Finite Group Databasebeta

← C7×#10323 #13187539

Subgroups — C7×#10323 #13187539

Order 1792.

664 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3187C211yesmaximal, characteristic
2C7×#2855C211yesmaximal, characteristic, complemented
3C7×#2854C211yesmaximal, characteristic
4C7×#1817C211yesmaximal, characteristic, complemented
5C7×#1520C211yesmaximal, characteristic, complemented
6C7×#1519C211yesmaximal, characteristic
7C7×#1289C211yesmaximal, characteristic, complemented
8C2×C7×#123—21no—
9C2×C7×C22⋊Q8C2211yescharacteristic
10C2×C7×#121C2211yescharacteristic
11C22×C42×C7C2211yescharacteristic
12C7×#483—21no—
13C7×#481—21no—
14C7×#465—22no—
15C7×#450—21no—
16C7×#448—21no—
17C2×C7×C22⋊C8C2211yescharacteristic
18C2×C7×C22⋊C8C2211yescharacteristic
19C7×#391—21no—
20C7×#390—21no—
21C7×#389—21no—
22C7×#389—21no—
23C7×#388—22no—
24C7×#387—22no—
25C7×#386—21no—
26C7×#385—21no—
27C7×#381—21no—
28C7×#380C2211yescharacteristic
29C7×#380C2211yescharacteristic
30C7×#336—22no—
31C7×#328—22no—
32C7×#327—22no—
33#10323C711yesmaximal, characteristic, direct product factor, Sylow
34C22×C7×Q8D411yescharacteristic
35C22×C7×D4—21no—
36C22×C7×D4D411yescharacteristic
37C23×C4×C7D411yescharacteristic
38C23×C4×C7D411yescharacteristic
39C23×C4×C7C2311yescharacteristic
40C2×C7×SD16—22no—
41C22×C7×C8—21no—
42C22×C7×C8D412yes—
43C7×#123—42no—
44C7×#123—42no—
45C7×C22⋊Q8—21no—
46C7×C22⋊Q8—21no—
47C7×C22⋊Q8—21no—
48C7×C22⋊Q8—21no—
49C7×#121—21no—
50C7×#121—21no—
51C7×#121—21no—
52C7×#121—21no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C22⋊C4—22no—
61C2×C7×C22⋊C4—22no—
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—21no—
66C2×C42×C7—21no—
67C2×C42×C7—21no—
68C2×C42×C7—21no—
69C2×C42×C7—21no—
70C2×C42×C7—21no—
71C2×C42×C7—21no—
72C7×#106—22no—
73C7×#103—22no—
74C7×#102—22no—
75C7×C22⋊C8—21no—
76C7×C22⋊C8—21no—
77C7×C22⋊C8—21no—
78C7×C22⋊C8—21no—
79C7×#95—22no—
80C7×#95—22no—
81C7×#95—22no—
82C7×#95—22no—
83C7×#95D412yes—
84C7×#95D412yes—
85#3187C2×C711yescharacteristic
86#2855C2×C711yescharacteristic, complemented
87#2854C2×C711yescharacteristic
88#1817C2×C711yescharacteristic, complemented
89#1520C2×C711yescharacteristic, complemented
90#1519C2×C711yescharacteristic
91#1289C2×C711yescharacteristic, complemented
92C24×C7—21no—
93C24×C7C2×D411yescharacteristic
94C2×C7×Q8—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8—42no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4—42no—
100C2×C7×D4—41no—
101C2×C7×D4—41no—
102C2×C7×D4—41no—
103C2×C7×D4—41no—
104C2×C7×D4—41no—
105C2×C7×D4—41no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—22no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—42no—
121C22×C4×C7—42no—
122C22×C4×C7SD1612yescomplemented
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2×D411yescharacteristic
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7#4111yescharacteristic
129C7×SD16—82no—
130C2×C7×C8—22no—
131C2×C7×C8—42no—
132C2×C7×C8—42no—
133C2×C7×C8—41no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—42no—
139C7×C4⋊C4—42no—
140C7×C4⋊C4—44no—
141C7×C4⋊C4—44no—
142C7×C4⋊C4—41no—
143C7×C4⋊C4—41no—
144C7×C22⋊C4—44no—
145C7×C22⋊C4—44no—
146C7×C22⋊C4—41no—
147C7×C22⋊C4—41no—
148C7×C22⋊C4—41no—
149C7×C22⋊C4—41no—
150C7×C22⋊C4—41no—
151C7×C22⋊C4—41no—
152C7×C22⋊C4—41no—
153C7×C22⋊C4—41no—
154C42×C7—41no—
155C42×C7—41no—
156C42×C7—41no—
157C42×C7—41no—
158C2×#123—21no—
159C2×C22⋊Q8C22×C711yescharacteristic
160C2×#121C22×C711yescharacteristic
161C22×C42C22×C711yescharacteristic
162#483—21no—
163#481—21no—
164#465—22no—
165#450—21no—
166#448—21no—
167C2×C22⋊C8C22×C711yescharacteristic
168C2×C22⋊C8C22×C711yescharacteristic
169#391—21no—
170#390—21no—
171#389—21no—
172#389—21no—
173#388—22no—
174#387—22no—
175#386—21no—
176#385—21no—
177#381—21no—
178#380C22×C711yescharacteristic
179#380C22×C711yescharacteristic
180#336—22no—
181#328—22no—
182#327—22no—
183C23×C7—41no—
184C23×C7—41no—
185C23×C7—42no—
186C23×C7—42no—
187C23×C7—41no—
188C23×C7—41no—
189C23×C7—41no—
190C23×C7#12011yescharacteristic
191C23×C7#12011yescharacteristic
192C23×C7#12111yescharacteristic
193C23×C7C2×SD1611yescharacteristic
194C23×C7#13511yescharacteristic
195C23×C7AGL(1,ℤ/8)11yescharacteristic
196C23×C7#13711yescharacteristic
197C7×Q8—42no—
198C7×Q8—81no—
199C7×D4—42no—
200C7×D4—81no—
201C7×D4—81no—
202C7×D4—81no—
203C7×D4—81no—
204C7×D4—81no—
