Finite Group Databasebeta

← C7×#13424 #13190640

Subgroups — C7×#13424 #13190640

Order 1792.

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

normal subgroups only

This group has 612 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
1C7×#2684C211yesmaximal, characteristic
2C2×C7×#354C211yesmaximal, characteristic, complemented
3C2×C7×AΣL(1,𝔽16)C211yesmaximal, characteristic
4C7×#1889C211yesmaximal, characteristic
5C7×#1885C211yesmaximal, characteristic, complemented
6C7×#1867C211yesmaximal, characteristic, complemented
7C7×#1390C211yesmaximal, characteristic, complemented
8C2×C7×#124C2211yescharacteristic
9C2×C7×#120C2211yescharacteristic
10C7×#518C2211yescharacteristic
11C7×#419—21no—
12C2×C7×#100C2211yescharacteristic
13C2×C7×#99C2211yescharacteristic
14C2×C7×#99C2211yescharacteristic
15C7×#397—21no—
16C7×#394—21no—
17C7×#393—21no—
18C7×#388—21no—
19C7×#386—21no—
20C7×#386C2211yescharacteristic
21C7×#385—21no—
22C7×#379—21no—
23C7×#379—21no—
24C7×#354—22no—
25C7×AΣL(1,𝔽16)—22no—
26C7×#342—21no—
27C7×#342—21no—
28C7×#339—21no—
29#13424C711yesmaximal, characteristic, direct product factor, Sylow
30C25×C7D411yescharacteristic
31C22×C7×Q8D411yescharacteristic
32C22×C7×D4—21no—
33C22×C7×D4C2311yescharacteristic
34C23×C4×C7D411yescharacteristic
35C2×C7×M16—21no—
36C7×#124—21no—
37C7×#124—21no—
38C7×#124—21no—
39C7×#124—21no—
40C7×#120—21no—
41C7×#120—21no—
42C7×#120—21no—
43C7×#120—21no—
44C4×C7×D4—41no—
45C4×C7×D4—41no—
46C7×#117—21no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4—41no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C7×C22⋊C4—21no—
53C2×C7×C22⋊C4—21no—
54C2×C7×C22⋊C4—21no—
55C2×C7×C22⋊C4—21no—
56C2×C7×C22⋊C4—21no—
57C2×C7×C22⋊C4—21no—
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—41no—
62C2×C7×C22⋊C4—41no—
63C2×C7×C22⋊C4D411yescharacteristic
64C2×C7×C22⋊C4D411yescharacteristic
65C2×C42×C7D411yescharacteristic
66C7×#103—41no—
67C7×#103—41no—
68C7×#100—21no—
69C7×#100—21no—
70C7×#99—21no—
71C7×#99—21no—
72C7×#99—21no—
73C7×#99D412yes—
74C7×#95—21no—
75C7×#95—21no—
76C7×#95—21no—
77C7×#95—21no—
78C7×#95—41no—
79#2684C2×C711yescharacteristic
80C2×#354C2×C711yescharacteristic, complemented
81C2×AΣL(1,𝔽16)C2×C711yescharacteristic
82#1889C2×C711yescharacteristic
83#1885C2×C711yescharacteristic, complemented
84#1867C2×C711yescharacteristic, complemented
85#1390C2×C711yescharacteristic, complemented
86C24×C7—21no—
87C24×C7—21no—
88C24×C7—21no—
89C24×C7—41no—
90C24×C7—41no—
91C24×C7—42no—
92C24×C7—81no—
93C24×C7C2×D411yescharacteristic
94C24×C7C2×D411yescharacteristic
95C2×C7×Q8—21no—
96C2×C7×Q8—21no—
97C2×C7×Q8—41no—
98C2×C7×Q8—41no—
99C2×C7×D4—41no—
100C2×C7×D4—41no—
101C2×C7×D4—42no—
102C2×C7×D4—41no—
103C2×C7×D4—41no—
104C2×C7×D4—81no—
105C2×C7×D4C2×D411yescharacteristic
106C2×C7×D4#4111yescharacteristic
107C2×C7×D4#4111yescharacteristic
108C2×C7×D4#4111yescharacteristic
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—41no—
119C22×C4×C7—41no—
120C22×C4×C7—41no—
121C22×C4×C7—81no—
122C22×C4×C7C2×D411yescharacteristic
123C7×M16—41no—
124C7×M16—41no—
125C2×C7×C8—41no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—21no—
128C7×C4⋊C4—81no—
129C7×C4⋊C4—82no—
