Finite Group Databasebeta

← C7×#15105 #13192321

Subgroups — C7×#15105 #13192321

Order 1792.

1226 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#387C211yesmaximal, characteristic
2C2×C4×C7×C4⋊C4C211yesmaximal, characteristic
3C7×#1445C211yesmaximal, characteristic
4C7×#1445C211yesmaximal, characteristic
5C7×#1444C211yesmaximal, characteristic
6C7×#1442C211yesmaximal, characteristic
7C7×#1441C212yesmaximal
8C7×#1441C212yesmaximal
9C7×#1436C212yesmaximal
10C7×#1436C212yesmaximal
11C7×C4⋊#95C211yesmaximal, characteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2212yes—
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2212yes—
18C22×C42×C7C2211yescharacteristic
19C22×C42×C7C2212yes—
20C7×#387—24no—
21C4×C7×C4⋊C4—24no—
22C2×C7×#95C2212yes—
23C2×C7×#95C2211yescharacteristic
24C2×C7×#95C2211yescharacteristic
25C2×C7×#95C2211yescharacteristic
26C2×C7×#95C2211yescharacteristic
27C2×C7×#95C2212yes—
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2211yescharacteristic
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2212yes—
33C2×C7×#95C2212yes—
34C2×C7×#95C2212yes—
35C2×C7×#95C2212yes—
36C2×C7×#95C2212yes—
37#15105C711yesmaximal, characteristic, direct product factor, Sylow
38C23×C4×C7C2311yescharacteristic
39C23×C4×C7C2312yes—
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2312yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—22no—
52C2×C7×C4⋊C4—24no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—24no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4—24no—
60C2×C7×C4⋊C4—42no—
61C2×C7×C4⋊C4—42no—
62C2×C7×C4⋊C4—42no—
63C2×C7×C4⋊C4Q814yes—
64C2×C7×C4⋊C4Q814yes—
65C2×C42×C7—24no—
66C2×C42×C7—24no—
67C2×C42×C7—42no—
68C2×C42×C7—42no—
69C2×C42×C7D414yes—
70C7×#95—24no—
71C7×#95—22no—
72C7×#95—22no—
73C7×#95—22no—
74C7×#95—22no—
75C7×#95—24no—
76C7×#95—22no—
77C7×#95—24no—
78C7×#95—24no—
79C7×#95—24no—
80C7×#95—22no—
81C7×#95—42no—
82C7×#95—42no—
83C7×#95—42no—
84C7×#95—42no—
85C2×#387C2×C711yescharacteristic
86C2×C4×C4⋊C4C2×C711yescharacteristic
87#1445C2×C711yescharacteristic
88#1445C2×C711yescharacteristic
89#1444C2×C711yescharacteristic
90#1442C2×C711yescharacteristic
91#1441C2×C712yes—
92#1441C2×C712yes—
93#1436C2×C712yes—
94#1436C2×C712yes—
95C4⋊#95C2×C711yescharacteristic
96C24×C7C2411yescharacteristic
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
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—21no—
117C22×C4×C7—22no—
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—22no—
125C22×C4×C7—22no—
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—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—21no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×Q812yes—
151C22×C4×C7C2×Q814yes—
152C22×C4×C7C2×Q812yes—
153C22×C4×C7C2×Q812yes—
154C22×C4×C7C2×Q812yes—
155C22×C4×C7#4114yes—
156C22×C4×C7#4114yes—
157C22×C4×C7#4112yes—
158C22×C4×C7#4112yes—
159C22×C4×C7#4114yes—
160C7×C4⋊C4—28no—
161C7×C4⋊C4—28no—
162C7×C4⋊C4—28no—
