Finite Group Databasebeta

← C73×#13422 #32922826

Subgroups — C73×#13422 #32922826

Order 18688.

734 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2669C211yesmaximal, characteristic
2C2×#353×C73C211yesmaximal, characteristic, complemented
3C2×AΣL(1,𝔽16)×C73C211yesmaximal, characteristic, complemented
4C73×#1889C211yesmaximal, characteristic, complemented
5C73×#1887C211yesmaximal, characteristic, complemented
6C73×#1867C211yesmaximal, characteristic, complemented
7C73×#1390C211yesmaximal, characteristic, complemented
8C2×#127×C73C2211yescharacteristic
9C2×#121×C73—21no—
10C2×#120×C73—21no—
11C2×#120×C73C2211yescharacteristic
12#518×C73C2211yescharacteristic
13#418×C73—21no—
14C2×#100×C73C2211yescharacteristic
15C2×#99×C73C2211yescharacteristic
16C2×#99×C73C2211yescharacteristic
17#394×C73—21no—
18#392×C73—21no—
19#392×C73—21no—
20#388×C73—21no—
21#386×C73C2211yescharacteristic
22#385×C73—21no—
23#379×C73—21no—
24#353×C73—22no—
25AΣL(1,𝔽16)×C73—22no—
26#342×C73—21no—
27#342×C73—21no—
28#339×C73—21no—
29C25×C73D411yescharacteristic
30C22×D4×C73—21no—
31C22×D4×C73—21no—
32C22×D4×C73—21no—
33C22×D4×C73—21no—
34C22×D4×C73—21no—
35C22×D4×C73D411yescharacteristic
36C22×D4×C73C2311yescharacteristic
37C23×C4×C73D411yescharacteristic
38C2×M16×C73—21no—
39#127×C73—21no—
40#127×C73—21no—
41#127×C73—21no—
42#127×C73—21no—
43#121×C73—44no—
44#120×C73—21no—
45#120×C73—21no—
46#120×C73—21no—
47#120×C73—21no—
48#120×C73—44no—
49C4×D4×C73—41no—
50C4×D4×C73—41no—
51#117×C73—21no—
52C2×C4⋊C4×C73—21no—
53C2×C4⋊C4×C73—41no—
54C2×C22⋊C4×C73—21no—
55C2×C22⋊C4×C73—21no—
56C2×C22⋊C4×C73—21no—
57C2×C22⋊C4×C73—21no—
58C2×C22⋊C4×C73—21no—
59C2×C22⋊C4×C73—21no—
60C2×C22⋊C4×C73—21no—
61C2×C22⋊C4×C73—21no—
62C2×C22⋊C4×C73—21no—
63C2×C22⋊C4×C73—21no—
64C2×C22⋊C4×C73—41no—
65C2×C22⋊C4×C73—41no—
66C2×C22⋊C4×C73D411yescharacteristic
67C2×C22⋊C4×C73D411yescharacteristic
68C2×C42×C73D411yescharacteristic
69#102×C73—41no—
70#102×C73—41no—
71#100×C73—21no—
72#100×C73—21no—
73#99×C73—21no—
74#99×C73—21no—
75#99×C73—21no—
76#99×C73D412yescomplemented
77#95×C73—21no—
78#95×C73—21no—
79#95×C73—41no—
80C24×C73—21no—
81C24×C73—21no—
82C24×C73—21no—
83C24×C73—21no—
84C24×C73—41no—
85C24×C73—42no—
86C24×C73—41no—
87C24×C73—81no—
88C24×C73C2×D411yescharacteristic
89C24×C73C2×D411yescharacteristic
90C2×D4×C73—21no—
91C2×D4×C73—21no—
92C2×D4×C73—41no—
93C2×D4×C73—41no—
94C2×D4×C73—41no—
95C2×D4×C73—41no—
96C2×D4×C73—42no—
97C2×D4×C73—42no—
98C2×D4×C73—42no—
99C2×D4×C73—42no—
100C2×D4×C73—42no—
101C2×D4×C73—42no—
102C2×D4×C73—41no—
103C2×D4×C73—41no—
104C2×D4×C73—41no—
105C2×D4×C73—41no—
106C2×D4×C73—41no—
107C2×D4×C73—41no—
108C2×D4×C73—81no—
109C2×D4×C73—81no—
110C2×D4×C73—81no—
111C2×D4×C73—81no—
112C2×D4×C73—81no—
113C2×D4×C73—81no—
114C2×D4×C73C2×D411yescharacteristic
115C2×D4×C73#4111yescharacteristic
116C2×D4×C73#4111yescharacteristic
117C2×D4×C73#4111yescharacteristic
118C22×C4×C73—21no—
119C22×C4×C73—21no—
120C22×C4×C73—21no—
121C22×C4×C73—21no—
122C22×C4×C73—21no—
123C22×C4×C73—21no—
124C22×C4×C73—21no—
125C22×C4×C73—21no—
126C22×C4×C73—42no—
127C22×C4×C73—41no—
128C22×C4×C73—41no—
129C22×C4×C73—41no—
130C22×C4×C73C2×D411yescharacteristic
131M16×C73—41no—
132M16×C73—41no—
133C2×C8×C73—41no—
134C4⋊C4×C73—21no—
135C4⋊C4×C73—21no—
136C4⋊C4×C73—42no—
137C4⋊C4×C73—82no—
138C22⋊C4×C73—24no—
139C22⋊C4×C73—22no—
140C22⋊C4×C73—21no—
141C22⋊C4×C73—21no—
142C22⋊C4×C73—42no—
143C22⋊C4×C73—42no—
144C22⋊C4×C73—42no—
145C22⋊C4×C73—41no—
146C22⋊C4×C73—41no—
147C22⋊C4×C73—41no—
