Finite Group Databasebeta

← C7×#19035 #13196251

Subgroups — C7×#19035 #13196251

Order 1792.

824 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3563C211yesmaximal, characteristic
2C7×#3239C211yesmaximal, characteristic
3C7×#3165C211yesmaximal, characteristic, complemented
4C2×C7×#491C211yesmaximal, characteristic, complemented
5C7×#1664C214yesmaximal, complemented
6C7×#1663C214yesmaximal, complemented
7C7×#1531C211yesmaximal, characteristic, complemented
8C7×#1528C211yesmaximal, characteristic, complemented
9C7×#1524C211yesmaximal, characteristic
10C7×#564C2212yes—
11C7×#563C2212yes—
12C7×#557—21no—
13C7×#554—21no—
14C7×#552—21no—
15C7×#552—22no—
16C7×#552—21no—
17C7×#544—21no—
18C7×#544—21no—
19C7×#541—21no—
20C7×#541—22no—
21C7×#539—21no—
22C7×#533C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic
24C7×#529C2211yescharacteristic
25C2×C7×#125C2211yescharacteristic
26C7×#501C2214yes—
27C7×#501C2212yes—
28C7×#500—21no—
29C7×#498—21no—
30C7×#491—22no—
31C7×#491C2214yes—
32C7×#489C2212yes—
33C7×#484—21no—
34C7×#481—21no—
35C7×#475—21no—
36C7×#474—21no—
37C7×#432C2211yescharacteristic
38C7×#428—22no—
39C2×C7×#103—21no—
40C2×C7×#103—21no—
41C4×C7×M16C2211yescharacteristic
42C2×C7×#97C2211yescharacteristic
43C7×#330C2212yes—
44C7×#330C2214yes—
45C7×#330C2214yes—
46C7×#329C2212yes—
47#19035C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—21no—
49C2×C7×M16—21no—
50C22×C7×C8—21no—
51C7×C4⋊Q8—22no—
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8D411yescharacteristic
58C7×C4⋊Q8D411yescharacteristic
59C7×C4⋊Q8C2312yes—
60C7×#126—22no—
61C7×#126—22no—
62C7×#126—22no—
63C7×#126—22no—
64C7×#126—21no—
65C7×#125—22no—
66C7×#125—22no—
67C7×#125—21no—
68C7×#125—22no—
69C7×#125—22no—
70C7×#125—22no—
71C7×#125—22no—
72C7×#125—22no—
73C7×#125—22no—
74C7×#125—21no—
75C7×#125—21no—
76C7×#125D412yes—
77C7×#125C2312yes—
78C7×#125C2311yescharacteristic
79C7×#125C2312yes—
80C7×#124C2311yescharacteristic
81C7×#123—22no—
82C7×#123—22no—
83C7×#123—21no—
84C7×C22⋊Q8—22no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—21no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—21no—
89C7×C22⋊Q8—21no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8D411yescharacteristic
93C7×#121D411yescharacteristic
94C4×C7×Q8—21no—
95C4×C7×Q8—22no—
96C4×C7×Q8—21no—
97C4×C7×Q8—21no—
98C4×C7×Q8—21no—
99C4×C7×Q8D411yescharacteristic
100C4×C7×D4—22no—
101C4×C7×D4—21no—
102C4×C7×D4D411yescharacteristic
103C7×#117—22no—
104C7×#117—21no—
105C7×#117—22no—
106C7×#117—21no—
107C7×#117D412yes—
108C2×C7×C4⋊C4—21no—
109C2×C7×C4⋊C4—21no—
110C2×C7×C4⋊C4—21no—
111C2×C7×C4⋊C4—21no—
112C2×C7×C4⋊C4—21no—
113C2×C7×C4⋊C4D411yescharacteristic
114C2×C7×C4⋊C4D411yescharacteristic
115C2×C42×C7C2311yescharacteristic
116C7×#107—22no—
117C7×#107—24no—
118C7×#106—22no—
119C7×#106—24no—
120C7×C4⋊C8—21no—
121C7×#103—22no—
122C7×#103—24no—
123C7×#103—24no—
124C7×#102—22no—
125C7×C22⋊C8—21no—
126C7×#97—21no—
127C7×#97C2312yes—
128C7×#97C2312yes—
129C7×#97C2312yes—
130C4×C7×C8—21no—
131#3563C2×C711yescharacteristic
132#3239C2×C711yescharacteristic
133#3165C2×C711yescharacteristic, complemented
134C2×#491C2×C711yescharacteristic, complemented
135#1664C2×C714yescomplemented
136#1663C2×C714yescomplemented
137#1531C2×C711yescharacteristic, complemented
