Finite Group Databasebeta

← C7×#9037 #13186253

Subgroups — C7×#9037 #13186253

Order 1792.

676 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3002C211yesmaximal, characteristic
2C7×#2976C211yesmaximal, characteristic, complemented
3C7×#1789C211yesmaximal, characteristic, complemented
4C7×#1780C211yesmaximal, characteristic, complemented
5C7×#1501C211yesmaximal, characteristic, complemented
6C7×#1298C211yesmaximal, characteristic, complemented
7C7×#1282C211yesmaximal, characteristic, complemented
8C22×C7×#41C2211yescharacteristic
9C2×C7×#135—21no—
10C7×#525C2211yescharacteristic
11C7×#452—21no—
12C7×#451C411yescharacteristic, complemented
13C7×#450C411yescharacteristic, complemented
14C7×#449—21no—
15C7×#448C411yescharacteristic, complemented
16C7×#447C411yescharacteristic, complemented
17C7×#436—21no—
18C7×#435—21no—
19C7×#435—21no—
20C7×C8×D4—21no—
21C7×#433—21no—
22C2×C7×#104—21no—
23C2×C7×#104—21no—
24C7×#416—21no—
25C7×#408—21no—
26C7×#407C2211yescharacteristic
27C2×C7×C22⋊C8—21no—
28C2×C7×C22⋊C8—21no—
29C2×C7×C22⋊C8C2211yescharacteristic
30C2×C7×C22⋊C8C2211yescharacteristic
31C2×C7×C22⋊C8C2211yescharacteristic
32C4×C7×C22⋊C4C2211yescharacteristic
33C7×#328—21no—
34C7×#327—21no—
35#9037C711yesmaximal, characteristic, direct product factor, Sylow
36C2×C7×#41—21no—
37C2×C7×#41—21no—
38C2×C7×#41—21no—
39C2×C7×#41—21no—
40C2×C7×#41—42no—
41C2×C7×#41—42no—
42C2×C7×#41D411yescharacteristic
43C22×C7×Q8C2×C411yescharacteristic
44C22×C7×D4—21no—
45C22×C7×D4C2×C411yescharacteristic
46C23×C4×C7—21no—
47C23×C4×C7C2311yescharacteristic
48C7×#135—42no—
49C7×#135—42no—
50C2×C7×Q16—21no—
51C2×C7×SD16—21no—
52C2×C7×SD16—21no—
53C2×C7×D8—21no—
54C2×C7×M16—21no—
55C2×C7×M16D411yescharacteristic
56C2×C7×M16D411yescharacteristic
57C22×C7×C8—21no—
58C22×C7×C8—21no—
59C22×C7×C8—21no—
60C22×C7×C8D411yescharacteristic
61C22×C7×C8D411yescharacteristic
62C7×#123—21no—
63C7×C22⋊Q8C2×C411yescharacteristic
64C7×C22⋊Q8D411yescharacteristic
65C7×#121C2×C411yescharacteristic
66C7×#121D411yescharacteristic
67C7×#120—21no—
68C4×C7×D4—21no—
69C4×C7×D4—21no—
70C7×#117D411yescharacteristic
71C2×C7×C22⋊C4—21no—
72C2×C42×C7—21no—
73C7×C4⋊C8—21no—
74C7×C4⋊C8—21no—
75C7×#104—42no—
76C7×#104—42no—
77C7×#103—21no—
78C7×#103—21no—
79C7×#102—21no—
80C7×#102—21no—
81C7×C22⋊C8—21no—
82C7×C22⋊C8—21no—
83C7×C22⋊C8—21no—
84C7×C22⋊C8—21no—
85C7×C22⋊C8—21no—
86C7×C22⋊C8—21no—
87C7×C22⋊C8—21no—
88C7×C22⋊C8—21no—
89C7×C22⋊C8—21no—
90C7×C22⋊C8—42no—
91C7×C22⋊C8—42no—
92C7×C22⋊C8C2×C411yescharacteristic
93C7×C22⋊C8C2×C411yescharacteristic
94C7×#97—21no—
95C4×C7×C8—21no—
96C7×#95—21no—
97#3002C2×C711yescharacteristic
98#2976C2×C711yescharacteristic, complemented
99#1789C2×C711yescharacteristic, complemented
100#1780C2×C711yescharacteristic, complemented
101#1501C2×C711yescharacteristic, complemented
102#1298C2×C711yescharacteristic, complemented
103#1282C2×C711yescharacteristic, complemented
104C24×C7—21no—
105C24×C7#4111yescharacteristic
106C7×#41—22no—
107C7×#41—22no—
108C7×#41—22no—
109C7×#41—22no—
110C7×#41—41no—
111C7×#41—41no—
112C7×#41—41no—
113C7×#41—41no—
114C7×#41—41no—
115C7×#41—41no—
116C7×#41—42no—
117C7×#41—42no—
118C7×#41—42no—
119C7×#41—42no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—42no—
123C2×C7×Q8C22⋊C411yescharacteristic
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—42no—
130C2×C7×D4—42no—
131C2×C7×D4—42no—
132C2×C7×D4—41no—
133C2×C7×D4—41no—
134C2×C7×D4C22⋊C411yescharacteristic
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—41no—
140C22×C4×C7—41no—
141C22×C4×C7—41no—
142C22×C4×C7—41no—
143C22×C4×C7—41no—
144C22×C4×C7—41no—
145C22×C4×C7—41no—
146C22×C4×C7C22×C411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7#4111yescharacteristic
