Finite Group Databasebeta

← C7×#13376 #13190592

Subgroups — C7×#13376 #13190592

Order 1792.

640 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2748C211yesmaximal, characteristic, complemented
2C7×#1792C211yesmaximal, characteristic
3C7×#1790C211yesmaximal, characteristic
4C7×#1790C211yesmaximal, characteristic
5C7×#1393C211yesmaximal, characteristic, complemented
6C7×#1390C211yesmaximal, characteristic, complemented
7C7×#1390C211yesmaximal, characteristic, complemented
8C2×C7×H(5,𝔽2)C2211yescharacteristic
9C2×C7×#104—21no—
10C7×#417—21no—
11C2×C7×#100C2211yescharacteristic
12C2×C7×#99—21no—
13C2×C7×#99—21no—
14C2×C7×#99C2211yescharacteristic
15C2×C7×#99C2211yescharacteristic
16C7×#399—21no—
17C7×#398—21no—
18C7×#396—21no—
19C7×#390C2211yescharacteristic
20C7×#387—21no—
21C7×#386—21no—
22C7×#386—21no—
23C7×#386C2211yescharacteristic
24C7×#386C2211yescharacteristic
25C7×#385—21no—
26C7×#380—21no—
27#13376C711yesmaximal, characteristic, direct product factor, Sylow
28C7×H(5,𝔽2)—21no—
29C7×H(5,𝔽2)—21no—
30C7×H(5,𝔽2)—21no—
31C7×H(5,𝔽2)—42no—
32C7×H(5,𝔽2)Q812yes—
33C2×C7×#41—21no—
34C2×C7×#41—41no—
35C22×C7×D4—21no—
36C22×C7×D4—21no—
37C22×C7×D4—41no—
38C22×C7×D4C2311yescharacteristic
39C23×C4×C7D411yescharacteristic
40C23×C4×C7D411yescharacteristic
41C2×C7×M16—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4D411yescharacteristic
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C7×C22⋊C4—21no—
53C2×C7×C22⋊C4—21no—
54C2×C7×C22⋊C4—41no—
55C2×C7×C22⋊C4—41no—
56C2×C7×C22⋊C4D411yescharacteristic
57C2×C7×C22⋊C4D411yescharacteristic
58C2×C42×C7D411yescharacteristic
59C7×#104—42no—
60C7×#103—41no—
61C7×#102—41no—
62C7×#100—21no—
63C7×#100—21no—
64C7×#99—21no—
65C7×#99—21no—
66C7×#99—21no—
67C7×#99—21no—
68C7×#99—42no—
69C7×#99—42no—
70C7×#95—21no—
71C7×#95—21no—
72C7×#95—21no—
73C7×#95—21no—
74C7×#95—21no—
75C7×#95—41no—
76C7×#95—41no—
77#2748C2×C711yescharacteristic, complemented
78#1792C2×C711yescharacteristic
79#1790C2×C711yescharacteristic
80#1790C2×C711yescharacteristic
81#1393C2×C711yescharacteristic, complemented
82#1390C2×C711yescharacteristic, complemented
83#1390C2×C711yescharacteristic, complemented
84C24×C7—21no—
85C24×C7—21no—
86C24×C7C2×D411yescharacteristic
87C24×C7C2×D411yescharacteristic
88C7×#41—22no—
89C7×#41—22no—
90C7×#41—42no—
91C7×#41—41no—
92C7×#41—42no—
93C7×#41—42no—
94C7×#41—42no—
95C7×#41—41no—
96C2×C7×Q8—21no—
97C2×C7×D4—21no—
98C2×C7×D4—22no—
99C2×C7×D4—22no—
100C2×C7×D4—22no—
101C2×C7×D4—22no—
102C2×C7×D4—41no—
103C2×C7×D4—41no—
104C2×C7×D4—42no—
105C2×C7×D4—42no—
106C2×C7×D4—41no—
107C2×C7×D4—41no—
108C2×C7×D4—41no—
109C2×C7×D4—41no—
110C2×C7×D4—42no—
111C2×C7×D4—42no—
112C2×C7×D4—41no—
113C2×C7×D4—41no—
114C2×C7×D4—41no—
115C2×C7×D4C2×Q811yescharacteristic
116C2×C7×D4#4111yescharacteristic
117C2×C7×D4#4111yescharacteristic
118C2×C7×D4#4111yescharacteristic
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—41no—
128C22×C4×C7—42no—
129C22×C4×C7—41no—
130C22×C4×C7—41no—
131C22×C4×C7—41no—
132C22×C4×C7—41no—
133C22×C4×C7—81no—
