Finite Group Databasebeta

← C79×#13422 #34125111

Subgroups — C79×#13422 #34125111

Order 20224.

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
1C79×#2669C211yesmaximal, characteristic
2C2×#353×C79C211yesmaximal, characteristic, complemented
3C2×AΣL(1,𝔽16)×C79C211yesmaximal, characteristic, complemented
4C79×#1889C211yesmaximal, characteristic, complemented
5C79×#1887C211yesmaximal, characteristic, complemented
6C79×#1867C211yesmaximal, characteristic, complemented
7C79×#1390C211yesmaximal, characteristic, complemented
8C2×#127×C79C2211yescharacteristic
9C2×#121×C79—21no—
10C2×#120×C79—21no—
11C2×#120×C79C2211yescharacteristic
12#518×C79C2211yescharacteristic
13#418×C79—21no—
14C2×#100×C79C2211yescharacteristic
15C2×#99×C79C2211yescharacteristic
16C2×#99×C79C2211yescharacteristic
17#394×C79—21no—
18#392×C79—21no—
19#392×C79—21no—
20#388×C79—21no—
21#386×C79C2211yescharacteristic
22#385×C79—21no—
23#379×C79—21no—
24#353×C79—22no—
25AΣL(1,𝔽16)×C79—22no—
26#342×C79—21no—
27#342×C79—21no—
28#339×C79—21no—
29C25×C79D411yescharacteristic
30C22×D4×C79—21no—
31C22×D4×C79—21no—
32C22×D4×C79—21no—
33C22×D4×C79—21no—
34C22×D4×C79—21no—
35C22×D4×C79D411yescharacteristic
36C22×D4×C79C2311yescharacteristic
37C23×C4×C79D411yescharacteristic
38C2×M16×C79—21no—
39#127×C79—21no—
40#127×C79—21no—
41#127×C79—21no—
42#127×C79—21no—
43#121×C79—44no—
44#120×C79—21no—
45#120×C79—21no—
46#120×C79—21no—
47#120×C79—21no—
48#120×C79—44no—
49C4×D4×C79—41no—
50C4×D4×C79—41no—
51#117×C79—21no—
52C2×C4⋊C4×C79—21no—
53C2×C4⋊C4×C79—41no—
54C2×C22⋊C4×C79—21no—
55C2×C22⋊C4×C79—21no—
56C2×C22⋊C4×C79—21no—
57C2×C22⋊C4×C79—21no—
58C2×C22⋊C4×C79—21no—
59C2×C22⋊C4×C79—21no—
60C2×C22⋊C4×C79—21no—
61C2×C22⋊C4×C79—21no—
62C2×C22⋊C4×C79—21no—
63C2×C22⋊C4×C79—21no—
64C2×C22⋊C4×C79—41no—
65C2×C22⋊C4×C79—41no—
66C2×C22⋊C4×C79D411yescharacteristic
67C2×C22⋊C4×C79D411yescharacteristic
68C2×C42×C79D411yescharacteristic
69#102×C79—41no—
70#102×C79—41no—
71#100×C79—21no—
72#100×C79—21no—
73#99×C79—21no—
74#99×C79—21no—
75#99×C79—21no—
76#99×C79D412yescomplemented
77#95×C79—21no—
78#95×C79—21no—
79#95×C79—41no—
80C24×C79—21no—
81C24×C79—21no—
82C24×C79—21no—
83C24×C79—21no—
84C24×C79—41no—
85C24×C79—42no—
86C24×C79—41no—
87C24×C79—81no—
88C24×C79C2×D411yescharacteristic
89C24×C79C2×D411yescharacteristic
90C2×D4×C79—21no—
91C2×D4×C79—21no—
92C2×D4×C79—41no—
93C2×D4×C79—41no—
94C2×D4×C79—41no—
95C2×D4×C79—41no—
96C2×D4×C79—42no—
97C2×D4×C79—42no—
98C2×D4×C79—42no—
99C2×D4×C79—42no—
100C2×D4×C79—42no—
101C2×D4×C79—42no—
102C2×D4×C79—41no—
103C2×D4×C79—41no—
104C2×D4×C79—41no—
105C2×D4×C79—41no—
106C2×D4×C79—41no—
107C2×D4×C79—41no—
108C2×D4×C79—81no—
109C2×D4×C79—81no—
110C2×D4×C79—81no—
111C2×D4×C79—81no—
112C2×D4×C79—81no—
113C2×D4×C79—81no—
114C2×D4×C79C2×D411yescharacteristic
115C2×D4×C79#4111yescharacteristic
116C2×D4×C79#4111yescharacteristic
117C2×D4×C79#4111yescharacteristic
118C22×C4×C79—21no—
119C22×C4×C79—21no—
120C22×C4×C79—21no—
121C22×C4×C79—21no—
122C22×C4×C79—21no—
123C22×C4×C79—21no—
124C22×C4×C79—21no—
125C22×C4×C79—21no—
126C22×C4×C79—42no—
127C22×C4×C79—41no—
128C22×C4×C79—41no—
129C22×C4×C79—41no—
130C22×C4×C79C2×D411yescharacteristic
