Finite Group Databasebeta

← C7×#18789 #13196005

Subgroups — C7×#18789 #13196005

Order 1792.

916 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×#41C211yesmaximal, characteristic
2C7×#3209C211yesmaximal, characteristic, complemented
3C7×#3079C211yesmaximal, characteristic
4C2×C7×#477C211yesmaximal, characteristic, complemented
5C7×#1637C212yesmaximal, complemented
6C7×#1636C212yesmaximal, complemented
7C7×#1633C212yesmaximal, complemented
8C7×#1632C212yesmaximal, complemented
9C7×#1550C211yesmaximal, characteristic, complemented
10C7×#1487C211yesmaximal, characteristic, complemented
11C7×#1477C211yesmaximal, characteristic
12C7×Q82C2211yescharacteristic
13C7×#557—21no—
14C7×#557C2211yescharacteristic
15C7×#554—22no—
16C7×#554—21no—
17C7×#554—21no—
18C7×#554C2211yescharacteristic
19C7×D4×Q8—21no—
20C7×D4×Q8C2211yescharacteristic
21C7×#533—22no—
22C7×#533—21no—
23C7×#533—21no—
24C7×#533C2211yescharacteristic
25C2×C7×C4⋊Q8C2211yescharacteristic
26C4×C7×#41—21no—
27C4×C7×#41C2211yescharacteristic
28C2×C4×C7×Q8—21no—
29C2×C4×C7×Q8C2211yescharacteristic
30C7×#494C2212yes—
31C7×#487C2212yes—
32C7×#477—21no—
33C7×#477—21no—
34C7×#477C2212yes—
35C7×#477C2212yes—
36C7×#477C2212yes—
37C7×#476C2212yes—
38C7×#475—21no—
39C7×#474—21no—
40C7×#462—22no—
41C7×#451—22no—
42C7×#431C2211yescharacteristic
43C7×#428—21no—
44C2×C7×#107—21no—
45C7×#423C2211yescharacteristic
46C2×C7×C4⋊C8C2211yescharacteristic
47C7×#417—21no—
48C2×C7×#103—21no—
49C7×#333C2212yes—
50C7×#332C2212yes—
51C7×Q8⋊C8C2212yes—
52C7×Q8⋊C8C2212yes—
53C7×Q8⋊C8C2212yes—
54C7×D4⋊C8C2212yes—
55#18789C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41D411yescharacteristic
57C22×C7×Q8—21no—
58C22×C7×Q8—21no—
59C22×C7×Q8D411yescharacteristic
60C2×C7×Q16—24no—
61C2×C7×M16—21no—
62C22×C7×C8—21no—
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8—21no—
67C7×C4⋊Q8—21no—
68C7×C4⋊Q8—21no—
69C7×C4⋊Q8—21no—
70C7×C4⋊Q8—21no—
71C7×C4⋊Q8—21no—
72C7×C4⋊Q8C2312yes—
73C7×C4⋊Q8C2311yescharacteristic
74C7×C4⋊Q8C2311yescharacteristic
75C7×#125—22no—
76C7×#125—22no—
77C7×#125—22no—
78C7×#125—21no—
79C7×#125—21no—
80C7×#125—21no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—22no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8—21no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—21no—
89C7×C22⋊Q8—21no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8D412yes—
92C4×C7×Q8—22no—
93C4×C7×Q8—22no—
94C4×C7×Q8—22no—
95C4×C7×Q8—21no—
96C4×C7×Q8—21no—
97C4×C7×Q8—21no—
98C4×C7×Q8—21no—
99C4×C7×Q8—21no—
100C4×C7×Q8—21no—
101C4×C7×Q8—21no—
102C4×C7×Q8D411yescharacteristic
103C4×C7×Q8D411yescharacteristic
104C4×C7×Q8D411yescharacteristic
105C4×C7×Q8D412yes—
106C4×C7×Q8C2311yescharacteristic
107C4×C7×Q8C2311yescharacteristic
108C4×C7×Q8C2311yescharacteristic
109C4×C7×D4—22no—
110C4×C7×D4—22no—
111C4×C7×D4—21no—
112C4×C7×D4—21no—
113C4×C7×D4—21no—
114C4×C7×D4—21no—
115C4×C7×D4D411yescharacteristic
116C4×C7×D4C2311yescharacteristic
117C7×#117—22no—
118C7×#117—21no—
119C7×#117—21no—
120C7×#117D411yescharacteristic
121C2×C7×C4⋊C4—22no—
122C2×C7×C4⋊C4—21no—
123C2×C7×C4⋊C4—21no—
124C2×C7×C4⋊C4—21no—
125C2×C7×C4⋊C4—21no—
126C2×C7×C4⋊C4D411yescharacteristic
