Finite Group Databasebeta

← C73×#60401 #32969805

Subgroups — C73×#60401 #32969805

Order 18688.

4854 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×H(7,𝔽2)C211yesmaximal, characteristic, complemented
2C73×#3485C213yesmaximal, complemented
3C73×#3028C214yesmaximal, complemented
4C73×#3026C218yesmaximal, complemented
5C73×#3025C214yesmaximal, complemented
6C2×#458×C73C218yesmaximal, complemented
7C73×#2883C213yesmaximal, complemented
8#585×C73—212no—
9#585×C73C2214yes—
10C2×H(5,𝔽2)×C73—212no—
11C2×H(5,𝔽2)×C73C2218yes—
12C2×H(5,𝔽2)×C73C2213yes—
13C22×#41×C73C2211yescharacteristic, complemented
14#547×C73—212no—
15#546×C73—212no—
16#536×C73C2216yescomplemented
17#534×C73C22112yescomplemented
18#525×C73C2216yescomplemented
19C2×#123×C73C22112yescomplemented
20#518×C73C2213yescomplemented
21#458×C73C22116yescomplemented
22#458×C73C22132yescomplemented
23B(4,𝔽2)×C73C22116yescomplemented
24#410×C73C22112yescomplemented
25C2×#99×C73C22124yescomplemented
262−1+4×C73—216no—
272−1+4×C73—212no—
28H(5,𝔽2)×C73—248no—
29H(5,𝔽2)×C73—212no—
30H(5,𝔽2)×C73—248no—
31H(5,𝔽2)×C73—224no—
32H(5,𝔽2)×C73C23116yes—
33C2×#41×C73—248no—
34C2×#41×C73—212no—
35C2×#41×C73—224no—
36C2×#41×C73—224no—
37C2×#41×C73D4112yescomplemented
38C2×#41×C73C2316yes—
39C22×Q8×C73C2311yescharacteristic, complemented
40C22×D4×C73—216no—
41C22×D4×C73—224no—
42C22×D4×C73D4112yescomplemented
43C22×D4×C73C23112yes—
44C22×D4×C73C2311yescharacteristic
45C22×D4×C73C2312yescomplemented
46C23×C4×C73C2312yescomplemented
47C23×C4×C73C2311yescharacteristic, complemented
48C4⋊Q8×C73—26no—
49#124×C73—26no—
50#123×C73—224no—
51#123×C73—212no—
52#123×C73C23112yescomplemented
53#123×C73C23124yescomplemented
54C22⋊Q8×C73—26no—
55C22⋊Q8×C73—212no—
56#121×C73—26no—
57#121×C73—224no—
58#120×C73—212no—
59#120×C73C23112yescomplemented
60C4×D4×C73—212no—
61C4×D4×C73—212no—
62#117×C73C2316yescomplemented
63C2×C4⋊C4×C73—212no—
64C2×C22⋊C4×C73—212no—
65C2×C22⋊C4×C73C23112yescomplemented
66#99×C73C23148yescomplemented
67C24×C73—28no—
68C24×C73C2×D4112yes—
69C24×C73C2411yescharacteristic
70C24×C73C2412yes—
71#41×C73—296no—
72#41×C73—248no—
73#41×C73—216no—
74#41×C73—248no—
75#41×C73—248no—
76#41×C73—224no—
77#41×C73—416no—
78C2×Q8×C73—224no—
79C2×Q8×C73—28no—
80C2×Q8×C73—26no—
81C2×Q8×C73C2×D416yes—
82C2×D4×C73—296no—
83C2×D4×C73—224no—
84C2×D4×C73—224no—
85C2×D4×C73—248no—
86C2×D4×C73—248no—
87C2×D4×C73—248no—
88C2×D4×C73—212no—
89C2×D4×C73—212no—
90C2×D4×C73C2×D4124yes—
91C2×D4×C73C2×D416yes—
92C2×D4×C73C2×D4112yes—
93C2×D4×C73C24112yes—
94C22×C4×C73—224no—
95C22×C4×C73—216no—
96C22×C4×C73—212no—
97C22×C4×C73—212no—
98C22×C4×C73—26no—
99C22×C4×C73C2×D4112yes—
100C22×C4×C73C2×D4112yes—
101C22×C4×C73C2411yescharacteristic
102C22×C4×C73C2412yes—
103C22×C4×C73C2411yescharacteristic
104C4⋊C4×C73—212no—
105C4⋊C4×C73—224no—
106C4⋊C4×C73—26no—
107C22⋊C4×C73—212no—
108C22⋊C4×C73—224no—
109C22⋊C4×C73—212no—
110C22⋊C4×C73—212no—
111C22⋊C4×C73C24112yes—
112C42×C73—26no—
113C23×C73—216no—
114C23×C73—224no—
115C23×C73—23no—
116C23×C73—82no—
117C23×C73—81no—
118C23×C73—812no—
119C23×C73—168no—
120C23×C73C22×D4112yes—
121C23×C73C22×D4112yes—
122C23×C73C2511yescharacteristic
123Q8×C73—216no—
124Q8×C73—212no—
