Finite Group Databasebeta

← C7×#15165 #13192381

Subgroups — C7×#15165 #13192381

Order 1792.

1614 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#396C212yesmaximal
2C2×C7×#389C211yesmaximal, characteristic, complemented
3C2×C4×C7×C22⋊C4C211yesmaximal, characteristic
4C7×#1444C211yesmaximal, characteristic, complemented
5C7×#1442C211yesmaximal, characteristic, complemented
6C7×#1441C212yesmaximal, complemented
7C7×#1439C211yesmaximal, characteristic
8C7×#1439C211yesmaximal, characteristic
9C7×#1436C212yesmaximal, complemented
10C7×C4⋊#95C211yesmaximal, characteristic, complemented
11C7×C22⋊#95C212yesmaximal
12C24×C4×C7C2211yescharacteristic
13C22×C7×C4⋊C4C2212yes—
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2212yes—
16C22×C7×C4⋊C4C2212yes—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4C2212yes—
19C22×C7×C22⋊C4C2212yes—
20C22×C7×C22⋊C4C2211yescharacteristic
21C22×C7×C22⋊C4C2211yescharacteristic
22C22×C42×C7C2211yescharacteristic
23C22×C42×C7C2211yescharacteristic
24C7×#396—28no—
25C7×#389—24no—
26C4×C7×C22⋊C4—24no—
27C2×C7×#95C2212yes—
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2211yescharacteristic
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2211yescharacteristic
33C2×C7×#95C2211yescharacteristic
34C2×C7×#95C2212yes—
35C2×C7×#95C2212yes—
36C2×C7×#95C2212yes—
37C2×C7×#95C2212yes—
38#15165C711yesmaximal, characteristic, direct product factor, Sylow
39C25×C7C2311yescharacteristic
40C23×C4×C7—22no—
41C23×C4×C7—22no—
42C23×C4×C7—42no—
43C23×C4×C7—42no—
44C23×C4×C7Q814yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C23×C4×C7C2312yes—
51C23×C4×C7C2312yes—
52C23×C4×C7C2311yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4—24no—
60C2×C7×C4⋊C4—24no—
61C2×C7×C4⋊C4—24no—
62C2×C7×C4⋊C4—24no—
63C2×C7×C4⋊C4—24no—
64C2×C7×C4⋊C4—24no—
65C2×C7×C4⋊C4—42no—
66C2×C7×C4⋊C4—42no—
67C2×C7×C4⋊C4—42no—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—24no—
73C2×C7×C22⋊C4—24no—
74C2×C7×C22⋊C4—24no—
75C2×C7×C22⋊C4—24no—
76C2×C7×C22⋊C4—42no—
77C2×C7×C22⋊C4—42no—
78C2×C7×C22⋊C4D414yes—
79C2×C7×C22⋊C4Q814yes—
80C2×C42×C7—22no—
81C2×C42×C7—42no—
82C2×C42×C7—42no—
83C2×C42×C7D414yes—
84C7×#95—22no—
85C7×#95—22no—
86C7×#95—24no—
87C7×#95—24no—
88C7×#95—22no—
89C7×#95—24no—
90C7×#95—24no—
91C7×#95—22no—
92C7×#95—42no—
93C7×#95—42no—
94C7×#95—42no—
95C2×#396C2×C712yes—
96C2×#389C2×C711yescharacteristic, complemented
97C2×C4×C22⋊C4C2×C711yescharacteristic
98#1444C2×C711yescharacteristic, complemented
99#1442C2×C711yescharacteristic, complemented
100#1441C2×C712yescomplemented
101#1439C2×C711yescharacteristic
102#1439C2×C711yescharacteristic
103#1436C2×C712yescomplemented
104C4⋊#95C2×C711yescharacteristic, complemented
105C22⋊#95C2×C712yes—
106C24×C7—21no—
107C24×C7—21no—
108C24×C7—21no—
109C24×C7—22no—
110C24×C7—22no—
111C24×C7—21no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7C2×D412yes—
115C24×C7C2×Q812yes—
116C24×C7C2×Q812yes—
117C24×C7C2411yescharacteristic
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—21no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—28no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—48no—
164C22×C4×C7—44no—
165C22×C4×C7—44no—
166C22×C4×C7—48no—
167C22×C4×C7C2×D412yes—
168C22×C4×C7C2×D412yes—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2×Q812yes—
173C22×C4×C7C2×Q812yes—
174C22×C4×C7C2×Q812yes—
175C22×C4×C7C2×Q812yes—
176C22×C4×C7#4114yes—
177C22×C4×C7#4114yes—
178C22×C4×C7#4112yes—
179C22×C4×C7#4112yes—
180C22×C4×C7#4112yes—
181C7×C4⋊C4—28no—
182C7×C4⋊C4—48no—
183C7×C4⋊C4—48no—