127C2×C42×C7—22no—
128C2×C42×C7—21no—
129C2×C42×C7—21no—
130C2×C42×C7C2311yescharacteristic
131C7×#107—22no—
132C7×#107—22no—
133C7×#107—22no—
134C7×#106—22no—
135C7×C4⋊C8—21no—
136C7×C4⋊C8—21no—
137C7×C4⋊C8—21no—
138C7×C4⋊C8C2312yes—
139C7×C4⋊C8C2312yes—
140C7×#103—22no—
141C7×#103—22no—
142C7×#103—22no—
143C7×#103—24no—
144C7×#102—22no—
145C7×C22⋊C8—21no—
146C4×C7×C8C2312yes—
147Q8×#41C2×C711yescharacteristic
148#3209C2×C711yescharacteristic, complemented
149#3079C2×C711yescharacteristic
150C2×#477C2×C711yescharacteristic, complemented
151#1637C2×C712yescomplemented
152#1636C2×C712yescomplemented
153#1633C2×C712yescomplemented
154#1632C2×C712yescomplemented
155#1550C2×C711yescharacteristic, complemented
156#1487C2×C711yescharacteristic, complemented
157#1477C2×C711yescharacteristic
158C7×#41—22no—
159C7×#41Q1614yescomplemented
160C2×C7×Q8—22no—
161C2×C7×Q8—22no—
162C2×C7×Q8—21no—
163C2×C7×Q8—21no—
164C2×C7×Q8—22no—
165C2×C7×Q8—22no—
166C2×C7×Q8—41no—
167C2×C7×Q8—42no—
168C2×C7×Q8—42no—
169C2×C7×Q8—41no—
170C2×C7×Q8—82no—
171C2×C7×Q8C2×D411yescharacteristic
172C2×C7×Q8C2×D411yescharacteristic
173C2×C7×Q8C2×D411yescharacteristic
174C2×C7×Q8C2×D412yes—
175C2×C7×D4—21no—
176C2×C7×D4C2×D411yescharacteristic
177C22×C4×C7—21no—
178C22×C4×C7—22no—
179C22×C4×C7—21no—
180C22×C4×C7—22no—
181C22×C4×C7—21no—
182C22×C4×C7—21no—
183C22×C4×C7—21no—
184C22×C4×C7—41no—
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C22×C4×C7C2×D411yescharacteristic
188C7×Q16—84no—
189C7×M16—42no—
190C2×C7×C8—22no—
191C2×C7×C8—22no—
192C2×C7×C8—22no—
193C2×C7×C8—42no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—21no—
214C7×C4⋊C4—22no—
215C7×C4⋊C4—22no—
216C7×C4⋊C4—41no—
217C7×C4⋊C4—41no—
218C7×C4⋊C4C2×D412yes—
219C7×C4⋊C4C2×D411yescharacteristic
220C7×C4⋊C4C2×D411yescharacteristic
221C7×C4⋊C4C2×D411yescharacteristic
222C7×C4⋊C4C2×D411yescharacteristic
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—21no—
229C42×C7—22no—
230C42×C7—22no—
231C42×C7—21no—
232C42×C7—22no—
233C42×C7—21no—
234C42×C7—21no—
235C42×C7—21no—
236C42×C7—21no—
237C42×C7—21no—
238C42×C7—21no—
239C42×C7C2×D411yescharacteristic
240C42×C7C2×D411yescharacteristic
241C42×C7C2×D411yescharacteristic
242C42×C7C2411yescharacteristic
243Q82C22×C711yescharacteristic
244#557—21no—
245#557C22×C711yescharacteristic
246#554—22no—
247#554—21no—
248#554—21no—
249#554C22×C711yescharacteristic
250D4×Q8—21no—
251D4×Q8C22×C711yescharacteristic
252#533—22no—
253#533—21no—
254#533—21no—
255#533C22×C711yescharacteristic
256C2×C4⋊Q8C22×C711yescharacteristic
257C4×#41—21no—
258C4×#41C22×C711yescharacteristic
259C2×C4×Q8—21no—
260C2×C4×Q8C22×C711yescharacteristic
261#494C22×C712yes—
262#487C22×C712yes—
263#477—21no—
264#477—21no—
265#477C22×C712yes—
266#477C22×C712yes—
267#477C22×C712yes—
268#476C22×C712yes—
269#475—21no—
270#474—21no—
271#462—22no—
272#451—22no—
273#431C22×C711yescharacteristic
274#428—21no—
275C2×#107—21no—
276#423C22×C711yescharacteristic
277C2×C4⋊C8C22×C711yescharacteristic
278#417—21no—
279C2×#103—21no—
280#333C22×C712yes—
281#332C22×C712yes—
282Q8⋊C8C22×C712yes—
283Q8⋊C8C22×C712yes—
284Q8⋊C8C22×C712yes—
285D4⋊C8C22×C712yes—
