Finite Group Databasebeta

← C73×#19274 #32928678

Subgroups — C73×#19274 #32928678

Order 18688.

722 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#486×C73C211yesmaximal, characteristic, complemented
2C73×#3079C211yesmaximal, characteristic, complemented
3C73×#3038C211yesmaximal, characteristic
4C8×#41×C73C211yesmaximal, characteristic, complemented
5C73×#1678C214yesmaximal, complemented
6C73×#1677C214yesmaximal, complemented
7C73×#1566C211yesmaximal, characteristic, complemented
8C73×#1548C211yesmaximal, characteristic, complemented
9C73×#1545C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)×C73—21no—
11C2×C4⋊Q8×C73C2211yescharacteristic
12C2×#127×C73C2211yescharacteristic
13C4×#41×C73C2211yescharacteristic
14#486×C73—21no—
15#486×C73—21no—
16#486×C73C2214yes—
17#477×C73C2212yescomplemented
18#476×C73C2212yescomplemented
19#475×C73—21no—
20#474×C73—21no—
21#460×C73—21no—
22#460×C73C2212yes—
23#459×C73—21no—
24#459×C73C2212yes—
25C8×Q8×C73C2212yescomplemented
26C8×D4×C73—22no—
27C8×D4×C73C2212yescomplemented
28#431×C73—21no—
29#431×C73C2211yescharacteristic
30#428×C73—21no—
31#423×C73C2211yescharacteristic
32#423×C73C2211yescharacteristic
33#418×C73—21no—
34#417×C73—21no—
35C2×#102×C73—22no—
36C2×C4×C8×C73—21no—
37C2×C4×C8×C73C2211yescharacteristic
38#334×C73C2214yescomplemented
39#333×C73C2214yescomplemented
40#331×C73C2214yes—
41C2×#41×C73D411yescharacteristic
42C22×Q8×C73—21no—
43C22×D4×C73—21no—
44C22×D4×C73—21no—
45C22×D4×C73—21no—
46AGL(1,ℤ/8)×C73—44no—
47C2×SD16×C73—22no—
48C2×D8×C73—22no—
49C2×M16×C73—21no—
50C2×M16×C73—21no—
51C22×C8×C73—22no—
52C22×C8×C73D412yes—
53C4⋊Q8×C73—21no—
54C4⋊Q8×C73—21no—
55C4⋊Q8×C73—21no—
56C4⋊Q8×C73C2312yes—
57#127×C73—21no—
58#127×C73—21no—
59#127×C73—21no—
60#127×C73C2312yes—
61C4×Q8×C73D411yescharacteristic
62C4×Q8×C73C2311yescharacteristic
63C4×D4×C73—21no—
64C4×D4×C73—21no—
65C4×D4×C73D411yescharacteristic
66C4×D4×C73C2311yescharacteristic
67#117×C73—21no—
68#117×C73D411yescharacteristic
69C2×C4⋊C4×C73—21no—
70C2×C4⋊C4×C73—21no—
71C2×C42×C73—21no—
72C2×C42×C73C2311yescharacteristic
73#107×C73—22no—
74#106×C73—22no—
75C4⋊C8×C73—22no—
76C4⋊C8×C73—21no—
77C4⋊C8×C73—21no—
78C4⋊C8×C73C2312yes—
79C4⋊C8×C73C2312yes—
80C4⋊C8×C73C2312yes—
81#103×C73—22no—
82#102×C73—22no—
83#102×C73—24no—
84#102×C73—24no—
85#102×C73—22no—
86#102×C73—22no—
87C22⋊C8×C73—21no—
88C22⋊C8×C73—22no—
89C4×C8×C73—22no—
90C4×C8×C73—21no—
91C4×C8×C73D411yescharacteristic
92C4×C8×C73D411yescharacteristic
93C4×C8×C73C2312yes—
94C24×C73—41no—
95#41×C73—22no—
96#41×C73SD1614yes—
97C2×Q8×C73—22no—
98C2×Q8×C73—41no—
99C2×Q8×C73—82no—
100C2×Q8×C73C2×D411yescharacteristic
101C2×D4×C73—21no—
102C2×D4×C73—22no—
103C2×D4×C73—22no—
104C2×D4×C73—22no—
105C2×D4×C73—44no—
106C2×D4×C73—41no—
107C2×D4×C73—41no—
108C2×D4×C73—41no—
109C2×D4×C73—42no—
110C2×D4×C73—42no—
111C2×D4×C73—82no—
112C2×D4×C73C2×D411yescharacteristic
113C22×C4×C73—21no—
114C22×C4×C73—21no—
115C22×C4×C73—21no—
116C22×C4×C73—41no—
117C22×C4×C73C2×D411yescharacteristic
118C22×C4×C73C2×D411yescharacteristic
119C22×C4×C73#4111yescharacteristic
120SD16×C73—44no—
121D8×C73—44no—
122M16×C73—42no—
123M16×C73—42no—
124C2×C8×C73—22no—
125C2×C8×C73—22no—
126C2×C8×C73—22no—
127C2×C8×C73—42no—
128C2×C8×C73—44no—
129C2×C8×C73C2×D412yes—
130C2×C8×C73C2×D412yes—
131C4⋊C4×C73—21no—
132C4⋊C4×C73—21no—
133C4⋊C4×C73—22no—
134C4⋊C4×C73—22no—
135C4⋊C4×C73—41no—
136C4⋊C4×C73—41no—
137C4⋊C4×C73C2×D411yescharacteristic
138C4⋊C4×C73C2×D411yescharacteristic
139C22⋊C4×C73—21no—
140C22⋊C4×C73—21no—
141C22⋊C4×C73—21no—
142C42×C73—21no—
143C42×C73—21no—
