Finite Group Databasebeta

← C7×#13361 #13190577

Subgroups — C7×#13361 #13190577

Order 1792.

686 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C7×#3200C211yesmaximal, characteristic, complemented
2C7×#3022C211yesmaximal, characteristic, complemented
3C7×#3021C211yesmaximal, characteristic
4C7×#2123C211yesmaximal, characteristic, complemented
5C7×#2122C211yesmaximal, characteristic, complemented
6C7×#1794C211yesmaximal, characteristic, complemented
7C7×#1392C211yesmaximal, characteristic, complemented
8C7×#585C2211yescharacteristic
9C2×C7×#137—21no—
10C2×C7×AGL(1,ℤ/8)—21no—
11C7×#536C2211yescharacteristic
12C7×#535C2211yescharacteristic
13C7×#485—21no—
14C7×#482—21no—
15C7×#468—21no—
16C7×#465—21no—
17C7×#456—22no—
18C7×#455—22no—
19C7×#454—22no—
20C7×#453—22no—
21C7×#450—21no—
22C7×#448—21no—
23C7×#421—21no—
24C7×#421—21no—
25C7×#421—21no—
26C7×#416—21no—
27C7×#413C2211yescharacteristic
28C7×#413C2211yescharacteristic
29C7×#410C2211yescharacteristic
30C7×#408C2211yescharacteristic
31C7×#352—22no—
32C7×AΣL(1,𝔽16)—22no—
33#13361C711yesmaximal, characteristic, direct product factor, Sylow
34C7×2−1+4—41no—
35C7×2−1+4D412yes—
36C7×H(5,𝔽2)—21no—
37C7×H(5,𝔽2)—21no—
38C7×H(5,𝔽2)—21no—
39C7×H(5,𝔽2)—41no—
40C2×C7×#41—21no—
41C2×C7×#41—21no—
42C2×C7×#41—21no—
43C2×C7×#41—42no—
44C2×C7×#41C2311yescharacteristic
45C22×C7×Q8D411yescharacteristic
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—21no—
53C2×C7×SD16—21no—
54C2×C7×SD16—41no—
55C2×C7×SD16—41no—
56C2×C7×D8—41no—
57C2×C7×M16—21no—
58C2×C7×M16—21no—
59C2×C7×M16D411yescharacteristic
60C22×C7×C8D411yescharacteristic
61C7×C4⋊Q8—21no—
62C7×#127—21no—
63C7×#124—21no—
64C7×#124—21no—
65C7×#123—21no—
66C7×#123—21no—
67C7×C22⋊Q8—21no—
68C7×C22⋊Q8—21no—
69C7×#121—21no—
70C7×#121—21no—
71C7×#120—21no—
72C7×#120—21no—
73C7×#117—21no—
74C7×#117D411yescharacteristic
75C7×#117D411yescharacteristic
76C7×#107—21no—
77C7×#106—21no—
78C7×#104—24no—
79C7×#104—24no—
80C7×#104—42no—
81C7×#103—21no—
82C7×#103—21no—
83C7×#103—41no—
84C7×#102—21no—
85C7×#102—21no—
86C7×#102—41no—
87C7×#101—21no—
88C7×#101—21no—
89C7×#100D412yes—
90C7×#100D412yes—
91C7×#99—21no—
92C7×#99—21no—
93C7×C22⋊C8—21no—
94C7×C22⋊C8—21no—
95#3200C2×C711yescharacteristic, complemented
96#3022C2×C711yescharacteristic, complemented
97#3021C2×C711yescharacteristic
98#2123C2×C711yescharacteristic, complemented
99#2122C2×C711yescharacteristic, complemented
100#1794C2×C711yescharacteristic, complemented
101#1392C2×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—41no—
113C7×#41—41no—
114C7×#41—42no—
115C7×#41—42no—
116C7×#41—42no—
117C7×#41—42no—
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—42no—
127C2×C7×Q8C2×D411yescharacteristic
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—41no—
134C2×C7×D4—41no—
135C2×C7×D4—42no—
136C2×C7×D4—42no—
137C2×C7×D4—42no—
138C2×C7×D4—41no—
139C2×C7×D4—41no—
140C2×C7×D4—41no—
141C2×C7×D4—41no—
142C2×C7×D4—41no—
143C2×C7×D4C2×D411yescharacteristic
144C2×C7×D4C2×D411yescharacteristic
145C2×C7×D4#4111yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—41no—
148C22×C4×C7—41no—
