Finite Group Databasebeta

← C73×#27135 #32936539

Subgroups — C73×#27135 #32936539

Order 18688.

670 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2512C211yesmaximal, characteristic
2C73×#2370C211yesmaximal, characteristic, complemented
3C73×#2100C211yesmaximal, characteristic, complemented
4C73×#2067C211yesmaximal, characteristic
5C73×#2059C211yesmaximal, characteristic, complemented
6C73×#2025C211yesmaximal, characteristic
7C73×#1947C211yesmaximal, characteristic, complemented
8C73×#1934C211yesmaximal, characteristic
9C73×#1926C211yesmaximal, characteristic
10C73×#1925C211yesmaximal, characteristic
11C73×#1925C211yesmaximal, characteristic
12C73×#1921C211yesmaximal, characteristic, complemented
13C73×#1920C211yesmaximal, characteristic, complemented
14C73×#1916C211yesmaximal, characteristic, complemented
15C4×#107×C73C211yesmaximal, characteristic, complemented
16C2×#126×C73—21no—
17C2×#125×C73C2211yescharacteristic
18C2×#107×C73C2211yescharacteristic
19C2×#107×C73C2211yescharacteristic
20C2×#107×C73C2211yescharacteristic
21C2×#106×C73C2211yescharacteristic
22C2×C4⋊C8×C73C2211yescharacteristic
23C2×#102×C73C2211yescharacteristic
24C2×#102×C73C2211yescharacteristic
25C2×#102×C73C2211yescharacteristic
26C2×#102×C73C2211yescharacteristic
27C2×C4×C8×C73C2211yescharacteristic
28#397×C73—21no—
29#396×C73—21no—
30#390×C73C2211yescharacteristic
31#390×C73C2211yescharacteristic
32#390×C73C2211yescharacteristic
33#389×C73—21no—
34#389×C73C2211yescharacteristic
35#389×C73C2211yescharacteristic
36#389×C73C2211yescharacteristic
37#388×C73—21no—
38#387×C73—21no—
39#387×C73—21no—
40#384×C73C2211yescharacteristic
41C4×C4⋊C4×C73—21no—
42C4×C4⋊C4×C73C2211yescharacteristic
43C4×C4⋊C4×C73C2211yescharacteristic
44C4×C4⋊C4×C73C2211yescharacteristic
45C4×C4⋊C4×C73C2211yescharacteristic
46C4×C4⋊C4×C73C2211yescharacteristic
47#340×C73C2211yescharacteristic
48#340×C73C2211yescharacteristic
49#340×C73C2211yescharacteristic
50#340×C73C2211yescharacteristic
51#340×C73C2211yescharacteristic
52#340×C73C2211yescharacteristic
53#340×C73C2211yescharacteristic
54#340×C73C2211yescharacteristic
55#336×C73C2211yescharacteristic
56#336×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×D4×C73C2311yescharacteristic
60C22×C8×C73C2311yescharacteristic
61C22×C8×C73C2311yescharacteristic
62C22×C8×C73C2311yescharacteristic
63C22×C8×C73C2311yescharacteristic
64#126×C73—44no—
65#125×C73—22no—
66#125×C73—22no—
67C2×C4⋊C4×C73—21no—
68C2×C4⋊C4×C73—21no—
69C2×C4⋊C4×C73—21no—
70C2×C4⋊C4×C73—21no—
71C2×C4⋊C4×C73—21no—
72C2×C4⋊C4×C73—21no—
73C2×C4⋊C4×C73—21no—
74C2×C4⋊C4×C73D411yescharacteristic
75C2×C4⋊C4×C73D411yescharacteristic
76C2×C4⋊C4×C73C2311yescharacteristic
77C2×C4⋊C4×C73C2311yescharacteristic
78C2×C4⋊C4×C73C2311yescharacteristic
79C2×C4⋊C4×C73C2311yescharacteristic
80C2×C4⋊C4×C73C2311yescharacteristic
81C2×C4⋊C4×C73C2311yescharacteristic
82C2×C4⋊C4×C73C2311yescharacteristic
83C2×C22⋊C4×C73—21no—
84C2×C22⋊C4×C73—21no—
85C2×C22⋊C4×C73—21no—
86C2×C22⋊C4×C73—21no—
87C2×C22⋊C4×C73—21no—
88C2×C22⋊C4×C73—21no—
89C2×C42×C73—21no—
90C2×C42×C73—21no—
91C2×C42×C73—21no—
92C2×C42×C73—21no—