144C42×C73—21no—
145C42×C73C2×D411yescharacteristic
146C42×C73C2×D411yescharacteristic
147C42×C73#4111yescharacteristic
148C42×C73C2411yescharacteristic
149C23×C73—21no—
150C23×C73—42no—
151C23×C73—82no—
152C23×C73—82no—
153C23×C73—82no—
154C23×C73#12111yescharacteristic
155Q8×C73—21no—
156Q8×C73—82no—
157Q8×C73—82no—
158Q8×C73C2×SD1612yes—
159D4×C73—24no—
160D4×C73—21no—
161D4×C73—42no—
162D4×C73—42no—
163D4×C73—44no—
164D4×C73—82no—
165D4×C73—82no—
166D4×C73—82no—
167D4×C73—82no—
168D4×C73C2×SD1612yes—
169C2×C4×C73—21no—
170C2×C4×C73—22no—
171C2×C4×C73—24no—
172C2×C4×C73—22no—
173C2×C4×C73—21no—
174C2×C4×C73—21no—
175C2×C4×C73—42no—
176C2×C4×C73—42no—
177C2×C4×C73—42no—
178C2×C4×C73—82no—
179C2×C4×C73#12111yescharacteristic
180C2×C4×C73#12111yescharacteristic
181C2×C4×C73#12111yescharacteristic
182C2×C4×C73C2×SD1612yes—
183C2×C4×C73#13512yes—
184C2×C4×C73C22×D411yescharacteristic
185C2×C4×C73C22×D411yescharacteristic
186C2×C4×C73C2×#4111yescharacteristic
187C8×C73—22no—
188C8×C73—22no—
189C8×C73—42no—
190C8×C73—42no—
191C8×C73—42no—
192C22×C73—21no—
193C22×C73—21no—
194C22×C73—22no—
195C22×C73—41no—
196C22×C73—41no—
197C22×C73—82no—
198C22×C73—82no—
199C22×C73—82no—
200C22×C73—164no—
201C22×C73#46512yes—
202C22×C73C2×#12111yescharacteristic
203C4×C73—21no—
204C4×C73—22no—
205C4×C73—21no—
206C4×C73—41no—
207C4×C73—41no—
208C4×C73—82no—
209C4×C73#46512yes—
210C4×C73#47212yes—
211C4×C73C22×SD1611yescharacteristic
212C4×C73C2×#13511yescharacteristic
213C4×C73#57512yes—
214#19274C7311yesmaximal, characteristic, direct product factor, Sylow
215C2×C73—22no—
216C2×C73—42no—
217C2×C73—162no—
218C2×C73#304011yescharacteristic
219C2×C73C2×#46511yescharacteristic
220C2×C73C2×#47211yescharacteristic
221C2×#486C2×C7311yescharacteristic, complemented
222#3079C2×C7311yescharacteristic, complemented
223#3038C2×C7311yescharacteristic
224C8×#41C2×C7311yescharacteristic, complemented
225#1678C2×C7314yescomplemented
226#1677C2×C7314yescomplemented
227#1566C2×C7311yescharacteristic, complemented
228#1548C2×C7311yescharacteristic, complemented
229#1545C2×C7311yescharacteristic, complemented
230C73#1927411yescharacteristic, direct product factor, Sylow
231C2×AGL(1,ℤ/8)—21no—
232C2×C4⋊Q8C22×C7311yescharacteristic
233C2×#127C22×C7311yescharacteristic
234C4×#41C22×C7311yescharacteristic
235#486—21no—
236#486—21no—
237#486C22×C7314yes—
238#477C22×C7312yescomplemented
239#476C22×C7312yescomplemented
240#475—21no—
241#474—21no—
242#460—21no—
243#460C22×C7312yes—
244#459—21no—
245#459C22×C7312yes—
246C8×Q8C22×C7312yescomplemented
247C8×D4—22no—
248C8×D4C22×C7312yescomplemented
249#431—21no—
250#431C22×C7311yescharacteristic
251#428—21no—
252#423C22×C7311yescharacteristic
253#423C22×C7311yescharacteristic
254#418—21no—
255#417—21no—
256C2×#102—22no—
257C2×C4×C8—21no—
258C2×C4×C8C22×C7311yescharacteristic
259#334C22×C7314yescomplemented
260#333C22×C7314yescomplemented
261#331C22×C7314yes—
262C2×#41D4×C7311yescharacteristic
263C22×Q8—21no—
264C22×D4—21no—
265C22×D4—21no—
266C22×D4—21no—
267AGL(1,ℤ/8)—44no—
268C2×SD16—22no—
269C2×D8—22no—
270C2×M16—21no—
271C2×M16—21no—
272C22×C8—22no—
273C22×C8D4×C7312yes—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8C23×C7312yes—
278#127—21no—
279#127—21no—
280#127—21no—
281#127C23×C7312yes—
282C4×Q8D4×C7311yescharacteristic
283C4×Q8C23×C7311yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4D4×C7311yescharacteristic
287C4×D4C23×C7311yescharacteristic
288#117—21no—
289#117D4×C7311yescharacteristic
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C42—21no—