138#1528C2×C711yescharacteristic, complemented
139#1524C2×C711yescharacteristic
140C2×C7×Q8—22no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—22no—
144C2×C7×Q8—44no—
145C2×C7×Q8—41no—
146C2×C7×Q8C2×D411yescharacteristic
147C2×C7×D4—21no—
148C2×C7×D4C2×D411yescharacteristic
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—41no—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C7×M16—24no—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C2×C7×C8—22no—
161C2×C7×C8—42no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—41no—
187C7×C4⋊C4—41no—
188C7×C4⋊C4C2×D412yes—
189C7×C4⋊C4C2×D412yes—
190C7×C4⋊C4C2×D412yes—
191C7×C4⋊C4C2×D411yescharacteristic
192C7×C4⋊C4C2×D412yes—
193C7×C4⋊C4C2×D411yescharacteristic
194C7×C22⋊C4—22no—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—21no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—21no—
200C7×C22⋊C4—21no—
201C7×C22⋊C4—21no—
202C7×C22⋊C4—21no—
203C42×C7—22no—
204C42×C7—22no—
205C42×C7—22no—
206C42×C7—22no—
207C42×C7—21no—
208C42×C7C2×D411yescharacteristic
209C42×C7C2×D411yescharacteristic
210C42×C7C2×D411yescharacteristic
211C42×C7C2411yescharacteristic
212#564C22×C712yes—
213#563C22×C712yes—
214#557—21no—
215#554—21no—
216#552—21no—
217#552—22no—
218#552—21no—
219#544—21no—
220#544—21no—
221#541—21no—
222#541—22no—
223#539—21no—
224#533C22×C711yescharacteristic
225C2×C4⋊Q8C22×C711yescharacteristic
226#529C22×C711yescharacteristic
227C2×#125C22×C711yescharacteristic
228#501C22×C714yes—
229#501C22×C712yes—
230#500—21no—
231#498—21no—
232#491—22no—
233#491C22×C714yes—
234#489C22×C712yes—
235#484—21no—
236#481—21no—
237#475—21no—
238#474—21no—
239#432C22×C711yescharacteristic
240#428—22no—
241C2×#103—21no—
242C2×#103—21no—
243C4×M16C22×C711yescharacteristic
244C2×#97C22×C711yescharacteristic
245#330C22×C712yes—
246#330C22×C714yes—
247#330C22×C714yes—
248#329C22×C712yes—
249C23×C7—21no—
250C23×C7#12011yescharacteristic
251C7×Q8—41no—
252C7×Q8—44no—
253C7×Q8—42no—
254C7×Q8—81no—
255C7×Q8—82no—
256C7×Q8—82no—
257C7×D4—41no—
258C7×D4—42no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—41no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—81no—
273C2×C4×C7—82no—
274C2×C4×C7—82no—
275C2×C4×C7#12011yescharacteristic
276C2×C4×C7#12011yescharacteristic
277C2×C4×C7#12011yescharacteristic
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C7×C8—24no—
282C7×C8—42no—
283C7×C8—42no—
284C22×Q8—21no—
285C2×M16—21no—
286C22×C8—21no—
287C4⋊Q8—22no—
288C4⋊Q8—21no—
289C4⋊Q8—22no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8C7×D411yescharacteristic
294C4⋊Q8C7×D411yescharacteristic
295C4⋊Q8C23×C712yes—
296#126—22no—
297#126—22no—
298#126—22no—
299#126—22no—
300#126—21no—
301#125—22no—
302#125—22no—
303#125—21no—
304#125—22no—
305#125—22no—
306#125—22no—
307#125—22no—
308#125—22no—
309#125—22no—
310#125—21no—
311#125—21no—
312#125C7×D412yes—
313#125C23×C712yes—
314#125C23×C711yescharacteristic
315#125C23×C712yes—
316#124C23×C711yescharacteristic
317#123—22no—
318#123—22no—
319#123—21no—
320C22⋊Q8—22no—
321C22⋊Q8—21no—
322C22⋊Q8—21no—
323C22⋊Q8—22no—
324C22⋊Q8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8C7×D411yescharacteristic
329#121C7×D411yescharacteristic
330C4×Q8—21no—
331C4×Q8—22no—
332C4×Q8—21no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8C7×D411yescharacteristic
