Finite Group Databasebeta

← C7×#24723 #13201939

Subgroups — C7×#24723 #13201939

Order 1792.

674 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3079C211yesmaximal, characteristic
2C2×C7×#475C211yesmaximal, characteristic, complemented
3C4×C7×#137C211yesmaximal, characteristic
4C7×#2925C211yesmaximal, characteristic
5C7×#2079C212yesmaximal
6C7×#2052C212yesmaximal
7C7×#1997C211yesmaximal, characteristic, complemented
8C7×#1951C212yesmaximal, complemented
9C7×#1950C211yesmaximal, characteristic, complemented
10C7×#1930C212yesmaximal, complemented
11C7×#1801C211yesmaximal, characteristic, complemented
12C2×C7×#137C2211yescharacteristic
13C2×C7×C4⋊Q8C2211yescharacteristic
14C4×C7×#41C2211yescharacteristic
15C2×C4×C7×Q8C2211yescharacteristic
16C7×#477—21no—
17C7×#476—21no—
18C7×#475—21no—
19C7×#475—21no—
20C7×#475—21no—
21C7×#475—22no—
22C7×#474—21no—
23C7×#441—21no—
24C7×#440—21no—
25C4×C7×Q16—21no—
26C4×C7×SD16—21no—
27C7×#433—22no—
28C7×#432—21no—
29C7×#431—21no—
30C2×C7×#108C2212yes—
31C7×#428C2211yescharacteristic
32C7×#428C2212yes—
33C2×C7×#106C2212yes—
34C7×#423C2211yescharacteristic
35C7×#423C2211yescharacteristic
36C2×C7×C4⋊C8C2211yescharacteristic
37C7×#419C2211yescharacteristic
38C7×#417C2211yescharacteristic
39C7×#417C2212yes—
40C2×C7×#103C2212yes—
41C2×C7×#101C2212yes—
42C7×#408C2212yes—
43C7×#405C2212yes—
44C4×C7×M16C2211yescharacteristic
45C7×#343C2212yes—
46C7×#343C2212yes—
47C7×#343C2212yes—
48C7×#341C2212yes—
49#24723C711yesmaximal, characteristic, direct product factor, Sylow
50C2×C7×#41C2311yescharacteristic
51C22×C7×Q8—21no—
52C22×C7×Q8C2311yescharacteristic
53C7×#137—22no—
54C7×#137—22no—
55C2×C7×Q16—21no—
56C2×C7×SD16—21no—
57C2×C7×M16C2311yescharacteristic
58C2×C7×M16C2312yes—
59C2×C7×M16C2312yes—
60C2×C7×M16C2311yescharacteristic
61C22×C7×C8C2312yes—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8—21no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8—21no—
66C4×C7×Q8—21no—
67C4×C7×Q8—21no—
68C4×C7×Q8—21no—
69C4×C7×Q8—21no—
70C4×C7×Q8D411yescharacteristic
71C4×C7×Q8D411yescharacteristic
72C4×C7×D4—21no—
73C4×C7×D4—21no—
74C4×C7×D4D411yescharacteristic
75C4×C7×D4D411yescharacteristic
76C7×#117—21no—
77C7×#117C2311yescharacteristic
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2312yes—
82C2×C42×C7—21no—
83C2×C42×C7—21no—
84C2×C42×C7C2311yescharacteristic
85C7×#108—24no—
86C7×#107—22no—
87C7×#107—21no—
88C7×#106—22no—
89C7×#106—22no—
90C7×#106—22no—
91C7×#106—21no—
92C7×C4⋊C8—21no—
93C7×C4⋊C8—21no—
94C7×C4⋊C8—21no—
95C7×C4⋊C8—21no—
96C7×C4⋊C8D411yescharacteristic
97C7×C4⋊C8D411yescharacteristic
98C7×C4⋊C8D412yes—
99C7×#103—22no—
100C7×#103—21no—
101C7×#103—22no—
102C7×#103—22no—
103C7×#103—21no—
104C7×#103—21no—
105C7×#102—22no—
106C7×#102—21no—
107C7×#101—24no—
108C7×C22⋊C8—22no—
109C7×C22⋊C8—22no—
110C7×#97—22no—
111C7×#97—21no—
112C4×C7×C8—22no—
113C4×C7×C8—21no—
114#3079C2×C711yescharacteristic
115C2×#475C2×C711yescharacteristic, complemented
116C4×#137C2×C711yescharacteristic
117#2925C2×C711yescharacteristic
118#2079C2×C712yes—
119#2052C2×C712yes—
120#1997C2×C711yescharacteristic, complemented
121#1951C2×C712yescomplemented
122#1950C2×C711yescharacteristic, complemented
123#1930C2×C712yescomplemented
124#1801C2×C711yescharacteristic, complemented
125C7×#41—21no—
126C7×#41—21no—
127C7×#41—21no—
128C7×#41—21no—
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8—21no—
133C2×C7×Q8—21no—
134C2×C7×Q8—41no—
135C2×C7×Q8—41no—
136C2×C7×Q8—82no—
137C2×C7×Q8C2×D411yescharacteristic
138C2×C7×Q8#4112yes—
139C2×C7×D4—21no—
140C2×C7×D4C2×D411yescharacteristic
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—22no—
147C22×C4×C7#4111yescharacteristic
