Finite Group Databasebeta

← C73×#61416 #32970820

Subgroups — C73×#61416 #32970820

Order 18688.

1850 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3507C212yesmaximal, complemented
2C73×#3401C218yesmaximal, complemented
3C73×#3398C218yesmaximal, complemented
4C73×#3156C212yesmaximal, complemented
5C73×#3150C211yesmaximal, characteristic
6C73×#3046C214yesmaximal, complemented
7C73×#3037C212yesmaximal, complemented
8C73×#2961C211yesmaximal, characteristic, complemented
9C73×#2952C212yesmaximal, complemented
10C2×#441×C73C211yesmaximal, characteristic, complemented
11C2×#137×C73C2214yes—
12C2×#135×C73C2212yes—
13C2×#135×C73C2212yes—
14C22×Q16×C73C2211yescharacteristic
15#559×C73C2214yes—
16#557×C73C2214yes—
17#556×C73—22no—
18#556×C73C2218yes—
19#554×C73—22no—
20#552×C73—24no—
21#539×C73—24no—
22#536×C73C2214yes—
23#533×C73C2211yescharacteristic
24#532×C73C2212yes—
25#529×C73—24no—
26#520×C73C2214yes—
27C2×C4×Q8×C73C2212yes—
28#497×C73C2214yes—
29#497×C73C2218yes—
30#496×C73C2214yes—
31#495×C73C2214yes—
32#494×C73C2214yes—
33#464×C73C2218yes—
34#464×C73C2218yes—
35#463×C73C2218yes—
36#462×C73C2214yes—
37#462×C73C2218yes—
38#461×C73C2218yes—
39#460×C73C2214yes—
40#446×C73C2218yes—
41#442×C73C2214yes—
42#441×C73C2214yes—
43#441×C73C2218yes—
44C4×SD16×C73C2218yes—
45C2×#106×C73C2211yescharacteristic
46#424×C73C2212yes—
47#423×C73C2211yescharacteristic
48#417×C73C2214yes—
49C2×#103×C73C2212yes—
50#405×C73C2212yes—
51C2×#97×C73C2211yescharacteristic
52C2×#41×C73C2314yes—
53C22×Q8×C73C2312yes—
54#137×C73—48no—
55#135×C73—28no—
56#135×C73—44no—
57C2×Q16×C73—24no—
58C2×Q16×C73C2312yes—
59C2×Q16×C73C2314yes—
60C2×Q16×C73C2312yes—
61C2×Q16×C73C2318yes—
62C2×SD16×C73C2314yes—
63C2×SD16×C73C2318yes—
64C2×SD16×C73C2314yes—
65C2×D8×C73C2312yes—
66C2×D8×C73C2312yes—
67C2×M16×C73C2312yes—
68C22×C8×C73C2311yescharacteristic
69C22×C8×C73C2311yescharacteristic
70C4⋊Q8×C73D412yes—
71C4⋊Q8×C73C2318yes—
72C4⋊Q8×C73C2312yes—
73#127×C73C2312yes—
74#126×C73—28no—
75#126×C73—28no—
76#125×C73—24no—
77#125×C73—28no—
78#125×C73D412yes—
79#124×C73—24no—
80#124×C73C2318yes—
81#124×C73C2314yes—
82#123×C73—28no—
83#123×C73—28no—
84C22⋊Q8×C73—28no—
85C22⋊Q8×C73—24no—
86C22⋊Q8×C73—24no—
87C22⋊Q8×C73D414yes—
88#121×C73—22no—
89#121×C73—22no—
90C4×Q8×C73—24no—
91C4×Q8×C73—22no—
92C4×Q8×C73—22no—
93C4×Q8×C73C2318yes—
94C4×Q8×C73C2314yes—
95C4×Q8×C73C2314yes—
96C4×Q8×C73C2314yes—
97C4×D4×C73—24no—
98C4×D4×C73—28no—
99C4×D4×C73—22no—
100C4×D4×C73—22no—
101C4×D4×C73C2318yes—
102#117×C73—24no—
103#117×C73—28no—
104#117×C73C2312yes—
105C2×C4⋊C4×C73—22no—
106C2×C4⋊C4×C73—24no—
107C2×C4⋊C4×C73C2312yes—
108C2×C42×C73—22no—
109C2×C42×C73C2311yescharacteristic