336C4×D4—22no—
337C4×D4—21no—
338C4×D4C7×D411yescharacteristic
339#117—22no—
340#117—21no—
341#117—22no—
342#117—21no—
343#117C7×D412yes—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C7×D411yescharacteristic
350C2×C4⋊C4C7×D411yescharacteristic
351C2×C42C23×C711yescharacteristic
352#107—24no—
353#107—22no—
354#106—24no—
355#106—22no—
356C4⋊C8—21no—
357#103—22no—
358#103—24no—
359#103—24no—
360#102—22no—
361C22⋊C8—21no—
362#97—21no—
363#97C23×C712yes—
364#97C23×C712yes—
365#97C23×C712yes—
366C4×C8—21no—
367C22×C7—21no—
368C22×C7—21no—
369C22×C7—41no—
370C22×C7—41no—
371C22×C7—41no—
372C22×C7#45612yes—
373C22×C7C2×#12011yescharacteristic
374C4×C7—21no—
375C4×C7—21no—
376C4×C7—21no—
377C4×C7—21no—
378C4×C7—21no—
379C4×C7—41no—
380C4×C7—42no—
381C4×C7—42no—
382C4×C7—41no—
383C4×C7—42no—
384C4×C7—42no—
385C4×C7—41no—
386C4×C7—82no—
387C4×C7#45412yes—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—41no—
394C2×Q8C2×C7×D411yescharacteristic
395C2×D4—21no—
396C2×D4C2×C7×D411yescharacteristic
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—41no—
402C22×C4C2×C7×D411yescharacteristic
403C22×C4C2×C7×D411yescharacteristic
404C22×C4C2×C7×D411yescharacteristic
405M16—24no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—42no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4C2×C7×D412yes—
437C4⋊C4C2×C7×D412yes—
438C4⋊C4C2×C7×D412yes—
439C4⋊C4C2×C7×D411yescharacteristic
440C4⋊C4C2×C7×D412yes—
441C4⋊C4C2×C7×D411yescharacteristic
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—21no—
446C22⋊C4—22no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—21no—
456C42C2×C7×D411yescharacteristic
457C42C2×C7×D411yescharacteristic
458C42C2×C7×D411yescharacteristic
459C42C24×C711yescharacteristic
460C2×C7—22no—
461C2×C7—81no—
462C2×C7C2×#45411yescharacteristic
463C2×C7C2×#45611yescharacteristic
464C2×C7#302211yescharacteristic
465C23—21no—
466C23C7×#12011yescharacteristic
467Q8—41no—
468Q8—44no—
469Q8—42no—
470Q8—81no—
471Q8—82no—
472Q8—82no—
473D4—41no—
474D4—42no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4C7×#12011yescharacteristic
492C2×C4C7×#12011yescharacteristic
493C2×C4C7×#12011yescharacteristic
494C2×C4C22×C7×D411yescharacteristic
495C2×C4C22×C7×D411yescharacteristic
496C2×C4C22×C7×D411yescharacteristic
497C8—24no—
498C8—42no—
499C8—42no—
500C7#1903511yescharacteristic, direct product factor, Sylow
501C22—21no—
502C22—21no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22C7×#45612yes—
507C22C2×C7×#12011yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—42no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—41no—
520C4—82no—
521C4C7×#45412yes—
522C2—22no—
523C2—81no—
524C2C2×C7×#45411yescharacteristic
525C2C2×C7×#45611yescharacteristic
526C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#3165C2semidirect
4. C2×C7×#491C2semidirect
5. C7×#1664C2semidirect
6. C7×#1663C2semidirect
7. C7×#1531C2semidirect
8. C7×#1528C2semidirect
47. #19035C7direct (N × H)
133. #3165C2×C7semidirect
134. C2×#491C2×C7semidirect
135. #1664C2×C7semidirect
136. #1663C2×C7semidirect
137. #1531C2×C7semidirect
138. #1528C2×C7semidirect
500. C7#19035direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35632
C7×#32392
C7×#31652
C2×C7×#4912
C7×#16642
C7×#16632
C7×#15312
C7×#15282
C7×#15242
#190357