148C22⋊C4×C73—41no—
149C22⋊C4×C73—41no—
150C22⋊C4×C73—41no—
151C22⋊C4×C73—41no—
152C22⋊C4×C73—41no—
153C22⋊C4×C73—41no—
154C22⋊C4×C73—81no—
155C22⋊C4×C73—82no—
156C22⋊C4×C73—82no—
157C42×C73—21no—
158C42×C73—21no—
159C42×C73—41no—
160C23×C73—21no—
161C23×C73—21no—
162C23×C73—21no—
163C23×C73—21no—
164C23×C73—22no—
165C23×C73—41no—
166C23×C73—41no—
167C23×C73—41no—
168C23×C73—41no—
169C23×C73—41no—
170C23×C73—41no—
171C23×C73—41no—
172C23×C73—42no—
173C23×C73—41no—
174C23×C73—41no—
175C23×C73—41no—
176C23×C73—41no—
177C23×C73—42no—
178C23×C73—42no—
179C23×C73—41no—
180C23×C73—41no—
181C23×C73—41no—
182C23×C73—41no—
183C23×C73—41no—
184C23×C73—41no—
185C23×C73—81no—
186C23×C73—82no—
187C23×C73—81no—
188C23×C73—82no—
189C23×C73—84no—
190C23×C73—81no—
191C23×C73#12011yescharacteristic
192C23×C73#12111yescharacteristic
193C23×C73#12111yescharacteristic
194C23×C73#12311yescharacteristic
195C23×C73#12411yescharacteristic
196D4×C73—41no—
197D4×C73—41no—
198D4×C73—41no—
199D4×C73—41no—
200D4×C73—42no—
201D4×C73—81no—
202D4×C73—82no—
203D4×C73—82no—
204D4×C73—82no—
205D4×C73—82no—
206D4×C73—84no—
207D4×C73—82no—
208D4×C73—82no—
209D4×C73—84no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—42no—
219C2×C4×C73—42no—
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—81no—
229C2×C4×C73—81no—
230C2×C4×C73—81no—
231C2×C4×C73—81no—
232C2×C4×C73—81no—
233C2×C4×C73—81no—
234C2×C4×C73—81no—
235C2×C4×C73—81no—
236C2×C4×C73—81no—
237C2×C4×C73—81no—
238C2×C4×C73#12111yescharacteristic
239C2×C4×C73#12311yescharacteristic
240C8×C73—81no—
241C22×C73—21no—
242C22×C73—21no—
243C22×C73—21no—
244C22×C73—21no—
245C22×C73—21no—
246C22×C73—21no—
247C22×C73—41no—
248C22×C73—41no—
249C22×C73—41no—
250C22×C73—41no—
251C22×C73—42no—
252C22×C73—42no—
253C22×C73—41no—
254C22×C73—41no—
255C22×C73—41no—
256C22×C73—41no—
257C22×C73—44no—
258C22×C73—42no—
259C22×C73—81no—
260C22×C73—81no—
261C22×C73—81no—
262C22×C73—81no—
263C22×C73—81no—
264C22×C73—81no—
265C22×C73—81no—
266C22×C73—81no—
267C22×C73—82no—
268C22×C73—82no—
269C22×C73—82no—
270C22×C73—81no—
271C22×C73—81no—
272C22×C73—81no—
273C22×C73—81no—
274C22×C73—81no—
275C22×C73#39411yescharacteristic
276C22×C73B(4,𝔽2)11yescharacteristic
277C22×C73#45811yescharacteristic
278C4×C73—21no—
279C4×C73—21no—
280C4×C73—42no—
281C4×C73—42no—
282C4×C73—81no—
283C4×C73—81no—
284C4×C73—82no—
285C4×C73—81no—
286C4×C73—81no—
287C4×C73—161no—
288#13422C7311yesmaximal, characteristic, direct product factor, Sylow
289C2×C73—21no—
290C2×C73—21no—
291C2×C73—41no—
292C2×C73—41no—
293C2×C73—41no—
294C2×C73—41no—
295C2×C73—82no—
296C2×C73—161no—
297C2×C73#204411yescharacteristic
298C2×C73#219611yescharacteristic
299C2×C73#220011yescharacteristic
300#2669C2×C7311yescharacteristic
301C2×#353C2×C7311yescharacteristic, complemented
302C2×AΣL(1,𝔽16)C2×C7311yescharacteristic, complemented
303#1889C2×C7311yescharacteristic, complemented
304#1887C2×C7311yescharacteristic, complemented
305#1867C2×C7311yescharacteristic, complemented
306#1390C2×C7311yescharacteristic, complemented
307C73#1342211yescharacteristic, direct product factor, Sylow
308C2×#127C22×C7311yescharacteristic
309C2×#121—21no—
310C2×#120—21no—
311C2×#120C22×C7311yescharacteristic
312#518C22×C7311yescharacteristic
313#418—21no—
314C2×#100C22×C7311yescharacteristic
315C2×#99C22×C7311yescharacteristic
316C2×#99C22×C7311yescharacteristic
317#394—21no—
318#392—21no—
319#392—21no—
320#388—21no—
321#386C22×C7311yescharacteristic
322#385—21no—
323#379—21no—
324#353—22no—
325AΣL(1,𝔽16)—22no—