125Q8×C73—44no—
126D4×C73—248no—
127D4×C73—248no—
128D4×C73—224no—
129D4×C73—212no—
130D4×C73—48no—
131D4×C73—812no—
132D4×C73#120116yes—
133C2×C4×C73—248no—
134C2×C4×C73—224no—
135C2×C4×C73—224no—
136C2×C4×C73—212no—
137C2×C4×C73—23no—
138C2×C4×C73—26no—
139C2×C4×C73—48no—
140C2×C4×C73—44no—
141C2×C4×C73—812no—
142C2×C4×C73C22×D4112yes—
143C2×C4×C73C22×D416yes—
144C22×C73—212no—
145C22×C73—412no—
146C22×C73—41no—
147C22×C73—412no—
148C22×C73—46no—
149C22×C73—41no—
150C22×C73—824no—
151C22×C73—816no—
152C22×C73C2×#12018yes—
153C22×C73C23×D413yes—
154C4×C73—212no—
155C4×C73—412no—
156C4×C73—41no—
157C4×C73—42no—
158C4×C73C2×#12014yes—
159#60401C7311yesmaximal, characteristic, direct product factor, Sylow
160C2×C73—23no—
161C2×C73—28no—
162C2×C73—412no—
163C2×C73—81no—
164C2×C73C22×#12011yescharacteristic
165H(7,𝔽2)C2×C7311yescharacteristic, complemented
166#3485C2×C7313yescomplemented
167#3028C2×C7314yescomplemented
168#3026C2×C7318yescomplemented
169#3025C2×C7314yescomplemented
170C2×#458C2×C7318yescomplemented
171#2883C2×C7313yescomplemented
172C73#6040111yescharacteristic, direct product factor, Sylow
173#585—212no—
174#585C22×C7314yes—
175C2×H(5,𝔽2)—212no—
176C2×H(5,𝔽2)C22×C7318yes—
177C2×H(5,𝔽2)C22×C7313yes—
178C22×#41C22×C7311yescharacteristic, complemented
179#547—212no—
180#546—212no—
181#536C22×C7316yescomplemented
182#534C22×C73112yescomplemented
183#525C22×C7316yescomplemented
184C2×#123C22×C73112yescomplemented
185#518C22×C7313yescomplemented
186#458C22×C73116yescomplemented
187#458C22×C73132yescomplemented
188B(4,𝔽2)C22×C73116yescomplemented
189#410C22×C73112yescomplemented
190C2×#99C22×C73124yescomplemented
1912−1+4—216no—
1922−1+4—212no—
193H(5,𝔽2)—248no—
194H(5,𝔽2)—212no—
195H(5,𝔽2)—248no—
196H(5,𝔽2)—224no—
197H(5,𝔽2)C23×C73116yes—
198C2×#41—248no—
199C2×#41—212no—
200C2×#41—224no—
201C2×#41—224no—
202C2×#41D4×C73112yescomplemented
203C2×#41C23×C7316yes—
204C22×Q8C23×C7311yescharacteristic, complemented
205C22×D4—216no—
206C22×D4—224no—
207C22×D4D4×C73112yescomplemented
208C22×D4C23×C73112yes—
209C22×D4C23×C7311yescharacteristic
210C22×D4C23×C7312yescomplemented
211C23×C4C23×C7312yescomplemented
212C23×C4C23×C7311yescharacteristic, complemented
213C4⋊Q8—26no—
214#124—26no—
215#123—224no—
216#123—212no—
217#123C23×C73112yescomplemented
218#123C23×C73124yescomplemented
219C22⋊Q8—26no—
220C22⋊Q8—212no—
221#121—26no—
222#121—224no—
223#120—212no—
224#120C23×C73112yescomplemented
225C4×D4—212no—
226C4×D4—212no—
227#117C23×C7316yescomplemented
228C2×C4⋊C4—212no—
229C2×C22⋊C4—212no—
230C2×C22⋊C4C23×C73112yescomplemented
231#99C23×C73148yescomplemented
232C24—28no—
233C24C2×D4×C73112yes—
234C24C24×C7311yescharacteristic
235C24C24×C7312yes—
236#41—296no—
237#41—248no—
238#41—216no—
239#41—248no—
240#41—248no—
241#41—224no—
242#41—416no—
243C2×Q8—224no—
244C2×Q8—28no—
245C2×Q8—26no—
246C2×Q8C2×D4×C7316yes—
247C2×D4—296no—
248C2×D4—224no—
249C2×D4—224no—
250C2×D4—248no—
251C2×D4—248no—
252C2×D4—248no—
253C2×D4—212no—
254C2×D4—212no—
255C2×D4C2×D4×C73124yes—
256C2×D4C2×D4×C7316yes—
257C2×D4C2×D4×C73112yes—
258C2×D4C24×C73112yes—
259C22×C4—224no—
260C22×C4—216no—
261C22×C4—212no—
262C22×C4—212no—
263C22×C4—26no—
