Finite Group Databasebeta

← C7×#24255 #13201471

Subgroups — C7×#24255 #13201471

Order 1792.

1392 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×D4×Q8C211yesmaximal, characteristic
2C2×C7×#462C211yesmaximal, characteristic, complemented
3C2×C7×#451C212yesmaximal, complemented
4C2×C7×#441C211yesmaximal, characteristic, complemented
5C7×#2536C211yesmaximal, characteristic
6C7×#2035C212yesmaximal, complemented
7C7×#2001C212yesmaximal, complemented
8C7×#1960C211yesmaximal, characteristic, complemented
9C7×#1955C211yesmaximal, characteristic
10C7×#1805C211yesmaximal, characteristic, complemented
11C7×#1787C212yesmaximal, complemented
12C23×C7×Q8C2212yes—
13C22×C7×Q16C2212yes—
14C22×C7×Q16C2211yescharacteristic
15C7×D4×Q8—24no—
16C7×D4×Q8—24no—
17C2×C7×C4⋊Q8—21no—
18C2×C7×C4⋊Q8C2211yescharacteristic
19C2×C7×C22⋊Q8—22no—
20C2×C7×C22⋊Q8C2212yes—
21C2×C7×C22⋊Q8C2212yes—
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×Q8C2211yescharacteristic
24C2×C4×C7×D4—21no—
25C2×C4×C7×D4C2211yescharacteristic
26C7×#462—24no—
27C7×#451—28no—
28C7×#441—24no—
29C2×C7×#106C2211yescharacteristic
30C2×C7×C4⋊C8C2211yescharacteristic
31C2×C7×#103C2211yescharacteristic
32C2×C7×#103C2211yescharacteristic
33C2×C7×#103C2212yes—
34C2×C7×#103C2212yes—
35C2×C7×#103C2212yes—
36C2×C7×#103C2212yes—
37C2×C7×C22⋊C8C2212yes—
38C2×C7×C22⋊C8C2212yes—
39C2×C7×#97C2211yescharacteristic
40C7×#398—21no—
41C7×#394—21no—
42C7×#388—22no—
43C7×#382C2211yescharacteristic
44C7×#380C2212yes—
45C7×#340C2212yes—
46#24255C711yesmaximal, characteristic, direct product factor, Sylow
47C22×C7×Q8—24no—
48C22×C7×Q8—24no—
49C22×C7×Q8—22no—
50C22×C7×Q8—24no—
51C22×C7×Q8—24no—
52C22×C7×Q8—42no—
53C22×C7×Q8—42no—
54C22×C7×Q8C2312yes—
55C22×C7×Q8C2311yescharacteristic
56C22×C7×Q8C2311yescharacteristic
57C22×C7×D4D411yescharacteristic
58C23×C4×C7—22no—
59C23×C4×C7C2312yes—
60C2×C7×Q16—22no—
61C2×C7×Q16—22no—
62C2×C7×Q16—24no—
63C2×C7×Q16—24no—
64C2×C7×Q16—42no—
65C2×C7×Q16D414yescomplemented
66C22×C7×C8C2312yes—
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C7×C4⋊Q8—28no—
70C7×C4⋊Q8—22no—
71C7×C4⋊Q8D414yes—
72C7×C22⋊Q8—216no—
73C7×C22⋊Q8—24no—
74C7×C22⋊Q8—24no—
75C7×C22⋊Q8—24no—
76C7×C22⋊Q8—42no—
77C4×C7×Q8—22no—
78C4×C7×Q8—22no—
79C4×C7×Q8—22no—
80C4×C7×Q8D414yes—
81C4×C7×D4—28no—
82C4×C7×D4—22no—
83C4×C7×D4—22no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—22no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4D411yescharacteristic
91C2×C7×C4⋊C4C2312yes—
92C2×C7×C4⋊C4C2311yescharacteristic
93C2×C7×C4⋊C4C2311yescharacteristic
94C2×C7×C22⋊C4—22no—
95C2×C7×C22⋊C4—22no—
96C2×C7×C22⋊C4—22no—
97C2×C7×C22⋊C4D412yes—
98C2×C42×C7—21no—
99C2×C42×C7—21no—
100C2×C42×C7C2311yescharacteristic
101C7×#106—22no—
102C7×C4⋊C8—22no—
103C7×#103—22no—
104C7×#103—24no—
105C7×#103—24no—
