Finite Group Databasebeta

← C73×#19226 #32928630

Subgroups — C73×#19226 #32928630

Order 18688.

562 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3233C212yesmaximal
2C73×#3170C211yesmaximal, characteristic, complemented
3C73×#2966C211yesmaximal, characteristic
4C73×#1696C211yesmaximal, characteristic, complemented
5C73×#1695C211yesmaximal, characteristic, complemented
6C73×#1690C211yesmaximal, characteristic, complemented
7C73×#1689C211yesmaximal, characteristic, complemented
8C73×#1684C212yesmaximal, complemented
9C73×#1683C212yesmaximal, complemented
10C73×#1564C211yesmaximal, characteristic, complemented
11C73×#1554C212yesmaximal, complemented
12#533×C73C2212yes—
13#529×C73C2211yescharacteristic
14#500×C73C2211yescharacteristic
15#499×C73—22no—
16#494×C73C2211yescharacteristic
17#488×C73—22no—
18#487×C73C2212yes—
19#485×C73—22no—
20#482×C73—22no—
21#477×C73C2212yes—
22#476×C73C2212yes—
23#475×C73C2212yes—
24#474×C73C2212yes—
25#467×C73—21no—
26#467×C73—21no—
27#446×C73C2211yescharacteristic
28C8×Q8×C73C2211yescharacteristic
29C4×Q16×C73—21no—
30C4×Q16×C73—21no—
31#436×C73C2211yescharacteristic
32#435×C73—21no—
33C8×D4×C73—21no—
34C8×D4×C73C2211yescharacteristic
35#433×C73—21no—
36#432×C73C2211yescharacteristic
37#431×C73C2212yes—
38#405×C73—21no—
39C2×C4×C8×C73C2211yescharacteristic
40#335×C73C2212yes—
41#334×C73C2211yescharacteristic
42#334×C73C2211yescharacteristic
43#333×C73C2212yes—
44#333×C73C2212yes—
45#332×C73C2212yes—
46#332×C73C2212yes—
47C2×Q16×C73—21no—
48C2×Q16×C73—21no—
49C2×M16×C73—21no—
50C22×C8×C73—21no—
51C22×C8×C73D411yescharacteristic
52C22×C8×C73D411yescharacteristic
53C4⋊Q8×C73C2312yes—
54C4⋊Q8×C73C2312yes—
55#126×C73—21no—
56#125×C73—22no—
57#125×C73D411yescharacteristic
58#124×C73D411yescharacteristic
59#123×C73—21no—
60C22⋊Q8×C73—22no—
61C22⋊Q8×C73—22no—
62C22⋊Q8×C73D411yescharacteristic
63#121×C73D411yescharacteristic
64C4×Q8×C73—22no—
65C4×Q8×C73—22no—
66C4×Q8×C73C2311yescharacteristic
67C4×D4×C73—21no—
68C4×D4×C73C2311yescharacteristic
69#117×C73—21no—
70#117×C73—22no—
71C2×C42×C73C2311yescharacteristic
72#107×C73—21no—
73#107×C73—21no—
74#107×C73—22no—
75#107×C73—22no—
76#106×C73—22no—
77#106×C73—22no—
78C4⋊C8×C73—21no—
79C4⋊C8×C73—21no—
80C4⋊C8×C73C2312yes—
81C4⋊C8×C73C2312yes—
82C4⋊C8×C73C2311yescharacteristic
83C4⋊C8×C73C2311yescharacteristic
84#103×C73—22no—
85#103×C73—22no—
86#103×C73—22no—
87#103×C73—22no—
88#102×C73—22no—
89#102×C73—22no—
90C22⋊C8×C73—22no—
91C22⋊C8×C73—21no—
92C22⋊C8×C73—21no—
93C22⋊C8×C73—21no—
94#97×C73—21no—
95#97×C73—21no—
96C4×C8×C73—21no—
97C4×C8×C73—22no—
98C4×C8×C73D411yescharacteristic
99C4×C8×C73D411yescharacteristic
100C4×C8×C73C2311yescharacteristic
101C4×C8×C73C2311yescharacteristic
102C2×Q8×C73—22no—
103C2×Q8×C73—22no—
104C2×Q8×C73C2×D411yescharacteristic
105C2×D4×C73—21no—
106C2×D4×C73C2×D411yescharacteristic
107C22×C4×C73—21no—
108C22×C4×C73C2×D411yescharacteristic
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73#4111yescharacteristic
111Q16×C73—42no—
112Q16×C73—42no—
113M16×C73—42no—
114C2×C8×C73—22no—
115C2×C8×C73—21no—
116C2×C8×C73—22no—
117C2×C8×C73—22no—
118C2×C8×C73—21no—
119C2×C8×C73—41no—
120C2×C8×C73—42no—
121C2×C8×C73C2×D411yescharacteristic
122C2×C8×C73C2×D411yescharacteristic
123C2×C8×C73C2×D411yescharacteristic
124C2×C8×C73C2×D411yescharacteristic
125C4⋊C4×C73—21no—
126C4⋊C4×C73—21no—
127C4⋊C4×C73—21no—
128C4⋊C4×C73—21no—
