Finite Group Databasebeta

← C73×#19239 #32928643

Subgroups — C73×#19239 #32928643

Order 18688.

658 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3246C211yesmaximal, characteristic
2C73×#3229C211yesmaximal, characteristic, complemented
3C73×#3152C211yesmaximal, characteristic
4C73×#2962C211yesmaximal, characteristic, complemented
5C73×#1690C212yesmaximal, complemented
6C73×#1686C212yesmaximal, complemented
7C73×#1670C212yesmaximal, complemented
8C73×#1666C212yesmaximal, complemented
9C73×#1565C211yesmaximal, characteristic, complemented
10C73×#1540C211yesmaximal, characteristic
11C2×#332×C73C211yesmaximal, characteristic, complemented
12C2×#137×C73—21no—
13C2×#137×C73—21no—
14#533×C73C2211yescharacteristic
15#532×C73C2211yescharacteristic
16C2×C4⋊Q8×C73C2211yescharacteristic
17#498×C73—21no—
18#497×C73—22no—
19#494×C73C2212yes—
20#492×C73C2212yes—
21#487×C73—21no—
22#483×C73—22no—
23#476×C73—21no—
24#475×C73C2212yes—
25#475×C73C2212yes—
26#475×C73C2212yes—
27#465×C73—22no—
28#461×C73—21no—
29#460×C73C2212yes—
30#446×C73—22no—
31#446×C73C2212yes—
32C8×Q8×C73C2212yes—
33#431×C73C2211yescharacteristic
34#424×C73—21no—
35C2×C4⋊C8×C73C2211yescharacteristic
36C2×C4⋊C8×C73C2211yescharacteristic
37#405×C73—21no—
38C4×M16×C73C2211yescharacteristic
39#335×C73C2212yes—
40#334×C73C2212yes—
41#332×C73C2212yes—
42#332×C73C2211yescharacteristic
43#332×C73C2212yes—
44#332×C73C2211yescharacteristic
45#331×C73C2211yescharacteristic
46#331×C73C2211yescharacteristic
47C2×#41×C73—21no—
48C22×Q8×C73—21no—
49#137×C73—44no—
50#137×C73—44no—
51C2×Q16×C73—22no—
52C2×Q16×C73—22no—
53C2×SD16×C73—22no—
54C2×SD16×C73—22no—
55C2×SD16×C73—22no—
56C2×M16×C73D411yescharacteristic
57C2×M16×C73D411yescharacteristic
58C22×C8×C73—21no—
59C22×C8×C73—21no—
60C4⋊Q8×C73—21no—
61C4⋊Q8×C73—21no—
62C4⋊Q8×C73—21no—
63C4⋊Q8×C73D411yescharacteristic
64C4⋊Q8×C73D411yescharacteristic
65C4⋊Q8×C73C2312yes—
66C4⋊Q8×C73C2311yescharacteristic
67#127×C73C2311yescharacteristic
68#125×C73—21no—
69#124×C73—21no—
70C22⋊Q8×C73—21no—
71C22⋊Q8×C73D412yes—
72#121×C73—21no—
73#121×C73—21no—
74C4×Q8×C73—21no—
75C4×Q8×C73C2312yes—
76C4×D4×C73—21no—
77C4×D4×C73—21no—
78#117×C73—21no—
79C2×C4⋊C4×C73—21no—
80C2×C4⋊C4×C73—21no—
81C2×C42×C73C2311yescharacteristic
82#106×C73—22no—
83#106×C73—22no—
84#106×C73—22no—
85C4⋊C8×C73—22no—
86C4⋊C8×C73—22no—
87C4⋊C8×C73—21no—
88C4⋊C8×C73—21no—
89C4⋊C8×C73C2311yescharacteristic
90C4⋊C8×C73C2311yescharacteristic
91C4⋊C8×C73C2312yes—
92C4⋊C8×C73C2311yescharacteristic
93C4⋊C8×C73C2311yescharacteristic
94#103×C73—22no—
95#103×C73—22no—
96#103×C73—22no—
97#102×C73—22no—
98C22⋊C8×C73—21no—
99#97×C73—22no—
100#97×C73D412yes—
101C4×C8×C73—22no—
102C4×C8×C73—21no—
103C4×C8×C73C2312yes—
104#41×C73—42no—
105#41×C73—42no—
106C2×Q8×C73—21no—
107C2×Q8×C73—21no—
108C2×Q8×C73—22no—
109C2×Q8×C73—41no—
110C2×Q8×C73—42no—
111C2×Q8×C73—42no—
112C2×Q8×C73C2×D412yes—
113C2×D4×C73—21no—
114C2×D4×C73—21no—
115C2×D4×C73—21no—
116C2×D4×C73—41no—
117C22×C4×C73—41no—
118C22×C4×C73—41no—
119C22×C4×C73C2×D411yescharacteristic
120C22×C4×C73C2×D411yescharacteristic
121C22×C4×C73#4111yescharacteristic
122Q16×C73—44no—
123Q16×C73—44no—
124SD16×C73—44no—
125SD16×C73—44no—
126SD16×C73—82no—
127M16×C73—22no—
128M16×C73SD1614yescomplemented
129C2×C8×C73—21no—
130C2×C8×C73—21no—
131C2×C8×C73—21no—
132C2×C8×C73—21no—
133C2×C8×C73—22no—
134C2×C8×C73—42no—
135C2×C8×C73—81no—
136C2×C8×C73C2×D412yes—
137C2×C8×C73C2×D412yes—
138C4⋊C4×C73—21no—
139C4⋊C4×C73—21no—
140C4⋊C4×C73—21no—
141C4⋊C4×C73—22no—
142C4⋊C4×C73—22no—