131M16×C79—41no—
132M16×C79—41no—
133C2×C8×C79—41no—
134C4⋊C4×C79—21no—
135C4⋊C4×C79—21no—
136C4⋊C4×C79—42no—
137C4⋊C4×C79—82no—
138C22⋊C4×C79—24no—
139C22⋊C4×C79—22no—
140C22⋊C4×C79—21no—
141C22⋊C4×C79—21no—
142C22⋊C4×C79—42no—
143C22⋊C4×C79—42no—
144C22⋊C4×C79—42no—
145C22⋊C4×C79—41no—
146C22⋊C4×C79—41no—
147C22⋊C4×C79—41no—
148C22⋊C4×C79—41no—
149C22⋊C4×C79—41no—
150C22⋊C4×C79—41no—
151C22⋊C4×C79—41no—
152C22⋊C4×C79—41no—
153C22⋊C4×C79—41no—
154C22⋊C4×C79—81no—
155C22⋊C4×C79—82no—
156C22⋊C4×C79—82no—
157C42×C79—21no—
158C42×C79—21no—
159C42×C79—41no—
160C23×C79—21no—
161C23×C79—21no—
162C23×C79—21no—
163C23×C79—21no—
164C23×C79—22no—
165C23×C79—41no—
166C23×C79—41no—
167C23×C79—41no—
168C23×C79—41no—
169C23×C79—41no—
170C23×C79—41no—
171C23×C79—41no—
172C23×C79—42no—
173C23×C79—41no—
174C23×C79—41no—
175C23×C79—41no—
176C23×C79—41no—
177C23×C79—42no—
178C23×C79—42no—
179C23×C79—41no—
180C23×C79—41no—
181C23×C79—41no—
182C23×C79—41no—
183C23×C79—41no—
184C23×C79—41no—
185C23×C79—81no—
186C23×C79—82no—
187C23×C79—81no—
188C23×C79—82no—
189C23×C79—84no—
190C23×C79—81no—
191C23×C79#12011yescharacteristic
192C23×C79#12111yescharacteristic
193C23×C79#12111yescharacteristic
194C23×C79#12311yescharacteristic
195C23×C79#12411yescharacteristic
196D4×C79—41no—
197D4×C79—41no—
198D4×C79—41no—
199D4×C79—41no—
200D4×C79—42no—
201D4×C79—81no—
202D4×C79—82no—
203D4×C79—82no—
204D4×C79—82no—
205D4×C79—82no—
206D4×C79—84no—
207D4×C79—82no—
208D4×C79—82no—
209D4×C79—84no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—22no—
213C2×C4×C79—22no—
214C2×C4×C79—41no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79—42no—
219C2×C4×C79—42no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—81no—
229C2×C4×C79—81no—
230C2×C4×C79—81no—
231C2×C4×C79—81no—
232C2×C4×C79—81no—
233C2×C4×C79—81no—
234C2×C4×C79—81no—
235C2×C4×C79—81no—
236C2×C4×C79—81no—
237C2×C4×C79—81no—
238C2×C4×C79#12111yescharacteristic
239C2×C4×C79#12311yescharacteristic
240C8×C79—81no—
241C22×C79—21no—
242C22×C79—21no—
243C22×C79—21no—
244C22×C79—21no—
245C22×C79—21no—
246C22×C79—21no—
247C22×C79—41no—
248C22×C79—41no—
249C22×C79—41no—
250C22×C79—41no—
251C22×C79—42no—
252C22×C79—42no—
253C22×C79—41no—
254C22×C79—41no—
255C22×C79—41no—
256C22×C79—41no—
257C22×C79—44no—
258C22×C79—42no—
259C22×C79—81no—
260C22×C79—81no—
261C22×C79—81no—
262C22×C79—81no—
263C22×C79—81no—
264C22×C79—81no—
265C22×C79—81no—
266C22×C79—81no—
267C22×C79—82no—
268C22×C79—82no—
269C22×C79—82no—
270C22×C79—81no—
271C22×C79—81no—
272C22×C79—81no—
273C22×C79—81no—
274C22×C79—81no—
275C22×C79#39411yescharacteristic
276C22×C79B(4,𝔽2)11yescharacteristic
277C22×C79#45811yescharacteristic
278C4×C79—21no—
279C4×C79—21no—
280C4×C79—42no—
281C4×C79—42no—
282C4×C79—81no—
283C4×C79—81no—
284C4×C79—82no—
285C4×C79—81no—
286C4×C79—81no—
287C4×C79—161no—
288#13422C7911yesmaximal, characteristic, direct product factor, Sylow
289C2×C79—21no—
290C2×C79—21no—
291C2×C79—41no—
292C2×C79—41no—
293C2×C79—41no—
294C2×C79—41no—
295C2×C79—82no—
296C2×C79—161no—
297C2×C79#204411yescharacteristic
298C2×C79#219611yescharacteristic
299C2×C79#220011yescharacteristic
300#2669C2×C7911yescharacteristic