149C22×C4×C7—41no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C7×Q16—44no—
154C7×SD16—44no—
155C7×SD16—44no—
156C7×SD16—81no—
157C7×SD16—81no—
158C7×D8—44no—
159C7×M16—22no—
160C7×M16—22no—
161C7×M16—41no—
162C7×M16—41no—
163C7×M16—41no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—41no—
167C2×C7×C8—41no—
168C2×C7×C8—41no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—41no—
172C7×C4⋊C4—41no—
173C7×C4⋊C4—41no—
174C7×C22⋊C4—21no—
175C7×C22⋊C4—21no—
176C7×C22⋊C4—41no—
177C7×C22⋊C4—41no—
178C7×C22⋊C4—41no—
179C7×C22⋊C4—41no—
180C7×C22⋊C4—41no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—41no—
184#585C22×C711yescharacteristic
185C2×#137—21no—
186C2×AGL(1,ℤ/8)—21no—
187#536C22×C711yescharacteristic
188#535C22×C711yescharacteristic
189#485—21no—
190#482—21no—
191#468—21no—
192#465—21no—
193#456—22no—
194#455—22no—
195#454—22no—
196#453—22no—
197#450—21no—
198#448—21no—
199#421—21no—
200#421—21no—
201#421—21no—
202#416—21no—
203#413C22×C711yescharacteristic
204#413C22×C711yescharacteristic
205#410C22×C711yescharacteristic
206#408C22×C711yescharacteristic
207#352—22no—
208AΣL(1,𝔽16)—22no—
209C23×C7—41no—
210C23×C7—41no—
211C23×C7—41no—
212C23×C7—41no—
213C23×C7—81no—
214C23×C7—161no—
215C23×C7#12111yescharacteristic
216C23×C7#12111yescharacteristic
217C23×C7#12711yescharacteristic
218C7×Q8—22no—
219C7×Q8—22no—
220C7×Q8—22no—
221C7×Q8—41no—
222C7×Q8—42no—
223C7×Q8—41no—
224C7×Q8—41no—
225C7×Q8—41no—
226C7×D4—22no—
227C7×D4—22no—
228C7×D4—22no—
229C7×D4—41no—
230C7×D4—41no—
231C7×D4—41no—
232C7×D4—42no—
233C7×D4—42no—
234C7×D4—42no—
235C7×D4—42no—
236C7×D4—42no—
237C7×D4—41no—
238C7×D4—41no—
239C7×D4—41no—
240C7×D4—161no—
241C7×D4—161no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—42no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—81no—
259C2×C4×C7—81no—
260C2×C4×C7#12011yescharacteristic
261C2×C4×C7#12011yescharacteristic
262C2×C4×C7#12011yescharacteristic
263C2×C4×C7#12111yescharacteristic
264C7×C8—42no—
265C7×C8—42no—
266C7×C8—41no—
267C7×C8—41no—
2682−1+4—41no—
2692−1+4C7×D412yes—
270H(5,𝔽2)—21no—
271H(5,𝔽2)—21no—
272H(5,𝔽2)—21no—
273H(5,𝔽2)—41no—
274C2×#41—21no—
275C2×#41—21no—
276C2×#41—21no—
277C2×#41—42no—
278C2×#41C23×C711yescharacteristic
279C22×Q8C7×D411yescharacteristic
280C22×D4C7×D411yescharacteristic
281#137—24no—
282#137—42no—
283AGL(1,ℤ/8)—24no—
284AGL(1,ℤ/8)—42no—
285C2×Q16—41no—
286C2×SD16—21no—
287C2×SD16—21no—
288C2×SD16—41no—
289C2×SD16—41no—
290C2×D8—41no—
291C2×M16—21no—
292C2×M16—21no—
293C2×M16C7×D411yescharacteristic
294C22×C8C7×D411yescharacteristic
295C4⋊Q8—21no—
296#127—21no—
297#124—21no—
298#124—21no—
299#123—21no—
300#123—21no—
301C22⋊Q8—21no—
302C22⋊Q8—21no—
303#121—21no—
304#121—21no—
305#120—21no—
306#120—21no—
307#117—21no—
308#117C7×D411yescharacteristic
309#117C7×D411yescharacteristic
310#107—21no—
311#106—21no—
312#104—24no—
313#104—24no—
314#104—42no—
315#103—21no—
316#103—21no—
317#103—41no—
318#102—21no—
319#102—21no—
320#102—41no—
321#101—21no—
322#101—21no—
323#100C7×D412yes—
324#100C7×D412yes—
325#99—21no—
326#99—21no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22×C7—21no—
