Finite Group Databasebeta

← C7×#19023 #13196239

Subgroups — C7×#19023 #13196239

Order 1792.

880 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3559C211yesmaximal, characteristic
2C7×#3239C211yesmaximal, characteristic, complemented
3C7×#3141C211yesmaximal, characteristic, complemented
4C2×C7×#491C211yesmaximal, characteristic, complemented
5C7×#1664C214yesmaximal, complemented
6C7×#1662C214yesmaximal, complemented
7C7×#1530C211yesmaximal, characteristic, complemented
8C7×#1528C211yesmaximal, characteristic, complemented
9C7×#1524C211yesmaximal, characteristic, complemented
10C7×#563C2212yes—
11C7×#562C2212yes—
12C7×#554—21no—
13C7×#553—21no—
14C7×#553—21no—
15C7×#547—22no—
16C7×#547—21no—
17C7×#544—21no—
18C7×#540—21no—
19C7×#539—22no—
20C7×#539—21no—
21C7×#537—21no—
22C7×#532C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic
24C7×#529C2211yescharacteristic
25C2×C7×#125C2211yescharacteristic
26C7×#501C2212yes—
27C7×#491—22no—
28C7×#491C2214yes—
29C7×#489C2212yes—
30C7×#489C2214yes—
31C7×#487—21no—
32C7×#486—21no—
33C7×#484—21no—
34C7×#481—21no—
35C7×#475—21no—
36C7×#474—21no—
37C7×#432C2211yescharacteristic
38C7×#417—22no—
39C2×C7×#103—21no—
40C2×C7×#103—21no—
41C4×C7×M16C2211yescharacteristic
42C2×C7×#97C2211yescharacteristic
43C7×#330C2212yes—
44C7×#330C2214yes—
45C7×#329C2212yes—
46C7×#329C2214yes—
47#19023C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—22no—
49C2×C7×#41D412yes—
50C22×C7×Q8—21no—
51C2×C7×M16—21no—
52C22×C7×C8—21no—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8—22no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8D411yescharacteristic
59C7×C4⋊Q8C2312yes—
60C7×#127D411yescharacteristic
61C7×#126—22no—
62C7×#126—22no—
63C7×#126—21no—
64C7×#125—22no—
65C7×#125—21no—
66C7×#125—21no—
67C7×#125—21no—
68C7×#125D412yes—
69C7×#125C2312yes—
70C7×#125C2311yescharacteristic
71C7×#124C2311yescharacteristic
72C7×#124C2312yes—
73C7×#123—22no—
74C7×#123—22no—
75C7×#123—22no—
76C7×#123—22no—
77C7×#123—21no—
78C7×#123—22no—
79C7×#123—22no—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—22no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8D411yescharacteristic
87C7×#121—22no—
88C7×#121—22no—
89C7×#121—21no—
90C7×#121—21no—
91C7×#121—22no—
92C7×#121—22no—
93C7×#121—21no—
94C7×#121—21no—
95C7×#121D411yescharacteristic
96C4×C7×Q8D411yescharacteristic
97C4×C7×D4—22no—
98C4×C7×D4—22no—
99C4×C7×D4—21no—
100C4×C7×D4—21no—
101C4×C7×D4—21no—
102C4×C7×D4—21no—
103C4×C7×D4—21no—
104C4×C7×D4D411yescharacteristic
105C7×#117—21no—
106C7×#117—21no—
107C2×C7×C4⋊C4—22no—
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×#106—22no—
118C7×C4⋊C8—21no—
119C7×#103—22no—
120C7×#103—24no—
121C7×#103—24no—
122C7×#103—24no—
123C7×#102—22no—
124C7×#102—24no—
125C7×C22⋊C8—21no—
126C7×#97—21no—
127C7×#97C2312yes—
128C7×#97C2312yes—
129C7×#97C2312yes—
130C4×C7×C8—21no—
131#3559C2×C711yescharacteristic
132#3239C2×C711yescharacteristic, complemented
133#3141C2×C711yescharacteristic, complemented
134C2×#491C2×C711yescharacteristic, complemented
135#1664C2×C714yescomplemented
136#1662C2×C714yescomplemented
137#1530C2×C711yescharacteristic, complemented
138#1528C2×C711yescharacteristic, complemented
139#1524C2×C711yescharacteristic, complemented
140C7×#41—44no—
141C7×#41—84no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C2×C7×Q8—44no—
145C2×C7×Q8—41no—
146C2×C7×Q8C2×D411yescharacteristic
147C2×C7×Q8C2×D412yes—