301C2×#353C2×C7911yescharacteristic, complemented
302C2×AΣL(1,𝔽16)C2×C7911yescharacteristic, complemented
303#1889C2×C7911yescharacteristic, complemented
304#1887C2×C7911yescharacteristic, complemented
305#1867C2×C7911yescharacteristic, complemented
306#1390C2×C7911yescharacteristic, complemented
307C79#1342211yescharacteristic, direct product factor, Sylow
308C2×#127C22×C7911yescharacteristic
309C2×#121—21no—
310C2×#120—21no—
311C2×#120C22×C7911yescharacteristic
312#518C22×C7911yescharacteristic
313#418—21no—
314C2×#100C22×C7911yescharacteristic
315C2×#99C22×C7911yescharacteristic
316C2×#99C22×C7911yescharacteristic
317#394—21no—
318#392—21no—
319#392—21no—
320#388—21no—
321#386C22×C7911yescharacteristic
322#385—21no—
323#379—21no—
324#353—22no—
325AΣL(1,𝔽16)—22no—
326#342—21no—
327#342—21no—
328#339—21no—
329C25D4×C7911yescharacteristic
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4D4×C7911yescharacteristic
336C22×D4C23×C7911yescharacteristic
337C23×C4D4×C7911yescharacteristic
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×C7911yescharacteristic
367C2×C22⋊C4D4×C7911yescharacteristic
368C2×C42D4×C7911yescharacteristic
369#102—41no—
370#102—41no—
371#100—21no—
372#100—21no—
373#99—21no—
374#99—21no—
375#99—21no—
376#99D4×C7912yescomplemented
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×C7911yescharacteristic
389C24C2×D4×C7911yescharacteristic
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×C7911yescharacteristic
415C2×D4#41×C7911yescharacteristic
416C2×D4#41×C7911yescharacteristic
417C2×D4#41×C7911yescharacteristic
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×C7911yescharacteristic
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×C7911yescharacteristic
492C23#121×C7911yescharacteristic
493C23#121×C7911yescharacteristic
494C23#123×C7911yescharacteristic
495C23#124×C7911yescharacteristic
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×C7911yescharacteristic
539C2×C4#123×C7911yescharacteristic
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×C7911yescharacteristic
576C22B(4,𝔽2)×C7911yescharacteristic
577C22#458×C7911yescharacteristic
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—
596C2C79×#204411yescharacteristic
597C2C79×#219611yescharacteristic
598C2C79×#220011yescharacteristic
subgroupG/H
center, socleC22×C79B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C79
prime radicalC79×#2669C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C79 ◁ C23×C79 ◁ C22×D4×C79 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#353×C79C2semidirect
3. C2×AΣL(1,𝔽16)×C79C2semidirect
4. C79×#1889C2semidirect
5. C79×#1887C2semidirect
6. C79×#1867C2semidirect
7. C79×#1390C2semidirect
76. #99×C79D4semidirect
288. #13422C79direct (N × H)
301. C2×#353C2×C79semidirect
302. C2×AΣL(1,𝔽16)C2×C79semidirect
303. #1889C2×C79semidirect
304. #1887C2×C79semidirect
305. #1867C2×C79semidirect
306. #1390C2×C79semidirect
307. C79#13422direct (N × H)
376. #99D4×C79semidirect

Maximal subgroups

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

subgroupindex
C79×#26692
C2×#353×C792
C2×AΣL(1,𝔽16)×C792
C79×#18892
C79×#18872
C79×#18672
C79×#13902
#1342279