330C22×C7—21no—
331C22×C7—21no—
332C22×C7—41no—
333C22×C7—41no—
334C22×C7—41no—
335C22×C7—42no—
336C22×C7—81no—
337C22×C7—81no—
338C22×C7—81no—
339C22×C7—161no—
340C22×C7—161no—
341C22×C7—161no—
342C22×C7—162no—
343C22×C7#39211yescharacteristic
344C4×C7—21no—
345C4×C7—21no—
346C4×C7—21no—
347C4×C7—42no—
348C4×C7—81no—
349C4×C7—81no—
350C4×C7—81no—
351C4×C7—161no—
352C4×C7B(4,𝔽2)11yescharacteristic
353C4×C7B(4,𝔽2)11yescharacteristic
354C24—21no—
355#41—22no—
356#41—22no—
357#41—22no—
358#41—22no—
359#41—22no—
360#41—22no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
365#41—41no—
366#41—42no—
367#41—42no—
368#41—42no—
369#41—42no—
370#41—41no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—42no—
379C2×Q8C2×C7×D411yescharacteristic
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4C2×C7×D411yescharacteristic
396C2×D4C2×C7×D411yescharacteristic
397C2×D4C7×#4111yescharacteristic
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4C2×C7×D411yescharacteristic
403C22×C4C2×C7×D411yescharacteristic
404C22×C4C2×C7×D411yescharacteristic
405Q16—44no—
406SD16—44no—
407SD16—44no—
408SD16—81no—
409SD16—81no—
410D8—44no—
411M16—22no—
412M16—22no—
413M16—41no—
414M16—41no—
415M16—41no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42—21no—
434C42—21no—
435C42—41no—
436C2×C7—21no—
437C2×C7—41no—
438C2×C7—41no—
439C2×C7—41no—
440C2×C7—82no—
441C2×C7—161no—
442C2×C7#202111yescharacteristic
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—81no—
448C23—161no—
449C23C7×#12111yescharacteristic
450C23C7×#12111yescharacteristic
451C23C7×#12711yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—22no—
455Q8—41no—
456Q8—42no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460D4—22no—
461D4—22no—
462D4—22no—
463D4—41no—
464D4—41no—
465D4—41no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—161no—
475D4—161no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4C7×#12011yescharacteristic
495C2×C4C7×#12011yescharacteristic
496C2×C4C7×#12011yescharacteristic
497C2×C4C7×#12111yescharacteristic
498C8—42no—
499C8—42no—
500C8—41no—
501C8—41no—
502C7#1336111yescharacteristic, direct product factor, Sylow
503C22—21no—
504C22—21no—
505C22—21no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22—162no—
517C22C7×#39211yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—161no—
526C4C7×B(4,𝔽2)11yescharacteristic
527C4C7×B(4,𝔽2)11yescharacteristic
528C2—21no—
529C2—41no—
530C2—41no—
531C2—41no—
532C2—82no—
533C2—161no—
534C2C7×#202111yescharacteristic
subgroupG/H
center, socleC2×C7#2021
commutator, Frattini, cosocleC2×#41C23×C7
prime radicalC7×#3021C2
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3200C2semidirect
2. C7×#3022C2semidirect
4. C7×#2123C2semidirect
5. C7×#2122C2semidirect
6. C7×#1794C2semidirect
7. C7×#1392C2semidirect
33. #13361C7direct (N × H)
95. #3200C2×C7semidirect
96. #3022C2×C7semidirect
98. #2123C2×C7semidirect
99. #2122C2×C7semidirect
100. #1794C2×C7semidirect
101. #1392C2×C7semidirect
502. C7#13361direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32002
C7×#30222
C7×#30212
C7×#21232
C7×#21222
C7×#17942
C7×#13922
#133617