Finite Group Databasebeta

← C7×#23616 #13200832

Subgroups — C7×#23616 #13200832

Order 1792.

1102 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#428C211yesmaximal, characteristic, complemented
2C2×C7×#424C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic
4C2×C7×#416C211yesmaximal, characteristic
5C7×#2317C211yesmaximal, characteristic, complemented
6C7×#2311C211yesmaximal, characteristic
7C7×#2068C212yesmaximal, complemented
8C7×#2067C212yesmaximal, complemented
9C7×#2066C212yesmaximal, complemented
10C7×#2065C212yesmaximal, complemented
11C2×C7×#340C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C2×C7×#117C2211yescharacteristic
16C2×C7×#117C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C7×#428—24no—
20C2×C7×#107C2212yes—
21C2×C7×#106C2212yes—
22C7×#424—24no—
23C2×C7×C4⋊C8C2212yes—
24C2×C7×C4⋊C8C2212yes—
25C7×#417—24no—
26C7×#416—24no—
27C2×C7×#103C2211yescharacteristic
28C2×C7×#103C2212yes—
29C2×C7×#103C2211yescharacteristic
30C2×C7×#102C2211yescharacteristic
31C2×C7×#102C2212yes—
32C2×C7×#102C2211yescharacteristic
33C7×#391C2212yes—
34C7×#390C2212yes—
35C7×#389C2212yes—
36C7×#389C2212yes—
37C7×#386—22no—
38C7×#385—22no—
39C7×#384—21no—
40C7×#382—21no—
41C7×#380—21no—
42C7×#380—21no—
43C7×#340C2212yes—
44C7×#340C2212yes—
45#23616C711yesmaximal, characteristic, direct product factor, Sylow
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41Q814yes—
52C22×C7×Q8C2311yescharacteristic
53C22×C7×D4—21no—
54C22×C7×D4C2311yescharacteristic
55C23×C4×C7—21no—
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7C2311yescharacteristic
59C2×C7×M16—22no—
60C2×C7×M16—22no—
61C2×C7×M16D414yes—
62C22×C7×C8—22no—
63C22×C7×C8—22no—
64C22×C7×C8—22no—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2312yes—
68C7×#117—22no—
69C7×#117—22no—
70C7×#117—22no—
71C7×#117D414yes—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4D411yescharacteristic
81C2×C7×C4⋊C4D411yescharacteristic
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C22⋊C4—21no—
87C2×C7×C22⋊C4—22no—
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4D412yes—
90C2×C42×C7—21no—
91C2×C42×C7C2312yes—
92C7×#107—24no—
93C7×#106—24no—
94C7×C4⋊C8—24no—
95C7×C4⋊C8—24no—
96C7×#103—24no—
97C7×#103—22no—
98C7×#103—22no—
99C7×#102—24no—
100C7×#102—22no—
101C7×#102—22no—
102C7×#95—22no—
103C7×#95—22no—
104C7×#95—22no—
105C7×#95—22no—
106C2×#428C2×C711yescharacteristic, complemented
107C2×#424C2×C711yescharacteristic, complemented
108C2×#417C2×C711yescharacteristic
109C2×#416C2×C711yescharacteristic
110#2317C2×C711yescharacteristic, complemented
111#2311C2×C711yescharacteristic
112#2068C2×C712yescomplemented
113#2067C2×C712yescomplemented
114#2066C2×C712yescomplemented
115#2065C2×C712yescomplemented
116C2×#340C2×C711yescharacteristic, complemented
117C24×C7—21no—
118C24×C7C2×D411yescharacteristic
119C7×#41—28no—
120C7×#41—28no—
121C7×#41—44no—
122C7×#41—44no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8C2×Q812yes—
129C2×C7×D4—24no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—42no—
136C2×C7×D4—42no—
137C2×C7×D4C2×Q812yes—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—24no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—42no—
145C22×C4×C7—42no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7—81no—
149C22×C4×C7—81no—
150C22×C4×C7—81no—
151C22×C4×C7—81no—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×Q812yes—
158C22×C4×C7#4112yes—
159C22×C4×C7#4112yes—
160C22×C4×C7#4112yes—
161C22×C4×C7C2411yescharacteristic
162C7×M16—28no—