152C7×Q16—42no—
153C7×SD16—42no—
154C7×SD16—42no—
155C7×D8—42no—
156C7×M16—41no—
157C7×M16—41no—
158C7×M16—81no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—41no—
166C2×C7×C8—41no—
167C2×C7×C8—41no—
168C2×C7×C8—41no—
169C2×C7×C8—41no—
170C2×C7×C8—41no—
171C2×C7×C8—41no—
172C2×C7×C8—81no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4C22⋊C411yescharacteristic
176C7×C4⋊C4C22⋊C411yescharacteristic
177C7×C22⋊C4—21no—
178C7×C22⋊C4—21no—
179C7×C22⋊C4—21no—
180C7×C22⋊C4—21no—
181C7×C22⋊C4—21no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—41no—
184C42×C7—21no—
185C42×C7—41no—
186C42×C7—41no—
187C22×#41C22×C711yescharacteristic
188C2×#135—21no—
189#525C22×C711yescharacteristic
190#452—21no—
191#451C4×C711yescharacteristic, complemented
192#450C4×C711yescharacteristic, complemented
193#449—21no—
194#448C4×C711yescharacteristic, complemented
195#447C4×C711yescharacteristic, complemented
196#436—21no—
197#435—21no—
198#435—21no—
199C8×D4—21no—
200#433—21no—
201C2×#104—21no—
202C2×#104—21no—
203#416—21no—
204#408—21no—
205#407C22×C711yescharacteristic
206C2×C22⋊C8—21no—
207C2×C22⋊C8—21no—
208C2×C22⋊C8C22×C711yescharacteristic
209C2×C22⋊C8C22×C711yescharacteristic
210C2×C22⋊C8C22×C711yescharacteristic
211C4×C22⋊C4C22×C711yescharacteristic
212#328—21no—
213#327—21no—
214C23×C7—21no—
215C23×C7—21no—
216C23×C7—22no—
217C23×C7—21no—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7C4×D411yescharacteristic
227C23×C7#12111yescharacteristic
228C23×C7#12311yescharacteristic
229C7×Q8—22no—
230C7×Q8—22no—
231C7×Q8—41no—
232C7×Q8—41no—
233C7×Q8—41no—
234C7×D4—22no—
235C7×D4—22no—
236C7×D4—41no—
237C7×D4—41no—
238C7×D4—41no—
239C7×D4—42no—
240C7×D4—42no—
241C7×D4—42no—
242C7×D4—42no—
243C7×D4—41no—
244C7×D4—41no—
245C7×D4—81no—
246C7×D4—81no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—22no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—81no—
270C2×C4×C7—81no—
271C2×C4×C7C2×C22⋊C411yescharacteristic
272C2×C4×C7C4×D411yescharacteristic
273C2×C4×C7#12011yescharacteristic
274C2×C4×C7#12111yescharacteristic
275C7×C8—41no—
276C7×C8—41no—
277C7×C8—41no—
278C7×C8—41no—
279C7×C8—41no—
280C7×C8—41no—
281C7×C8—81no—
282C2×#41—21no—
283C2×#41—21no—
284C2×#41—21no—
285C2×#41—21no—
286C2×#41—42no—
287C2×#41—42no—
288C2×#41C7×D411yescharacteristic
289C22×Q8C2×C4×C711yescharacteristic
290C22×D4—21no—
291C22×D4C2×C4×C711yescharacteristic
292C23×C4—21no—
293C23×C4C23×C711yescharacteristic
294#135—42no—
295#135—42no—
296C2×Q16—21no—
297C2×SD16—21no—
298C2×SD16—21no—
299C2×D8—21no—
300C2×M16—21no—
301C2×M16C7×D411yescharacteristic
302C2×M16C7×D411yescharacteristic
303C22×C8—21no—
304C22×C8—21no—
305C22×C8—21no—
306C22×C8C7×D411yescharacteristic
307C22×C8C7×D411yescharacteristic
308#123—21no—
309C22⋊Q8C2×C4×C711yescharacteristic
310C22⋊Q8C7×D411yescharacteristic
311#121C2×C4×C711yescharacteristic
312#121C7×D411yescharacteristic
313#120—21no—
314C4×D4—21no—
315C4×D4—21no—
316#117C7×D411yescharacteristic
317C2×C22⋊C4—21no—
318C2×C42—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321#104—42no—
322#104—42no—
323#103—21no—
324#103—21no—
325#102—21no—
326#102—21no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332C22⋊C8—21no—
333C22⋊C8—21no—
334C22⋊C8—21no—
335C22⋊C8—21no—
336C22⋊C8—42no—
337C22⋊C8—42no—
338C22⋊C8C2×C4×C711yescharacteristic
339C22⋊C8C2×C4×C711yescharacteristic
340#97—21no—
341C4×C8—21no—
342#95—21no—
343C22×C7—21no—
344C22×C7—21no—
345C22×C7—21no—
346C22×C7—21no—
347C22×C7—21no—
348C22×C7—21no—
349C22×C7—21no—
350C22×C7—21no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—41no—
354C22×C7—41no—
355C22×C7—42no—
356C22×C7—41no—
357C22×C7—81no—