130C7×C22⋊C4—22no—
131C7×C22⋊C4—21no—
132C7×C22⋊C4—21no—
133C7×C22⋊C4—41no—
134C7×C22⋊C4—42no—
135C7×C22⋊C4—41no—
136C7×C22⋊C4—41no—
137C7×C22⋊C4—41no—
138C7×C22⋊C4—41no—
139C7×C22⋊C4—41no—
140C7×C22⋊C4—41no—
141C7×C22⋊C4—41no—
142C7×C22⋊C4—41no—
143C7×C22⋊C4—41no—
144C7×C22⋊C4—41no—
145C7×C22⋊C4—41no—
146C7×C22⋊C4—41no—
147C7×C22⋊C4—81no—
148C7×C22⋊C4—81no—
149C7×C22⋊C4—81no—
150C7×C22⋊C4—81no—
151C7×C22⋊C4—82no—
152C7×C22⋊C4—82no—
153C42×C7—21no—
154C42×C7—21no—
155C42×C7—41no—
156C2×#124C22×C711yescharacteristic
157C2×#120C22×C711yescharacteristic
158#518C22×C711yescharacteristic
159#419—21no—
160C2×#100C22×C711yescharacteristic
161C2×#99C22×C711yescharacteristic
162C2×#99C22×C711yescharacteristic
163#397—21no—
164#394—21no—
165#393—21no—
166#388—21no—
167#386—21no—
168#386C22×C711yescharacteristic
169#385—21no—
170#379—21no—
171#379—21no—
172#354—22no—
173AΣL(1,𝔽16)—22no—
174#342—21no—
175#342—21no—
176#339—21no—
177C23×C7—21no—
178C23×C7—21no—
179C23×C7—21no—
180C23×C7—21no—
181C23×C7—22no—
182C23×C7—41no—
183C23×C7—41no—
184C23×C7—41no—
185C23×C7—41no—
186C23×C7—41no—
187C23×C7—41no—
188C23×C7—41no—
189C23×C7—42no—
190C23×C7—42no—
191C23×C7—42no—
192C23×C7—41no—
193C23×C7—41no—
194C23×C7—41no—
195C23×C7—81no—
196C23×C7—82no—
197C23×C7—81no—
198C23×C7—82no—
199C23×C7—84no—
200C23×C7—81no—
201C23×C7#12011yescharacteristic
202C23×C7#12111yescharacteristic
203C23×C7#12111yescharacteristic
204C23×C7#12311yescharacteristic
205C23×C7#12411yescharacteristic
206C7×Q8—42no—
207C7×Q8—81no—
208C7×D4—41no—
209C7×D4—41no—
210C7×D4—41no—
211C7×D4—41no—
212C7×D4—82no—
213C7×D4—82no—
214C2×C4×C7—21no—
215C2×C4×C7—21no—
216C2×C4×C7—42no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—41no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7—41no—
223C2×C4×C7—41no—
224C2×C4×C7—41no—
225C2×C4×C7—42no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7—81no—
236C2×C4×C7—81no—
237C2×C4×C7—81no—
238C2×C4×C7—81no—
239C2×C4×C7—81no—
240C2×C4×C7—81no—
241C2×C4×C7—81no—
242C2×C4×C7—81no—
243C2×C4×C7—81no—
244C2×C4×C7—81no—
245C2×C4×C7#12111yescharacteristic
246C2×C4×C7#12311yescharacteristic
247C7×C8—81no—
248C25C7×D411yescharacteristic
249C22×Q8C7×D411yescharacteristic
250C22×D4—21no—
251C22×D4C23×C711yescharacteristic
252C23×C4C7×D411yescharacteristic
253C2×M16—21no—
254#124—21no—
255#124—21no—
256#124—21no—
257#124—21no—
258#120—21no—
259#120—21no—
260#120—21no—
261#120—21no—
262C4×D4—41no—
263C4×D4—41no—
264#117—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—41no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4C7×D411yescharacteristic
282C2×C22⋊C4C7×D411yescharacteristic
283C2×C42C7×D411yescharacteristic
284#103—41no—
285#103—41no—
286#100—21no—
287#100—21no—
288#99—21no—
289#99—21no—
290#99—21no—
291#99C7×D412yes—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—41no—
297C22×C7—21no—
298C22×C7—21no—
299C22×C7—21no—
300C22×C7—21no—
301C22×C7—21no—
302C22×C7—21no—
303C22×C7—41no—
304C22×C7—41no—
305C22×C7—41no—
306C22×C7—41no—
307C22×C7—42no—
308C22×C7—42no—
309C22×C7—41no—
310C22×C7—41no—
311C22×C7—41no—
312C22×C7—41no—
313C22×C7—44no—
314C22×C7—42no—
315C22×C7—81no—
316C22×C7—81no—
317C22×C7—81no—
