Finite Group Databasebeta

← C7×#13358 #13190574

Subgroups — C7×#13358 #13190574

Order 1792.

722 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#456C211yesmaximal, characteristic, complemented
2C2×C7×#454C211yesmaximal, characteristic
3C7×#2731C211yesmaximal, characteristic, complemented
4C2×C7×#356C211yesmaximal, characteristic, complemented
5C2×C7×#355C211yesmaximal, characteristic, complemented
6C7×#1793C211yesmaximal, characteristic, complemented
7C7×#1394C211yesmaximal, characteristic, complemented
8C2×C7×2−1+4C2211yescharacteristic
9C2×C7×#137—21no—
10C2×C7×AGL(1,ℤ/8)—21no—
11C2×C7×C4⋊Q8C2211yescharacteristic
12C2×C7×#124C2211yescharacteristic
13C7×#456—22no—
14C7×#456—22no—
15C7×#454—22no—
16C7×#454—22no—
17C2×C7×#104—21no—
18C2×C7×#104—21no—
19C2×C7×#104—21no—
20C7×#417—21no—
21C2×C7×#101C2211yescharacteristic
22C2×C7×#101C2211yescharacteristic
23C2×C7×#101C2211yescharacteristic
24C7×#396—21no—
25C7×#394—21no—
26C7×#393—21no—
27C7×#391C2211yescharacteristic
28C7×#388—21no—
29C7×#388—21no—
30C7×#387—21no—
31C7×#356—22no—
32C7×#355—22no—
33#13358C711yesmaximal, characteristic, direct product factor, Sylow
34C7×2−1+4—21no—
35C7×2−1+4—21no—
36C7×2−1+4—21no—
37C7×2−1+4—42no—
38C7×2−1+4D412yes—
39C2×C7×#41—21no—
40C2×C7×#41—21no—
41C2×C7×#41—21no—
42C2×C7×#41—41no—
43C22×C7×Q8—41no—
44C22×C7×Q8D411yescharacteristic
45C22×C7×Q8C2311yescharacteristic
46C22×C7×D4D411yescharacteristic
47C7×#137—24no—
48C7×#137—42no—
49C7×AGL(1,ℤ/8)—24no—
50C7×AGL(1,ℤ/8)—42no—
51C2×C7×Q16—41no—
52C2×C7×SD16—41no—
53C2×C7×SD16—41no—
54C2×C7×D8—41no—
55C2×C7×M16—21no—
56C2×C7×M16—21no—
57C2×C7×M16—21no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8—21no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8—21no—
62C7×#124—21no—
63C7×#124—21no—
64C7×#124—21no—
65C7×#124—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4D411yescharacteristic
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C42×C7D411yescharacteristic
77C2×C42×C7D411yescharacteristic
78C2×C42×C7D411yescharacteristic
79C7×#104—24no—
80C7×#104—24no—
81C7×#104—42no—
82C7×#103—41no—
83C7×#102—41no—
84C7×#101—21no—
85C7×#101—21no—
86C7×#101—21no—
87C7×#101—21no—
88C7×#101D412yes—
89C7×#101D412yes—
90C7×#95—21no—
91C7×#95—21no—
92C7×#95—21no—
93C7×#95—21no—
94C7×#95—21no—
95C2×#456C2×C711yescharacteristic, complemented
96C2×#454C2×C711yescharacteristic
97#2731C2×C711yescharacteristic, complemented
98C2×#356C2×C711yescharacteristic, complemented
99C2×#355C2×C711yescharacteristic, complemented
100#1793C2×C711yescharacteristic, complemented
101#1394C2×C711yescharacteristic, complemented
102C24×C7—21no—
103C7×#41—22no—
104C7×#41—22no—
105C7×#41—22no—
106C7×#41—22no—
107C7×#41—22no—
108C7×#41—22no—
109C7×#41—41no—
110C7×#41—41no—
111C7×#41—41no—
112C7×#41—42no—
113C7×#41—42no—
114C7×#41—42no—
115C7×#41—42no—
116C7×#41—41no—
117C7×#41—41no—
118C7×#41—41no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—41no—
125C2×C7×Q8—41no—
126C2×C7×Q8—41no—
127C2×C7×Q8—41no—
128C2×C7×Q8—42no—
129C2×C7×Q8—42no—
130C2×C7×Q8—42no—
131C2×C7×Q8—42no—
132C2×C7×Q8C2×D411yescharacteristic
133C2×C7×Q8C2×D411yescharacteristic