106C7×#103—24no—
107C7×#103—22no—
108C7×#103—42no—
109C7×C22⋊C8—24no—
110C7×C22⋊C8—42no—
111C7×#97—22no—
112C7×#95—22no—
113C7×#95—21no—
114C7×#95—21no—
115C2×D4×Q8C2×C711yescharacteristic
116C2×#462C2×C711yescharacteristic, complemented
117C2×#451C2×C712yescomplemented
118C2×#441C2×C711yescharacteristic, complemented
119#2536C2×C711yescharacteristic
120#2035C2×C712yescomplemented
121#2001C2×C712yescomplemented
122#1960C2×C711yescharacteristic, complemented
123#1955C2×C711yescharacteristic
124#1805C2×C711yescharacteristic, complemented
125#1787C2×C712yescomplemented
126C24×C7C2×D412yes—
127C2×C7×Q8—28no—
128C2×C7×Q8—21no—
129C2×C7×Q8—22no—
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8—21no—
133C2×C7×Q8—21no—
134C2×C7×Q8—22no—
135C2×C7×Q8—21no—
136C2×C7×Q8—21no—
137C2×C7×Q8—22no—
138C2×C7×Q8—22no—
139C2×C7×Q8—21no—
140C2×C7×Q8—22no—
141C2×C7×Q8—44no—
142C2×C7×Q8—44no—
143C2×C7×Q8—48no—
144C2×C7×Q8—48no—
145C2×C7×Q8—48no—
146C2×C7×Q8—48no—
147C2×C7×Q8C2×D414yes—
148C2×C7×Q8C2×D412yes—
149C2×C7×Q8C2×D412yes—
150C2×C7×Q8C2×D412yes—
151C2×C7×Q8#4112yes—
152C2×C7×D4—24no—
153C2×C7×D4—22no—
154C22×C4×C7—21no—
155C22×C4×C7—28no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—21no—
167C22×C4×C7—44no—
168C22×C4×C7—44no—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2411yescharacteristic
173C7×Q16—28no—
174C7×Q16—48no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C2×C7×C8—21no—
181C2×C7×C8—22no—
182C2×C7×C8—21no—
183C2×C7×C8C2×D412yes—
184C7×C4⋊C4—28no—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—24no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—44no—
195C7×C4⋊C4—42no—
196C7×C4⋊C4C2×D412yes—
197C7×C22⋊C4—28no—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—44no—
200C7×C22⋊C4—44no—
201C42×C7—24no—
202C42×C7—42no—
203C42×C7C2×D412yes—
204C42×C7C2×D412yes—
205C23×Q8C22×C712yes—
206C22×Q16C22×C712yes—
207C22×Q16C22×C711yescharacteristic
208D4×Q8—24no—
209D4×Q8—24no—
210C2×C4⋊Q8—21no—
211C2×C4⋊Q8C22×C711yescharacteristic
212C2×C22⋊Q8—22no—
213C2×C22⋊Q8C22×C712yes—
214C2×C22⋊Q8C22×C712yes—
215C2×C4×Q8C22×C711yescharacteristic
216C2×C4×Q8C22×C711yescharacteristic
217C2×C4×D4—21no—
218C2×C4×D4C22×C711yescharacteristic
219#462—24no—
220#451—28no—
221#441—24no—
222C2×#106C22×C711yescharacteristic
223C2×C4⋊C8C22×C711yescharacteristic
224C2×#103C22×C712yes—
225C2×#103C22×C711yescharacteristic
226C2×#103C22×C711yescharacteristic
227C2×#103C22×C712yes—
228C2×#103C22×C712yes—
229C2×#103C22×C712yes—
230C2×C22⋊C8C22×C712yes—
231C2×C22⋊C8C22×C712yes—
232C2×#97C22×C711yescharacteristic
233#398—21no—
234#394—21no—
235#388—22no—
236#382C22×C711yescharacteristic
237#380C22×C712yes—
238#340C22×C712yes—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7#13714yes—
246C23×C7C22×D411yescharacteristic
247C7×Q8—22no—
248C7×Q8—22no—