293C2×C42C23×C7311yescharacteristic
294#107—22no—
295#106—22no—
296C4⋊C8—22no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8C23×C7312yes—
300C4⋊C8C23×C7312yes—
301C4⋊C8C23×C7312yes—
302#103—22no—
303#102—22no—
304#102—24no—
305#102—24no—
306#102—22no—
307#102—22no—
308C22⋊C8—21no—
309C22⋊C8—22no—
310C4×C8—22no—
311C4×C8—21no—
312C4×C8D4×C7311yescharacteristic
313C4×C8D4×C7311yescharacteristic
314C4×C8C23×C7312yes—
315C24—41no—
316#41—22no—
317#41SD16×C7314yes—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—82no—
321C2×Q8C2×D4×C7311yescharacteristic
322C2×D4—21no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—44no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—82no—
333C2×D4C2×D4×C7311yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4C2×D4×C7311yescharacteristic
339C22×C4C2×D4×C7311yescharacteristic
340C22×C4#41×C7311yescharacteristic
341SD16—44no—
342D8—44no—
343M16—42no—
344M16—42no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—42no—
349C2×C8—44no—
350C2×C8C2×D4×C7312yes—
351C2×C8C2×D4×C7312yes—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4C2×D4×C7311yescharacteristic
359C4⋊C4C2×D4×C7311yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42C2×D4×C7311yescharacteristic
367C42C2×D4×C7311yescharacteristic
368C42#41×C7311yescharacteristic
369C42C24×C7311yescharacteristic
370C23—21no—
371C23—42no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23#121×C7311yescharacteristic
376Q8—21no—
377Q8—82no—
378Q8—82no—
379Q8C2×SD16×C7312yes—
380D4—24no—
381D4—21no—
382D4—42no—
383D4—42no—
384D4—44no—
385D4—82no—
386D4—82no—
387D4—82no—
388D4—82no—
389D4C2×SD16×C7312yes—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4#121×C7311yescharacteristic
401C2×C4#121×C7311yescharacteristic
402C2×C4#121×C7311yescharacteristic
403C2×C4C2×SD16×C7312yes—
404C2×C4#135×C7312yes—
405C2×C4C22×D4×C7311yescharacteristic
406C2×C4C22×D4×C7311yescharacteristic
407C2×C4C2×#41×C7311yescharacteristic
408C8—22no—
409C8—22no—
410C8—42no—
411C8—42no—
412C8—42no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—164no—
422C22#465×C7312yes—
423C22C2×#121×C7311yescharacteristic
424C4—21no—
425C4—22no—
426C4—21no—
427C4—41no—
428C4—41no—
429C4—82no—
430C4#465×C7312yes—
431C4#472×C7312yes—
432C4C22×SD16×C7311yescharacteristic
433C4C2×#135×C7311yescharacteristic
434C4#575×C7312yes—
435C2—22no—
436C2—42no—
437C2—162no—
438C2C73×#304011yescharacteristic
439C2C2×#465×C7311yescharacteristic
440C2C2×#472×C7311yescharacteristic
subgroupG/H
center, socleC22×C73C2×#121
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#3038C2
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
1. C2×#486×C73C2semidirect
2. C73×#3079C2semidirect
4. C8×#41×C73C2semidirect
5. C73×#1678C2semidirect
6. C73×#1677C2semidirect
7. C73×#1566C2semidirect
8. C73×#1548C2semidirect
9. C73×#1545C2semidirect
17. #477×C73C22semidirect
18. #476×C73C22semidirect
25. C8×Q8×C73C22semidirect
27. C8×D4×C73C22semidirect
38. #334×C73C22semidirect
39. #333×C73C22semidirect
214. #19274C73direct (N × H)
221. C2×#486C2×C73semidirect
222. #3079C2×C73semidirect
224. C8×#41C2×C73semidirect
225. #1678C2×C73semidirect
226. #1677C2×C73semidirect
227. #1566C2×C73semidirect
228. #1548C2×C73semidirect
229. #1545C2×C73semidirect
230. C73#19274direct (N × H)
238. #477C22×C73semidirect
239. #476C22×C73semidirect
246. C8×Q8C22×C73semidirect
248. C8×D4C22×C73semidirect
259. #334C22×C73semidirect
260. #333C22×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#486×C732
C73×#30792
C73×#30382
C8×#41×C732
C73×#16782
C73×#16772
C73×#15662
C73×#15482
C73×#15452
#1927473