Finite Group Databasebeta

← C7×#15916 #13193132

Subgroups — C7×#15916 #13193132

Order 1792.

1386 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#399C211yesmaximal, characteristic
2C2×C7×#381C211yesmaximal, characteristic
3C7×#1443C212yesmaximal, complemented
4C7×#1442C212yesmaximal, complemented
5C7×#1442C212yesmaximal, complemented
6C7×#1440C212yesmaximal
7C7×#1437C211yesmaximal, characteristic
8C7×#1436C212yesmaximal, complemented
9C7×C22⋊#95C212yesmaximal
10C24×C4×C7C2211yescharacteristic
11C22×C7×C4⋊C4C2212yes—
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2212yes—
15C22×C7×C22⋊C4C2212yes—
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C42×C7C2211yescharacteristic
18C7×#399—24no—
19C7×#381—24no—
20C2×C7×#95C2211yescharacteristic
21C2×C7×#95C2212yes—
22C2×C7×#95C2212yes—
23C2×C7×#95C2212yes—
24C2×C7×#95C2212yes—
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2211yescharacteristic
32C2×C7×#95C2212yes—
33#15916C711yesmaximal, characteristic, direct product factor, Sylow
34C25×C7C2311yescharacteristic
35C23×C4×C7—22no—
36C23×C4×C7—24no—
37C23×C4×C7—24no—
38C23×C4×C7—41no—
39C23×C4×C7—41no—
40C23×C4×C7C2312yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2312yes—
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2312yes—
48C2×C7×C4⋊C4—24no—
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4—24no—
51C2×C7×C4⋊C4—42no—
52C2×C7×C4⋊C4—41no—
53C2×C7×C4⋊C4—41no—
54C2×C7×C22⋊C4—22no—
55C2×C7×C22⋊C4—24no—
56C2×C7×C22⋊C4—24no—
57C2×C7×C22⋊C4—24no—
58C2×C7×C22⋊C4—24no—
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—41no—
61C2×C7×C22⋊C4—41no—
62C2×C7×C22⋊C4—42no—
63C2×C7×C22⋊C4—42no—
64C2×C7×C22⋊C4D414yes—
65C2×C42×C7—41no—
66C2×C42×C7—42no—
67C7×#95—24no—
68C7×#95—24no—
69C7×#95—22no—
70C7×#95—24no—
71C7×#95—24no—
72C7×#95—24no—
73C7×#95—24no—
74C7×#95—24no—
75C7×#95—42no—
76C7×#95—42no—
77C7×#95—41no—
78C7×#95—42no—
79C7×#95—42no—
80C2×#399C2×C711yescharacteristic
81C2×#381C2×C711yescharacteristic
82#1443C2×C712yescomplemented
83#1442C2×C712yescomplemented
84#1442C2×C712yescomplemented
85#1440C2×C712yes—
86#1437C2×C711yescharacteristic
87#1436C2×C712yescomplemented
88C22⋊#95C2×C712yes—
89C24×C7—22no—
90C24×C7—22no—
91C24×C7—22no—
92C24×C7—22no—
93C24×C7—21no—
94C24×C7—21no—
95C24×C7—22no—
96C24×C7C2×D412yes—
97C24×C7#4114yes—
98C24×C7C2411yescharacteristic
99C22×C4×C7—28no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—21no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—21no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
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—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—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—48no—
148C22×C4×C7—48no—
149C22×C4×C7—44no—
150C22×C4×C7—44no—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7#4114yes—
154C22×C4×C7#4114yes—
155C22×C4×C7#4114yes—
156C7×C4⋊C4—48no—
157C7×C4⋊C4—44no—
158C7×C22⋊C4—44no—
159C7×C22⋊C4—48no—
160C7×C22⋊C4—48no—
161C7×C22⋊C4—44no—
162C42×C7—44no—
163C24×C4C22×C711yescharacteristic
164C22×C4⋊C4C22×C712yes—
165C22×C4⋊C4C22×C711yescharacteristic
166C22×C22⋊C4C22×C711yescharacteristic
167C22×C22⋊C4C22×C712yes—
168C22×C22⋊C4C22×C712yes—
169C22×C22⋊C4C22×C711yescharacteristic
170C22×C42C22×C711yescharacteristic
171#399—24no—
172#381—24no—
173C2×#95C22×C711yescharacteristic
174C2×#95C22×C712yes—
175C2×#95C22×C712yes—
176C2×#95C22×C712yes—
177C2×#95C22×C712yes—
178C2×#95C22×C712yes—
179C2×#95C22×C712yes—
180C2×#95C22×C712yes—
181C2×#95C22×C712yes—
182C2×#95C22×C712yes—
183C2×#95C22×C712yes—
184C2×#95C22×C711yescharacteristic
185C2×#95C22×C712yes—
186C23×C7—22no—
187C23×C7—24no—
188C23×C7—22no—
189C23×C7—24no—
190C23×C7—24no—
191C23×C7—24no—
192C23×C7—24no—
193C23×C7—24no—
194C23×C7—24no—
195C23×C7—24no—
196C23×C7—42no—
197C23×C7—42no—
198C23×C7—42no—
199C23×C7—42no—
200C23×C7—42no—
201C23×C7—42no—
202C23×C7—41no—
203C23×C7—41no—
204C23×C7—41no—
205C23×C7—41no—
206C23×C7#12414yes—
207C23×C7C22×D411yescharacteristic
208C23×C7C2×#4112yes—
209C23×C7C2×#4112yes—
210C23×C7C2×#4112yes—
211C23×C7C2×#4112yes—
212C23×C7H(5,𝔽2)11yescharacteristic
213C23×C7H(5,𝔽2)11yescharacteristic
214C23×C7H(5,𝔽2)12yes—
215C23×C72−1+412yes—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—24no—
