Finite Group Databasebeta

← C7×#15174 #13192390

Subgroups — C7×#15174 #13192390

Order 1792.

1418 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#398C211yesmaximal, characteristic, complemented
2C2×C7×#397C211yesmaximal, characteristic
3C2×C7×#388C211yesmaximal, characteristic
4C2×C4×C7×C4⋊C4C211yesmaximal, characteristic, complemented
5C7×#1443C212yesmaximal, complemented
6C7×#1441C211yesmaximal, characteristic, complemented
7C7×#1439C211yesmaximal, characteristic
8C7×#1439C212yesmaximal
9C7×#1437C212yesmaximal
10C7×#1436C212yesmaximal, complemented
11C7×#1434C211yesmaximal, characteristic, complemented
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2212yes—
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C2212yes—
20C22×C7×C22⋊C4C2212yes—
21C22×C42×C7C2211yescharacteristic
22C22×C42×C7C2212yes—
23C7×#398—24no—
24C7×#397—24no—
25C7×#388—24no—
26C4×C7×C4⋊C4—24no—
27C2×C7×#95C2211yescharacteristic
28C2×C7×#95C2212yes—
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—
38C2×C7×#95C2212yes—
39#15174C711yesmaximal, characteristic, direct product factor, Sylow
40C25×C7C2311yescharacteristic
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2312yes—
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2312yes—
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C2×C7×C4⋊C4—22no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—42no—
61C2×C7×C4⋊C4D414yes—
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—24no—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—24no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—24no—
71C2×C7×C22⋊C4—24no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—24no—
74C2×C7×C22⋊C4—42no—
75C2×C7×C22⋊C4—42no—
76C2×C42×C7—24no—
77C2×C42×C7—24no—
78C2×C42×C7—42no—
79C2×C42×C7—42no—
80C2×C42×C7D414yes—
81C7×#95—24no—
82C7×#95—22no—
83C7×#95—22no—
84C7×#95—24no—
85C7×#95—22no—
86C7×#95—22no—
87C7×#95—24no—
88C7×#95—24no—
89C7×#95—24no—
90C7×#95—24no—
91C7×#95—24no—
92C7×#95—42no—
93C2×#398C2×C711yescharacteristic, complemented
94C2×#397C2×C711yescharacteristic
95C2×#388C2×C711yescharacteristic
96C2×C4×C4⋊C4C2×C711yescharacteristic, complemented
97#1443C2×C712yescomplemented
98#1441C2×C711yescharacteristic, complemented
99#1439C2×C711yescharacteristic
100#1439C2×C712yes—
101#1437C2×C712yes—
102#1436C2×C712yescomplemented
103#1434C2×C711yescharacteristic, complemented
104C24×C7—21no—
105C24×C7—21no—
106C24×C7—21no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—22no—
110C24×C7—21no—
111C24×C7—21no—
112C24×C7—21no—
113C24×C7—22no—
114C24×C7#4112yes—
115C24×C7C2411yescharacteristic
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—21no—
119C22×C4×C7—21no—
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—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—21no—
148C22×C4×C7—22no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—22no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—22no—
159C22×C4×C7—21no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—21no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×D414yes—
168C22×C4×C7C2×D412yes—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7#4114yes—
172C22×C4×C7#4114yes—
173C22×C4×C7#4112yes—
174C22×C4×C7#4112yes—
175C22×C4×C7#4114yes—
176C7×C4⋊C4—28no—
177C7×C4⋊C4—28no—
178C7×C4⋊C4—48no—
179C7×C4⋊C4—44no—
180C7×C22⋊C4—28no—
181C7×C22⋊C4—48no—
182C7×C22⋊C4—48no—
183C7×C22⋊C4—44no—