143C4⋊C4×C73—22no—
144C4⋊C4×C73—21no—
145C4⋊C4×C73—22no—
146C4⋊C4×C73—21no—
147C4⋊C4×C73—41no—
148C4⋊C4×C73—41no—
149C4⋊C4×C73C2×D412yes—
150C22⋊C4×C73—22no—
151C22⋊C4×C73—41no—
152C22⋊C4×C73—41no—
153C42×C73—22no—
154C42×C73C2×D411yescharacteristic
155C42×C73C2×D411yescharacteristic
156C42×C73#4111yescharacteristic
157C42×C73C2411yescharacteristic
158C23×C73—41no—
159C23×C73#12111yescharacteristic
160Q8×C73—22no—
161Q8×C73—22no—
162Q8×C73—42no—
163Q8×C73—41no—
164Q8×C73—42no—
165Q8×C73—41no—
166Q8×C73—81no—
167Q8×C73—82no—
168D4×C73—41no—
169D4×C73—41no—
170D4×C73—41no—
171D4×C73—41no—
172D4×C73—81no—
173D4×C73—82no—
174C2×C4×C73—21no—
175C2×C4×C73—21no—
176C2×C4×C73—22no—
177C2×C4×C73—22no—
178C2×C4×C73—41no—
179C2×C4×C73—41no—
180C2×C4×C73—42no—
181C2×C4×C73—82no—
182C2×C4×C73—82no—
183C2×C4×C73#12111yescharacteristic
184C2×C4×C73#12111yescharacteristic
185C2×C4×C73#12111yescharacteristic
186C2×C4×C73C2×SD1612yes—
187C2×C4×C73AGL(1,ℤ/8)12yes—
188C2×C4×C73C22×D411yescharacteristic
189C2×C4×C73C22×D411yescharacteristic
190C2×C4×C73C2×#4111yescharacteristic
191C8×C73—22no—
192C8×C73—41no—
193C8×C73—41no—
194C8×C73—41no—
195C8×C73—42no—
196C8×C73—41no—
197C8×C73C2×SD1614yes—
198C22×C73—21no—
199C22×C73—21no—
200C22×C73—81no—
201C22×C73—81no—
202C22×C73—81no—
203C22×C73#46012yes—
204C22×C73C2×#12111yescharacteristic
205C4×C73—21no—
206C4×C73—21no—
207C4×C73—21no—
208C4×C73—42no—
209C4×C73—42no—
210C4×C73—81no—
211C4×C73—82no—
212C4×C73#46012yes—
213C4×C73C22×SD1611yescharacteristic
214C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
215C4×C73#57711yescharacteristic
216C4×C73#57811yescharacteristic
217#19239C7311yesmaximal, characteristic, direct product factor, Sylow
218C2×C73—22no—
219C2×C73—161no—
220C2×C73C2×#46011yescharacteristic
221C2×C73#306311yescharacteristic
222C2×C73#306911yescharacteristic
223#3246C2×C7311yescharacteristic
224#3229C2×C7311yescharacteristic, complemented
225#3152C2×C7311yescharacteristic
226#2962C2×C7311yescharacteristic, complemented
227#1690C2×C7312yescomplemented
228#1686C2×C7312yescomplemented
229#1670C2×C7312yescomplemented
230#1666C2×C7312yescomplemented
231#1565C2×C7311yescharacteristic, complemented
232#1540C2×C7311yescharacteristic
233C2×#332C2×C7311yescharacteristic, complemented
234C73#1923911yescharacteristic, direct product factor, Sylow
235C2×#137—21no—
236C2×#137—21no—
237#533C22×C7311yescharacteristic
238#532C22×C7311yescharacteristic
239C2×C4⋊Q8C22×C7311yescharacteristic
240#498—21no—
241#497—22no—
242#494C22×C7312yes—
243#492C22×C7312yes—
244#487—21no—
245#483—22no—
246#476—21no—
247#475C22×C7312yes—
248#475C22×C7312yes—
249#475C22×C7312yes—
250#465—22no—
251#461—21no—
252#460C22×C7312yes—
253#446—22no—
254#446C22×C7312yes—
255C8×Q8C22×C7312yes—
256#431C22×C7311yescharacteristic
257#424—21no—
258C2×C4⋊C8C22×C7311yescharacteristic
259C2×C4⋊C8C22×C7311yescharacteristic
260#405—21no—
261C4×M16C22×C7311yescharacteristic
262#335C22×C7312yes—
263#334C22×C7312yes—
264#332C22×C7312yes—
265#332C22×C7312yes—
266#332C22×C7311yescharacteristic
267#332C22×C7311yescharacteristic
268#331C22×C7311yescharacteristic
269#331C22×C7311yescharacteristic
270C2×#41—21no—
271C22×Q8—21no—
272#137—44no—
273#137—44no—
274C2×Q16—22no—
275C2×Q16—22no—
276C2×SD16—22no—
277C2×SD16—22no—
278C2×SD16—22no—
279C2×M16D4×C7311yescharacteristic
280C2×M16D4×C7311yescharacteristic
281C22×C8—21no—
282C22×C8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8D4×C7311yescharacteristic
287C4⋊Q8D4×C7311yescharacteristic
288C4⋊Q8C23×C7312yes—
