C73semidirect
163. C73#60070direct (N × H)
166. C2×2−1+4C4×C73semidirect
168. C2×H(5,𝔽2)C4×C73semidirect
170. C22×#41C4×C73semidirect
174. #519C22×C73semidirect
175. #518C22×C73semidirect
177. C2×#117C22×C73semidirect
178. #410C22×C73semidirect
179. #410C22×C73semidirect
180. C2×#99C22×C73semidirect
191. C2×#41C2×C4×C73semidirect
192. C2×#41C2×C4×C73semidirect
193. C2×#41C2×C4×C73semidirect
194. C2×#41C2×C4×C73semidirect
199. C22×Q8C2×C4×C73semidirect
200. C22×Q8C2×C4×C73semidirect
202. C22×D4C2×C4×C73semidirect
203. C22×D4C2×C4×C73semidirect
206. C23×C4C2×C4×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#585×C732
C73×#34282
C73×#28842
C73×#28832
C2×#410×C732
#6007073