Finite Group Databasebeta

← C73×#60143 #32969547

Subgroups — C73×#60143 #32969547

Order 18688.

2706 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#568×C73C212yesmaximal, complemented
2C73×#3449C211yesmaximal, characteristic
3C73×#2933C2116yesmaximal, complemented
4C73×#2932C214yesmaximal, complemented
5C73×#2931C214yesmaximal, complemented
6C73×#2905C211yesmaximal, characteristic, complemented
7C2×#408×C73C212yesmaximal, complemented
8C2×#405×C73C211yesmaximal, characteristic, complemented
9C22×#41×C73C2212yes—
10#568×C73—216no—
11C2×#131×C73C2218yes—
12C2×#131×C73C2218yescomplemented
13C22×M16×C73C2212yes—
14C22×M16×C73C2211yescharacteristic
15C22×M16×C73C2218yescomplemented
16C23×C8×C73C2211yescharacteristic
17#537×C73C414yes—
18#536×C73C414yes—
19#535×C73C414yes—
20#534×C73C414yes—
21#532×C73C2214yes—
22#525×C73C2218yes—
23C2×#117×C73C2211yescharacteristic
24#436×C73C22116yescomplemented
25#435×C73C22132yescomplemented
26C8×D4×C73C22116yescomplemented
27#424×C73C2218yescomplemented
28#423×C73C2214yescomplemented
29#408×C73C2218yes—
30#408×C73C2214yescomplemented
31#408×C73C2214yescomplemented
32C2×C22⋊C8×C73C2218yescomplemented
33#405×C73C2218yescomplemented
34C2×#97×C73C2212yescomplemented
35C2×C4×C8×C73C2212yescomplemented
36C2×#41×C73—28no—
37C2×#41×C73—28no—
38C2×#41×C73C2×C414yes—
39C2×#41×C73C2×C414yes—
40C2×#41×C73C2314yes—
41C2×#41×C73C2314yes—
42C22×Q8×C73C2×C412yes—
43C22×D4×C73C2×C414yes—
44C22×D4×C73C2×C412yes—
45C23×C4×C73C2314yes—
46C23×C4×C73C2311yescharacteristic
47#131×C73—232no—
48#131×C73—232no—
49C2×M16×C73—28no—
50C2×M16×C73—232no—
51C2×M16×C73D418yescomplemented
52C2×M16×C73C2314yes—
53C2×M16×C73C2314yes—
54C2×M16×C73C23116yes—
55C2×M16×C73C2314yes—
56C22×C8×C73—24no—
57C22×C8×C73C2311yescharacteristic
58C22×C8×C73C2311yescharacteristic
59C22×C8×C73C2314yes—
60C22×C8×C73C2314yes—
61C22×C8×C73C23116yes—
62C22×C8×C73C2312yes—
63C4⋊Q8×C73C2×C414yes—
64#127×C73C2×C414yes—
65#124×C73C2×C418yes—
66#123×C73C2×C4116yes—
67C22⋊Q8×C73C2×C4116yes—
68#121×C73C2×C4116yes—
69#121×C73C2×C4116yes—
70#120×C73C2×C4116yes—
71C4×D4×C73C23116yes—
72#117×C73C2×C414yes—
73#117×C73C2314yes—
74C2×C4⋊C4×C73C2×C412yes—
75C2×C22⋊C4×C73C2×C412yes—
76C2×C42×C73C2312yes—
77C4⋊C8×C73C23116yes—
78C22⋊C8×C73C23116yes—
79C22⋊C8×C73C23116yes—
80#97×C73C2318yes—
81C4×C8×C73C2318yes—
82C24×C73C22×C414yes—
83C24×C73C22×C411yescharacteristic
84#41×C73—216no—
85#41×C73—216no—
86#41×C73—28no—
87#41×C73—28no—
88#41×C73—28no—
89#41×C73—28no—
90C2×Q8×C73—28no—
91C2×Q8×C73C22×C414yes—
92C2×Q8×C73C22×C414yes—
93C2×D4×C73—216no—
94C2×D4×C73—28no—
95C2×D4×C73C22×C418yes—
96C2×D4×C73C22×C418yes—
97C2×D4×C73C22×C414yes—