110#106×C73C2314yes—
111C4⋊C8×C73C2314yes—
112C4⋊C8×C73C2318yes—
113#103×C73C2318yes—
114#103×C73C2318yes—
115#102×C73C2318yes—
116#97×C73C2314yes—
117#97×C73C2314yes—
118C4×C8×C73C2314yes—
119#41×C73—28no—
120#41×C73—28no—
121C2×Q8×C73—22no—
122C2×Q8×C73—24no—
123C2×Q8×C73—24no—
124C2×Q8×C73C2×D414yes—
125C2×Q8×C73C2414yes—
126C2×Q8×C73C2414yes—
127C2×D4×C73—24no—
128C2×D4×C73—22no—
129C2×D4×C73—22no—
130C2×D4×C73C2414yes—
131C22×C4×C73—22no—
132C22×C4×C73—22no—
133C22×C4×C73—24no—
134C22×C4×C73—28no—
135C22×C4×C73C2×D412yes—
136C22×C4×C73C2411yescharacteristic
137Q16×C73—28no—
138Q16×C73—24no—
139Q16×C73—42no—
140Q16×C73—48no—
141SD16×C73—28no—
142SD16×C73—44no—
143SD16×C73—48no—
144D8×C73—24no—
145D8×C73—42no—
146M16×C73—24no—
147C2×C8×C73—22no—
148C2×C8×C73#4114yes—
149C2×C8×C73C2414yes—
150C2×C8×C73C2412yes—
151C2×C8×C73C2412yes—
152C4⋊C4×C73—22no—
153C4⋊C4×C73—28no—
154C4⋊C4×C73—24no—
155C4⋊C4×C73—24no—
156C4⋊C4×C73—28no—
157C4⋊C4×C73—22no—
158C4⋊C4×C73—22no—
159C4⋊C4×C73—28no—
160C4⋊C4×C73C2×D414yes—
161C4⋊C4×C73C2×D418yes—
162C4⋊C4×C73C2×D414yes—
163C4⋊C4×C73C2414yes—
164C22⋊C4×C73—24no—
165C22⋊C4×C73—28no—
166C22⋊C4×C73—28no—
167C22⋊C4×C73—24no—
168C22⋊C4×C73—28no—
169C22⋊C4×C73C2×D414yes—
170C42×C73—24no—
171C42×C73—24no—
172C42×C73—28no—
173C42×C73C2×D411yescharacteristic
174C42×C73C2×D411yescharacteristic
175C42×C73C2414yes—
176C42×C73C2412yes—
177C23×C73—24no—
178C23×C73C22×D411yescharacteristic
179Q8×C73—24no—
180Q8×C73—24no—
181Q8×C73—24no—
182Q8×C73—24no—
183Q8×C73—44no—
184Q8×C73—44no—
185D4×C73—24no—
186D4×C73—24no—
187D4×C73—44no—
188D4×C73—48no—
189D4×C73—82no—
190C2×C4×C73—24no—
191C2×C4×C73—24no—
192C2×C4×C73—24no—
193C2×C4×C73—42no—
194C2×C4×C73—44no—
195C2×C4×C73—48no—
196C2×C4×C73—88no—
197C2×C4×C73—82no—
198C2×C4×C73C22×D412yes—
199C2×C4×C73C22×D412yes—
200C2×C4×C73C22×D418yes—
201C2×C4×C73C22×D411yescharacteristic
202C2×C4×C73C2×#4112yes—
203C2×C4×C73H(5,𝔽2)12yes—
204C2×C4×C73C2511yescharacteristic
205C8×C73—22no—
206C8×C73—24no—
207C8×C73C2×#4114yes—
208C22×C73—22no—
209C22×C73—44no—
210C22×C73—44no—
211C22×C73—44no—
212C22×C73#55012yes—
213C22×C73C23×D411yescharacteristic
214C4×C73—22no—
215C4×C73—22no—
216C4×C73—28no—
217C4×C73—44no—
218C4×C73—44no—
219C4×C73—44no—
220C4×C73#55012yes—
221C4×C73C22×#4111yescharacteristic
222C4×C73C2×H(5,𝔽2)11yescharacteristic
223#61416C7311yesmaximal, characteristic, direct product factor, Sylow
224C2×C73—22no—
225C2×C73—84no—
226C2×C73C2×#55011yescharacteristic
227C2×C73#358612yes—
228#3507C2×C7312yescomplemented
229#3401C2×C7318yescomplemented
230#3398C2×C7318yescomplemented
231#3156C2×C7312yescomplemented
232#3150C2×C7311yescharacteristic