148C22×C4×C7#4111yescharacteristic
149C22×C4×C7C2411yescharacteristic
150C7×Q16—24no—
151C7×SD16—24no—
152C7×M16—21no—
153C7×M16—22no—
154C7×M16—22no—
155C7×M16—22no—
156C7×M16—21no—
157C7×M16—21no—
158C7×M16—22no—
159C7×M16#4112yes—
160C2×C7×C8—21no—
161C2×C7×C8—22no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C2×C7×C8—21no—
165C2×C7×C8—22no—
166C2×C7×C8C2×D412yes—
167C2×C7×C8C2×D412yes—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—41no—
175C7×C4⋊C4—41no—
176C7×C4⋊C4—41no—
177C7×C4⋊C4—41no—
178C7×C4⋊C4C2×D411yescharacteristic
179C7×C4⋊C4C2×D411yescharacteristic
180C7×C22⋊C4—21no—
181C7×C22⋊C4—21no—
182C7×C22⋊C4—21no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—41no—
187C42×C7—41no—
188C42×C7C2×D411yescharacteristic
189C42×C7C2×D411yescharacteristic
190C42×C7C2×D411yescharacteristic
191C42×C7C2×D411yescharacteristic
192C2×#137C22×C711yescharacteristic
193C2×C4⋊Q8C22×C711yescharacteristic
194C4×#41C22×C711yescharacteristic
195C2×C4×Q8C22×C711yescharacteristic
196#477—21no—
197#476—21no—
198#475—21no—
199#475—21no—
200#475—21no—
201#475—22no—
202#474—21no—
203#441—21no—
204#440—21no—
205C4×Q16—21no—
206C4×SD16—21no—
207#433—22no—
208#432—21no—
209#431—21no—
210C2×#108C22×C712yes—
211#428C22×C712yes—
212#428C22×C711yescharacteristic
213C2×#106C22×C712yes—
214#423C22×C711yescharacteristic
215#423C22×C711yescharacteristic
216C2×C4⋊C8C22×C711yescharacteristic
217#419C22×C711yescharacteristic
218#417C22×C712yes—
219#417C22×C711yescharacteristic
220C2×#103C22×C712yes—
221C2×#101C22×C712yes—
222#408C22×C712yes—
223#405C22×C712yes—
224C4×M16C22×C711yescharacteristic
225#343C22×C712yes—
226#343C22×C712yes—
227#343C22×C712yes—
228#341C22×C712yes—
229C23×C7—21no—
230C23×C7C2×#4111yescharacteristic
231C7×Q8—22no—
232C7×Q8—21no—
233C7×Q8—21no—
234C7×Q8—22no—
235C7×Q8—41no—
236C7×Q8—41no—
237C7×Q8—82no—
238C7×Q8—82no—
239C7×D4—21no—
240C7×D4—21no—
241C7×D4—41no—
242C7×D4—41no—
243C2×C4×C7—22no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—22no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—41no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—42no—
254C2×C4×C7—41no—
255C2×C4×C7—42no—
256C2×C4×C7—81no—
257C2×C4×C7—81no—
258C2×C4×C7—82no—
259C2×C4×C7#12111yescharacteristic
260C2×C4×C7#12111yescharacteristic
261C2×C4×C7#12111yescharacteristic
262C2×C4×C7#12111yescharacteristic
263C2×C4×C7C22×D411yescharacteristic
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C2×#4111yescharacteristic
266C2×C4×C7C2×#4111yescharacteristic
267C2×C4×C72−1+411yescharacteristic
268C2×C4×C72−1+411yescharacteristic
269C7×C8—22no—
270C7×C8—22no—
271C7×C8—22no—
272C7×C8—42no—
273C7×C8—41no—
274C7×C8—42no—
275C2×#41C23×C711yescharacteristic
276C22×Q8—21no—
277C22×Q8C23×C711yescharacteristic
278#137—22no—
279#137—22no—
280C2×Q16—21no—
281C2×SD16—21no—
282C2×M16C23×C711yescharacteristic
283C2×M16C23×C712yes—
284C2×M16C23×C712yes—
285C2×M16C23×C711yescharacteristic
286C22×C8C23×C712yes—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8C7×D411yescharacteristic
296C4×Q8C7×D411yescharacteristic
297C4×D4—21no—
298C4×D4—21no—
299C4×D4C7×D411yescharacteristic
300C4×D4C7×D411yescharacteristic
301#117—21no—
302#117C23×C711yescharacteristic
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4C23×C711yescharacteristic
306C2×C4⋊C4C23×C712yes—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42C23×C711yescharacteristic
310#108—24no—
311#107—21no—
312#107—22no—
313#106—22no—
314#106—22no—
315#106—21no—
316#106—22no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8C7×D411yescharacteristic
322C4⋊C8C7×D411yescharacteristic
323C4⋊C8C7×D412yes—
324#103—21no—
325#103—22no—
326#103—22no—