264C22×C4C2×D4×C73112yes—
265C22×C4C2×D4×C73112yes—
266C22×C4C24×C7311yescharacteristic
267C22×C4C24×C7312yes—
268C22×C4C24×C7311yescharacteristic
269C4⋊C4—212no—
270C4⋊C4—224no—
271C4⋊C4—26no—
272C22⋊C4—212no—
273C22⋊C4—224no—
274C22⋊C4—212no—
275C22⋊C4—212no—
276C22⋊C4C24×C73112yes—
277C42—26no—
278C23—216no—
279C23—224no—
280C23—23no—
281C23—82no—
282C23—81no—
283C23—812no—
284C23—168no—
285C23C22×D4×C73112yes—
286C23C22×D4×C73112yes—
287C23C25×C7311yescharacteristic
288Q8—216no—
289Q8—212no—
290Q8—44no—
291D4—248no—
292D4—248no—
293D4—224no—
294D4—212no—
295D4—48no—
296D4—812no—
297D4#120×C73116yes—
298C2×C4—248no—
299C2×C4—224no—
300C2×C4—224no—
301C2×C4—212no—
302C2×C4—23no—
303C2×C4—26no—
304C2×C4—48no—
305C2×C4—44no—
306C2×C4—812no—
307C2×C4C22×D4×C73112yes—
308C2×C4C22×D4×C7316yes—
309C22—212no—
310C22—412no—
311C22—41no—
312C22—412no—
313C22—46no—
314C22—41no—
315C22—824no—
316C22—816no—
317C22C2×#120×C7318yes—
318C22C23×D4×C7313yes—
319C4—212no—
320C4—412no—
321C4—41no—
322C4—42no—
323C4C2×#120×C7314yes—
324C2—23no—
325C2—28no—
326C2—412no—
327C2—81no—
328C2C22×#120×C7311yescharacteristic
subgroupG/H
center, socleC2×C73C22×#120
commutator, Frattini, cosocleC23C25×C73
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C73 ◁ C22×D4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×H(7,𝔽2)C2semidirect
2. C73×#3485C2semidirect
3. C73×#3028C2semidirect
4. C73×#3026C2semidirect
5. C73×#3025C2semidirect
6. C2×#458×C73C2semidirect
7. C73×#2883C2semidirect
13. C22×#41×C73C22semidirect
16. #536×C73C22semidirect
17. #534×C73C22semidirect
18. #525×C73C22semidirect
19. C2×#123×C73C22semidirect
20. #518×C73C22semidirect
21. #458×C73C22semidirect
22. #458×C73C22semidirect
23. B(4,𝔽2)×C73C22semidirect
24. #410×C73C22semidirect
25. C2×#99×C73C22semidirect
37. C2×#41×C73D4semidirect
39. C22×Q8×C73C23semidirect
42. C22×D4×C73D4semidirect
45. C22×D4×C73C23semidirect
46. C23×C4×C73C23semidirect
47. C23×C4×C73C23semidirect
52. #123×C73C23semidirect
53. #123×C73C23semidirect
59. #120×C73C23semidirect
62. #117×C73C23semidirect
65. C2×C22⋊C4×C73C23semidirect
66. #99×C73C23semidirect
159. #60401C73direct (N × H)
165. H(7,𝔽2)C2×C73semidirect
166. #3485C2×C73semidirect
167. #3028C2×C73semidirect
168. #3026C2×C73semidirect
169. #3025C2×C73semidirect
170. C2×#458C2×C73semidirect
171. #2883C2×C73semidirect
172. C73#60401direct (N × H)
178. C22×#41C22×C73semidirect
181. #536C22×C73semidirect
182. #534C22×C73semidirect
183. #525C22×C73semidirect
184. C2×#123C22×C73semidirect
185. #518C22×C73semidirect
186. #458C22×C73semidirect
187. #458C22×C73semidirect
188. B(4,𝔽2)C22×C73semidirect
189. #410C22×C73semidirect
190. C2×#99C22×C73semidirect
202. C2×#41D4×C73semidirect
204. C22×Q8C23×C73semidirect
207. C22×D4D4×C73semidirect
210. C22×D4C23×C73semidirect
211. C23×C4C23×C73semidirect
212. C23×C4C23×C73semidirect
217. #123C23×C73semidirect
218. #123C23×C73semidirect
224. #120C23×C73semidirect
227. #117C23×C73semidirect
230. C2×C22⋊C4C23×C73semidirect
231. #99C23×C73semidirect

Maximal subgroups

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

subgroupindex
C73×H(7,𝔽2)2
C73×#34852
C73×#30282
C73×#30262
C73×#30252
C2×#458×C732
C73×#28832
#6040173