286C23×C7—21no—
287C23×C7#12011yescharacteristic
288C7×Q8—21no—
289C7×Q8—21no—
290C7×Q8—21no—
291C7×Q8—21no—
292C7×Q8—42no—
293C7×Q8—42no—
294C7×Q8—44no—
295C7×Q8—44no—
296C7×Q8—41no—
297C7×Q8—41no—
298C7×Q8—82no—
299C7×Q8—82no—
300C7×Q8C2×Q1612yes—
301C7×Q8#13712yes—
302C7×D4—24no—
303C7×D4—21no—
304C7×D4C2×Q1612yes—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—24no—
310C2×C4×C7—21no—
311C2×C4×C7—21no—
312C2×C4×C7—22no—
313C2×C4×C7—22no—
314C2×C4×C7—21no—
315C2×C4×C7—21no—
316C2×C4×C7—21no—
317C2×C4×C7—41no—
318C2×C4×C7—42no—
319C2×C4×C7—44no—
320C2×C4×C7—41no—
321C2×C4×C7—41no—
322C2×C4×C7—42no—
323C2×C4×C7—81no—
324C2×C4×C7—81no—
325C2×C4×C7—82no—
326C2×C4×C7—82no—
327C2×C4×C7#12011yescharacteristic
328C2×C4×C7#12011yescharacteristic
329C2×C4×C7#12011yescharacteristic
330C2×C4×C7C2×Q1612yes—
331C2×C4×C7#13712yes—
332C2×C4×C7C22×D411yescharacteristic
333C2×C4×C7C22×D411yescharacteristic
334C2×C4×C7C22×D411yescharacteristic
335C7×C8—42no—
336C7×C8—42no—
337C7×C8—42no—
338C2×#41C7×D411yescharacteristic
339C22×Q8—21no—
340C22×Q8—21no—
341C22×Q8C7×D411yescharacteristic
342C2×Q16—24no—
343C2×M16—21no—
344C22×C8—21no—
345C4⋊Q8—22no—
346C4⋊Q8—21no—
347C4⋊Q8—22no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—21no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8C23×C712yes—
355C4⋊Q8C23×C711yescharacteristic
356C4⋊Q8C23×C711yescharacteristic
357#125—22no—
358#125—22no—
359#125—22no—
360#125—21no—
361#125—21no—
362#125—21no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—21no—
369C22⋊Q8—22no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—22no—
373C22⋊Q8C7×D412yes—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8—21no—
384C4×Q8C7×D411yescharacteristic
385C4×Q8C7×D411yescharacteristic
386C4×Q8C7×D411yescharacteristic
387C4×Q8C7×D412yes—
388C4×Q8C23×C711yescharacteristic
389C4×Q8C23×C711yescharacteristic
390C4×Q8C23×C711yescharacteristic
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4C7×D411yescharacteristic
398C4×D4C23×C711yescharacteristic
399#117—22no—
400#117—21no—
401#117—21no—
402#117C7×D411yescharacteristic
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4C7×D411yescharacteristic
409C2×C42—22no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42C23×C711yescharacteristic
413#107—22no—
414#107—22no—
415#107—22no—
416#106—22no—
417C4⋊C8—21no—
418C4⋊C8—21no—
419C4⋊C8—21no—
420C4⋊C8C23×C712yes—
421C4⋊C8C23×C712yes—
422#103—22no—
423#103—22no—
424#103—22no—
425#103—24no—
426#102—22no—
427C22⋊C8—21no—
428C4×C8C23×C712yes—
429C22×C7—21no—
430C22×C7—21no—
431C22×C7—22no—
432C22×C7—41no—
433C22×C7—41no—
434C22×C7#45112yes—
435C22×C7C2×#12011yescharacteristic
436C4×C7—22no—
437C4×C7—21no—
438C4×C7—22no—
439C4×C7—21no—
440C4×C7—21no—
441C4×C7—41no—
442C4×C7—42no—
443C4×C7—41no—
444C4×C7—41no—
445C4×C7—42no—
446C4×C7—41no—
447C4×C7—41no—
448C4×C7—82no—
449C4×C7#45112yes—
450C4×C7C22×Q1611yescharacteristic
451C4×C7C2×#13711yescharacteristic
452C4×C7C2×#13711yescharacteristic
453C4×C7#57511yescharacteristic
454#41—22no—
455#41C7×Q1614yescomplemented
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—41no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—41no—