134C22×C4×C7C2×D411yescharacteristic
135C7×M16—41no—
136C7×M16—41no—
137C2×C7×C8—41no—
138C7×C4⋊C4—41no—
139C7×C4⋊C4—42no—
140C7×C4⋊C4—81no—
141C7×C4⋊C4—81no—
142C7×C4⋊C4—81no—
143C7×C4⋊C4—81no—
144C7×C22⋊C4—42no—
145C7×C22⋊C4—41no—
146C7×C22⋊C4—41no—
147C7×C22⋊C4—41no—
148C7×C22⋊C4—41no—
149C7×C22⋊C4—41no—
150C7×C22⋊C4—41no—
151C7×C22⋊C4—81no—
152C7×C22⋊C4—81no—
153C7×C22⋊C4—81no—
154C7×C22⋊C4—82no—
155C7×C22⋊C4—82no—
156C42×C7—41no—
157C2×H(5,𝔽2)C22×C711yescharacteristic
158C2×#104—21no—
159#417—21no—
160C2×#100C22×C711yescharacteristic
161C2×#99—21no—
162C2×#99—21no—
163C2×#99C22×C711yescharacteristic
164C2×#99C22×C711yescharacteristic
165#399—21no—
166#398—21no—
167#396—21no—
168#390C22×C711yescharacteristic
169#387—21no—
170#386—21no—
171#386—21no—
172#386C22×C711yescharacteristic
173#386C22×C711yescharacteristic
174#385—21no—
175#380—21no—
176C23×C7—21no—
177C23×C7—21no—
178C23×C7—21no—
179C23×C7—21no—
180C23×C7—22no—
181C23×C7—22no—
182C23×C7—41no—
183C23×C7—41no—
184C23×C7—41no—
185C23×C7—41no—
186C23×C7—41no—
187C23×C7—41no—
188C23×C7—41no—
189C23×C7—81no—
190C23×C7—81no—
191C23×C7—81no—
192C23×C7—81no—
193C23×C7#12011yescharacteristic
194C23×C7C22⋊Q811yescharacteristic
195C23×C7C22⋊Q811yescharacteristic
196C23×C7#12411yescharacteristic
197C23×C7#12411yescharacteristic
198C7×Q8—22no—
199C7×Q8—41no—
200C7×D4—22no—
201C7×D4—41no—
202C7×D4—41no—
203C7×D4—41no—
204C7×D4—41no—
205C7×D4—41no—
206C7×D4—42no—
207C7×D4—42no—
208C7×D4—42no—
209C7×D4—42no—
210C7×D4—42no—
211C7×D4—42no—
212C2×C4×C7—21no—
213C2×C4×C7—21no—
214C2×C4×C7—22no—
215C2×C4×C7—22no—
216C2×C4×C7—22no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7—41no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7—41no—
223C2×C4×C7—41no—
224C2×C4×C7—42no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7—42no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—41no—
241C2×C4×C7—81no—
242C2×C4×C7—81no—
243C2×C4×C7—81no—
244C2×C4×C7—81no—
245C2×C4×C7—81no—
246C2×C4×C7—81no—
247C2×C4×C7—81no—
248C2×C4×C7—81no—
249C2×C4×C7C22⋊Q811yescharacteristic
250C2×C4×C7#12411yescharacteristic
251C7×C8—81no—
252H(5,𝔽2)—21no—
253H(5,𝔽2)—21no—
254H(5,𝔽2)—21no—
255H(5,𝔽2)—42no—
256H(5,𝔽2)C7×Q812yes—
257C2×#41—21no—
258C2×#41—41no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4—41no—
262C22×D4C23×C711yescharacteristic
263C23×C4C7×D411yescharacteristic
264C23×C4C7×D411yescharacteristic
265C2×M16—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C7×D411yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4C7×D411yescharacteristic
281C2×C22⋊C4C7×D411yescharacteristic
282C2×C42C7×D411yescharacteristic
283#104—42no—
284#103—41no—
285#102—41no—
286#100—21no—
287#100—21no—
288#99—21no—
289#99—21no—
290#99—21no—
291#99—21no—
292#99—42no—
293#99—42no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—41no—
300#95—41no—
301C22×C7—21no—
302C22×C7—21no—
303C22×C7—21no—
304C22×C7—21no—
305C22×C7—21no—
306C22×C7—21no—
307C22×C7—41no—
308C22×C7—41no—
309C22×C7—41no—
310C22×C7—41no—
311C22×C7—41no—
312C22×C7—41no—
313C22×C7—41no—