289C4⋊Q8C23×C7311yescharacteristic
290#127C23×C7311yescharacteristic
291#125—21no—
292#124—21no—
293C22⋊Q8—21no—
294C22⋊Q8D4×C7312yes—
295#121—21no—
296#121—21no—
297C4×Q8—21no—
298C4×Q8C23×C7312yes—
299C4×D4—21no—
300C4×D4—21no—
301#117—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C42C23×C7311yescharacteristic
305#106—22no—
306#106—22no—
307#106—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8C23×C7311yescharacteristic
313C4⋊C8C23×C7311yescharacteristic
314C4⋊C8C23×C7311yescharacteristic
315C4⋊C8C23×C7311yescharacteristic
316C4⋊C8C23×C7312yes—
317#103—22no—
318#103—22no—
319#103—22no—
320#102—22no—
321C22⋊C8—21no—
322#97—22no—
323#97D4×C7312yes—
324C4×C8—22no—
325C4×C8—21no—
326C4×C8C23×C7312yes—
327#41—42no—
328#41—42no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—22no—
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8C2×D4×C7312yes—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4C2×D4×C7311yescharacteristic
343C22×C4C2×D4×C7311yescharacteristic
344C22×C4#41×C7311yescharacteristic
345Q16—44no—
346Q16—44no—
347SD16—44no—
348SD16—44no—
349SD16—82no—
350M16—22no—
351M16SD16×C7314yescomplemented
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—22no—
357C2×C8—42no—
358C2×C8—81no—
359C2×C8C2×D4×C7312yes—
360C2×C8C2×D4×C7312yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4C2×D4×C7312yes—
373C22⋊C4—22no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C42—22no—
377C42C2×D4×C7311yescharacteristic
378C42C2×D4×C7311yescharacteristic
379C42#41×C7311yescharacteristic
380C42C24×C7311yescharacteristic
381C23—41no—
382C23#121×C7311yescharacteristic
383Q8—22no—
384Q8—22no—
385Q8—42no—
386Q8—41no—
387Q8—42no—
388Q8—41no—
389Q8—81no—
390Q8—82no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—81no—
396D4—82no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#121×C7311yescharacteristic
407C2×C4#121×C7311yescharacteristic
408C2×C4#121×C7311yescharacteristic
409C2×C4C2×SD16×C7312yes—
410C2×C4AGL(1,ℤ/8)×C7312yes—
411C2×C4C22×D4×C7311yescharacteristic
412C2×C4C22×D4×C7311yescharacteristic
413C2×C4C2×#41×C7311yescharacteristic
414C8—22no—
415C8—41no—
416C8—41no—
417C8—42no—
418C8—41no—
419C8—41no—
420C8C2×SD16×C7314yes—
421C22—21no—
422C22—21no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22#460×C7312yes—
427C22C2×#121×C7311yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—42no—
433C4—81no—
434C4—82no—
435C4#460×C7312yes—
436C4C22×SD16×C7311yescharacteristic
437C4C2×AGL(1,ℤ/8)×C7311yescharacteristic
438C4#577×C7311yescharacteristic
439C4#578×C7311yescharacteristic
440C2—22no—
441C2—161no—
442C2C2×#460×C7311yescharacteristic
443C2C73×#306311yescharacteristic
444C2C73×#306911yescharacteristic
subgroupG/H
center, socleC22×C73C2×#121
commutator, Frattini, cosocleC42C24×C73
prime radical#532×C73C22
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×#3229C2semidirect
4. C73×#2962C2semidirect
5. C73×#1690C2semidirect
6. C73×#1686C2semidirect
7. C73×#1670C2semidirect
8. C73×#1666C2semidirect
9. C73×#1565C2semidirect
11. C2×#332×C73C2semidirect
128. M16×C73SD16semidirect
217. #19239C73direct (N × H)
224. #3229C2×C73semidirect
226. #2962C2×C73semidirect
227. #1690C2×C73semidirect
228. #1686C2×C73semidirect
229. #1670C2×C73semidirect
230. #1666C2×C73semidirect
231. #1565C2×C73semidirect
233. C2×#332C2×C73semidirect
234. C73#19239direct (N × H)
351. M16SD16×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#32462
C73×#32292
C73×#31522
C73×#29622
C73×#16902
C73×#16862
C73×#16702
C73×#16662
C73×#15652
C73×#15402
C2×#332×C732
#1923973