93C2×C42×C73C2311yescharacteristic
94C2×C42×C73C2311yescharacteristic
95C2×C42×C73C2311yescharacteristic
96#107×C73—22no—
97#107×C73—22no—
98#107×C73—22no—
99#106×C73—22no—
100C4⋊C8×C73—22no—
101#102×C73—22no—
102#102×C73—22no—
103#102×C73—22no—
104#102×C73—22no—
105C4×C8×C73—22no—
106#95×C73—21no—
107#95×C73—21no—
108#95×C73—21no—
109#95×C73—21no—
110#95×C73—21no—
111#95×C73—21no—
112#95×C73—21no—
113#95×C73D411yescharacteristic
114#95×C73D411yescharacteristic
115C24×C73—21no—
116C2×D4×C73—21no—
117C2×D4×C73—21no—
118C2×D4×C73—21no—
119C2×D4×C73—21no—
120C2×D4×C73—21no—
121C2×D4×C73—21no—
122C22×C4×C73—21no—
123C22×C4×C73—21no—
124C22×C4×C73—21no—
125C22×C4×C73—21no—
126C22×C4×C73—21no—
127C22×C4×C73—21no—
128C22×C4×C73—21no—
129C22×C4×C73C2×D411yescharacteristic
130C22×C4×C73C2×D411yescharacteristic
131C22×C4×C73C2×D411yescharacteristic
132C22×C4×C73C2×D411yescharacteristic
133C22×C4×C73C2×D411yescharacteristic
134C22×C4×C73C2×D411yescharacteristic
135C22×C4×C73C2411yescharacteristic
136C2×C8×C73—21no—
137C2×C8×C73—21no—
138C2×C8×C73—21no—
139C2×C8×C73—21no—
140C2×C8×C73—21no—
141C2×C8×C73—21no—
142C2×C8×C73—21no—
143C2×C8×C73—21no—
144C2×C8×C73—21no—
145C2×C8×C73—21no—
146C2×C8×C73—21no—
147C2×C8×C73#4112yes—
148C4⋊C4×C73—24no—
149C4⋊C4×C73—24no—
150C4⋊C4×C73—22no—
151C4⋊C4×C73—22no—
152C4⋊C4×C73—21no—
153C4⋊C4×C73—21no—
154C4⋊C4×C73—21no—
155C4⋊C4×C73—21no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—21no—
159C4⋊C4×C73—21no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—42no—
164C4⋊C4×C73—42no—
165C4⋊C4×C73—42no—
166C4⋊C4×C73—81no—
167C4⋊C4×C73—81no—
168C4⋊C4×C73#4112yes—
169C4⋊C4×C73#4112yes—
170C4⋊C4×C73#4112yes—
171C22⋊C4×C73—42no—
172C22⋊C4×C73—42no—
173C22⋊C4×C73—42no—
174C22⋊C4×C73—42no—
175C22⋊C4×C73—81no—
176C22⋊C4×C73—81no—
177C42×C73—21no—
178C42×C73—21no—
179C42×C73—21no—
180C42×C73—21no—
181C42×C73—21no—
182C42×C73—21no—
183C42×C73—42no—
184C42×C73—42no—
185C42×C73—81no—
186C42×C73—81no—
187C23×C73—41no—
188C23×C73—41no—
189C23×C73—41no—
190C23×C73—41no—
191C23×C73—41no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73C22×D411yescharacteristic
195D4×C73—41no—
196D4×C73—41no—
197D4×C73—41no—
198D4×C73—41no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—22no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73#13512yes—
237C2×C4×C73AGL(1,ℤ/8)12yes—
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C732−1+411yescharacteristic
243C2×C4×C732−1+411yescharacteristic
244C8×C73—22no—
245C8×C73—41no—
246C8×C73—41no—
247C8×C73—41no—
248C22×C73—81no—
249C22×C73—81no—
250C22×C73—81no—
251C22×C73—81no—
252C22×C73—81no—
253C22×C73—81no—
254C22×C73—81no—
255C22×C73#52511yescharacteristic
256C22×C73#54711yescharacteristic
257C22×C73#54811yescharacteristic
258C22×C73C2×#13511yescharacteristic
259C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C73#57611yescharacteristic
261C22×C73#57811yescharacteristic
262C4×C73—22no—
263C4×C73—22no—
264C4×C73—41no—
265C4×C73—41no—
266C4×C73—41no—
267C4×C73—42no—
268C4×C73—42no—
269C4×C73—41no—
270C4×C73—42no—
271C4×C73—81no—
272C4×C73—81no—
273C4×C73—81no—
274C4×C73—81no—
275C4×C73#52911yescharacteristic
