Finite Group Databasebeta

← C73×#32840 #32942244

Subgroups — C73×#32840 #32942244

Order 18688.

648 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×#2862C211yesmaximal, characteristic, complemented
2C73×#2860C211yesmaximal, characteristic
3C73×#2101C212yesmaximal, complemented
4C73×#2092C212yesmaximal
5C73×#2085C212yesmaximal, complemented
6C73×#2073C212yesmaximal
7C73×#2058C211yesmaximal, characteristic, complemented
8C73×#2057C211yesmaximal, characteristic, complemented
9C73×#2023C211yesmaximal, characteristic
10C73×#2023C211yesmaximal, characteristic
11C73×#1839C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8×C73C2211yescharacteristic
13C2×C4⋊Q8×C73C2211yescharacteristic
14C2×#126×C73—21no—
15C2×#107×C73C2212yes—
16C2×#106×C73C2212yes—
17C2×#102×C73C2212yes—
18C2×#102×C73C2212yes—
19#391×C73—21no—
20#391×C73—21no—
21#391×C73—21no—
22#391×C73—21no—
23#390×C73C2211yescharacteristic
24#390×C73C2211yescharacteristic
25#390×C73C2211yescharacteristic
26#389×C73C2211yescharacteristic
27#389×C73C2211yescharacteristic
28#389×C73C2211yescharacteristic
29#388×C73—21no—
30#388×C73—21no—
31#388×C73—21no—
32#382×C73C2212yes—
33#382×C73C2212yes—
34C43×C73C2211yescharacteristic
35#340×C73C2212yes—
36#340×C73C2212yes—
37#340×C73C2212yes—
38#340×C73C2212yes—
39#336×C73C2211yescharacteristic
40#336×C73C2211yescharacteristic
41#336×C73C2212yes—
42#336×C73C2212yes—
43C22×Q8×C73—21no—
44C22×Q8×C73—21no—
45C22×D4×C73C2311yescharacteristic
46C22×C8×C73C2312yes—
47C22×C8×C73C2312yes—
48C4⋊Q8×C73—42no—
49C4⋊Q8×C73—42no—
50#126×C73—44no—
51C2×C4⋊C4×C73—21no—
52C2×C4⋊C4×C73—21no—
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73—21no—
55C2×C4⋊C4×C73—21no—
56C2×C4⋊C4×C73C2312yes—
57C2×C4⋊C4×C73C2312yes—
58C2×C4⋊C4×C73C2311yescharacteristic
59C2×C4⋊C4×C73C2311yescharacteristic
60C2×C4⋊C4×C73C2311yescharacteristic
61C2×C22⋊C4×C73—21no—
62C2×C22⋊C4×C73—21no—
63C2×C22⋊C4×C73—21no—
64C2×C22⋊C4×C73—21no—
65C2×C22⋊C4×C73—21no—
66C2×C22⋊C4×C73—21no—
67C2×C42×C73D411yescharacteristic
68C2×C42×C73D411yescharacteristic
69C2×C42×C73D411yescharacteristic
70C2×C42×C73D411yescharacteristic
71C2×C42×C73C2311yescharacteristic
72C2×C42×C73C2311yescharacteristic
73C2×C42×C73C2311yescharacteristic
74#107×C73—42no—
75#106×C73—42no—
76#102×C73—42no—
77#102×C73—42no—
78#95×C73—22no—
79#95×C73—22no—
80#95×C73—22no—
81#95×C73—22no—
82#95×C73—21no—
83#95×C73—21no—
84#95×C73—21no—
85C24×C73—21no—
86C2×Q8×C73—42no—
87C2×Q8×C73—42no—
88C2×Q8×C73—42no—
89C2×Q8×C73—42no—
90C2×Q8×C73—42no—
91C2×Q8×C73—42no—
92C2×D4×C73—22no—
93C2×D4×C73—22no—
94C2×D4×C73—21no—
95C2×D4×C73—21no—
96C22×C4×C73—22no—
97C22×C4×C73—22no—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73—21no—
101C22×C4×C73C2×D411yescharacteristic
102C22×C4×C73C2×D411yescharacteristic
103C22×C4×C73C2×D411yescharacteristic
104C22×C4×C73C2×D411yescharacteristic
105C22×C4×C73C2×D411yescharacteristic
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2411yescharacteristic
108C2×C8×C73—22no—
109C2×C8×C73—22no—
110C2×C8×C73—22no—
111C2×C8×C73—22no—
112C2×C8×C73—22no—
113C2×C8×C73—22no—
114C4⋊C4×C73—22no—
115C4⋊C4×C73—22no—
116C4⋊C4×C73—22no—
117C4⋊C4×C73—22no—
118C4⋊C4×C73—22no—
119C4⋊C4×C73—22no—
120C4⋊C4×C73—21no—
121C4⋊C4×C73—21no—
122C4⋊C4×C73—42no—
123C4⋊C4×C73—42no—