233#3046C2×C7314yescomplemented
234#3037C2×C7312yescomplemented
235#2961C2×C7311yescharacteristic, complemented
236#2952C2×C7312yescomplemented
237C2×#441C2×C7311yescharacteristic, complemented
238C73#6141611yescharacteristic, direct product factor, Sylow
239C2×#137C22×C7314yes—
240C2×#135C22×C7312yes—
241C2×#135C22×C7312yes—
242C22×Q16C22×C7311yescharacteristic
243#559C22×C7314yes—
244#557C22×C7314yes—
245#556—22no—
246#556C22×C7318yes—
247#554—22no—
248#552—24no—
249#539—24no—
250#536C22×C7314yes—
251#533C22×C7311yescharacteristic
252#532C22×C7312yes—
253#529—24no—
254#520C22×C7314yes—
255C2×C4×Q8C22×C7312yes—
256#497C22×C7314yes—
257#497C22×C7318yes—
258#496C22×C7314yes—
259#495C22×C7314yes—
260#494C22×C7314yes—
261#464C22×C7318yes—
262#464C22×C7318yes—
263#463C22×C7318yes—
264#462C22×C7314yes—
265#462C22×C7318yes—
266#461C22×C7318yes—
267#460C22×C7314yes—
268#446C22×C7318yes—
269#442C22×C7314yes—
270#441C22×C7314yes—
271#441C22×C7318yes—
272C4×SD16C22×C7318yes—
273C2×#106C22×C7311yescharacteristic
274#424C22×C7312yes—
275#423C22×C7311yescharacteristic
276#417C22×C7314yes—
277C2×#103C22×C7312yes—
278#405C22×C7312yes—
279C2×#97C22×C7311yescharacteristic
280C2×#41C23×C7314yes—
281C22×Q8C23×C7312yes—
282#137—48no—
283#135—28no—
284#135—44no—
285C2×Q16—24no—
286C2×Q16C23×C7314yes—
287C2×Q16C23×C7312yes—
288C2×Q16C23×C7312yes—
289C2×Q16C23×C7318yes—
290C2×SD16C23×C7314yes—
291C2×SD16C23×C7314yes—
292C2×SD16C23×C7318yes—
293C2×D8C23×C7312yes—
294C2×D8C23×C7312yes—
295C2×M16C23×C7312yes—
296C22×C8C23×C7311yescharacteristic
297C22×C8C23×C7311yescharacteristic
298C4⋊Q8D4×C7312yes—
299C4⋊Q8C23×C7312yes—
300C4⋊Q8C23×C7318yes—
301#127C23×C7312yes—
302#126—28no—
303#126—28no—
304#125—24no—
305#125—28no—
306#125D4×C7312yes—
307#124—24no—
308#124C23×C7314yes—
309#124C23×C7318yes—
310#123—28no—
311#123—28no—
312C22⋊Q8—28no—
313C22⋊Q8—24no—
314C22⋊Q8—24no—
315C22⋊Q8D4×C7314yes—
316#121—22no—
317#121—22no—
318C4×Q8—24no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8C23×C7314yes—
322C4×Q8C23×C7314yes—
323C4×Q8C23×C7318yes—
324C4×Q8C23×C7314yes—
325C4×D4—24no—
326C4×D4—28no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4C23×C7318yes—
330#117—24no—
331#117—28no—
332#117C23×C7312yes—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4C23×C7312yes—
336C2×C42—22no—
337C2×C42C23×C7311yescharacteristic
338#106C23×C7314yes—
339C4⋊C8C23×C7314yes—
340C4⋊C8C23×C7318yes—
341#103C23×C7318yes—
342#103C23×C7318yes—
343#102C23×C7318yes—
344#97C23×C7314yes—
345#97C23×C7314yes—
346C4×C8C23×C7314yes—
347#41—28no—
348#41—28no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8C2×D4×C7314yes—
353C2×Q8C24×C7314yes—