358C22×C7—81no—
359C22×C7—81no—
360C22×C7—81no—
361C22×C7—81no—
362C22×C7—81no—
363C22×C7#38611yescharacteristic
364C22×C7#44311yescharacteristic
365C22×C7#44411yescharacteristic
366C4×C7—21no—
367C4×C7—21no—
368C4×C7—21no—
369C4×C7—41no—
370C4×C7—41no—
371C4×C7—42no—
372C4×C7—41no—
373C4×C7—82no—
374C4×C7B(4,𝔽2)11yescharacteristic
375C4×C7B(4,𝔽2)11yescharacteristic
376C24—21no—
377C24C7×#4111yescharacteristic
378#41—22no—
379#41—22no—
380#41—22no—
381#41—22no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388#41—42no—
389#41—42no—
390#41—42no—
391#41—42no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—42no—
395C2×Q8C7×C22⋊C411yescharacteristic
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4C7×C22⋊C411yescharacteristic
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4C22×C4×C711yescharacteristic
419C22×C4C2×C7×D411yescharacteristic
420C22×C4C2×C7×D411yescharacteristic
421C22×C4C2×C7×D411yescharacteristic
422C22×C4C2×C7×D411yescharacteristic
423C22×C4C7×#4111yescharacteristic
424Q16—42no—
425SD16—42no—
426SD16—42no—
427D8—42no—
428M16—41no—
429M16—41no—
430M16—81no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—81no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4C7×C22⋊C411yescharacteristic
448C4⋊C4C7×C22⋊C411yescharacteristic
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—22no—
455C22⋊C4—41no—
456C42—21no—
457C42—41no—
458C42—41no—
459C2×C7—21no—
460C2×C7—21no—
461C2×C7—41no—
462C2×C7—41no—
463C2×C7—81no—
464C2×C7—82no—
465C2×C7#187611yescharacteristic
466C2×C7#188111yescharacteristic
467C2×C7#188911yescharacteristic
468C23—21no—
469C23—21no—
470C23—22no—
471C23—21no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23C4×C7×D411yescharacteristic
481C23C7×#12111yescharacteristic
482C23C7×#12311yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—41no—
486Q8—41no—
487Q8—41no—
488D4—22no—
489D4—22no—
490D4—41no—
491D4—41no—
492D4—41no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—41no—
498D4—41no—
499D4—81no—
500D4—81no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4C2×C7×C22⋊C411yescharacteristic
526C2×C4C4×C7×D411yescharacteristic
527C2×C4C7×#12011yescharacteristic
528C2×C4C7×#12111yescharacteristic
529C8—41no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—81no—
536C7#903711yescharacteristic, direct product factor, Sylow
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22—42no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C7×#38611yescharacteristic
558C22C7×#44311yescharacteristic
559C22C7×#44411yescharacteristic
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—82no—
568C4C7×B(4,𝔽2)11yescharacteristic
569C4C7×B(4,𝔽2)11yescharacteristic
570C2—21no—
571C2—21no—
572C2—41no—
573C2—41no—
574C2—82no—
575C2—81no—
576C2C7×#187611yescharacteristic
577C2C7×#188111yescharacteristic
578C2C7×#188911yescharacteristic
subgroupG/H
centerC2×C4×C7#120
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
socleC22×C7#386
prime radicalC7×#3002C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2976C2semidirect
3. C7×#1789C2semidirect
4. C7×#1780C2semidirect
5. C7×#1501C2semidirect
6. C7×#1298C2semidirect
7. C7×#1282C2semidirect
12. C7×#451C4semidirect
13. C7×#450C4semidirect
15. C7×#448C4semidirect
16. C7×#447C4semidirect
35. #9037C7direct (N × H)
98. #2976C2×C7semidirect
99. #1789C2×C7semidirect
100. #1780C2×C7semidirect
101. #1501C2×C7semidirect
102. #1298C2×C7semidirect
103. #1282C2×C7semidirect
191. #451C4×C7semidirect
192. #450C4×C7semidirect
194. #448C4×C7semidirect
195. #447C4×C7semidirect
536. C7#9037direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#30022
C7×#29762
C7×#17892
C7×#17802
C7×#15012
C7×#12982
C7×#12822
#90377