205C2×C4×C7—22no—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7—22no—
212C2×C4×C7—22no—
213C2×C4×C7—22no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—41no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7—41no—
223C2×C4×C7—41no—
224C2×C4×C7—41no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—42no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7—41no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7#12011yescharacteristic
239C2×C4×C7#12111yescharacteristic
240C2×C4×C7#12111yescharacteristic
241C2×C4×C7#12711yescharacteristic
242C7×C8—42no—
243C7×C8—81no—
244C22×Q8C7×D411yescharacteristic
245C22×D4—21no—
246C22×D4C7×D411yescharacteristic
247C23×C4C7×D411yescharacteristic
248C23×C4C7×D411yescharacteristic
249C23×C4C23×C711yescharacteristic
250C2×SD16—22no—
251C22×C8—21no—
252C22×C8C7×D412yes—
253#123—42no—
254#123—42no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C7×D411yescharacteristic
269C2×C4⋊C4C7×D411yescharacteristic
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282#106—22no—
283#103—22no—
284#102—22no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95C7×D412yes—
294#95C7×D412yes—
295C22×C7—21no—
296C22×C7—21no—
297C22×C7—21no—
298C22×C7—21no—
299C22×C7—21no—
300C22×C7—21no—
301C22×C7—41no—
302C22×C7—41no—
303C22×C7—41no—
304C22×C7—41no—
305C22×C7—82no—
306C22×C7—82no—
307C22×C7—81no—
308C22×C7—81no—
309C22×C7—81no—
310C22×C7#39211yescharacteristic
311C22×C7#44811yescharacteristic
312C22×C7#44911yescharacteristic
313C22×C7#45011yescharacteristic
314C22×C7#45211yescharacteristic
315C22×C7#46511yescharacteristic
316C22×C7#46811yescharacteristic
317C4×C7—21no—
318C4×C7—21no—
319C4×C7—21no—
320C4×C7—42no—
321C4×C7—42no—
322C4×C7—82no—
323C4×C7—82no—
324C4×C7—81no—
325C4×C7—81no—
326C4×C7—81no—
327C4×C7#45812yes—
328C24—21no—
329C24C2×C7×D411yescharacteristic
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—42no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4C7×SD1612yescomplemented
359C22×C4C2×C7×D411yescharacteristic
360C22×C4C2×C7×D411yescharacteristic
361C22×C4C2×C7×D411yescharacteristic
362C22×C4C2×C7×D411yescharacteristic
363C22×C4C2×C7×D411yescharacteristic
364C22×C4C7×#4111yescharacteristic
365SD16—82no—
366C2×C8—22no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—41no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C2×C7—21no—
395C2×C7—21no—
396C2×C7—21no—
397C2×C7—21no—
398C2×C7—21no—
399C2×C7—21no—
400C2×C7—161no—
401C2×C7#200011yescharacteristic
402C2×C7#201011yescharacteristic
403C2×C7#202211yescharacteristic
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23C7×#12011yescharacteristic
412C23C7×#12011yescharacteristic
413C23C7×#12111yescharacteristic
414C23C2×C7×SD1611yescharacteristic
415C23C7×#13511yescharacteristic
416C23C7×AGL(1,ℤ/8)11yescharacteristic
417C23C7×#13711yescharacteristic
418Q8—42no—
419Q8—81no—
420D4—42no—
421D4—81no—
422D4—81no—
423D4—81no—
424D4—81no—
425D4—81no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4C7×#12011yescharacteristic
460C2×C4C7×#12111yescharacteristic
461C2×C4C7×#12111yescharacteristic
462C2×C4C7×#12711yescharacteristic
463C8—42no—
464C8—81no—
465C7#1032311yescharacteristic, direct product factor, Sylow
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—82no—
477C22—82no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22C7×#39211yescharacteristic
482C22C7×#44811yescharacteristic
483C22C7×#44911yescharacteristic
484C22C7×#45011yescharacteristic
485C22C7×#45211yescharacteristic
486C22C7×#46511yescharacteristic
487C22C7×#46811yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—42no—
492C4—42no—
493C4—82no—
494C4—82no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4C7×#45812yes—
499C2—21no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—161no—
506C2C7×#200011yescharacteristic
507C2C7×#201011yescharacteristic
508C2C7×#202211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC2×C7×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2855C2semidirect
4. C7×#1817C2semidirect
5. C7×#1520C2semidirect
7. C7×#1289C2semidirect
33. #10323C7direct (N × H)
86. #2855C2×C7semidirect
88. #1817C2×C7semidirect
89. #1520C2×C7semidirect
91. #1289C2×C7semidirect
122. C22×C4×C7SD16semidirect
358. C22×C4C7×SD16semidirect
465. C7#10323direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31872
C7×#28552
C7×#28542
C7×#18172
C7×#15202
C7×#15192
C7×#12892
#103237