163C7×C4⋊C4—48no—
164C7×C4⋊C4—48no—
165C7×C4⋊C4—44no—
166C42×C7—28no—
167C42×C7—48no—
168C22×C4⋊C4C22×C711yescharacteristic
169C22×C4⋊C4C22×C711yescharacteristic
170C22×C4⋊C4C22×C712yes—
171C22×C4⋊C4C22×C711yescharacteristic
172C22×C4⋊C4C22×C711yescharacteristic
173C22×C4⋊C4C22×C712yes—
174C22×C42C22×C711yescharacteristic
175C22×C42C22×C712yes—
176#387—24no—
177C4×C4⋊C4—24no—
178C2×#95C22×C712yes—
179C2×#95C22×C711yescharacteristic
180C2×#95C22×C711yescharacteristic
181C2×#95C22×C711yescharacteristic
182C2×#95C22×C712yes—
183C2×#95C22×C712yes—
184C2×#95C22×C712yes—
185C2×#95C22×C712yes—
186C2×#95C22×C711yescharacteristic
187C2×#95C22×C711yescharacteristic
188C2×#95C22×C711yescharacteristic
189C2×#95C22×C712yes—
190C2×#95C22×C712yes—
191C2×#95C22×C712yes—
192C2×#95C22×C712yes—
193C23×C7C22×D411yescharacteristic
194C23×C7C22×Q811yescharacteristic
195C23×C7C22×Q811yescharacteristic
196C23×C7C2×#4111yescharacteristic
197C23×C7C2×#4111yescharacteristic
198C23×C7C2×#4112yes—
199C23×C7C2×#4112yes—
200C23×C7C2×#4112yes—
201C23×C7H(5,𝔽2)12yes—
202C23×C7H(5,𝔽2)11yescharacteristic
203C23×C72−1+411yescharacteristic
204C2×C4×C7—22no—
205C2×C4×C7—22no—
206C2×C4×C7—22no—
207C2×C4×C7—24no—
208C2×C4×C7—22no—
209C2×C4×C7—24no—
210C2×C4×C7—22no—
211C2×C4×C7—22no—
212C2×C4×C7—24no—
213C2×C4×C7—24no—
214C2×C4×C7—24no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—22no—
218C2×C4×C7—22no—
219C2×C4×C7—24no—
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—41no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—41no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—41no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
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×C7C22⋊Q814yes—
265C2×C4×C7C22⋊Q814yes—
266C2×C4×C7#12314yes—
267C23×C4C23×C711yescharacteristic
268C23×C4C23×C711yescharacteristic
269C23×C4C23×C712yes—
270C23×C4C23×C711yescharacteristic
271C23×C4C23×C711yescharacteristic
272C23×C4C23×C712yes—
273C23×C4C23×C712yes—
274C23×C4C23×C712yes—
275C23×C4C23×C711yescharacteristic
276C23×C4C23×C711yescharacteristic
277C23×C4C23×C711yescharacteristic
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4C7×Q814yes—
293C2×C4⋊C4C7×Q814yes—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42C7×D414yes—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—42no—
311#95—42no—
312#95—42no—
313#95—42no—
314C22×C7C2×C22⋊Q811yescharacteristic
315C22×C7C2×C22⋊Q811yescharacteristic
316C22×C7C2×#12311yescharacteristic
317C22×C7#52511yescharacteristic
318C22×C7#52912yes—
319C22×C7#52912yes—
320C22×C7#53211yescharacteristic
321C22×C7#53312yes—
322C22×C7#53312yes—
323C22×C7#53511yescharacteristic
324C22×C7#53711yescharacteristic
325C22×C7#53812yes—
326C22×C7#53912yes—
327C22×C7#55112yes—
328C22×C7#55212yes—
329C22×C7#55312yes—
330C22×C7#55312yes—
331C22×C7#55412yes—
332C22×C7#55412yes—
333C22×C7#55711yescharacteristic
334C22×C7#55711yescharacteristic
335C22×C7Q8211yescharacteristic
336C22×C7#55911yescharacteristic
