Finite Group Databasebeta

← C7×#19034 #13196250

Subgroups — C7×#19034 #13196250

Order 1792.

836 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3563C211yesmaximal, characteristic
2C7×#3218C211yesmaximal, characteristic
3C7×#3165C211yesmaximal, characteristic, complemented
4C2×C7×#501C211yesmaximal, characteristic, complemented
5C7×#1663C214yesmaximal, complemented
6C7×#1660C214yesmaximal, 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×#501—22no—
27C7×#501C2214yes—
28C7×#501C2214yes—
29C7×#500—21no—
30C7×#498—21no—
31C7×#497C2212yes—
32C7×#491C2212yes—
33C7×#462—21no—
34C7×#461—21no—
35C7×#451—21no—
36C7×#448—21no—
37C7×#432C2211yescharacteristic
38C7×#428—22no—
39C2×C7×#107—21no—
40C2×C7×#106—21no—
41C4×C7×M16C2211yescharacteristic
42C2×C7×#97C2211yescharacteristic
43C7×#330C2212yes—
44C7×#330C2214yes—
45C7×#330C2214yes—
46C7×#329C2212yes—
47#19034C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—21no—
49C2×C7×Q16—22no—
50C2×C7×SD16—22no—
51C2×C7×M16—21no—
52C22×C7×C8—21no—
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8D411yescharacteristic
60C7×C4⋊Q8D411yescharacteristic
61C7×C4⋊Q8C2312yes—
62C7×#126—22no—
63C7×#126—22no—
64C7×#126—22no—
65C7×#126—22no—
66C7×#126—21no—
67C7×#125—22no—
68C7×#125—22no—
69C7×#125—21no—
70C7×#125—22no—
71C7×#125—22no—
72C7×#125—22no—
73C7×#125—22no—
74C7×#125—22no—
75C7×#125—22no—
76C7×#125—21no—
77C7×#125—21no—
78C7×#125D412yes—
79C7×#125C2311yescharacteristic
80C7×#125C2312yes—
81C7×#125C2312yes—
82C7×#124C2311yescharacteristic
83C7×#123—22no—
84C7×#123—22no—
85C7×#123—21no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8—21no—
88C7×C22⋊Q8—21no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8—21no—
91C7×C22⋊Q8—21no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8D411yescharacteristic
95C7×#121D411yescharacteristic
96C4×C7×Q8—21no—
97C4×C7×Q8—22no—
98C4×C7×Q8—21no—
99C4×C7×Q8—21no—
100C4×C7×Q8—21no—
101C4×C7×Q8D411yescharacteristic
102C4×C7×D4—22no—
103C4×C7×D4—21no—
104C4×C7×D4D411yescharacteristic
105C7×#117—22no—
106C7×#117—21no—
107C7×#117—22no—
108C7×#117—21no—
109C7×#117D412yes—
110C2×C7×C4⋊C4—21no—
111C2×C7×C4⋊C4—21no—
112C2×C7×C4⋊C4—21no—
113C2×C7×C4⋊C4—21no—
114C2×C7×C4⋊C4—21no—
115C2×C7×C4⋊C4D411yescharacteristic
116C2×C7×C4⋊C4D411yescharacteristic
117C2×C42×C7C2311yescharacteristic
118C7×#107—24no—
119C7×#107—24no—
120C7×#106—24no—
121C7×#106—24no—
122C7×C4⋊C8—21no—
123C7×#103—22no—
124C7×#103—22no—
125C7×C22⋊C8—21no—
126C7×#97—21no—
127C7×#97C2312yes—
128C7×#97C2312yes—
129C7×#97C2312yes—
130C4×C7×C8—21no—
131#3563C2×C711yescharacteristic
132#3218C2×C711yescharacteristic
133#3165C2×C711yescharacteristic, complemented
134C2×#501C2×C711yescharacteristic, complemented
135#1663C2×C714yescomplemented
136#1660C2×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—41no—
145C2×C7×Q8—82no—
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×Q16—82no—
158C7×SD16—82no—
159C7×M16—24no—
160C2×C7×C8—24no—
161C2×C7×C8—22no—
162C2×C7×C8—22no—
163C2×C7×C8—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—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—41no—