184C7×C22⋊C4—44no—
185C42×C7—28no—
186C42×C7—48no—
187C22×C4⋊C4C22×C711yescharacteristic
188C22×C4⋊C4C22×C711yescharacteristic
189C22×C4⋊C4C22×C712yes—
190C22×C4⋊C4C22×C711yescharacteristic
191C22×C22⋊C4C22×C711yescharacteristic
192C22×C22⋊C4C22×C711yescharacteristic
193C22×C22⋊C4C22×C711yescharacteristic
194C22×C22⋊C4C22×C712yes—
195C22×C22⋊C4C22×C712yes—
196C22×C42C22×C711yescharacteristic
197C22×C42C22×C712yes—
198#398—24no—
199#397—24no—
200#388—24no—
201C4×C4⋊C4—24no—
202C2×#95C22×C711yescharacteristic
203C2×#95C22×C712yes—
204C2×#95C22×C711yescharacteristic
205C2×#95C22×C712yes—
206C2×#95C22×C712yes—
207C2×#95C22×C712yes—
208C2×#95C22×C711yescharacteristic
209C2×#95C22×C712yes—
210C2×#95C22×C712yes—
211C2×#95C22×C712yes—
212C2×#95C22×C711yescharacteristic
213C2×#95C22×C712yes—
214C23×C7—22no—
215C23×C7—24no—
216C23×C7—22no—
217C23×C7—24no—
218C23×C7—22no—
219C23×C7—41no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—41no—
225C23×C7—42no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—41no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—41no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7C22×D411yescharacteristic
238C23×C7C22×D411yescharacteristic
239C23×C7C2×#4111yescharacteristic
240C23×C7C2×#4112yes—
241C23×C7C2×#4111yescharacteristic
242C23×C7C2×#4111yescharacteristic
243C23×C7C2×#4112yes—
244C23×C7C2×#4112yes—
245C23×C7H(5,𝔽2)11yescharacteristic
246C23×C7H(5,𝔽2)11yescharacteristic
247C23×C72−1+412yes—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—22no—
263C2×C4×C7—24no—
264C2×C4×C7—22no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—22no—
271C2×C4×C7—41no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—41no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—41no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—41no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—41no—
297C2×C4×C7—42no—
298C2×C4×C7—41no—
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#12114yes—
307C2×C4×C7#12314yes—
308C25C23×C711yescharacteristic
309C23×C4C23×C711yescharacteristic
310C23×C4C23×C711yescharacteristic
311C23×C4C23×C712yes—
312C23×C4C23×C712yes—
313C23×C4C23×C711yescharacteristic
314C23×C4C23×C711yescharacteristic
315C23×C4C23×C712yes—
316C23×C4C23×C712yes—
317C23×C4C23×C711yescharacteristic
318C23×C4C23×C711yescharacteristic
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4C7×D414yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—42no—
343C2×C22⋊C4—42no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—42no—
347C2×C42—42no—
348C2×C42C7×D414yes—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—42no—
361C22×C7—44no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—44no—
366C22×C7—82no—
367C22×C7—82no—
368C22×C7—81no—
369C22×C7—81no—
370C22×C7—81no—
371C22×C7—81no—
372C22×C7C2×#12111yescharacteristic
373C22×C7C2×#12311yescharacteristic
374C22×C7#52511yescharacteristic
375C22×C7#52511yescharacteristic
376C22×C7#52912yes—
377C22×C7#52912yes—
378C22×C7#52912yes—
379C22×C7#53212yes—
380C22×C7#53211yescharacteristic
381C22×C7#53511yescharacteristic
382C22×C7#53611yescharacteristic
383C22×C7#53912yes—
384C22×C7#53912yes—
385C22×C7#54611yescharacteristic
386C22×C7#54611yescharacteristic
387C22×C7#54812yes—
388C22×C7#55112yes—
389C22×C7#55212yes—
390C22×C7#55311yescharacteristic
391C22×C7#55311yescharacteristic
392C22×C7#55312yes—
393C22×C7#55612yes—
394C22×C7#55612yes—