129C4⋊C4×C73—22no—
130C4⋊C4×C73—22no—
131C4⋊C4×C73—22no—
132C4⋊C4×C73—22no—
133C4⋊C4×C73—42no—
134C4⋊C4×C73—42no—
135C4⋊C4×C73C2×D411yescharacteristic
136C4⋊C4×C73C2×D411yescharacteristic
137C22⋊C4×C73—21no—
138C22⋊C4×C73—21no—
139C22⋊C4×C73—21no—
140C22⋊C4×C73—21no—
141C22⋊C4×C73—21no—
142C22⋊C4×C73—42no—
143C42×C73—21no—
144C42×C73—42no—
145C42×C73C2×D411yescharacteristic
146C42×C73C2×D411yescharacteristic
147C42×C73#4111yescharacteristic
148C42×C73C2411yescharacteristic
149C23×C73—21no—
150C23×C73#12111yescharacteristic
151Q8×C73—22no—
152Q8×C73—42no—
153Q8×C73—42no—
154Q8×C73—82no—
155D4×C73—22no—
156D4×C73—42no—
157C2×C4×C73—21no—
158C2×C4×C73—21no—
159C2×C4×C73—22no—
160C2×C4×C73—21no—
161C2×C4×C73—21no—
162C2×C4×C73—21no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—42no—
167C2×C4×C73—42no—
168C2×C4×C73#12111yescharacteristic
169C2×C4×C73#12111yescharacteristic
170C2×C4×C73#12111yescharacteristic
171C2×C4×C73C22×D411yescharacteristic
172C2×C4×C73C22×D411yescharacteristic
173C2×C4×C73C2×#4111yescharacteristic
174C8×C73—21no—
175C8×C73—21no—
176C8×C73—21no—
177C8×C73—41no—
178C8×C73—41no—
179C8×C73—42no—
180C8×C73—42no—
181C8×C73#13512yes—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—21no—
185C22×C73—41no—
186C22×C73—41no—
187C22×C73—41no—
188C22×C73C2×#12111yescharacteristic
189C4×C73—21no—
190C4×C73—21no—
191C4×C73—21no—
192C4×C73—21no—
193C4×C73—41no—
194C4×C73—41no—
195C4×C73—41no—
196C4×C73—82no—
197C4×C73—82no—
198C4×C73C2×#13511yescharacteristic
199C4×C73#57511yescharacteristic
200C4×C73#57812yes—
201#19226C7311yesmaximal, characteristic, direct product factor, Sylow
202C2×C73—41no—
203C2×C73—81no—
204C2×C73#303511yescharacteristic
205C2×C73#305911yescharacteristic
206C2×C73#306711yescharacteristic
207#3233C2×C7312yes—
208#3170C2×C7311yescharacteristic, complemented
209#2966C2×C7311yescharacteristic
210#1696C2×C7311yescharacteristic, complemented
211#1695C2×C7311yescharacteristic, complemented
212#1690C2×C7311yescharacteristic, complemented
213#1689C2×C7311yescharacteristic, complemented
214#1684C2×C7312yescomplemented
215#1683C2×C7312yescomplemented
216#1564C2×C7311yescharacteristic, complemented
217#1554C2×C7312yescomplemented
218C73#1922611yescharacteristic, direct product factor, Sylow
219#533C22×C7312yes—
220#529C22×C7311yescharacteristic
221#500C22×C7311yescharacteristic
222#499—22no—
223#494C22×C7311yescharacteristic
224#488—22no—
225#487C22×C7312yes—
226#485—22no—
227#482—22no—
228#477C22×C7312yes—
229#476C22×C7312yes—
230#475C22×C7312yes—
231#474C22×C7312yes—
232#467—21no—
233#467—21no—
234#446C22×C7311yescharacteristic
235C8×Q8C22×C7311yescharacteristic
236C4×Q16—21no—
237C4×Q16—21no—
238#436C22×C7311yescharacteristic
239#435—21no—
240C8×D4—21no—
241C8×D4C22×C7311yescharacteristic
242#433—21no—
243#432C22×C7311yescharacteristic
244#431C22×C7312yes—
245#405—21no—
246C2×C4×C8C22×C7311yescharacteristic
247#335C22×C7312yes—
248#334C22×C7311yescharacteristic
249#334C22×C7311yescharacteristic
250#333C22×C7312yes—
251#333C22×C7312yes—
252#332C22×C7312yes—
253#332C22×C7312yes—
254C2×Q16—21no—
255C2×Q16—21no—
256C2×M16—21no—
257C22×C8—21no—
258C22×C8D4×C7311yescharacteristic
259C22×C8D4×C7311yescharacteristic
260C4⋊Q8C23×C7312yes—
261C4⋊Q8C23×C7312yes—
262#126—21no—
263#125—22no—
264#125D4×C7311yescharacteristic
265#124D4×C7311yescharacteristic
266#123—21no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8D4×C7311yescharacteristic