337C4×C7—24no—
338C4×C7—24no—
339C4×C7—24no—
340C4×C7—44no—
341C4×C7—44no—
342C4×C7—44no—
343C4×C7—42no—
344C4×C7—42no—
345C4×C7—82no—
346C4×C7—81no—
347C4×C7—81no—
348C24C24×C711yescharacteristic
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
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—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4C2×C7×D412yes—
400C22×C4C2×C7×D412yes—
401C22×C4C2×C7×D412yes—
402C22×C4C2×C7×Q812yes—
403C22×C4C2×C7×Q812yes—
404C22×C4C2×C7×Q814yes—
405C22×C4C2×C7×Q812yes—
406C22×C4C2×C7×Q812yes—
407C22×C4C7×#4114yes—
408C22×C4C7×#4114yes—
409C22×C4C7×#4112yes—
410C22×C4C7×#4112yes—
411C22×C4C7×#4114yes—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C42—28no—
419C42—48no—
420C2×C7#239611yescharacteristic
421C2×C7#248711yescharacteristic
422C2×C7#249811yescharacteristic
423C2×C7#256611yescharacteristic
424C2×C7#258211yescharacteristic
425C2×C7#258512yes—
426C2×C7#258612yes—
427C2×C7#259012yes—
428C2×C7#259712yes—
429C2×C7#263611yescharacteristic
430C2×C7#285511yescharacteristic
431C23C22×C7×D411yescharacteristic
432C23C22×C7×Q811yescharacteristic
433C23C22×C7×Q811yescharacteristic
434C23C2×C7×#4111yescharacteristic
435C23C2×C7×#4111yescharacteristic
436C23C2×C7×#4112yes—
437C23C2×C7×#4112yes—
438C23C2×C7×#4112yes—
439C23C7×H(5,𝔽2)12yes—
440C23C7×H(5,𝔽2)11yescharacteristic
441C23C7×2−1+411yescharacteristic
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4C7×C22⋊Q814yes—
503C2×C4C7×C22⋊Q814yes—
504C2×C4C7×#12314yes—
505C7#1510511yescharacteristic, direct product factor, Sylow
506C22C2×C7×C22⋊Q811yescharacteristic
507C22C2×C7×C22⋊Q811yescharacteristic
508C22C2×C7×#12311yescharacteristic
509C22C7×#52511yescharacteristic
510C22C7×#52912yes—
511C22C7×#52912yes—
512C22C7×#53211yescharacteristic
513C22C7×#53312yes—
514C22C7×#53312yes—
515C22C7×#53511yescharacteristic
516C22C7×#53711yescharacteristic
517C22C7×#53812yes—
518C22C7×#53912yes—
519C22C7×#55112yes—
520C22C7×#55212yes—
521C22C7×#55312yes—
522C22C7×#55312yes—
523C22C7×#55412yes—
524C22C7×#55412yes—
525C22C7×#55711yescharacteristic
526C22C7×#55711yescharacteristic
527C22C7×Q8211yescharacteristic
528C22C7×#55911yescharacteristic
529C4—24no—
530C4—24no—
531C4—24no—
532C4—42no—
533C4—42no—
534C4—44no—
535C4—44no—
536C4—44no—
537C4—81no—
538C4—82no—
539C4—81no—
540C2C7×#239611yescharacteristic
541C2C7×#248711yescharacteristic
542C2C7×#249811yescharacteristic
543C2C7×#256611yescharacteristic
544C2C7×#258211yescharacteristic
545C2C7×#258512yes—
546C2C7×#258612yes—
547C2C7×#259012yes—
548C2C7×#259712yes—
549C2C7×#263611yescharacteristic
550C2C7×#285511yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C7C24
commutator, Frattini, cosocleC24C24×C7
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
37. #15105C7direct (N × H)
505. C7#15105direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3872
C2×C4×C7×C4⋊C42
C7×#14452
C7×#14442
C7×#14422
C7×#14412
C7×#14362
C7×C4⋊#952
#151057