Finite Group Databasebeta

← C7×#25945 #13203161

Subgroups — C7×#25945 #13203161

Order 1792.

1072 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2C2×C7×#491C211yesmaximal, characteristic, complemented
3C2×C7×#461C211yesmaximal, characteristic
4C7×#2568C211yesmaximal, characteristic, complemented
5C7×#2026C212yesmaximal, complemented
6C7×#2024C212yesmaximal, complemented
7C7×#1989C211yesmaximal, characteristic, complemented
8C7×#1985C211yesmaximal, characteristic, complemented
9C7×#1877C212yesmaximal
10C7×#1877C212yesmaximal
11C7×#1762C211yesmaximal, characteristic
12C22×C7×SD16C2212yes—
13C7×#554—24no—
14C7×#554—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#125C2211yescharacteristic
18C2×C7×#125C2211yescharacteristic
19C2×C7×C22⋊Q8—21no—
20C2×C7×C22⋊Q8—21no—
21C2×C7×C22⋊Q8—21no—
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C2×C4×C7×D4C2211yescharacteristic
25C2×C4×C7×D4C2211yescharacteristic
26C22×C7×C4⋊C4—21no—
27C7×#491—24no—
28C7×#461—24no—
29C2×C7×C4⋊C8C2211yescharacteristic
30C2×C7×#103C2212yes—
31C2×C7×#103C2212yes—
32C2×C7×#103C2212yes—
33C2×C7×#102C2212yes—
34C2×C7×#97C2211yescharacteristic
35C7×#400—21no—
36C7×#400—21no—
37C7×#391—21no—
38C7×#391C2212yes—
39C7×#391C2212yes—
40C7×#382—21no—
41C4×C7×C4⋊C4C2211yescharacteristic
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#336C2212yes—
46C7×#336C2212yes—
47#25945C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8D411yescharacteristic
50C22×C7×Q8C2312yes—
51C22×C7×D4C2311yescharacteristic
52C23×C4×C7—21no—
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C2×C7×SD16—24no—
56C2×C7×SD16—28no—
57C22×C7×C8C2312yes—
58C22×C7×C8C2312yes—
59C7×C4⋊Q8—22no—
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8—22no—
62C7×C4⋊Q8—22no—
63C7×#125—22no—
64C7×#125—22no—
65C7×#125—22no—
66C7×#125D414yes—
67C7×C22⋊Q8—28no—
68C7×C22⋊Q8—28no—
69C7×C22⋊Q8—28no—
70C4×C7×Q8—22no—
71C4×C7×Q8—22no—
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C4×C7×D4—22no—
77C4×C7×D4D414yes—
78C2×C7×C4⋊C4—28no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4D411yescharacteristic
87C2×C7×C4⋊C4D411yescharacteristic
88C2×C7×C4⋊C4D411yescharacteristic
89C2×C7×C4⋊C4C2312yes—
90C2×C7×C4⋊C4C2311yescharacteristic
91C2×C7×C4⋊C4C2311yescharacteristic
92C2×C7×C4⋊C4C2311yescharacteristic
93C2×C7×C22⋊C4—21no—
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—21no—
96C2×C42×C7—21no—
97C2×C42×C7C2311yescharacteristic
98C2×C42×C7C2311yescharacteristic
99C2×C42×C7C2311yescharacteristic
100C7×C4⋊C8—22no—
101C7×#103—24no—
102C7×#103—24no—
103C7×#103—24no—
104C7×#102—24no—
105C7×#97—22no—
106C7×#95—21no—
107C7×#95—22no—
108C7×#95—22no—
109C7×#95—22no—
110C7×#95—22no—
111C2×#554C2×C711yescharacteristic
112C2×#491C2×C711yescharacteristic, complemented
113C2×#461C2×C711yescharacteristic
114#2568C2×C711yescharacteristic, complemented
115#2026C2×C712yescomplemented
116#2024C2×C712yescomplemented
117#1989C2×C711yescharacteristic, complemented
118#1985C2×C711yescharacteristic, complemented
119#1877C2×C712yes—
120#1877C2×C712yes—
121#1762C2×C711yescharacteristic
122C24×C7—21no—
123C2×C7×Q8—24no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—24no—
129C2×C7×Q8—24no—