276C4×C73#54111yescharacteristic
277C4×C73#55611yescharacteristic
278C4×C73#55611yescharacteristic
279#27135C7311yesmaximal, characteristic, direct product factor, Sylow
280C2×C73—161no—
281C2×C73#246911yescharacteristic
282C2×C73#310611yescharacteristic
283C2×C73#312411yescharacteristic
284C2×C73#333911yescharacteristic
285C2×C73#334311yescharacteristic
286C2×C73#336411yescharacteristic
287C2×C73#336611yescharacteristic
288#2512C2×C7311yescharacteristic
289#2370C2×C7311yescharacteristic, complemented
290#2100C2×C7311yescharacteristic, complemented
291#2067C2×C7311yescharacteristic
292#2059C2×C7311yescharacteristic, complemented
293#2025C2×C7311yescharacteristic
294#1947C2×C7311yescharacteristic, complemented
295#1934C2×C7311yescharacteristic
296#1926C2×C7311yescharacteristic
297#1925C2×C7311yescharacteristic
298#1925C2×C7311yescharacteristic
299#1921C2×C7311yescharacteristic, complemented
300#1920C2×C7311yescharacteristic, complemented
301#1916C2×C7311yescharacteristic, complemented
302C4×#107C2×C7311yescharacteristic, complemented
303C73#2713511yescharacteristic, direct product factor, Sylow
304C2×#126—21no—
305C2×#125C22×C7311yescharacteristic
306C2×#107C22×C7311yescharacteristic
307C2×#107C22×C7311yescharacteristic
308C2×#107C22×C7311yescharacteristic
309C2×#106C22×C7311yescharacteristic
310C2×C4⋊C8C22×C7311yescharacteristic
311C2×#102C22×C7311yescharacteristic
312C2×#102C22×C7311yescharacteristic
313C2×#102C22×C7311yescharacteristic
314C2×#102C22×C7311yescharacteristic
315C2×C4×C8C22×C7311yescharacteristic
316#397—21no—
317#396—21no—
318#390C22×C7311yescharacteristic
319#390C22×C7311yescharacteristic
320#390C22×C7311yescharacteristic
321#389—21no—
322#389C22×C7311yescharacteristic
323#389C22×C7311yescharacteristic
324#389C22×C7311yescharacteristic
325#388—21no—
326#387—21no—
327#387—21no—
328#384C22×C7311yescharacteristic
329C4×C4⋊C4—21no—
330C4×C4⋊C4C22×C7311yescharacteristic
331C4×C4⋊C4C22×C7311yescharacteristic
332C4×C4⋊C4C22×C7311yescharacteristic
333C4×C4⋊C4C22×C7311yescharacteristic
334C4×C4⋊C4C22×C7311yescharacteristic
335#340C22×C7311yescharacteristic
336#340C22×C7311yescharacteristic
337#340C22×C7311yescharacteristic
338#340C22×C7311yescharacteristic
339#340C22×C7311yescharacteristic
340#340C22×C7311yescharacteristic
341#340C22×C7311yescharacteristic
342#340C22×C7311yescharacteristic
343#336C22×C7311yescharacteristic
344#336C22×C7311yescharacteristic
345#336C22×C7311yescharacteristic
346#336C22×C7311yescharacteristic
347C22×D4C23×C7311yescharacteristic
348C22×C8C23×C7311yescharacteristic
349C22×C8C23×C7311yescharacteristic
350C22×C8C23×C7311yescharacteristic
351C22×C8C23×C7311yescharacteristic
352#126—44no—
353#125—22no—
354#125—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4D4×C7311yescharacteristic
363C2×C4⋊C4D4×C7311yescharacteristic
364C2×C4⋊C4C23×C7311yescharacteristic
365C2×C4⋊C4C23×C7311yescharacteristic
366C2×C4⋊C4C23×C7311yescharacteristic
367C2×C4⋊C4C23×C7311yescharacteristic
368C2×C4⋊C4C23×C7311yescharacteristic
369C2×C4⋊C4C23×C7311yescharacteristic
370C2×C4⋊C4C23×C7311yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42C23×C7311yescharacteristic