134C2×C7×Q8C2×D411yescharacteristic
135C2×C7×Q8#4111yescharacteristic
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—41no—
142C2×C7×D4—41no—
143C2×C7×D4—41no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—41no—
150C22×C4×C7—41no—
151C22×C4×C7—41no—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C7×Q16—44no—
156C7×SD16—44no—
157C7×SD16—44no—
158C7×D8—44no—
159C7×M16—22no—
160C7×M16—22no—
161C7×M16—41no—
162C7×M16—41no—
163C7×M16—41no—
164C7×M16—41no—
165C2×C7×C8—41no—
166C2×C7×C8—41no—
167C2×C7×C8—41no—
168C7×C4⋊C4—41no—
169C7×C4⋊C4—42no—
170C7×C4⋊C4—41no—
171C7×C4⋊C4—41no—
172C7×C4⋊C4—41no—
173C7×C4⋊C4—41no—
174C7×C22⋊C4—42no—
175C7×C22⋊C4—42no—
176C7×C22⋊C4—41no—
177C7×C22⋊C4—41no—
178C7×C22⋊C4—41no—
179C7×C22⋊C4—41no—
180C7×C22⋊C4—81no—
181C7×C22⋊C4—81no—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—41no—
187C2×2−1+4C22×C711yescharacteristic
188C2×#137—21no—
189C2×AGL(1,ℤ/8)—21no—
190C2×C4⋊Q8C22×C711yescharacteristic
191C2×#124C22×C711yescharacteristic
192#456—22no—
193#456—22no—
194#454—22no—
195#454—22no—
196C2×#104—21no—
197C2×#104—21no—
198C2×#104—21no—
199#417—21no—
200C2×#101C22×C711yescharacteristic
201C2×#101C22×C711yescharacteristic
202C2×#101C22×C711yescharacteristic
203#396—21no—
204#394—21no—
205#393—21no—
206#391C22×C711yescharacteristic
207#388—21no—
208#388—21no—
209#387—21no—
210#356—22no—
211#355—22no—
212C23×C7—41no—
213C23×C7—41no—
214C23×C7—41no—
215C23×C7—41no—
216C23×C7—41no—
217C23×C7—41no—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7#12011yescharacteristic
221C7×Q8—22no—
222C7×Q8—22no—
223C7×Q8—22no—
224C7×Q8—41no—
225C7×Q8—41no—
226C7×Q8—41no—
227C7×Q8—41no—
228C7×Q8—41no—
229C7×Q8—41no—
230C7×Q8—41no—
231C7×Q8—42no—
232C7×Q8—81no—
233C7×D4—22no—
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—81no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—22no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7#12011yescharacteristic
272C2×C4×C7#12011yescharacteristic
273C2×C4×C7#12111yescharacteristic
274C2×C4×C7#12111yescharacteristic
275C2×C4×C7#12111yescharacteristic
276C2×C4×C7#12711yescharacteristic
277C7×C8—42no—
278C7×C8—42no—
279C7×C8—81no—
2802−1+4—21no—
2812−1+4—21no—
2822−1+4—21no—
2832−1+4—42no—
2842−1+4C7×D412yes—
285C2×#41—21no—
286C2×#41—21no—
287C2×#41—21no—
288C2×#41—41no—
289C22×Q8—41no—
290C22×Q8C7×D411yescharacteristic
291C22×Q8C23×C711yescharacteristic
292C22×D4C7×D411yescharacteristic
293#137—24no—
294#137—42no—
295AGL(1,ℤ/8)—24no—
296AGL(1,ℤ/8)—42no—
297C2×Q16—41no—
298C2×SD16—41no—
299C2×SD16—41no—
300C2×D8—41no—
301C2×M16—21no—
302C2×M16—21no—
303C2×M16—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308#124—21no—
309#124—21no—
310#124—21no—
311#124—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4C7×D411yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C42C7×D411yescharacteristic
323C2×C42C7×D411yescharacteristic
324C2×C42C7×D411yescharacteristic
325#104—24no—
326#104—24no—
327#104—42no—
328#103—41no—
329#102—41no—
330#101—21no—
331#101—21no—
332#101—21no—
333#101—21no—
334#101C7×D412yes—
335#101C7×D412yes—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341C22×C7—21no—