124C4⋊C4×C73—42no—
125C4⋊C4×C73—42no—
126C4⋊C4×C73—42no—
127C22⋊C4×C73—42no—
128C22⋊C4×C73—42no—
129C22⋊C4×C73—42no—
130C22⋊C4×C73—42no—
131C22⋊C4×C73—42no—
132C22⋊C4×C73—42no—
133C42×C73—22no—
134C42×C73—22no—
135C42×C73—22no—
136C42×C73—22no—
137C42×C73—21no—
138C42×C73—21no—
139C42×C73—41no—
140C42×C73—41no—
141C23×C73—41no—
142C23×C73—41no—
143C23×C73—41no—
144C23×C73—41no—
145C23×C73—41no—
146C23×C73—41no—
147C23×C73—41no—
148C23×C73C22×D411yescharacteristic
149Q8×C73—84no—
150Q8×C73—84no—
151D4×C73—42no—
152D4×C73—42no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—21no—
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—21no—
161C2×C4×C73—21no—
162C2×C4×C73—21no—
163C2×C4×C73—21no—
164C2×C4×C73—21no—
165C2×C4×C73—21no—
166C2×C4×C73—21no—
167C2×C4×C73—21no—
168C2×C4×C73—21no—
169C2×C4×C73—42no—
170C2×C4×C73—42no—
171C2×C4×C73—41no—
172C2×C4×C73—41no—
173C2×C4×C73—41no—
174C2×C4×C73—41no—
175C2×C4×C73—42no—
176C2×C4×C73—41no—
177C2×C4×C73—41no—
178C2×C4×C73—41no—
179C2×C4×C73—42no—
180C2×C4×C73—41no—
181C2×C4×C73—41no—
182C2×C4×C73—42no—
183C2×C4×C73—42no—
184C2×C4×C73#13512yes—
185C2×C4×C73AGL(1,ℤ/8)12yes—
186C2×C4×C73H(5,𝔽2)12yes—
187C2×C4×C732−1+412yes—
188C2×C4×C732−1+411yescharacteristic
189C2×C4×C732−1+411yescharacteristic
190C8×C73—42no—
191C8×C73—42no—
192C22×C73—81no—
193C22×C73—81no—
194C22×C73—81no—
195C22×C73—81no—
196C22×C73—81no—
197C22×C73—81no—
198C22×C73—81no—
199C22×C73#53511yescharacteristic
200C22×C73#53611yescharacteristic
201C22×C73#53611yescharacteristic
202C22×C73C2×#13511yescharacteristic
203C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
204C22×C73#57511yescharacteristic
205C22×C73#57511yescharacteristic
206C4×C73—22no—
207C4×C73—22no—
208C4×C73—41no—
209C4×C73—41no—
210C4×C73—41no—
211C4×C73—41no—
212C4×C73—82no—
213C4×C73—81no—
214C4×C73—82no—
215C4×C73—81no—
216C4×C73—81no—
217C4×C73#56312yes—
218C4×C73#56312yes—
219#32840C7311yesmaximal, characteristic, direct product factor, Sylow
220C2×C73—161no—
221C2×C73#282111yescharacteristic
222C2×C73#321111yescharacteristic
223C2×C73#321311yescharacteristic
224C2×C73#322811yescharacteristic
225C2×C73#323311yescharacteristic
226C2×C73#323511yescharacteristic
227C2×C73#324011yescharacteristic
228#2862C2×C7311yescharacteristic, complemented
229#2860C2×C7311yescharacteristic
230#2101C2×C7312yescomplemented
231#2092C2×C7312yes—
232#2085C2×C7312yescomplemented
233#2073C2×C7312yes—
234#2058C2×C7311yescharacteristic, complemented
235#2057C2×C7311yescharacteristic, complemented
236#2023C2×C7311yescharacteristic
237#2023C2×C7311yescharacteristic
238#1839C2×C7311yescharacteristic, complemented
239C73#3284011yescharacteristic, direct product factor, Sylow
240C2×C4⋊Q8C22×C7311yescharacteristic
241C2×C4⋊Q8C22×C7311yescharacteristic
242C2×#126—21no—
243C2×#107C22×C7312yes—
244C2×#106C22×C7312yes—
245C2×#102C22×C7312yes—
246C2×#102C22×C7312yes—
247#391—21no—
248#391—21no—
249#391—21no—
250#391—21no—
251#390C22×C7311yescharacteristic
252#390C22×C7311yescharacteristic
253#390C22×C7311yescharacteristic
254#389C22×C7311yescharacteristic
255#389C22×C7311yescharacteristic
256#389C22×C7311yescharacteristic
257#388—21no—
258#388—21no—
259#388—21no—
260#382C22×C7312yes—
261#382C22×C7312yes—
262C43C22×C7311yescharacteristic
263#340C22×C7312yes—