189C7×C4⋊C4—41no—
190C7×C4⋊C4C2×D411yescharacteristic
191C7×C4⋊C4C2×D411yescharacteristic
192C7×C4⋊C4C2×D412yes—
193C7×C4⋊C4C2×D412yes—
194C7×C4⋊C4C2×D412yes—
195C7×C4⋊C4C2×D412yes—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—21no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—21no—
202C7×C22⋊C4—21no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—21no—
205C42×C7—22no—
206C42×C7—22no—
207C42×C7—22no—
208C42×C7—22no—
209C42×C7—21no—
210C42×C7C2×D411yescharacteristic
211C42×C7C2×D411yescharacteristic
212C42×C7C2×D411yescharacteristic
213C42×C7C2411yescharacteristic
214#564C22×C712yes—
215#563C22×C712yes—
216#557—21no—
217#554—21no—
218#552—21no—
219#552—22no—
220#552—21no—
221#544—21no—
222#544—21no—
223#541—21no—
224#541—22no—
225#539—21no—
226#533C22×C711yescharacteristic
227C2×C4⋊Q8C22×C711yescharacteristic
228#529C22×C711yescharacteristic
229C2×#125C22×C711yescharacteristic
230#501—22no—
231#501C22×C714yes—
232#501C22×C714yes—
233#500—21no—
234#498—21no—
235#497C22×C712yes—
236#491C22×C712yes—
237#462—21no—
238#461—21no—
239#451—21no—
240#448—21no—
241#432C22×C711yescharacteristic
242#428—22no—
243C2×#107—21no—
244C2×#106—21no—
245C4×M16C22×C711yescharacteristic
246C2×#97C22×C711yescharacteristic
247#330C22×C714yes—
248#330C22×C714yes—
249#330C22×C712yes—
250#329C22×C712yes—
251C23×C7—21no—
252C23×C7#12011yescharacteristic
253C7×Q8—41no—
254C7×Q8—42no—
255C7×Q8—42no—
256C7×Q8—42no—
257C7×Q8—81no—
258C7×Q8—82no—
259C7×Q8—82no—
260C7×D4—41no—
261C7×D4—42no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—41no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—81no—
276C2×C4×C7—82no—
277C2×C4×C7—82no—
278C2×C4×C7#12011yescharacteristic
279C2×C4×C7#12011yescharacteristic
280C2×C4×C7#12011yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C22×D411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C7×C8—24no—
285C7×C8—24no—
286C7×C8—42no—
287C22×Q8—21no—
288C2×Q16—22no—
289C2×SD16—22no—
290C2×M16—21no—
291C22×C8—21no—
292C4⋊Q8—22no—
293C4⋊Q8—21no—
294C4⋊Q8—22no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8C7×D411yescharacteristic
299C4⋊Q8C7×D411yescharacteristic
300C4⋊Q8C23×C712yes—
301#126—22no—
302#126—22no—
303#126—22no—
304#126—22no—
305#126—21no—
306#125—22no—
307#125—22no—
308#125—21no—
309#125—22no—
310#125—22no—
311#125—22no—
312#125—22no—
313#125—22no—
314#125—22no—
315#125—21no—
316#125—21no—
317#125C7×D412yes—
318#125C23×C712yes—
319#125C23×C711yescharacteristic
320#125C23×C712yes—
321#124C23×C711yescharacteristic
322#123—22no—
323#123—22no—
324#123—21no—
325C22⋊Q8—22no—
326C22⋊Q8—21no—
327C22⋊Q8—21no—
328C22⋊Q8—22no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8C7×D411yescharacteristic
334#121C7×D411yescharacteristic
335C4×Q8—21no—
336C4×Q8—22no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—21no—
340C4×Q8C7×D411yescharacteristic
341C4×D4—22no—
342C4×D4—21no—
343C4×D4C7×D411yescharacteristic
344#117—22no—
345#117—21no—
346#117—22no—
347#117—21no—