130C2×C7×Q8—41no—
131C2×C7×Q8—82no—
132C2×C7×D4—21no—
133C2×C7×D4—22no—
134C2×C7×D4C2×D412yes—
135C2×C7×D4#4114yes—
136C22×C4×C7—24no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—24no—
141C22×C4×C7—24no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—42no—
145C22×C4×C7—42no—
146C22×C4×C7—82no—
147C22×C4×C7—82no—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C7×SD16—48no—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—24no—
160C2×C7×C8—22no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4—42no—
176C7×C4⋊C4C2×D412yes—
177C7×C4⋊C4C2×D412yes—
178C7×C4⋊C4C2×D412yes—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—24no—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—42no—
187C42×C7C2×D412yes—
188C42×C7C2×D412yes—
189C22×SD16C22×C712yes—
190#554—24no—
191#554—24no—
192C2×C4⋊Q8C22×C711yescharacteristic
193C2×C4⋊Q8C22×C711yescharacteristic
194C2×#125C22×C711yescharacteristic
195C2×#125C22×C711yescharacteristic
196C2×C22⋊Q8—21no—
197C2×C22⋊Q8—21no—
198C2×C22⋊Q8—21no—
199C2×C4×Q8C22×C711yescharacteristic
200C2×C4×D4C22×C711yescharacteristic
201C2×C4×D4C22×C711yescharacteristic
202C2×C4×D4C22×C711yescharacteristic
203C22×C4⋊C4—21no—
204#491—24no—
205#461—24no—
206C2×C4⋊C8C22×C711yescharacteristic
207C2×#103C22×C712yes—
208C2×#103C22×C712yes—
209C2×#103C22×C712yes—
210C2×#102C22×C712yes—
211C2×#97C22×C711yescharacteristic
212#400—21no—
213#400—21no—
214#391—21no—
215#391C22×C712yes—
216#391C22×C712yes—
217#382—21no—
218C4×C4⋊C4C22×C711yescharacteristic
219#340C22×C712yes—
220#340C22×C712yes—
221#340C22×C712yes—
222#336C22×C712yes—
223#336C22×C712yes—
224C23×C7—22no—
225C23×C7—24no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—42no—
229C23×C7C22×D411yescharacteristic
230C7×Q8—44no—
231C7×Q8—42no—
232C7×Q8—42no—
233C7×Q8—44no—
234C7×Q8—84no—
235C7×D4—22no—
236C7×D4—24no—
237C7×D4#12414yes—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—22no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—48no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—44no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—84no—
264C2×C4×C7—84no—
265C2×C4×C7#12012yes—
266C2×C4×C7#12012yes—
267C2×C4×C7#13712yes—
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C2×#4112yes—
271C2×C4×C72−1+412yes—
272C7×C8—42no—
273C7×C8—42no—
274C22×Q8—21no—
275C22×Q8C7×D411yescharacteristic
276C22×Q8C23×C712yes—
277C22×D4C23×C711yescharacteristic
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—21no—
281C2×SD16—24no—
282C2×SD16—28no—
283C22×C8C23×C712yes—
284C22×C8C23×C712yes—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289#125—22no—
290#125—22no—
291#125—22no—
292#125C7×D414yes—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4C7×D414yes—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4C7×D411yescharacteristic
313C2×C4⋊C4C7×D411yescharacteristic
314C2×C4⋊C4C7×D411yescharacteristic
315C2×C4⋊C4C23×C712yes—
316C2×C4⋊C4C23×C711yescharacteristic
317C2×C4⋊C4C23×C711yescharacteristic
318C2×C4⋊C4C23×C711yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C42—21no—
323C2×C42C23×C711yescharacteristic
324C2×C42C23×C711yescharacteristic
325C2×C42C23×C711yescharacteristic
326C4⋊C8—22no—
327#103—24no—