382C2×C42C23×C7311yescharacteristic
383C2×C42C23×C7311yescharacteristic
384#107—22no—
385#107—22no—
386#107—22no—
387#106—22no—
388C4⋊C8—22no—
389#102—22no—
390#102—22no—
391#102—22no—
392#102—22no—
393C4×C8—22no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95D4×C7311yescharacteristic
402#95D4×C7311yescharacteristic
403C24—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4C2×D4×C7311yescharacteristic
418C22×C4C2×D4×C7311yescharacteristic
419C22×C4C2×D4×C7311yescharacteristic
420C22×C4C2×D4×C7311yescharacteristic
421C22×C4C2×D4×C7311yescharacteristic
422C22×C4C2×D4×C7311yescharacteristic
423C22×C4C24×C7311yescharacteristic
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8#41×C7312yes—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4#41×C7312yes—
457C4⋊C4#41×C7312yes—
458C4⋊C4#41×C7312yes—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42—42no—
473C42—81no—
474C42—81no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23C22×D4×C7311yescharacteristic
483D4—41no—
484D4—41no—
485D4—41no—
486D4—41no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4#135×C7312yes—
525C2×C4AGL(1,ℤ/8)×C7312yes—
526C2×C4C2×#41×C7311yescharacteristic
527C2×C4C2×#41×C7311yescharacteristic
528C2×C4C2×#41×C7311yescharacteristic
529C2×C4C2×#41×C7311yescharacteristic
530C2×C42−1+4×C7311yescharacteristic
531C2×C42−1+4×C7311yescharacteristic
532C8—22no—
533C8—41no—
534C8—41no—
535C8—41no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22#525×C7311yescharacteristic
544C22#547×C7311yescharacteristic
545C22#548×C7311yescharacteristic
546C22C2×#135×C7311yescharacteristic
547C22C2×AGL(1,ℤ/8)×C7311yescharacteristic
548C22#576×C7311yescharacteristic
549C22#578×C7311yescharacteristic
550C4—22no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—41no—
558C4—42no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4#529×C7311yescharacteristic
564C4#541×C7311yescharacteristic
565C4#556×C7311yescharacteristic
566C4#556×C7311yescharacteristic
567C2—161no—
568C2C73×#246911yescharacteristic
569C2C73×#310611yescharacteristic
570C2C73×#312411yescharacteristic
571C2C73×#333911yescharacteristic
572C2C73×#334311yescharacteristic
573C2C73×#336411yescharacteristic
574C2C73×#336611yescharacteristic
subgroupG/H
center, socleC23×C73C22×D4
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC22×D4×C73C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C73 ◁ C4×C4⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#2370C2semidirect
3. C73×#2100C2semidirect
5. C73×#2059C2semidirect
7. C73×#1947C2semidirect
12. C73×#1921C2semidirect
13. C73×#1920C2semidirect
14. C73×#1916C2semidirect
15. C4×#107×C73C2semidirect
279. #27135C73direct (N × H)
289. #2370C2×C73semidirect
290. #2100C2×C73semidirect
292. #2059C2×C73semidirect
294. #1947C2×C73semidirect
299. #1921C2×C73semidirect
300. #1920C2×C73semidirect
301. #1916C2×C73semidirect
302. C4×#107C2×C73semidirect
303. C73#27135direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#25122
C73×#23702
C73×#21002
C73×#20672
C73×#20592
C73×#20252
C73×#19472
C73×#19342
C73×#19262
C73×#19252
C73×#19212
C73×#19202
C73×#19162
C4×#107×C732
#2713573