348#117C7×D412yes—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4C7×D411yescharacteristic
355C2×C4⋊C4C7×D411yescharacteristic
356C2×C42C23×C711yescharacteristic
357#107—24no—
358#107—24no—
359#106—24no—
360#106—24no—
361C4⋊C8—21no—
362#103—22no—
363#103—22no—
364C22⋊C8—21no—
365#97—21no—
366#97C23×C712yes—
367#97C23×C712yes—
368#97C23×C712yes—
369C4×C8—21no—
370C22×C7—21no—
371C22×C7—21no—
372C22×C7—41no—
373C22×C7—41no—
374C22×C7—41no—
375C22×C7#45612yes—
376C22×C7C2×#12011yescharacteristic
377C4×C7—21no—
378C4×C7—21no—
379C4×C7—21no—
380C4×C7—21no—
381C4×C7—21no—
382C4×C7—42no—
383C4×C7—41no—
384C4×C7—41no—
385C4×C7—42no—
386C4×C7—42no—
387C4×C7—42no—
388C4×C7—41no—
389C4×C7—82no—
390C4×C7#45612yes—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—41no—
396C2×Q8—82no—
397C2×Q8C2×C7×D411yescharacteristic
398C2×D4—21no—
399C2×D4C2×C7×D411yescharacteristic
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—41no—
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C2×C7×D411yescharacteristic
407C22×C4C2×C7×D411yescharacteristic
408Q16—82no—
409SD16—82no—
410M16—24no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4C2×C7×D411yescharacteristic
442C4⋊C4C2×C7×D411yescharacteristic
443C4⋊C4C2×C7×D412yes—
444C4⋊C4C2×C7×D412yes—
445C4⋊C4C2×C7×D412yes—
446C4⋊C4C2×C7×D412yes—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—21no—
451C22⋊C4—22no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—21no—
461C42C2×C7×D411yescharacteristic
462C42C2×C7×D411yescharacteristic
463C42C2×C7×D411yescharacteristic
464C42C24×C711yescharacteristic
465C2×C7—22no—
466C2×C7—81no—
467C2×C7C2×#45611yescharacteristic
468C2×C7C2×#45611yescharacteristic
469C2×C7#302111yescharacteristic
470C23—21no—
471C23C7×#12011yescharacteristic
472Q8—41no—
473Q8—42no—
474Q8—42no—
475Q8—42no—
476Q8—81no—
477Q8—82no—
478Q8—82no—
479D4—41no—
480D4—42no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4C7×#12011yescharacteristic
498C2×C4C7×#12011yescharacteristic
499C2×C4C7×#12011yescharacteristic
500C2×C4C22×C7×D411yescharacteristic
501C2×C4C22×C7×D411yescharacteristic
502C2×C4C22×C7×D411yescharacteristic
503C8—24no—
504C8—24no—
505C8—42no—
506C7#1903411yescharacteristic, direct product factor, Sylow
507C22—21no—
508C22—21no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22C7×#45612yes—
513C22C2×C7×#12011yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—41no—
526C4—82no—
527C4C7×#45612yes—
528C2—22no—
529C2—81no—
530C2C2×C7×#45611yescharacteristic
531C2C2×C7×#45611yescharacteristic
532C2C7×#302111yescharacteristic
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×#501C2semidirect
5. C7×#1663C2semidirect
6. C7×#1660C2semidirect
7. C7×#1531C2semidirect
8. C7×#1528C2semidirect
47. #19034C7direct (N × H)
133. #3165C2×C7semidirect
134. C2×#501C2×C7semidirect
135. #1663C2×C7semidirect
136. #1660C2×C7semidirect
137. #1531C2×C7semidirect
138. #1528C2×C7semidirect
506. C7#19034direct (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×#32182
C7×#31652
C2×C7×#5012
C7×#16632
C7×#16602
C7×#15312
C7×#15282
C7×#15242
#190347