98C2×D4×C73C22×C414yes—
99C2×D4×C73C22×C4116yes—
100C22×C4×C73—28no—
101C22×C4×C73—28no—
102C22×C4×C73C22×C418yes—
103C22×C4×C73C22×C414yes—
104C22×C4×C73C22×C414yes—
105C22×C4×C73C22×C411yescharacteristic
106C22×C4×C73C22×C412yes—
107C22×C4×C73C2×D414yes—
108C22×C4×C73#4114yes—
109C22×C4×C73C2418yes—
110C22×C4×C73C2411yescharacteristic
111C22×C4×C73C2412yes—
112M16×C73—28no—
113M16×C73—28no—
114M16×C73—232no—
115M16×C73C2×D4116yes—
116C2×C8×C73—28no—
117C2×C8×C73—28no—
118C2×C8×C73—22no—
119C2×C8×C73—22no—
120C2×C8×C73—232no—
121C2×C8×C73C2×D418yes—
122C2×C8×C73C2412yes—
123C2×C8×C73C2412yes—
124C2×C8×C73C2418yes—
125C2×C8×C73C2414yes—
126C4⋊C4×C73C22×C418yes—
127C4⋊C4×C73C22×C4116yes—
128C22⋊C4×C73C22×C418yes—
129C22⋊C4×C73C22×C4132yes—
130C42×C73C22×C414yes—
131C42×C73C2414yes—
132C23×C73—28no—
133C23×C73—42no—
134C23×C73—48no—
135C23×C73C4×D414yes—
136C23×C73C23×C418yes—
137C23×C73C23×C412yes—
138C23×C73C23×C411yescharacteristic
139Q8×C73—28no—
140Q8×C73—28no—
141D4×C73—216no—
142D4×C73—216no—
143D4×C73—28no—
144D4×C73—28no—
145D4×C73—416no—
146C2×C4×C73—28no—
147C2×C4×C73—216no—
148C2×C4×C73—28no—
149C2×C4×C73—28no—
150C2×C4×C73—44no—
151C2×C4×C73C4×D418yes—
152C2×C4×C73C4×D414yes—
153C2×C4×C73C23×C418yes—
154C2×C4×C73C23×C418yes—
155C2×C4×C73C23×C411yescharacteristic
156C2×C4×C73C23×C412yes—
157C2×C4×C73C22×D414yes—
158C2×C4×C73C22×D412yes—
159C2×C4×C73C2×#4114yes—
160C2×C4×C73C2×#4112yes—
161C2×C4×C73C2511yescharacteristic
162C8×C73—28no—
163C8×C73—22no—
164C8×C73—22no—
165C8×C73C22×D418yes—
166C22×C73—22no—
167C22×C73—24no—
168C22×C73—28no—
169C22×C73—216no—
170C22×C73—44no—
171C22×C73—88no—
172C22×C73C2×C4×D414yes—
173C22×C73C2×C4×D412yes—
174C22×C73C24×C411yescharacteristic
175C4×C73—28no—
176C4×C73—28no—
177C4×C73C2×C4×D414yes—
178C4×C73C2×C4×D412yes—
179C4×C73C23×D411yescharacteristic
180C4×C73C22×#4111yescharacteristic
181#60143C7311yesmaximal, characteristic, direct product factor, Sylow
182C2×C73—24no—
183C2×C73—22no—
184C2×C73—48no—
185C2×C73C22×C4×D411yescharacteristic
186C2×C73#357312yes—
187C2×#568C2×C7312yescomplemented
188#3449C2×C7311yescharacteristic
189#2933C2×C73116yescomplemented
190#2932C2×C7314yescomplemented
191#2931C2×C7314yescomplemented
192#2905C2×C7311yescharacteristic, complemented
193C2×#408C2×C7312yescomplemented
194C2×#405C2×C7311yescharacteristic, complemented
195C73#6014311yescharacteristic, direct product factor, Sylow
196C22×#41C22×C7312yes—
197#568—216no—
198C2×#131C22×C7318yes—
199C2×#131C22×C7318yescomplemented
200C22×M16C22×C7312yes—
201C22×M16C22×C7311yescharacteristic
202C22×M16C22×C7318yescomplemented
203C23×C8C22×C7311yescharacteristic
204#537C4×C7314yes—