249C7×Q8—22no—
250C7×Q8—22no—
251C7×Q8—22no—
252C7×Q8—24no—
253C7×Q8—24no—
254C7×Q8—416no—
255C7×Q8—42no—
256C7×Q8—41no—
257C7×Q8—42no—
258C7×Q8—41no—
259C7×Q8#12114yes—
260C7×D4—28no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—24no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—416no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—41no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
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—42no—
287C2×C4×C7#12012yes—
288C2×C4×C7#12012yes—
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7H(5,𝔽2)11yescharacteristic
294C2×C4×C7H(5,𝔽2)11yescharacteristic
295C7×C8—22no—
296C7×C8—41no—
297C7×C8—42no—
298C22×Q8—24no—
299C22×Q8—24no—
300C22×Q8—22no—
301C22×Q8—24no—
302C22×Q8—24no—
303C22×Q8—42no—
304C22×Q8—42no—
305C22×Q8C23×C711yescharacteristic
306C22×Q8C23×C712yes—
307C22×Q8C23×C711yescharacteristic
308C22×D4C7×D411yescharacteristic
309C23×C4—22no—
310C23×C4C23×C712yes—
311C2×Q16—22no—
312C2×Q16—22no—
313C2×Q16—24no—
314C2×Q16—24no—
315C2×Q16—42no—
316C2×Q16C7×D414yescomplemented
317C22×C8C23×C712yes—
318C22×C8C23×C711yescharacteristic
319C22×C8C23×C711yescharacteristic
320C4⋊Q8—28no—
321C4⋊Q8—22no—
322C4⋊Q8C7×D414yes—
323C22⋊Q8—216no—
324C22⋊Q8—24no—
325C22⋊Q8—24no—
326C22⋊Q8—24no—
327C22⋊Q8—42no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8C7×D414yes—
332C4×D4—28no—
333C4×D4—22no—
334C4×D4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4C7×D411yescharacteristic
342C2×C4⋊C4C23×C711yescharacteristic
343C2×C4⋊C4C23×C712yes—
344C2×C4⋊C4C23×C711yescharacteristic
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4C7×D412yes—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42C23×C711yescharacteristic
352#106—22no—
353C4⋊C8—22no—
354#103—22no—
355#103—24no—
356#103—24no—
357#103—24no—
358#103—22no—
359#103—42no—
360C22⋊C8—24no—
361C22⋊C8—42no—
362#97—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366C22×C7—24no—
367C22×C7—24no—
368C22×C7—24no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7—42no—
373C22×C7C2×#12011yescharacteristic
374C22×C7D4211yescharacteristic
375C22×C7#54611yescharacteristic
376C22×C7C2×#13712yes—
377C22×C7#57812yes—
378C4×C7—24no—
379C4×C7—22no—
380C4×C7—22no—
381C4×C7—42no—
382C4×C7—44no—
383C4×C7—42no—
384C4×C7—42no—
385C4×C7—81no—
386C4×C7—82no—
387C4×C7C2×#12111yescharacteristic
388C4×C7#53511yescharacteristic
389C4×C7D4211yescharacteristic
390C4×C7#54611yescharacteristic
391C24C2×C7×D412yes—
392C2×Q8—28no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—48no—
409C2×Q8—48no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8C2×C7×D414yes—
413C2×Q8C2×C7×D412yes—
414C2×Q8C2×C7×D412yes—
415C2×Q8C2×C7×D412yes—
416C2×Q8C7×#4112yes—
417C2×D4—24no—
418C2×D4—22no—
419C22×C4—21no—
