Finite Group Databasebeta

← C7×#25892 #13203108

Subgroups — C7×#25892 #13203108

Order 1792.

1292 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic
2C2×C7×#491C211yesmaximal, characteristic, complemented
3C2×C7×#478C211yesmaximal, characteristic, complemented
4C7×#2562C211yesmaximal, characteristic, complemented
5C7×#2036C212yesmaximal, complemented
6C7×#2035C212yesmaximal, complemented
7C7×#1981C211yesmaximal, characteristic, complemented
8C7×#1955C211yesmaximal, characteristic, complemented
9C7×#1788C212yesmaximal, complemented
10C7×#1788C212yesmaximal, complemented
11C7×#1762C211yesmaximal, characteristic, complemented
12C22×C7×#41C2212yes—
13C7×#547—24no—
14C7×#547—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125C2211yescharacteristic
17C2×C7×#123—22no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C7×#121—21no—
20C2×C4×C7×D4C2211yescharacteristic
21C2×C4×C7×D4C2211yescharacteristic
22C22×C7×C4⋊C4—21no—
23C7×#491—24no—
24C7×#478—24no—
25C2×C7×#107C2211yescharacteristic
26C2×C7×#106C2211yescharacteristic
27C2×C7×C4⋊C8C2211yescharacteristic
28C2×C7×#103C2212yes—
29C2×C7×#103C2212yes—
30C2×C7×#103C2212yes—
31C2×C7×#103C2212yes—
32C2×C7×#102C2212yes—
33C2×C7×#102C2211yescharacteristic
34C2×C7×#102C2211yescharacteristic
35C2×C7×C22⋊C8C2212yes—
36C2×C7×C22⋊C8C2212yes—
37C2×C7×#97C2211yescharacteristic
38C7×#400—21no—
39C7×#394—21no—
40C7×#388—22no—
41C7×#380C2212yes—
42C7×#380C2212yes—
43C4×C7×C4⋊C4C2211yescharacteristic
44C7×#340C2212yes—
45#25892C711yesmaximal, characteristic, direct product factor, Sylow
46C2×C7×#41—24no—
47C2×C7×#41—24no—
48C2×C7×#41—24no—
49C2×C7×#41—24no—
50C2×C7×#41—42no—
51C2×C7×#41—42no—
52C22×C7×Q8C2312yes—
53C22×C7×D4—22no—
54C22×C7×D4D411yescharacteristic
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—22no—
57C23×C4×C7—21no—
58C23×C4×C7C2312yes—
59C22×C7×C8C2312yes—
60C22×C7×C8C2311yescharacteristic
61C22×C7×C8C2311yescharacteristic
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8D414yes—
64C7×#125—22no—
65C7×#125—22no—
66C7×#123—216no—
67C7×C22⋊Q8—24no—
68C7×C22⋊Q8—24no—
69C7×#121—28no—
70C4×C7×D4—22no—
71C4×C7×D4—22no—
72C4×C7×D4—22no—
73C4×C7×D4D414yes—
74C2×C7×C4⋊C4—28no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4D411yescharacteristic
81C2×C7×C4⋊C4C2312yes—
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—21no—
87C2×C7×C22⋊C4—22no—
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4D412yes—
90C2×C42×C7—21no—
91C2×C42×C7C2311yescharacteristic
92C7×#107—22no—
93C7×#106—22no—
94C7×C4⋊C8—22no—
95C7×#103—24no—
96C7×#103—24no—
97C7×#103—42no—
98C7×#103—42no—
99C7×#102—22no—
100C7×#102—22no—
101C7×#102—24no—
102C7×C22⋊C8—42no—
103C7×C22⋊C8—42no—
104C7×#97—22no—
105C7×#95—21no—
106C7×#95—22no—
107C7×#95—22no—
108C2×#547C2×C711yescharacteristic
109C2×#491C2×C711yescharacteristic, complemented
110C2×#478C2×C711yescharacteristic, complemented
111#2562C2×C711yescharacteristic, complemented
112#2036C2×C712yescomplemented
113#2035C2×C712yescomplemented
114#1981C2×C711yescharacteristic, complemented
115#1955C2×C711yescharacteristic, complemented
116#1788C2×C712yescomplemented
117#1788C2×C712yescomplemented
118#1762C2×C711yescharacteristic, complemented
119C24×C7—21no—
120C24×C7C2×D412yes—
121C7×#41—48no—
122C7×#41—48no—
123C7×#41—48no—
124C7×#41—48no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8C2×D414yes—