326#342—21no—
327#342—21no—
328#339—21no—
329C25D4×C7311yescharacteristic
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4D4×C7311yescharacteristic
336C22×D4C23×C7311yescharacteristic
337C23×C4D4×C7311yescharacteristic
338C2×M16—21no—
339#127—21no—
340#127—21no—
341#127—21no—
342#127—21no—
343#121—44no—
344#120—21no—
345#120—21no—
346#120—21no—
347#120—21no—
348#120—44no—
349C4×D4—41no—
350C4×D4—41no—
351#117—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—41no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4—41no—
366C2×C22⋊C4D4×C7311yescharacteristic
367C2×C22⋊C4D4×C7311yescharacteristic
368C2×C42D4×C7311yescharacteristic
369#102—41no—
370#102—41no—
371#100—21no—
372#100—21no—
373#99—21no—
374#99—21no—
375#99—21no—
376#99D4×C7312yescomplemented
377#95—21no—
378#95—21no—
379#95—41no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—41no—
385C24—42no—
386C24—41no—
387C24—81no—
388C24C2×D4×C7311yescharacteristic
389C24C2×D4×C7311yescharacteristic
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4—81no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4C2×D4×C7311yescharacteristic
415C2×D4#41×C7311yescharacteristic
416C2×D4#41×C7311yescharacteristic
417C2×D4#41×C7311yescharacteristic
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4C2×D4×C7311yescharacteristic
431M16—41no—
432M16—41no—
433C2×C8—41no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—82no—
438C22⋊C4—24no—
439C22⋊C4—22no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—81no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C42—21no—
458C42—21no—
459C42—41no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—81no—
486C23—82no—
487C23—81no—
488C23—82no—
489C23—84no—
490C23—81no—
491C23#120×C7311yescharacteristic
492C23#121×C7311yescharacteristic
493C23#121×C7311yescharacteristic
494C23#123×C7311yescharacteristic
495C23#124×C7311yescharacteristic
496D4—41no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—42no—
501D4—81no—
502D4—82no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—84no—
507D4—82no—
508D4—82no—
509D4—84no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4#121×C7311yescharacteristic
539C2×C4#123×C7311yescharacteristic
540C8—81no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—44no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22#394×C7311yescharacteristic
576C22B(4,𝔽2)×C7311yescharacteristic
577C22#458×C7311yescharacteristic
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—82no—
585C4—81no—
586C4—81no—
587C4—161no—
588C2—21no—
589C2—21no—
590C2—41no—
591C2—41no—
592C2—41no—
593C2—41no—
594C2—82no—
595C2—161no—
596C2C73×#204411yescharacteristic
597C2C73×#219611yescharacteristic
598C2C73×#220011yescharacteristic
subgroupG/H
center, socleC22×C73B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C73
prime radicalC73×#2669C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C73 ◁ C22×D4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#353×C73C2semidirect
3. C2×AΣL(1,𝔽16)×C73C2semidirect
4. C73×#1889C2semidirect
5. C73×#1887C2semidirect
6. C73×#1867C2semidirect
7. C73×#1390C2semidirect
76. #99×C73D4semidirect
288. #13422C73direct (N × H)
301. C2×#353C2×C73semidirect
302. C2×AΣL(1,𝔽16)C2×C73semidirect
303. #1889C2×C73semidirect
304. #1887C2×C73semidirect
305. #1867C2×C73semidirect
306. #1390C2×C73semidirect
307. C73#13422direct (N × H)
376. #99D4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#26692
C2×#353×C732
C2×AΣL(1,𝔽16)×C732
C73×#18892
C73×#18872
C73×#18672
C73×#13902
#1342273