205#536C4×C7314yes—
206#535C4×C7314yes—
207#534C4×C7314yes—
208#532C22×C7314yes—
209#525C22×C7318yes—
210C2×#117C22×C7311yescharacteristic
211#436C22×C73116yescomplemented
212#435C22×C73132yescomplemented
213C8×D4C22×C73116yescomplemented
214#424C22×C7318yescomplemented
215#423C22×C7314yescomplemented
216#408C22×C7318yes—
217#408C22×C7314yescomplemented
218#408C22×C7314yescomplemented
219C2×C22⋊C8C22×C7318yescomplemented
220#405C22×C7318yescomplemented
221C2×#97C22×C7312yescomplemented
222C2×C4×C8C22×C7312yescomplemented
223C2×#41—28no—
224C2×#41—28no—
225C2×#41C2×C4×C7314yes—
226C2×#41C2×C4×C7314yes—
227C2×#41C23×C7314yes—
228C2×#41C23×C7314yes—
229C22×Q8C2×C4×C7312yes—
230C22×D4C2×C4×C7314yes—
231C22×D4C2×C4×C7312yes—
232C23×C4C23×C7314yes—
233C23×C4C23×C7311yescharacteristic
234#131—232no—
235#131—232no—
236C2×M16—28no—
237C2×M16—232no—
238C2×M16D4×C7318yescomplemented
239C2×M16C23×C7314yes—
240C2×M16C23×C7314yes—
241C2×M16C23×C73116yes—
242C2×M16C23×C7314yes—
243C22×C8—24no—
244C22×C8C23×C7311yescharacteristic
245C22×C8C23×C7311yescharacteristic
246C22×C8C23×C7314yes—
247C22×C8C23×C7314yes—
248C22×C8C23×C73116yes—
249C22×C8C23×C7312yes—
250C4⋊Q8C2×C4×C7314yes—
251#127C2×C4×C7314yes—
252#124C2×C4×C7318yes—
253#123C2×C4×C73116yes—
254C22⋊Q8C2×C4×C73116yes—
255#121C2×C4×C73116yes—
256#121C2×C4×C73116yes—
257#120C2×C4×C73116yes—
258C4×D4C23×C73116yes—
259#117C2×C4×C7314yes—
260#117C23×C7314yes—
261C2×C4⋊C4C2×C4×C7312yes—
262C2×C22⋊C4C2×C4×C7312yes—
263C2×C42C23×C7312yes—
264C4⋊C8C23×C73116yes—
265C22⋊C8C23×C73116yes—
266C22⋊C8C23×C73116yes—
267#97C23×C7318yes—
268C4×C8C23×C7318yes—
269C24C22×C4×C7314yes—
270C24C22×C4×C7311yescharacteristic
271#41—216no—
272#41—216no—
273#41—28no—
274#41—28no—
275#41—28no—
276#41—28no—
277C2×Q8—28no—
278C2×Q8C22×C4×C7314yes—
279C2×Q8C22×C4×C7314yes—
280C2×D4—216no—
281C2×D4—28no—
282C2×D4C22×C4×C7318yes—
283C2×D4C22×C4×C7318yes—
284C2×D4C22×C4×C7314yes—
285C2×D4C22×C4×C7314yes—
286C2×D4C22×C4×C73116yes—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4C22×C4×C7318yes—
290C22×C4C22×C4×C7314yes—
291C22×C4C22×C4×C7314yes—
292C22×C4C22×C4×C7311yescharacteristic
293C22×C4C22×C4×C7312yes—
294C22×C4C2×D4×C7314yes—
295C22×C4#41×C7314yes—
296C22×C4C24×C7318yes—
297C22×C4C24×C7311yescharacteristic
298C22×C4C24×C7312yes—
299M16—28no—
300M16—28no—
301M16—232no—
302M16C2×D4×C73116yes—
303C2×C8—28no—
304C2×C8—28no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—232no—
308C2×C8C2×D4×C7318yes—
309C2×C8C24×C7312yes—
310C2×C8C24×C7312yes—
311C2×C8C24×C7318yes—
312C2×C8C24×C7314yes—
313C4⋊C4C22×C4×C7318yes—
314C4⋊C4C22×C4×C73116yes—
315C22⋊C4C22×C4×C7318yes—
316C22⋊C4C22×C4×C73132yes—