420C22×C4—28no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4C2×C7×D412yes—
435C22×C4C2×C7×D411yescharacteristic
436C22×C4C2×C7×D411yescharacteristic
437C22×C4C24×C711yescharacteristic
438Q16—28no—
439Q16—48no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—21no—
446C2×C8—22no—
447C2×C8—21no—
448C2×C8C2×C7×D412yes—
449C4⋊C4—28no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4C2×C7×D412yes—
462C22⋊C4—28no—
463C22⋊C4—24no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C42—24no—
467C42—42no—
468C42C2×C7×D412yes—
469C42C2×C7×D412yes—
470C2×C7—44no—
471C2×C7#240811yescharacteristic
472C2×C7#300411yescharacteristic
473C2×C7#301111yescharacteristic
474C2×C7#327012yes—
475C2×C7#331212yes—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23C7×#13714yes—
483C23C22×C7×D411yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—22no—
487Q8—22no—
488Q8—22no—
489Q8—24no—
490Q8—24no—
491Q8—416no—
492Q8—42no—
493Q8—41no—
494Q8—42no—
495Q8—41no—
496Q8C7×#12114yes—
497D4—28no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—416no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4C7×#12012yes—
525C2×C4C7×#12012yes—
526C2×C4C22×C7×D411yescharacteristic
527C2×C4C22×C7×D411yescharacteristic
528C2×C4C22×C7×D411yescharacteristic
529C2×C4C2×C7×#4111yescharacteristic
530C2×C4C7×H(5,𝔽2)11yescharacteristic
531C2×C4C7×H(5,𝔽2)11yescharacteristic
532C8—22no—
533C8—41no—
534C8—42no—
535C7#2425511yescharacteristic, direct product factor, Sylow
536C22—24no—
537C22—24no—
538C22—24no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22C2×C7×#12011yescharacteristic
544C22C7×D4211yescharacteristic
545C22C7×#54611yescharacteristic
546C22C2×C7×#13712yes—
547C22C7×#57812yes—
548C4—24no—
549C4—22no—
550C4—22no—
551C4—42no—
552C4—44no—
553C4—42no—
554C4—42no—
555C4—81no—
556C4—82no—
557C4C2×C7×#12111yescharacteristic
558C4C7×#53511yescharacteristic
559C4C7×D4211yescharacteristic
560C4C7×#54611yescharacteristic
561C2—44no—
562C2C7×#240811yescharacteristic
563C2C7×#300411yescharacteristic
564C2C7×#301111yescharacteristic
565C2C7×#327012yes—
566C2C7×#331212yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#462C2semidirect
3. C2×C7×#451C2semidirect
4. C2×C7×#441C2semidirect
6. C7×#2035C2semidirect
7. C7×#2001C2semidirect
8. C7×#1960C2semidirect
10. C7×#1805C2semidirect
11. C7×#1787C2semidirect
46. #24255C7direct (N × H)
65. C2×C7×Q16D4semidirect
116. C2×#462C2×C7semidirect
117. C2×#451C2×C7semidirect
118. C2×#441C2×C7semidirect
120. #2035C2×C7semidirect
121. #2001C2×C7semidirect
122. #1960C2×C7semidirect
124. #1805C2×C7semidirect
125. #1787C2×C7semidirect
316. C2×Q16C7×D4semidirect
535. C7#24255direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×D4×Q82
C2×C7×#4622
C2×C7×#4512
C2×C7×#4412
C7×#25362
C7×#20352
C7×#20012
C7×#19602
C7×#19552
C7×#18052
C7×#17872
#242557