354C2×Q8C24×C7314yes—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4C24×C7314yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4C2×D4×C7312yes—
364C22×C4C24×C7311yescharacteristic
365Q16—28no—
366Q16—24no—
367Q16—42no—
368Q16—48no—
369SD16—28no—
370SD16—44no—
371SD16—48no—
372D8—24no—
373D8—42no—
374M16—24no—
375C2×C8—22no—
376C2×C8#41×C7314yes—
377C2×C8C24×C7314yes—
378C2×C8C24×C7312yes—
379C2×C8C24×C7312yes—
380C4⋊C4—22no—
381C4⋊C4—28no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—28no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—28no—
388C4⋊C4C2×D4×C7314yes—
389C4⋊C4C2×D4×C7318yes—
390C4⋊C4C2×D4×C7314yes—
391C4⋊C4C24×C7314yes—
392C22⋊C4—24no—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—24no—
396C22⋊C4—28no—
397C22⋊C4C2×D4×C7314yes—
398C42—24no—
399C42—24no—
400C42—28no—
401C42C2×D4×C7311yescharacteristic
402C42C2×D4×C7311yescharacteristic
403C42C24×C7314yes—
404C42C24×C7312yes—
405C23—24no—
406C23C22×D4×C7311yescharacteristic
407Q8—24no—
408Q8—24no—
409Q8—24no—
410Q8—24no—
411Q8—44no—
412Q8—44no—
413D4—24no—
414D4—24no—
415D4—44no—
416D4—48no—
417D4—82no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—48no—
424C2×C4—88no—
425C2×C4—82no—
426C2×C4C22×D4×C7312yes—
427C2×C4C22×D4×C7312yes—
428C2×C4C22×D4×C7318yes—
429C2×C4C22×D4×C7311yescharacteristic
430C2×C4C2×#41×C7312yes—
431C2×C4H(5,𝔽2)×C7312yes—
432C2×C4C25×C7311yescharacteristic
433C8—22no—
434C8—24no—
435C8C2×#41×C7314yes—
436C22—22no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22#550×C7312yes—
441C22C23×D4×C7311yescharacteristic
442C4—22no—
443C4—22no—
444C4—28no—
445C4—44no—
446C4—44no—
447C4—44no—
448C4#550×C7312yes—
449C4C22×#41×C7311yescharacteristic
450C4C2×H(5,𝔽2)×C7311yescharacteristic
451C2—22no—
452C2—84no—
453C2C2×#550×C7311yescharacteristic
454C2C73×#358612yes—
subgroupG/H
center, socleC22×C73C23×D4
commutator, Frattini, cosocleC2×C4C25×C73
prime radicalC73×#3150C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ #533×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3507C2semidirect
2. C73×#3401C2semidirect
3. C73×#3398C2semidirect
4. C73×#3156C2semidirect
6. C73×#3046C2semidirect
7. C73×#3037C2semidirect
8. C73×#2961C2semidirect
9. C73×#2952C2semidirect
10. C2×#441×C73C2semidirect
223. #61416C73direct (N × H)
228. #3507C2×C73semidirect
229. #3401C2×C73semidirect
230. #3398C2×C73semidirect
231. #3156C2×C73semidirect
233. #3046C2×C73semidirect
234. #3037C2×C73semidirect
235. #2961C2×C73semidirect
236. #2952C2×C73semidirect
237. C2×#441C2×C73semidirect
238. C73#61416direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#35072
C73×#34012
C73×#33982
C73×#31562
C73×#31502
C73×#30462
C73×#30372
C73×#29612
C73×#29522
C2×#441×C732
#6141673