148C2×C7×D4—22no—
149C2×C7×D4—21no—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4C2×D411yescharacteristic
155C2×C7×D4C2×D412yes—
156C22×C4×C7—21no—
157C22×C4×C7—22no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—22no—
163C22×C4×C7—41no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C7×M16—42no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8—42no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—41no—
191C7×C4⋊C4—41no—
192C7×C4⋊C4C2×D412yes—
193C7×C4⋊C4C2×D412yes—
194C7×C4⋊C4C2×D411yescharacteristic
195C7×C4⋊C4C2×D411yescharacteristic
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—21no—
202C7×C22⋊C4—21no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—21no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—21no—
208C42×C7—22no—
209C42×C7—21no—
210C42×C7C2×D411yescharacteristic
211C42×C7C2×D411yescharacteristic
212C42×C7C2×D411yescharacteristic
213C42×C7C2411yescharacteristic
214#563C22×C712yes—
215#562C22×C712yes—
216#554—21no—
217#553—21no—
218#553—21no—
219#547—22no—
220#547—21no—
221#544—21no—
222#540—21no—
223#539—22no—
224#539—21no—
225#537—21no—
226#532C22×C711yescharacteristic
227C2×C4⋊Q8C22×C711yescharacteristic
228#529C22×C711yescharacteristic
229C2×#125C22×C711yescharacteristic
230#501C22×C712yes—
231#491—22no—
232#491C22×C714yes—
233#489C22×C714yes—
234#489C22×C712yes—
235#487—21no—
236#486—21no—
237#484—21no—
238#481—21no—
239#475—21no—
240#474—21no—
241#432C22×C711yescharacteristic
242#417—22no—
243C2×#103—21no—
244C2×#103—21no—
245C4×M16C22×C711yescharacteristic
246C2×#97C22×C711yescharacteristic
247#330C22×C714yes—
248#330C22×C712yes—
249#329C22×C712yes—
250#329C22×C714yes—
251C23×C7—22no—
252C23×C7—21no—
253C23×C7#12011yescharacteristic
254C7×Q8—41no—
255C7×Q8—42no—
256C7×Q8—44no—
257C7×Q8—82no—
258C7×Q8—82no—
259C7×D4—41no—
260C7×D4—42no—
261C7×D4—44no—
262C7×D4—42no—
263C7×D4—42no—
264C7×D4—81no—
265C7×D4—82no—
266C7×D4—82no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—21no—
274C2×C4×C7—22no—
275C2×C4×C7—41no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—82no—
280C2×C4×C7—81no—
281C2×C4×C7—82no—
282C2×C4×C7—82no—
283C2×C4×C7—82no—
284C2×C4×C7—82no—
285C2×C4×C7#12011yescharacteristic
286C2×C4×C7#12011yescharacteristic
287C2×C4×C7#12011yescharacteristic
288C2×C4×C7C22×D411yescharacteristic
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C7×C8—42no—
292C7×C8—42no—
293C7×C8—42no—
294C2×#41—22no—
295C2×#41C7×D412yes—
296C22×Q8—21no—
297C2×M16—21no—
298C22×C8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—22no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8C7×D411yescharacteristic
305C4⋊Q8C23×C712yes—
306#127C7×D411yescharacteristic
307#126—22no—
308#126—22no—
309#126—21no—
310#125—22no—
311#125—21no—
312#125—21no—
313#125—21no—
314#125C7×D412yes—
315#125C23×C712yes—
316#125C23×C711yescharacteristic
317#124C23×C711yescharacteristic
318#124C23×C712yes—
319#123—22no—
320#123—22no—
321#123—22no—
322#123—22no—
323#123—21no—
324#123—22no—
325#123—22no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8C7×D411yescharacteristic
333#121—22no—
334#121—22no—
335#121—21no—
336#121—21no—
337#121—22no—
338#121—22no—
339#121—21no—
340#121—21no—
341#121C7×D411yescharacteristic
342C4×Q8C7×D411yescharacteristic
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—21no—
350C4×D4C7×D411yescharacteristic
351#117—21no—
352#117—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4C7×D411yescharacteristic