342C22×C7—21no—
343C22×C7—42no—
344C22×C7—81no—
345C22×C7—81no—
346C22×C7—81no—
347C22×C7—81no—
348C22×C7—81no—
349C22×C7—81no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7—81no—
353C22×C7#39211yescharacteristic
354C22×C7B(4,𝔽2)11yescharacteristic
355C22×C7B(4,𝔽2)11yescharacteristic
356C4×C7—21no—
357C4×C7—21no—
358C4×C7—21no—
359C4×C7—21no—
360C4×C7—21no—
361C4×C7—21no—
362C4×C7—42no—
363C4×C7—42no—
364C4×C7—81no—
365C4×C7—81no—
366C4×C7—81no—
367C4×C7—81no—
368C24—21no—
369#41—22no—
370#41—22no—
371#41—22no—
372#41—22no—
373#41—22no—
374#41—22no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—42no—
379#41—42no—
380#41—42no—
381#41—42no—
382#41—41no—
383#41—41no—
384#41—41no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8C2×C7×D411yescharacteristic
399C2×Q8C2×C7×D411yescharacteristic
400C2×Q8C2×C7×D411yescharacteristic
401C2×Q8C7×#4111yescharacteristic
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4C2×C7×D411yescharacteristic
419C22×C4C2×C7×D411yescharacteristic
420C22×C4C2×C7×D411yescharacteristic
421Q16—44no—
422SD16—44no—
423SD16—44no—
424D8—44no—
425M16—22no—
426M16—22no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C4⋊C4—41no—
435C4⋊C4—42no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—41no—
453C2×C7—21no—
454C2×C7—21no—
455C2×C7—82no—
456C2×C7—161no—
457C2×C7#202111yescharacteristic
458C2×C7#220311yescharacteristic
459C2×C7#220411yescharacteristic
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23C7×#12011yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—22no—
472Q8—41no—
473Q8—41no—
474Q8—41no—
475Q8—41no—
476Q8—41no—
477Q8—41no—
478Q8—41no—
479Q8—42no—
480Q8—81no—
481D4—22no—
482D4—22no—
483D4—22no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—81no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4C7×#12011yescharacteristic
520C2×C4C7×#12011yescharacteristic
521C2×C4C7×#12111yescharacteristic
522C2×C4C7×#12111yescharacteristic
523C2×C4C7×#12111yescharacteristic
524C2×C4C7×#12711yescharacteristic
525C8—42no—
526C8—42no—
527C8—81no—
528C7#1335811yescharacteristic, direct product factor, Sylow
529C22—21no—
530C22—21no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22C7×#39211yescharacteristic
542C22C7×B(4,𝔽2)11yescharacteristic
543C22C7×B(4,𝔽2)11yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—42no—
551C4—42no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—81no—
556C2—21no—
557C2—21no—
558C2—82no—
559C2—161no—
560C2C7×#202111yescharacteristic
561C2C7×#220311yescharacteristic
562C2C7×#220411yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23×C7
prime radicalC2×C7×#454C2
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#456C2semidirect
3. C7×#2731C2semidirect
4. C2×C7×#356C2semidirect
5. C2×C7×#355C2semidirect
6. C7×#1793C2semidirect
7. C7×#1394C2semidirect
33. #13358C7direct (N × H)
95. C2×#456C2×C7semidirect
97. #2731C2×C7semidirect
98. C2×#356C2×C7semidirect
99. C2×#355C2×C7semidirect
100. #1793C2×C7semidirect
101. #1394C2×C7semidirect
528. C7#13358direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4562
C2×C7×#4542
C7×#27312
C2×C7×#3562
C2×C7×#3552
C7×#17932
C7×#13942
#133587