264#340C22×C7312yes—
265#340C22×C7312yes—
266#340C22×C7312yes—
267#336C22×C7311yescharacteristic
268#336C22×C7311yescharacteristic
269#336C22×C7312yes—
270#336C22×C7312yes—
271C22×Q8—21no—
272C22×Q8—21no—
273C22×D4C23×C7311yescharacteristic
274C22×C8C23×C7312yes—
275C22×C8C23×C7312yes—
276C4⋊Q8—42no—
277C4⋊Q8—42no—
278#126—44no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C23×C7312yes—
285C2×C4⋊C4C23×C7312yes—
286C2×C4⋊C4C23×C7311yescharacteristic
287C2×C4⋊C4C23×C7311yescharacteristic
288C2×C4⋊C4C23×C7311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C42D4×C7311yescharacteristic
296C2×C42D4×C7311yescharacteristic
297C2×C42D4×C7311yescharacteristic
298C2×C42D4×C7311yescharacteristic
299C2×C42C23×C7311yescharacteristic
300C2×C42C23×C7311yescharacteristic
301C2×C42C23×C7311yescharacteristic
302#107—42no—
303#106—42no—
304#102—42no—
305#102—42no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—21no—
311#95—21no—
312#95—21no—
313C24—21no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—21no—
323C2×D4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4C2×D4×C7311yescharacteristic
330C22×C4C2×D4×C7311yescharacteristic
331C22×C4C2×D4×C7311yescharacteristic
332C22×C4C2×D4×C7311yescharacteristic
333C22×C4C2×D4×C7311yescharacteristic
334C22×C4C2×D4×C7311yescharacteristic
335C22×C4C24×C7311yescharacteristic
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—21no—
366C42—21no—
367C42—41no—
368C42—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23C22×D4×C7311yescharacteristic
377Q8—84no—
378Q8—84no—
379D4—42no—
380D4—42no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4#135×C7312yes—
413C2×C4AGL(1,ℤ/8)×C7312yes—
414C2×C4H(5,𝔽2)×C7312yes—
415C2×C42−1+4×C7312yes—
416C2×C42−1+4×C7311yescharacteristic
417C2×C42−1+4×C7311yescharacteristic
418C8—42no—
419C8—42no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22#535×C7311yescharacteristic
428C22#536×C7311yescharacteristic
429C22#536×C7311yescharacteristic
430C22C2×#135×C7311yescharacteristic
431C22C2×AGL(1,ℤ/8)×C7311yescharacteristic
432C22#575×C7311yescharacteristic
433C22#575×C7311yescharacteristic
434C4—22no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—82no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4#563×C7312yes—
446C4#563×C7312yes—
447C2—161no—
448C2C73×#282111yescharacteristic
449C2C73×#321111yescharacteristic
450C2C73×#321311yescharacteristic
451C2C73×#322811yescharacteristic
452C2C73×#323311yescharacteristic
453C2C73×#323511yescharacteristic
454C2C73×#324011yescharacteristic
subgroupG/H
center, socleC23×C73C22×D4
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC22×D4×C73C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C73 ◁ C43×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#2862C2semidirect
3. C73×#2101C2semidirect
5. C73×#2085C2semidirect
7. C73×#2058C2semidirect
8. C73×#2057C2semidirect
11. C73×#1839C2semidirect
219. #32840C73direct (N × H)
228. #2862C2×C73semidirect
230. #2101C2×C73semidirect
232. #2085C2×C73semidirect
234. #2058C2×C73semidirect
235. #2057C2×C73semidirect
238. #1839C2×C73semidirect
239. C73#32840direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#28622
C73×#28602
C73×#21012
C73×#20922
C73×#20852
C73×#20732
C73×#20582
C73×#20572
C73×#20232
C73×#18392
#3284073