184C7×C4⋊C4—44no—
185C7×C4⋊C4—48no—
186C7×C22⋊C4—28no—
187C7×C22⋊C4—28no—
188C7×C22⋊C4—48no—
189C7×C22⋊C4—48no—
190C42×C7—44no—
191C42×C7—44no—
192C24×C4C22×C711yescharacteristic
193C22×C4⋊C4C22×C712yes—
194C22×C4⋊C4C22×C711yescharacteristic
195C22×C4⋊C4C22×C712yes—
196C22×C4⋊C4C22×C712yes—
197C22×C4⋊C4C22×C711yescharacteristic
198C22×C22⋊C4C22×C712yes—
199C22×C22⋊C4C22×C712yes—
200C22×C22⋊C4C22×C711yescharacteristic
201C22×C22⋊C4C22×C711yescharacteristic
202C22×C42C22×C711yescharacteristic
203C22×C42C22×C711yescharacteristic
204#396—28no—
205#389—24no—
206C4×C22⋊C4—24no—
207C2×#95C22×C712yes—
208C2×#95C22×C712yes—
209C2×#95C22×C712yes—
210C2×#95C22×C711yescharacteristic
211C2×#95C22×C711yescharacteristic
212C2×#95C22×C712yes—
213C2×#95C22×C711yescharacteristic
214C2×#95C22×C711yescharacteristic
215C2×#95C22×C712yes—
216C2×#95C22×C712yes—
217C2×#95C22×C712yes—
218C23×C7—22no—
219C23×C7—22no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—24no—
225C23×C7—24no—
226C23×C7—24no—
227C23×C7—24no—
228C23×C7—24no—
229C23×C7—24no—
230C23×C7—41no—
231C23×C7—42no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7C4⋊Q814yes—
241C23×C7C22×D411yescharacteristic
242C23×C7C22×D411yescharacteristic
243C23×C7C22×Q811yescharacteristic
244C23×C7C22×Q811yescharacteristic
245C23×C7C2×#4111yescharacteristic
246C23×C7C2×#4111yescharacteristic
247C23×C7C2×#4111yescharacteristic
248C23×C7C2×#4112yes—
249C23×C7C2×#4112yes—
250C23×C7H(5,𝔽2)12yes—
251C23×C7H(5,𝔽2)12yes—
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—24no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—24no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—416no—
277C2×C4×C7—42no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—42no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—42no—
285C2×C4×C7—41no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7#12114yes—
309C2×C4×C7C22⋊Q814yes—
310C2×C4×C7C22⋊Q814yes—
311C25C23×C711yescharacteristic
312C23×C4—22no—
313C23×C4—22no—
314C23×C4—42no—
315C23×C4—42no—
316C23×C4C7×Q814yes—
317C23×C4C23×C712yes—
318C23×C4C23×C712yes—
319C23×C4C23×C711yescharacteristic
320C23×C4C23×C711yescharacteristic
321C23×C4C23×C711yescharacteristic
322C23×C4C23×C712yes—
323C23×C4C23×C712yes—
324C23×C4C23×C711yescharacteristic
325C23×C4C23×C711yescharacteristic
326C23×C4C23×C711yescharacteristic
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—42no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—42no—
349C2×C22⋊C4—42no—
350C2×C22⋊C4C7×D414yes—
351C2×C22⋊C4C7×Q814yes—
352C2×C42—22no—
353C2×C42—42no—
354C2×C42—42no—
355C2×C42C7×D414yes—
356#95—22no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—42no—
365#95—42no—
366#95—42no—
367C22×C7—24no—
368C22×C7—28no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—44no—
372C22×C7—44no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—44no—
378C22×C7C2×#12111yescharacteristic
379C22×C7C2×C22⋊Q811yescharacteristic
380C22×C7C2×C22⋊Q811yescharacteristic
381C22×C7#52511yescharacteristic
382C22×C7#52912yes—
383C22×C7C2×C4⋊Q811yescharacteristic
384C22×C7#53211yescharacteristic
385C22×C7#53211yescharacteristic
386C22×C7#53311yescharacteristic
387C22×C7#53311yescharacteristic
388C22×C7#53312yes—
389C22×C7#53411yescharacteristic
390C22×C7#53511yescharacteristic
391C22×C7#53812yes—
392C22×C7#54312yes—
393C22×C7#54612yes—
394C22×C7#54612yes—
395C22×C7#55312yes—
396C22×C7#55312yes—
397C22×C7#55312yes—
398C22×C7#55312yes—
399C22×C7#55412yes—
400C22×C7#55412yes—
401C4×C7—24no—
402C4×C7—24no—
403C4×C7—24no—
404C4×C7—42no—