314C22×C7—41no—
315C22×C7—42no—
316C22×C7—81no—
317C22×C7—81no—
318C22×C7—82no—
319C22×C7—82no—
320C22×C7—81no—
321C22×C7—81no—
322C22×C7—161no—
323C22×C7—161no—
324C22×C7#39311yescharacteristic
325C22×C7B(4,𝔽2)11yescharacteristic
326C22×C7B(4,𝔽2)11yescharacteristic
327C4×C7—21no—
328C4×C7—21no—
329C4×C7—42no—
330C4×C7—42no—
331C4×C7—81no—
332C4×C7—81no—
333C4×C7—81no—
334C4×C7—81no—
335C4×C7—81no—
336C4×C7—161no—
337C4×C7—161no—
338C24—21no—
339C24—21no—
340C24C2×C7×D411yescharacteristic
341C24C2×C7×D411yescharacteristic
342#41—22no—
343#41—22no—
344#41—42no—
345#41—41no—
346#41—42no—
347#41—42no—
348#41—42no—
349#41—41no—
350C2×Q8—21no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4C2×C7×Q811yescharacteristic
370C2×D4C7×#4111yescharacteristic
371C2×D4C7×#4111yescharacteristic
372C2×D4C7×#4111yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—81no—
388C22×C4C2×C7×D411yescharacteristic
389M16—41no—
390M16—41no—
391C2×C8—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—81no—
395C4⋊C4—81no—
396C4⋊C4—81no—
397C4⋊C4—81no—
398C22⋊C4—42no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—81no—
406C22⋊C4—81no—
407C22⋊C4—81no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C42—41no—
411C2×C7—21no—
412C2×C7—21no—
413C2×C7—41no—
414C2×C7—41no—
415C2×C7—41no—
416C2×C7—41no—
417C2×C7—82no—
418C2×C7#202911yescharacteristic
419C2×C7#219711yescharacteristic
420C2×C7#220011yescharacteristic
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—22no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23C7×#12011yescharacteristic
439C23C7×C22⋊Q811yescharacteristic
440C23C7×C22⋊Q811yescharacteristic
441C23C7×#12411yescharacteristic
442C23C7×#12411yescharacteristic
443Q8—22no—
444Q8—41no—
445D4—22no—
446D4—41no—
447D4—41no—
448D4—41no—
449D4—41no—
450D4—41no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4C7×C22⋊Q811yescharacteristic
495C2×C4C7×#12411yescharacteristic
496C8—81no—
497C7#1337611yescharacteristic, direct product factor, Sylow
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—42no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22—82no—
517C22—81no—
518C22—81no—
519C22—161no—
520C22—161no—
521C22C7×#39311yescharacteristic
522C22C7×B(4,𝔽2)11yescharacteristic
523C22C7×B(4,𝔽2)11yescharacteristic
524C4—21no—
525C4—21no—
526C4—42no—
527C4—42no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4—161no—
534C4—161no—
535C2—21no—
536C2—21no—
537C2—41no—
538C2—41no—
539C2—41no—
540C2—41no—
541C2—82no—
542C2C7×#202911yescharacteristic
543C2C7×#219711yescharacteristic
544C2C7×#220011yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C7
prime radicalC2×C7×H(5,𝔽2)C22
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2748C2semidirect
5. C7×#1393C2semidirect
6. C7×#1390C2semidirect
7. C7×#1390C2semidirect
27. #13376C7direct (N × H)
77. #2748C2×C7semidirect
81. #1393C2×C7semidirect
82. #1390C2×C7semidirect
83. #1390C2×C7semidirect
497. C7#13376direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#27482
C7×#17922
C7×#17902
C7×#13932
C7×#13902
#133767