466C2×Q8—82no—
467C2×Q8C2×C7×D411yescharacteristic
468C2×Q8C2×C7×D411yescharacteristic
469C2×Q8C2×C7×D411yescharacteristic
470C2×Q8C2×C7×D412yes—
471C2×D4—21no—
472C2×D4C2×C7×D411yescharacteristic
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—41no—
481C22×C4C2×C7×D411yescharacteristic
482C22×C4C2×C7×D411yescharacteristic
483C22×C4C2×C7×D411yescharacteristic
484Q16—84no—
485M16—42no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—42no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4C2×C7×D412yes—
515C4⋊C4C2×C7×D411yescharacteristic
516C4⋊C4C2×C7×D411yescharacteristic
517C4⋊C4C2×C7×D411yescharacteristic
518C4⋊C4C2×C7×D411yescharacteristic
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—22no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42C2×C7×D411yescharacteristic
536C42C2×C7×D411yescharacteristic
537C42C2×C7×D411yescharacteristic
538C42C24×C711yescharacteristic
539C2×C7—22no—
540C2×C7—42no—
541C2×C7C2×#45111yescharacteristic
542C2×C7#300711yescharacteristic
543C2×C7#302111yescharacteristic
544C23—21no—
545C23C7×#12011yescharacteristic
546Q8—21no—
547Q8—21no—
548Q8—21no—
549Q8—21no—
550Q8—42no—
551Q8—42no—
552Q8—44no—
553Q8—44no—
554Q8—41no—
555Q8—41no—
556Q8—82no—
557Q8—82no—
558Q8C2×C7×Q1612yes—
559Q8C7×#13712yes—
560D4—24no—
561D4—21no—
562D4C2×C7×Q1612yes—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—24no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4C7×#12011yescharacteristic
586C2×C4C7×#12011yescharacteristic
587C2×C4C7×#12011yescharacteristic
588C2×C4C2×C7×Q1612yes—
589C2×C4C7×#13712yes—
590C2×C4C22×C7×D411yescharacteristic
591C2×C4C22×C7×D411yescharacteristic
592C2×C4C22×C7×D411yescharacteristic
593C8—42no—
594C8—42no—
595C8—42no—
596C7#1878911yescharacteristic, direct product factor, Sylow
597C22—21no—
598C22—21no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22C7×#45112yes—
603C22C2×C7×#12011yescharacteristic
604C4—22no—
605C4—21no—
606C4—22no—
607C4—21no—
608C4—21no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—82no—
617C4C7×#45112yes—
618C4C22×C7×Q1611yescharacteristic
619C4C2×C7×#13711yescharacteristic
620C4C2×C7×#13711yescharacteristic
621C4C7×#57511yescharacteristic
622C2—22no—
623C2—42no—
624C2C2×C7×#45111yescharacteristic
625C2C7×#300711yescharacteristic
626C2C7×#302111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC2×C7×#41D4
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×#3209C2semidirect
4. C2×C7×#477C2semidirect
5. C7×#1637C2semidirect
6. C7×#1636C2semidirect
7. C7×#1633C2semidirect
8. C7×#1632C2semidirect
9. C7×#1550C2semidirect
10. C7×#1487C2semidirect
55. #18789C7direct (N × H)
148. #3209C2×C7semidirect
150. C2×#477C2×C7semidirect
151. #1637C2×C7semidirect
152. #1636C2×C7semidirect
153. #1633C2×C7semidirect
154. #1632C2×C7semidirect
155. #1550C2×C7semidirect
156. #1487C2×C7semidirect
159. C7×#41Q16semidirect
455. #41C7×Q16semidirect
596. C7#18789direct (N × H)

Maximal subgroups

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

subgroupindex
C7×Q8×#412
C7×#32092
C7×#30792
C2×C7×#4772
C7×#16372
C7×#16362
C7×#16332
C7×#16322
C7×#15502
C7×#14872
C7×#14772
#187897