270#121D4×C7311yescharacteristic
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8C23×C7311yescharacteristic
274C4×D4—21no—
275C4×D4C23×C7311yescharacteristic
276#117—21no—
277#117—22no—
278C2×C42C23×C7311yescharacteristic
279#107—21no—
280#107—21no—
281#107—22no—
282#107—22no—
283#106—22no—
284#106—22no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8C23×C7311yescharacteristic
288C4⋊C8C23×C7311yescharacteristic
289C4⋊C8C23×C7312yes—
290C4⋊C8C23×C7312yes—
291#103—22no—
292#103—22no—
293#103—22no—
294#103—22no—
295#102—22no—
296#102—22no—
297C22⋊C8—22no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301#97—21no—
302#97—21no—
303C4×C8—21no—
304C4×C8—22no—
305C4×C8D4×C7311yescharacteristic
306C4×C8D4×C7311yescharacteristic
307C4×C8C23×C7311yescharacteristic
308C4×C8C23×C7311yescharacteristic
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8C2×D4×C7311yescharacteristic
312C2×D4—21no—
313C2×D4C2×D4×C7311yescharacteristic
314C22×C4—21no—
315C22×C4C2×D4×C7311yescharacteristic
316C22×C4C2×D4×C7311yescharacteristic
317C22×C4#41×C7311yescharacteristic
318Q16—42no—
319Q16—42no—
320M16—42no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8C2×D4×C7311yescharacteristic
329C2×C8C2×D4×C7311yescharacteristic
330C2×C8C2×D4×C7311yescharacteristic
331C2×C8C2×D4×C7311yescharacteristic
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4C2×D4×C7311yescharacteristic
343C4⋊C4C2×D4×C7311yescharacteristic
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—42no—
350C42—21no—
351C42—42no—
352C42C2×D4×C7311yescharacteristic
353C42C2×D4×C7311yescharacteristic
354C42#41×C7311yescharacteristic
355C42C24×C7311yescharacteristic
356C23—21no—
357C23#121×C7311yescharacteristic
358Q8—22no—
359Q8—42no—
360Q8—42no—
361Q8—82no—
362D4—22no—
363D4—42no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4#121×C7311yescharacteristic
376C2×C4#121×C7311yescharacteristic
377C2×C4#121×C7311yescharacteristic
378C2×C4C22×D4×C7311yescharacteristic
379C2×C4C22×D4×C7311yescharacteristic
380C2×C4C2×#41×C7311yescharacteristic
381C8—21no—
382C8—21no—
383C8—21no—
384C8—41no—
385C8—41no—
386C8—42no—
387C8—42no—
388C8#135×C7312yes—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22C2×#121×C7311yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—82no—
404C4—82no—
405C4C2×#135×C7311yescharacteristic
406C4#575×C7311yescharacteristic
407C4#578×C7312yes—
408C2—41no—
409C2—81no—
410C2C73×#303511yescharacteristic
411C2C73×#305911yescharacteristic
412C2C73×#306711yescharacteristic
subgroupG/H
center, socleC22×C73C2×#121
commutator, Frattini, cosocleC42C24×C73
prime radical#121×C73D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#3170C2semidirect
4. C73×#1696C2semidirect
5. C73×#1695C2semidirect
6. C73×#1690C2semidirect
7. C73×#1689C2semidirect
8. C73×#1684C2semidirect
9. C73×#1683C2semidirect
10. C73×#1564C2semidirect
11. C73×#1554C2semidirect
201. #19226C73direct (N × H)
208. #3170C2×C73semidirect
210. #1696C2×C73semidirect
211. #1695C2×C73semidirect
212. #1690C2×C73semidirect
213. #1689C2×C73semidirect
214. #1684C2×C73semidirect
215. #1683C2×C73semidirect
216. #1564C2×C73semidirect
217. #1554C2×C73semidirect
218. C73#19226direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#32332
C73×#31702
C73×#29662
C73×#16962
C73×#16952
C73×#16902
C73×#16892
C73×#16842
C73×#16832
C73×#15642
C73×#15542
#1922673