131C2×C7×D4—28no—
132C2×C7×D4—22no—
133C2×C7×D4—21no—
134C2×C7×D4—22no—
135C2×C7×D4—42no—
136C2×C7×D4—44no—
137C2×C7×D4—44no—
138C2×C7×D4C2×D412yes—
139C2×C7×D4#4114yes—
140C22×C4×C7—28no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—24no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—44no—
154C22×C4×C7—44no—
155C22×C4×C7—44no—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C2×C7×C8—22no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C7×C4⋊C4—28no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4C2×D412yes—
181C7×C22⋊C4—28no—
182C7×C22⋊C4—24no—
183C7×C22⋊C4—24no—
184C7×C22⋊C4—44no—
185C7×C22⋊C4—82no—
186C42×C7—81no—
187C42×C7C2×D412yes—
188C42×C7C2×D412yes—
189C22×#41C22×C712yes—
190#547—24no—
191#547—24no—
192C2×C4⋊Q8C22×C711yescharacteristic
193C2×#125C22×C711yescharacteristic
194C2×#123—22no—
195C2×C22⋊Q8C22×C712yes—
196C2×#121—21no—
197C2×C4×D4C22×C711yescharacteristic
198C2×C4×D4C22×C711yescharacteristic
199C22×C4⋊C4—21no—
200#491—24no—
201#478—24no—
202C2×#107C22×C711yescharacteristic
203C2×#106C22×C711yescharacteristic
204C2×C4⋊C8C22×C711yescharacteristic
205C2×#103C22×C712yes—
206C2×#103C22×C712yes—
207C2×#103C22×C712yes—
208C2×#103C22×C712yes—
209C2×#102C22×C711yescharacteristic
210C2×#102C22×C711yescharacteristic
211C2×#102C22×C712yes—
212C2×C22⋊C8C22×C712yes—
213C2×C22⋊C8C22×C712yes—
214C2×#97C22×C711yescharacteristic
215#400—21no—
216#394—21no—
217#388—22no—
218#380C22×C712yes—
219#380C22×C712yes—
220C4×C4⋊C4C22×C711yescharacteristic
221#340C22×C712yes—
222C23×C7—22no—
223C23×C7—24no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—42no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7C22×D411yescharacteristic
235C7×Q8—24no—
236C7×Q8—24no—
237C7×Q8—42no—
238C7×Q8—42no—
239C7×D4—22no—
240C7×D4—24no—
241C7×D4—44no—
242C7×D4—416no—
243C7×D4#12414yes—
244C2×C4×C7—22no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—48no—
253C2×C4×C7—416no—
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—41no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7#12012yes—
270C2×C4×C7#12012yes—
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C2×#4112yes—
274C2×C4×C7H(5,𝔽2)12yes—
275C7×C8—41no—
276C7×C8—41no—
277C7×C8—42no—
278C2×#41—24no—
279C2×#41—24no—
280C2×#41—24no—
281C2×#41—24no—
282C2×#41—42no—
283C2×#41—42no—
284C22×Q8C23×C712yes—
285C22×D4—22no—
286C22×D4C7×D411yescharacteristic
287C22×D4C23×C711yescharacteristic
288C23×C4—21no—
289C23×C4—22no—
290C23×C4C23×C712yes—
291C22×C8C23×C712yes—
292C22×C8C23×C711yescharacteristic
293C22×C8C23×C711yescharacteristic
294C4⋊Q8—22no—
295C4⋊Q8C7×D414yes—
296#125—22no—
297#125—22no—
298#123—216no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
301#121—28no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4C7×D414yes—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4C7×D411yescharacteristic
313C2×C4⋊C4C23×C711yescharacteristic
314C2×C4⋊C4C23×C711yescharacteristic
315C2×C4⋊C4C23×C712yes—
316C2×C4⋊C4C23×C711yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4C7×D412yes—
322C2×C42—21no—
323C2×C42C23×C711yescharacteristic
324#107—22no—
325#106—22no—
326C4⋊C8—22no—
327#103—24no—
328#103—24no—
329#103—42no—
330#103—42no—
331#102—22no—
332#102—22no—
333#102—24no—
334C22⋊C8—42no—
335C22⋊C8—42no—
336#97—22no—
337#95—21no—
338#95—22no—
339#95—22no—