221C2×C4×C7—24no—
222C2×C4×C7—24no—
223C2×C4×C7—24no—
224C2×C4×C7—24no—
225C2×C4×C7—24no—
226C2×C4×C7—24no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—416no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—41no—
244C2×C4×C7—41no—
245C2×C4×C7—41no—
246C2×C4×C7—42no—
247C2×C4×C7—41no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C25C23×C711yescharacteristic
270C23×C4—22no—
271C23×C4—24no—
272C23×C4—24no—
273C23×C4—41no—
274C23×C4—41no—
275C23×C4C23×C712yes—
276C23×C4C23×C712yes—
277C23×C4C23×C712yes—
278C23×C4C23×C711yescharacteristic
279C23×C4C23×C711yescharacteristic
280C23×C4C23×C712yes—
281C23×C4C23×C712yes—
282C23×C4C23×C712yes—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—41no—
288C2×C4⋊C4—41no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4C7×D414yes—
300C2×C42—41no—
301C2×C42—42no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—42no—
311#95—42no—
312#95—41no—
313#95—42no—
314#95—42no—
315C22×C7—24no—
316C22×C7—28no—
317C22×C7—44no—
318C22×C7—42no—
319C22×C7—42no—
320C22×C7—44no—
321C22×C7—44no—
322C22×C7—44no—
323C22×C7—44no—
324C22×C7#52511yescharacteristic
325C22×C7C2×#12411yescharacteristic
326C22×C7#52912yes—
327C22×C7#52912yes—
328C22×C7#53411yescharacteristic
329C22×C7#53912yes—
330C22×C7#53912yes—
331C22×C7#54012yes—
332C22×C7#54212yes—
333C22×C7#54612yes—
334C22×C7#54712yes—
335C22×C7#55112yes—
336C22×C7#55112yes—
337C22×C7#55112yes—
338C22×C7#55212yes—
339C22×C7#55212yes—
340C22×C7#55312yes—
341C22×C7#55911yescharacteristic
342C22×C7#55911yescharacteristic
343C22×C7#56011yescharacteristic
344C22×C7#56311yescharacteristic
345C4×C7—44no—
346C4×C7—42no—
347C4×C7—42no—
348C4×C7—44no—
349C4×C7—44no—
350C4×C7—82no—
351C4×C7—82no—
352C4×C7—82no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24C2×C7×D412yes—
361C24C7×#4114yes—
362C24C24×C711yescharacteristic
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4C2×C7×D412yes—
416C22×C4C2×C7×D412yes—
417C22×C4C7×#4114yes—
418C22×C4C7×#4114yes—
419C22×C4C7×#4114yes—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—44no—
426C42—44no—
427C2×C7—44no—
428C2×C7#247012yes—
429C2×C7#248812yes—
430C2×C7#251612yes—
431C2×C7#256111yescharacteristic
432C2×C7#257512yes—
433C2×C7#270911yescharacteristic
434C2×C7#272411yescharacteristic
435C2×C7#273812yes—
436C2×C7#274612yes—
437C23—22no—
438C23—24no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23C7×#12414yes—
458C23C22×C7×D411yescharacteristic
459C23C2×C7×#4112yes—
460C23C2×C7×#4112yes—
461C23C2×C7×#4112yes—
462C23C2×C7×#4112yes—
463C23C7×H(5,𝔽2)11yescharacteristic
464C23C7×H(5,𝔽2)11yescharacteristic
465C23C7×H(5,𝔽2)12yes—
466C23C7×2−1+412yes—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—416no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C7#1591611yescharacteristic, direct product factor, Sylow
521C22—28no—
522C22—24no—
523C22—42no—
524C22—42no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22C7×#52511yescharacteristic
531C22C2×C7×#12411yescharacteristic
532C22C7×#52912yes—
533C22C7×#52912yes—
534C22C7×#53411yescharacteristic
535C22C7×#53912yes—
536C22C7×#53912yes—
537C22C7×#54012yes—
538C22C7×#54212yes—
539C22C7×#54612yes—
540C22C7×#54712yes—
541C22C7×#55112yes—
542C22C7×#55112yes—
543C22C7×#55112yes—
544C22C7×#55212yes—
545C22C7×#55212yes—
546C22C7×#55312yes—
547C22C7×#55911yescharacteristic
548C22C7×#55911yescharacteristic
549C22C7×#56011yescharacteristic
550C22C7×#56311yescharacteristic
551C4—42no—
552C4—42no—
553C4—44no—
554C4—44no—
555C4—44no—
556C4—82no—
557C4—82no—
558C4—82no—
559C2—44no—
560C2C7×#247012yes—
561C2C7×#248812yes—
562C2C7×#251612yes—
563C2C7×#256111yescharacteristic
564C2C7×#257512yes—
565C2C7×#270911yescharacteristic
566C2C7×#272411yescharacteristic
567C2C7×#273812yes—
568C2C7×#274612yes—
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
3. C7×#1443C2semidirect
4. C7×#1442C2semidirect
5. C7×#1442C2semidirect
8. C7×#1436C2semidirect
33. #15916C7direct (N × H)
82. #1443C2×C7semidirect
83. #1442C2×C7semidirect
84. #1442C2×C7semidirect
87. #1436C2×C7semidirect
520. C7#15916direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3992
C2×C7×#3812
C7×#14432
C7×#14422
C7×#14402
C7×#14372
C7×#14362
C7×C22⋊#952
#159167