327#103—22no—
328#103—21no—
329#103—21no—
330#102—21no—
331#102—22no—
332#101—24no—
333C22⋊C8—22no—
334C22⋊C8—22no—
335#97—22no—
336#97—21no—
337C4×C8—22no—
338C4×C8—21no—
339C22×C7—21no—
340C22×C7—21no—
341C22×C7—41no—
342C22×C7—41no—
343C22×C7—41no—
344C22×C7C2×#12111yescharacteristic
345C22×C7#55211yescharacteristic
346C22×C7#55611yescharacteristic
347C4×C7—21no—
348C4×C7—21no—
349C4×C7—21no—
350C4×C7—41no—
351C4×C7—42no—
352C4×C7—41no—
353C4×C7—41no—
354C4×C7—82no—
355C4×C7—81no—
356C4×C7#47112yes—
357C4×C7#52511yescharacteristic
358C4×C7#53611yescharacteristic
359C4×C7#54711yescharacteristic
360C4×C7#54811yescharacteristic
361#41—21no—
362#41—21no—
363#41—21no—
364#41—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—82no—
373C2×Q8C2×C7×D411yescharacteristic
374C2×Q8C7×#4112yes—
375C2×D4—21no—
376C2×D4C2×C7×D411yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4C7×#4111yescharacteristic
384C22×C4C7×#4111yescharacteristic
385C22×C4C24×C711yescharacteristic
386Q16—24no—
387SD16—24no—
388M16—21no—
389M16—22no—
390M16—22no—
391M16—22no—
392M16—21no—
393M16—21no—
394M16—22no—
395M16C7×#4112yes—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—21no—
401C2×C8—22no—
402C2×C8C2×C7×D412yes—
403C2×C8C2×C7×D412yes—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4C2×C7×D411yescharacteristic
415C4⋊C4C2×C7×D411yescharacteristic
416C22⋊C4—21no—
417C22⋊C4—21no—
418C22⋊C4—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—41no—
423C42—41no—
424C42C2×C7×D411yescharacteristic
425C42C2×C7×D411yescharacteristic
426C42C2×C7×D411yescharacteristic
427C42C2×C7×D411yescharacteristic
428C2×C7—21no—
429C2×C7—21no—
430C2×C7—81no—
431C2×C7#241511yescharacteristic
432C2×C7C2×#47111yescharacteristic
433C2×C7#306911yescharacteristic
434C23—21no—
435C23C2×C7×#4111yescharacteristic
436Q8—22no—
437Q8—21no—
438Q8—21no—
439Q8—22no—
440Q8—41no—
441Q8—41no—
442Q8—82no—
443Q8—82no—
444D4—21no—
445D4—21no—
446D4—41no—
447D4—41no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4C7×#12111yescharacteristic
465C2×C4C7×#12111yescharacteristic
466C2×C4C7×#12111yescharacteristic
467C2×C4C7×#12111yescharacteristic
468C2×C4C22×C7×D411yescharacteristic
469C2×C4C22×C7×D411yescharacteristic
470C2×C4C2×C7×#4111yescharacteristic
471C2×C4C2×C7×#4111yescharacteristic
472C2×C4C7×2−1+411yescharacteristic
473C2×C4C7×2−1+411yescharacteristic
474C8—22no—
475C8—22no—
476C8—22no—
477C8—42no—
478C8—41no—
479C8—42no—
480C7#2472311yescharacteristic, direct product factor, Sylow
481C22—21no—
482C22—21no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22C2×C7×#12111yescharacteristic
487C22C7×#55211yescharacteristic
488C22C7×#55611yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—41no—
493C4—42no—
494C4—41no—
495C4—41no—
496C4—82no—
497C4—81no—
498C4C7×#47112yes—
499C4C7×#52511yescharacteristic
500C4C7×#53611yescharacteristic
501C4C7×#54711yescharacteristic
502C4C7×#54811yescharacteristic
503C2—21no—
504C2—21no—
505C2—81no—
506C2C7×#241511yescharacteristic
507C2C2×C7×#47111yescharacteristic
508C2C7×#306911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#121
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#475C2semidirect
7. C7×#1997C2semidirect
8. C7×#1951C2semidirect
9. C7×#1950C2semidirect
10. C7×#1930C2semidirect
11. C7×#1801C2semidirect
49. #24723C7direct (N × H)
115. C2×#475C2×C7semidirect
120. #1997C2×C7semidirect
121. #1951C2×C7semidirect
122. #1950C2×C7semidirect
123. #1930C2×C7semidirect
124. #1801C2×C7semidirect
480. C7#24723direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#30792
C2×C7×#4752
C4×C7×#1372
C7×#29252
C7×#20792
C7×#20522
C7×#19972
C7×#19512
C7×#19502
C7×#19302
C7×#18012
#247237