395C4×C7—24no—
396C4×C7—24no—
397C4×C7—44no—
398C4×C7—44no—
399C4×C7—42no—
400C4×C7—44no—
401C4×C7—42no—
402C4×C7—81no—
403C4×C7—81no—
404C4×C7—82no—
405C24—21no—
406C24—21no—
407C24—21no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—21no—
412C24—21no—
413C24—21no—
414C24—22no—
415C24C7×#4112yes—
416C24C24×C711yescharacteristic
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4C2×C7×D412yes—
468C22×C4C2×C7×D414yes—
469C22×C4C2×C7×D412yes—
470C22×C4C2×C7×D412yes—
471C22×C4C2×C7×D412yes—
472C22×C4C7×#4114yes—
473C22×C4C7×#4114yes—
474C22×C4C7×#4112yes—
475C22×C4C7×#4112yes—
476C22×C4C7×#4114yes—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C22⋊C4—28no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C42—28no—
487C42—48no—
488C2×C7—82no—
489C2×C7#239511yescharacteristic
490C2×C7#253511yescharacteristic
491C2×C7#254511yescharacteristic
492C2×C7#255812yes—
493C2×C7#256012yes—
494C2×C7#257712yes—
495C2×C7#259412yes—
496C2×C7#259911yescharacteristic
497C2×C7#260311yescharacteristic
498C2×C7#262311yescharacteristic
499C2×C7#285411yescharacteristic
500C23—22no—
501C23—24no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C22×C7×D411yescharacteristic
524C23C22×C7×D411yescharacteristic
525C23C2×C7×#4111yescharacteristic
526C23C2×C7×#4112yes—
527C23C2×C7×#4111yescharacteristic
528C23C2×C7×#4111yescharacteristic
529C23C2×C7×#4112yes—
530C23C2×C7×#4112yes—
531C23C7×H(5,𝔽2)11yescharacteristic
532C23C7×H(5,𝔽2)11yescharacteristic
533C23C7×2−1+412yes—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4C7×#12114yes—
593C2×C4C7×#12314yes—
594C7#1517411yescharacteristic, direct product factor, Sylow
595C22—42no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22—81no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22C2×C7×#12111yescharacteristic
607C22C2×C7×#12311yescharacteristic
608C22C7×#52511yescharacteristic
609C22C7×#52511yescharacteristic
610C22C7×#52912yes—
611C22C7×#52912yes—
612C22C7×#52912yes—
613C22C7×#53211yescharacteristic
614C22C7×#53212yes—
615C22C7×#53511yescharacteristic
616C22C7×#53611yescharacteristic
617C22C7×#53912yes—
618C22C7×#53912yes—
619C22C7×#54611yescharacteristic
620C22C7×#54611yescharacteristic
621C22C7×#54812yes—
622C22C7×#55112yes—
623C22C7×#55212yes—
624C22C7×#55311yescharacteristic
625C22C7×#55311yescharacteristic
626C22C7×#55312yes—
627C22C7×#55612yes—
628C22C7×#55612yes—
629C4—24no—
630C4—24no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—44no—
635C4—44no—
636C4—81no—
637C4—82no—
638C4—81no—
639C2—82no—
640C2C7×#239511yescharacteristic
641C2C7×#253511yescharacteristic
642C2C7×#254511yescharacteristic
643C2C7×#255812yes—
644C2C7×#256012yes—
645C2C7×#257712yes—
646C2C7×#259412yes—
647C2C7×#259911yescharacteristic
648C2C7×#260311yescharacteristic
649C2C7×#262311yescharacteristic
650C2C7×#285411yescharacteristic
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
1. C2×C7×#398C2semidirect
4. C2×C4×C7×C4⋊C4C2semidirect
5. C7×#1443C2semidirect
6. C7×#1441C2semidirect
10. C7×#1436C2semidirect
11. C7×#1434C2semidirect
39. #15174C7direct (N × H)
93. C2×#398C2×C7semidirect
96. C2×C4×C4⋊C4C2×C7semidirect
97. #1443C2×C7semidirect
98. #1441C2×C7semidirect
102. #1436C2×C7semidirect
103. #1434C2×C7semidirect
594. C7#15174direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3982
C2×C7×#3972
C2×C7×#3882
C2×C4×C7×C4⋊C42
C7×#14432
C7×#14412
C7×#14392
C7×#14372
C7×#14362
C7×#14342
#151747