328#103—24no—
329#103—24no—
330#102—24no—
331#97—22no—
332#95—21no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337C22×C7—24no—
338C22×C7—44no—
339C22×C7—44no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7C2×#12011yescharacteristic
343C22×C7#54812yes—
344C22×C7C2×#13711yescharacteristic
345C22×C7#57511yescharacteristic
346C22×C7#57712yes—
347C4×C7—22no—
348C4×C7—22no—
349C4×C7—22no—
350C4×C7—41no—
351C4×C7—42no—
352C4×C7—41no—
353C4×C7—41no—
354C4×C7—42no—
355C4×C7—42no—
356C4×C7—82no—
357C4×C7—82no—
358C4×C7C2×#12411yescharacteristic
359C4×C7#53611yescharacteristic
360C4×C7#54812yes—
361C24—21no—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—41no—
370C2×Q8—82no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4C2×C7×D412yes—
374C2×D4C7×#4114yes—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4C2×C7×D411yescharacteristic
388C22×C4C2×C7×D411yescharacteristic
389C22×C4C2×C7×D411yescharacteristic
390C22×C4C2×C7×D411yescharacteristic
391C22×C4C2×C7×D411yescharacteristic
392C22×C4C2×C7×D411yescharacteristic
393C22×C4C24×C711yescharacteristic
394SD16—48no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4C2×C7×D412yes—
416C4⋊C4C2×C7×D412yes—
417C4⋊C4C2×C7×D412yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—42no—
426C42C2×C7×D412yes—
427C42C2×C7×D412yes—
428C2×C7—44no—
429C2×C7#243611yescharacteristic
430C2×C7#300711yescharacteristic
431C2×C7#300911yescharacteristic
432C2×C7#337212yes—
433C2×C7#337612yes—
434C23—22no—
435C23—24no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23C22×C7×D411yescharacteristic
440Q8—44no—
441Q8—42no—
442Q8—42no—
443Q8—44no—
444Q8—84no—
445D4—22no—
446D4—24no—
447D4C7×#12414yes—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—48no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4C7×#12012yes—
476C2×C4C7×#12012yes—
477C2×C4C7×#13712yes—
478C2×C4C22×C7×D411yescharacteristic
479C2×C4C22×C7×D411yescharacteristic
480C2×C4C2×C7×#4112yes—
481C2×C4C7×2−1+412yes—
482C8—42no—
483C8—42no—
484C7#2594511yescharacteristic, direct product factor, Sylow
485C22—24no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22C2×C7×#12011yescharacteristic
491C22C7×#54812yes—
492C22C2×C7×#13711yescharacteristic
493C22C7×#57511yescharacteristic
494C22C7×#57712yes—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—42no—
500C4—41no—
501C4—41no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—82no—
506C4C2×C7×#12411yescharacteristic
507C4C7×#53611yescharacteristic
508C4C7×#54812yes—
509C2—44no—
510C2C7×#243611yescharacteristic
511C2C7×#300711yescharacteristic
512C2C7×#300911yescharacteristic
513C2C7×#337212yes—
514C2C7×#337612yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#491C2semidirect
4. C7×#2568C2semidirect
5. C7×#2026C2semidirect
6. C7×#2024C2semidirect
7. C7×#1989C2semidirect
8. C7×#1985C2semidirect
47. #25945C7direct (N × H)
112. C2×#491C2×C7semidirect
114. #2568C2×C7semidirect
115. #2026C2×C7semidirect
116. #2024C2×C7semidirect
117. #1989C2×C7semidirect
118. #1985C2×C7semidirect
484. C7#25945direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C2×C7×#4912
C2×C7×#4612
C7×#25682
C7×#20262
C7×#20242
C7×#19892
C7×#19852
C7×#18772
C7×#17622
#259457