340C22×C7—24no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—44no—
344C22×C7—44no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7C2×#12011yescharacteristic
353C22×C7#54612yes—
354C22×C7#57512yes—
355C22×C7#57711yescharacteristic
356C22×C7#57811yescharacteristic
357C4×C7—22no—
358C4×C7—22no—
359C4×C7—42no—
360C4×C7—44no—
361C4×C7—42no—
362C4×C7—81no—
363C4×C7—81no—
364C4×C7—82no—
365C4×C7C2×#12411yescharacteristic
366C4×C7#53511yescharacteristic
367C4×C7#54612yes—
368C24—21no—
369C24C2×C7×D412yes—
370#41—48no—
371#41—48no—
372#41—48no—
373#41—48no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8C2×C7×D414yes—
380C2×D4—28no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4C2×C7×D412yes—
388C2×D4C7×#4114yes—
389C22×C4—28no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4C2×C7×D412yes—
406C22×C4C2×C7×D411yescharacteristic
407C22×C4C2×C7×D411yescharacteristic
408C22×C4C24×C711yescharacteristic
409C2×C8—22no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C4⋊C4—28no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4C2×C7×D412yes—
430C22⋊C4—28no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—44no—
434C22⋊C4—82no—
435C42—81no—
436C42C2×C7×D412yes—
437C42C2×C7×D412yes—
438C2×C7—44no—
439C2×C7—82no—
440C2×C7#243511yescharacteristic
441C2×C7#300511yescharacteristic
442C2×C7#301311yescharacteristic
443C2×C7#330712yes—
444C2×C7#331512yes—
445C23—22no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23C22×C7×D411yescharacteristic
458Q8—24no—
459Q8—24no—
460Q8—42no—
461Q8—42no—
462D4—22no—
463D4—24no—
464D4—44no—
465D4—416no—
466D4C7×#12414yes—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—48no—
476C2×C4—416no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4C7×#12012yes—
493C2×C4C7×#12012yes—
494C2×C4C22×C7×D411yescharacteristic
495C2×C4C22×C7×D411yescharacteristic
496C2×C4C2×C7×#4112yes—
497C2×C4C7×H(5,𝔽2)12yes—
498C8—41no—
499C8—41no—
500C8—42no—
501C7#2589211yescharacteristic, direct product factor, Sylow
502C22—24no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22C2×C7×#12011yescharacteristic
515C22C7×#54612yes—
516C22C7×#57512yes—
517C22C7×#57711yescharacteristic
518C22C7×#57811yescharacteristic
519C4—22no—
520C4—22no—
521C4—42no—
522C4—44no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—82no—
527C4C2×C7×#12411yescharacteristic
528C4C7×#53511yescharacteristic
529C4C7×#54612yes—
530C2—44no—
531C2—82no—
532C2C7×#243511yescharacteristic
533C2C7×#300511yescharacteristic
534C2C7×#301311yescharacteristic
535C2C7×#330712yes—
536C2C7×#331512yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#547C2
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×#491C2semidirect
3. C2×C7×#478C2semidirect
4. C7×#2562C2semidirect
5. C7×#2036C2semidirect
6. C7×#2035C2semidirect
7. C7×#1981C2semidirect
8. C7×#1955C2semidirect
9. C7×#1788C2semidirect
10. C7×#1788C2semidirect
11. C7×#1762C2semidirect
45. #25892C7direct (N × H)
109. C2×#491C2×C7semidirect
110. C2×#478C2×C7semidirect
111. #2562C2×C7semidirect
112. #2036C2×C7semidirect
113. #2035C2×C7semidirect
114. #1981C2×C7semidirect
115. #1955C2×C7semidirect
116. #1788C2×C7semidirect
117. #1788C2×C7semidirect
118. #1762C2×C7semidirect
501. C7#25892direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C2×C7×#4912
C2×C7×#4782
C7×#25622
C7×#20362
C7×#20352
C7×#19812
C7×#19552
C7×#17882
C7×#17622
#258927