360C2×C4⋊C4C7×D411yescharacteristic
361C2×C42C23×C711yescharacteristic
362#107—22no—
363#106—22no—
364C4⋊C8—21no—
365#103—24no—
366#103—22no—
367#103—24no—
368#103—24no—
369#102—24no—
370#102—22no—
371C22⋊C8—21no—
372#97—21no—
373#97C23×C712yes—
374#97C23×C712yes—
375#97C23×C712yes—
376C4×C8—21no—
377C22×C7—21no—
378C22×C7—21no—
379C22×C7—42no—
380C22×C7—41no—
381C22×C7—41no—
382C22×C7—42no—
383C22×C7—42no—
384C22×C7—41no—
385C22×C7#45512yes—
386C22×C7C2×#12011yescharacteristic
387C4×C7—21no—
388C4×C7—21no—
389C4×C7—21no—
390C4×C7—21no—
391C4×C7—21no—
392C4×C7—42no—
393C4×C7—42no—
394C4×C7—41no—
395C4×C7—41no—
396C4×C7—42no—
397C4×C7—41no—
398C4×C7—82no—
399C4×C7#45412yes—
400#41—44no—
401#41—84no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—41no—
406C2×Q8C2×C7×D411yescharacteristic
407C2×Q8C2×C7×D412yes—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C2×C7×D411yescharacteristic
415C2×D4C2×C7×D412yes—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4C2×C7×D411yescharacteristic
425C22×C4C2×C7×D411yescharacteristic
426C22×C4C2×C7×D411yescharacteristic
427M16—42no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—42no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4C2×C7×D412yes—
453C4⋊C4C2×C7×D412yes—
454C4⋊C4C2×C7×D411yescharacteristic
455C4⋊C4C2×C7×D411yescharacteristic
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—21no—
468C42—22no—
469C42—21no—
470C42C2×C7×D411yescharacteristic
471C42C2×C7×D411yescharacteristic
472C42C2×C7×D411yescharacteristic
473C42C24×C711yescharacteristic
474C2×C7—22no—
475C2×C7—82no—
476C2×C7—81no—
477C2×C7C2×#45511yescharacteristic
478C2×C7C2×#45411yescharacteristic
479C2×C7#302211yescharacteristic
480C23—22no—
481C23—21no—
482C23C7×#12011yescharacteristic
483Q8—41no—
484Q8—42no—
485Q8—44no—
486Q8—82no—
487Q8—82no—
488D4—41no—
489D4—42no—
490D4—44no—
491D4—42no—
492D4—42no—
493D4—81no—
494D4—82no—
495D4—82no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4C7×#12011yescharacteristic
515C2×C4C7×#12011yescharacteristic
516C2×C4C7×#12011yescharacteristic
517C2×C4C22×C7×D411yescharacteristic
518C2×C4C22×C7×D411yescharacteristic
519C2×C4C22×C7×D411yescharacteristic
520C8—42no—
521C8—42no—
522C8—42no—
523C7#1902311yescharacteristic, direct product factor, Sylow
524C22—21no—
525C22—21no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—42no—
530C22—42no—
531C22—41no—
532C22C7×#45512yes—
533C22C2×C7×#12011yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—42no—
540C4—42no—
541C4—41no—
542C4—41no—
543C4—42no—
544C4—41no—
545C4—82no—
546C4C7×#45412yes—
547C2—22no—
548C2—82no—
549C2—81no—
550C2C2×C7×#45511yescharacteristic
551C2C2×C7×#45411yescharacteristic
552C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3559C2
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
2. C7×#3239C2semidirect
3. C7×#3141C2semidirect
4. C2×C7×#491C2semidirect
5. C7×#1664C2semidirect
6. C7×#1662C2semidirect
7. C7×#1530C2semidirect
8. C7×#1528C2semidirect
9. C7×#1524C2semidirect
47. #19023C7direct (N × H)
132. #3239C2×C7semidirect
133. #3141C2×C7semidirect
134. C2×#491C2×C7semidirect
135. #1664C2×C7semidirect
136. #1662C2×C7semidirect
137. #1530C2×C7semidirect
138. #1528C2×C7semidirect
139. #1524C2×C7semidirect
523. C7#19023direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35592
C7×#32392
C7×#31412
C2×C7×#4912
C7×#16642
C7×#16622
C7×#15302
C7×#15282
C7×#15242
#190237