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—44no—
172C2×C7×C8C2×D414yes—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—42no—
182C7×C4⋊C4—42no—
183C7×C4⋊C4—82no—
184C7×C4⋊C4—82no—
185C7×C4⋊C4C2×D412yes—
186C7×C4⋊C4C2×D412yes—
187C7×C22⋊C4—24no—
188C7×C22⋊C4—24no—
189C7×C22⋊C4—82no—
190C7×C22⋊C4—82no—
191C42×C7—24no—
192C42×C7—22no—
193C42×C7—22no—
194C22×#41C22×C711yescharacteristic
195C22×M16C22×C711yescharacteristic
196C23×C8C22×C711yescharacteristic
197C2×#117C22×C711yescharacteristic
198C2×#117C22×C711yescharacteristic
199C22×C4⋊C4C22×C711yescharacteristic
200C22×C4⋊C4C22×C711yescharacteristic
201#428—24no—
202C2×#107C22×C712yes—
203C2×#106C22×C712yes—
204#424—24no—
205C2×C4⋊C8C22×C712yes—
206C2×C4⋊C8C22×C712yes—
207#417—24no—
208#416—24no—
209C2×#103C22×C711yescharacteristic
210C2×#103C22×C712yes—
211C2×#103C22×C711yescharacteristic
212C2×#102C22×C711yescharacteristic
213C2×#102C22×C712yes—
214C2×#102C22×C711yescharacteristic
215#391C22×C712yes—
216#390C22×C712yes—
217#389C22×C712yes—
218#389C22×C712yes—
219#386—22no—
220#385—22no—
221#384—21no—
222#382—21no—
223#380—21no—
224#380—21no—
225#340C22×C712yes—
226#340C22×C712yes—
227C23×C7—21no—
228C23×C7—21no—
229C23×C7—21no—
230C23×C7—21no—
231C23×C7—22no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7#12112yes—
239C23×C7#12112yes—
240C23×C7C22⋊Q812yes—
241C23×C7C22×D411yescharacteristic
242C7×Q8—22no—
243C7×Q8—22no—
244C7×Q8—41no—
245C7×Q8—41no—
246C7×D4—22no—
247C7×D4—22no—
248C7×D4—48no—
249C7×D4—41no—
250C7×D4—41no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—48no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—84no—
274C2×C4×C7—84no—
275C2×C4×C7#12114yes—
276C2×C4×C7#12112yes—
277C2×C4×C7#12112yes—
278C2×C4×C7C22⋊Q814yes—
279C2×C4×C7C22⋊Q814yes—
280C2×C4×C7C22⋊Q812yes—
281C2×C4×C7#12614yes—
282C2×C4×C7C22×D411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×Q811yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C7×C8—24no—
289C7×C8—41no—
290C7×C8—41no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41C7×Q814yes—
297C22×Q8C23×C711yescharacteristic
298C22×D4—21no—
299C22×D4C23×C711yescharacteristic
300C23×C4—21no—
301C23×C4—21no—
302C23×C4—21no—
303C23×C4C23×C711yescharacteristic
304C2×M16—22no—
305C2×M16—22no—
306C2×M16C7×D414yes—
307C22×C8—22no—
308C22×C8—22no—
309C22×C8—22no—
310C22×C8C23×C711yescharacteristic
311C22×C8C23×C711yescharacteristic
312C22×C8C23×C712yes—
313#117—22no—
314#117—22no—
315#117—22no—
316#117C7×D414yes—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4C7×D411yescharacteristic
326C2×C4⋊C4C7×D411yescharacteristic
327C2×C4⋊C4C23×C712yes—
328C2×C4⋊C4C23×C712yes—
329C2×C4⋊C4C23×C711yescharacteristic
330C2×C4⋊C4C23×C711yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4C7×D412yes—
335C2×C42—21no—
336C2×C42C23×C712yes—
337#107—24no—
338#106—24no—
339C4⋊C8—24no—
340C4⋊C8—24no—
341#103—24no—
342#103—22no—
343#103—22no—
344#102—24no—
345#102—22no—
346#102—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—22no—
355C22×C7—22no—
356C22×C7—22no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—81no—
361C22×C7—81no—
362C22×C7—81no—
363C22×C7—81no—
364C22×C7#39614yes—
365C22×C7C2×#12111yescharacteristic
366C22×C7C2×#12111yescharacteristic
367C22×C7C2×C22⋊Q811yescharacteristic
368C22×C7#57611yescharacteristic
369C22×C7#57711yescharacteristic
370C22×C7#57711yescharacteristic
371C22×C7#57811yescharacteristic
372C4×C7—42no—
373C4×C7—42no—
374C4×C7—42no—
375C4×C7—42no—
376C4×C7—42no—
377C4×C7—82no—