405C4×C7—42no—
406C4×C7—42no—
407C4×C7—44no—
408C4×C7—44no—
409C4×C7—82no—
410C4×C7—82no—
411C24—21no—
412C24—21no—
413C24—21no—
414C24—22no—
415C24—22no—
416C24—21no—
417C24—22no—
418C24—22no—
419C24C2×C7×D412yes—
420C24C2×C7×Q812yes—
421C24C2×C7×Q812yes—
422C24C24×C711yescharacteristic
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4C2×C7×D412yes—
473C22×C4C2×C7×D412yes—
474C22×C4C2×C7×D412yes—
475C22×C4C2×C7×D412yes—
476C22×C4C2×C7×D412yes—
477C22×C4C2×C7×Q812yes—
478C22×C4C2×C7×Q812yes—
479C22×C4C2×C7×Q812yes—
480C22×C4C2×C7×Q812yes—
481C22×C4C7×#4114yes—
482C22×C4C7×#4114yes—
483C22×C4C7×#4112yes—
484C22×C4C7×#4112yes—
485C22×C4C7×#4112yes—
486C4⋊C4—28no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C42—44no—
496C42—44no—
497C2×C7—44no—
498C2×C7#239611yescharacteristic
499C2×C7#253511yescharacteristic
500C2×C7#254311yescharacteristic
501C2×C7#254912yes—
502C2×C7#255912yes—
503C2×C7#256611yescharacteristic
504C2×C7#257911yescharacteristic
505C2×C7#258712yes—
506C2×C7#260812yes—
507C2×C7#262611yescharacteristic
508C2×C7#285711yescharacteristic
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—41no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C7×C4⋊Q814yes—
532C23C22×C7×D411yescharacteristic
533C23C22×C7×D411yescharacteristic
534C23C22×C7×Q811yescharacteristic
535C23C22×C7×Q811yescharacteristic
536C23C2×C7×#4111yescharacteristic
537C23C2×C7×#4111yescharacteristic
538C23C2×C7×#4111yescharacteristic
539C23C2×C7×#4112yes—
540C23C2×C7×#4112yes—
541C23C7×H(5,𝔽2)12yes—
542C23C7×H(5,𝔽2)12yes—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—416no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4C7×#12114yes—
600C2×C4C7×C22⋊Q814yes—
601C2×C4C7×C22⋊Q814yes—
602C7#1516511yescharacteristic, direct product factor, Sylow
603C22—28no—
604C22—24no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22C2×C7×#12111yescharacteristic
615C22C2×C7×C22⋊Q811yescharacteristic
616C22C2×C7×C22⋊Q811yescharacteristic
617C22C7×#52511yescharacteristic
618C22C7×#52912yes—
619C22C2×C7×C4⋊Q811yescharacteristic
620C22C7×#53211yescharacteristic
621C22C7×#53211yescharacteristic
622C22C7×#53312yes—
623C22C7×#53311yescharacteristic
624C22C7×#53311yescharacteristic
625C22C7×#53411yescharacteristic
626C22C7×#53511yescharacteristic
627C22C7×#53812yes—
628C22C7×#54312yes—
629C22C7×#54612yes—
630C22C7×#54612yes—
631C22C7×#55312yes—
632C22C7×#55312yes—
633C22C7×#55312yes—
634C22C7×#55312yes—
635C22C7×#55412yes—
636C22C7×#55412yes—
637C4—24no—
638C4—24no—
639C4—24no—
640C4—42no—
641C4—44no—
642C4—44no—
643C4—42no—
644C4—42no—
645C4—82no—
646C4—82no—
647C2—44no—
648C2C7×#239611yescharacteristic
649C2C7×#253511yescharacteristic
650C2C7×#254311yescharacteristic
651C2C7×#254912yes—
652C2C7×#255912yes—
653C2C7×#256611yescharacteristic
654C2C7×#257911yescharacteristic
655C2C7×#258712yes—
656C2C7×#260812yes—
657C2C7×#262611yescharacteristic
658C2C7×#285711yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC24C24×C7
prime radicalC25×C7C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#389C2semidirect
4. C7×#1444C2semidirect
5. C7×#1442C2semidirect
6. C7×#1441C2semidirect
9. C7×#1436C2semidirect
10. C7×C4⋊#95C2semidirect
38. #15165C7direct (N × H)
96. C2×#389C2×C7semidirect
98. #1444C2×C7semidirect
99. #1442C2×C7semidirect
100. #1441C2×C7semidirect
103. #1436C2×C7semidirect
104. C4⋊#95C2×C7semidirect
602. C7#15165direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3962
C2×C7×#3892
C2×C4×C7×C22⋊C42
C7×#14442
C7×#14422
C7×#14412
C7×#14392
C7×#14362
C7×C4⋊#952
C7×C22⋊#952
#151657