317C42C22×C4×C7314yes—
318C42C24×C7314yes—
319C23—28no—
320C23—42no—
321C23—48no—
322C23C4×D4×C7314yes—
323C23C23×C4×C7318yes—
324C23C23×C4×C7312yes—
325C23C23×C4×C7311yescharacteristic
326Q8—28no—
327Q8—28no—
328D4—216no—
329D4—216no—
330D4—28no—
331D4—28no—
332D4—416no—
333C2×C4—28no—
334C2×C4—216no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—44no—
338C2×C4C4×D4×C7318yes—
339C2×C4C4×D4×C7314yes—
340C2×C4C23×C4×C7318yes—
341C2×C4C23×C4×C7318yes—
342C2×C4C23×C4×C7311yescharacteristic
343C2×C4C23×C4×C7312yes—
344C2×C4C22×D4×C7314yes—
345C2×C4C22×D4×C7312yes—
346C2×C4C2×#41×C7314yes—
347C2×C4C2×#41×C7312yes—
348C2×C4C25×C7311yescharacteristic
349C8—28no—
350C8—22no—
351C8—22no—
352C8C22×D4×C7318yes—
353C22—22no—
354C22—24no—
355C22—216no—
356C22—28no—
357C22—44no—
358C22—88no—
359C22C2×C4×D4×C7314yes—
360C22C2×C4×D4×C7312yes—
361C22C24×C4×C7311yescharacteristic
362C4—28no—
363C4—28no—
364C4C2×C4×D4×C7314yes—
365C4C2×C4×D4×C7312yes—
366C4C23×D4×C7311yescharacteristic
367C4C22×#41×C7311yescharacteristic
368C2—24no—
369C2—22no—
370C2—48no—
371C2C22×C4×D4×C7311yescharacteristic
372C2C73×#357312yes—
subgroupG/H
centerC2×C4×C73C25
commutatorC22C24×C4×C73
Frattini, cosocleC2×C4C25×C73
socleC22×C73C24×C4
prime radicalC73×#3449C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#568×C73C2semidirect
3. C73×#2933C2semidirect
4. C73×#2932C2semidirect
5. C73×#2931C2semidirect
6. C73×#2905C2semidirect
7. C2×#408×C73C2semidirect
8. C2×#405×C73C2semidirect
12. C2×#131×C73C22semidirect
15. C22×M16×C73C22semidirect
24. #436×C73C22semidirect
25. #435×C73C22semidirect
26. C8×D4×C73C22semidirect
27. #424×C73C22semidirect
28. #423×C73C22semidirect
30. #408×C73C22semidirect
31. #408×C73C22semidirect
32. C2×C22⋊C8×C73C22semidirect
33. #405×C73C22semidirect
34. C2×#97×C73C22semidirect
35. C2×C4×C8×C73C22semidirect
51. C2×M16×C73D4semidirect
181. #60143C73direct (N × H)
187. C2×#568C2×C73semidirect
189. #2933C2×C73semidirect
190. #2932C2×C73semidirect
191. #2931C2×C73semidirect
192. #2905C2×C73semidirect
193. C2×#408C2×C73semidirect
194. C2×#405C2×C73semidirect
195. C73#60143direct (N × H)
199. C2×#131C22×C73semidirect
202. C22×M16C22×C73semidirect
211. #436C22×C73semidirect
212. #435C22×C73semidirect
213. C8×D4C22×C73semidirect
214. #424C22×C73semidirect
215. #423C22×C73semidirect
217. #408C22×C73semidirect
218. #408C22×C73semidirect
219. C2×C22⋊C8C22×C73semidirect
220. #405C22×C73semidirect
221. C2×#97C22×C73semidirect
222. C2×C4×C8C22×C73semidirect
238. C2×M16D4×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#568×C732
C73×#34492
C73×#29332
C73×#29322
C73×#29312
C73×#29052
C2×#408×C732
C2×#405×C732
#6014373