378C4×C7—82no—
379C4×C7#39614yes—
380C4×C7C2×#12111yescharacteristic
381C4×C7C2×C22⋊Q811yescharacteristic
382C4×C7C2×C22⋊Q811yescharacteristic
383C4×C7C2×#12611yescharacteristic
384C24—21no—
385C24C2×C7×D411yescharacteristic
386#41—28no—
387#41—28no—
388#41—44no—
389#41—44no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8C2×C7×Q812yes—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4C2×C7×Q812yes—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4C2×C7×D412yes—
420C22×C4C2×C7×D412yes—
421C22×C4C2×C7×D411yescharacteristic
422C22×C4C2×C7×D412yes—
423C22×C4C2×C7×D412yes—
424C22×C4C2×C7×Q812yes—
425C22×C4C7×#4112yes—
426C22×C4C7×#4112yes—
427C22×C4C7×#4112yes—
428C22×C4C24×C711yescharacteristic
429M16—28no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—44no—
439C2×C8C2×C7×D414yes—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4C2×C7×D412yes—
453C4⋊C4C2×C7×D412yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C42—24no—
459C42—22no—
460C42—22no—
461C2×C7—24no—
462C2×C7—82no—
463C2×C7C2×#39611yescharacteristic
464C2×C7#304511yescharacteristic
465C2×C7#304611yescharacteristic
466C2×C7#306211yescharacteristic
467C2×C7#306311yescharacteristic
468C2×C7#308311yescharacteristic
469C2×C7#308411yescharacteristic
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23C7×#12112yes—
482C23C7×#12112yes—
483C23C7×C22⋊Q812yes—
484C23C22×C7×D411yescharacteristic
485Q8—22no—
486Q8—22no—
487Q8—41no—
488Q8—41no—
489D4—22no—
490D4—22no—
491D4—48no—
492D4—41no—
493D4—41no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4C7×#12114yes—
519C2×C4C7×#12112yes—
520C2×C4C7×#12112yes—
521C2×C4C7×C22⋊Q814yes—
522C2×C4C7×C22⋊Q814yes—
523C2×C4C7×C22⋊Q812yes—
524C2×C4C7×#12614yes—
525C2×C4C22×C7×D411yescharacteristic
526C2×C4C22×C7×D411yescharacteristic
527C2×C4C22×C7×Q811yescharacteristic
528C2×C4C2×C7×#4111yescharacteristic
529C2×C4C2×C7×#4111yescharacteristic
530C2×C4C2×C7×#4111yescharacteristic
531C8—24no—
532C8—41no—
533C8—41no—
534C7#2361611yescharacteristic, direct product factor, Sylow
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22C7×#39614yes—
549C22C2×C7×#12111yescharacteristic
550C22C2×C7×#12111yescharacteristic
551C22C2×C7×C22⋊Q811yescharacteristic
552C22C7×#57611yescharacteristic
553C22C7×#57711yescharacteristic
554C22C7×#57711yescharacteristic
555C22C7×#57811yescharacteristic
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—82no—
562C4—82no—
563C4C7×#39614yes—
564C4C2×C7×#12111yescharacteristic
565C4C2×C7×C22⋊Q811yescharacteristic
566C4C2×C7×C22⋊Q811yescharacteristic
567C4C2×C7×#12611yescharacteristic
568C2—24no—
569C2—82no—
570C2C2×C7×#39611yescharacteristic
571C2C7×#304511yescharacteristic
572C2C7×#304611yescharacteristic
573C2C7×#306211yescharacteristic
574C2C7×#306311yescharacteristic
575C2C7×#308311yescharacteristic
576C2C7×#308411yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#428C2semidirect
2. C2×C7×#424C2semidirect
5. C7×#2317C2semidirect
7. C7×#2068C2semidirect
8. C7×#2067C2semidirect
9. C7×#2066C2semidirect
10. C7×#2065C2semidirect
11. C2×C7×#340C2semidirect
45. #23616C7direct (N × H)
106. C2×#428C2×C7semidirect
107. C2×#424C2×C7semidirect
110. #2317C2×C7semidirect
112. #2068C2×C7semidirect
113. #2067C2×C7semidirect
114. #2066C2×C7semidirect
115. #2065C2×C7semidirect
116. C2×#340C2×C7semidirect
534. C7#23616direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4282
C2×C7×#4242
C2×C7×#4172
C2×C7×#4162
C7×#23172
C7×#23112
C7×#20682
C7×#20672
C7×#20662
C7×#20652
C2×C7×#3402
#236167