318C22×C7—81no—
319C22×C7—82no—
320C22×C7—82no—
321C22×C7—82no—
322C22×C7—81no—
323C22×C7—81no—
324C22×C7#39411yescharacteristic
325C22×C7B(4,𝔽2)11yescharacteristic
326C22×C7#45811yescharacteristic
327C4×C7—21no—
328C4×C7—21no—
329C4×C7—81no—
330C4×C7—81no—
331C4×C7—81no—
332C4×C7—82no—
333C4×C7—81no—
334C4×C7—81no—
335C4×C7—81no—
336C4×C7—81no—
337C4×C7—161no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—41no—
342C24—41no—
343C24—42no—
344C24—81no—
345C24C2×C7×D411yescharacteristic
346C24C2×C7×D411yescharacteristic
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—42no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—81no—
357C2×D4C2×C7×D411yescharacteristic
358C2×D4C7×#4111yescharacteristic
359C2×D4C7×#4111yescharacteristic
360C2×D4C7×#4111yescharacteristic
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—81no—
374C22×C4C2×C7×D411yescharacteristic
375M16—41no—
376M16—41no—
377C2×C8—41no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—81no—
381C4⋊C4—82no—
382C22⋊C4—22no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—42no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—81no—
400C22⋊C4—81no—
401C22⋊C4—81no—
402C22⋊C4—81no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C42—21no—
406C42—21no—
407C42—41no—
408C2×C7—21no—
409C2×C7—21no—
410C2×C7—41no—
411C2×C7—41no—
412C2×C7—41no—
413C2×C7—41no—
414C2×C7—82no—
415C2×C7#204411yescharacteristic
416C2×C7#219711yescharacteristic
417C2×C7#219911yescharacteristic
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—81no—
437C23—82no—
438C23—81no—
439C23—82no—
440C23—84no—
441C23—81no—
442C23C7×#12011yescharacteristic
443C23C7×#12111yescharacteristic
444C23C7×#12111yescharacteristic
445C23C7×#12311yescharacteristic
446C23C7×#12411yescharacteristic
447Q8—42no—
448Q8—81no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—82no—
454D4—82no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4C7×#12111yescharacteristic
487C2×C4C7×#12311yescharacteristic
488C8—81no—
489C7#1342411yescharacteristic, direct product factor, Sylow
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—42no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—44no—
507C22—42no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—81no—
516C22—81no—
517C22C7×#39411yescharacteristic
518C22C7×B(4,𝔽2)11yescharacteristic
519C22C7×#45811yescharacteristic
520C4—21no—
521C4—21no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—82no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—161no—
531C2—21no—
532C2—21no—
533C2—41no—
534C2—41no—
535C2—41no—
536C2—41no—
537C2—82no—
538C2C7×#204411yescharacteristic
539C2C7×#219711yescharacteristic
540C2C7×#219911yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C7
prime radicalC2×C7×#120C22
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#354C2semidirect
5. C7×#1885C2semidirect
6. C7×#1867C2semidirect
7. C7×#1390C2semidirect
29. #13424C7direct (N × H)
80. C2×#354C2×C7semidirect
83. #1885C2×C7semidirect
84. #1867C2×C7semidirect
85. #1390C2×C7semidirect
489. C7#13424direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26842
C2×C7×#3542
C2×C7×AΣL(1,𝔽16)2
C7×#18892
C7×#18852
C7×#18672
C7×#13902
#134247