Finite Group Databasebeta

← C7×#10232 #13187448

Subgroups — C7×#10232 #13187448

Order 1792.

644 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#475C211yesmaximal, characteristic, complemented
2C2×C7×#461C211yesmaximal, characteristic
3C7×#2538C211yesmaximal, characteristic
4C7×#1990C211yesmaximal, characteristic, complemented
5C2×C7×Q8⋊C8C211yesmaximal, characteristic, complemented
6C2×C7×D4⋊C8C211yesmaximal, characteristic
7C7×#1296C211yesmaximal, characteristic, complemented
8C22×C7×SD16—21no—
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×C22⋊Q8—21no—
11C2×C4×C7×Q8C2211yescharacteristic
12C2×C4×C7×D4C2211yescharacteristic
13C7×#475—22no—
14C7×#475—22no—
15C7×#461—22no—
16C7×#461—22no—
17C2×C7×#106—21no—
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×#103—21no—
21C2×C7×#103—21no—
22C2×C4×C7×C8C2211yescharacteristic
23C7×#396—22no—
24C7×#390—21no—
25C7×#389—21no—
26C7×#384C2211yescharacteristic
27C7×#380—21no—
28C7×#340—22no—
29C7×Q8⋊C8—22no—
30C7×D4⋊C8—22no—
31#10232C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8—21no—
33C22×C7×Q8D411yescharacteristic
34C22×C7×D4D411yescharacteristic
35C23×C4×C7—21no—
36C2×C7×SD16—24no—
37C2×C7×SD16—42no—
38C2×C7×SD16—42no—
39C22×C7×C8—21no—
40C22×C7×C8—21no—
41C22×C7×C8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8D412yes—
46C7×C22⋊Q8—42no—
47C7×C22⋊Q8—42no—
48C4×C7×Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C4×C7×D4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4D411yescharacteristic
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C7×C4⋊C4D412yes—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—22no—
68C2×C42×C7—21no—
69C2×C42×C7C2311yescharacteristic
70C7×#106—24no—
71C7×C4⋊C8—21no—
72C7×C4⋊C8—21no—
73C7×C4⋊C8D412yes—
74C7×C4⋊C8D412yes—
75C7×#103—24no—
76C7×#103—24no—
77C4×C7×C8—21no—
78C4×C7×C8—21no—
79C7×#95—22no—
80C2×#475C2×C711yescharacteristic, complemented
81C2×#461C2×C711yescharacteristic
82#2538C2×C711yescharacteristic
83#1990C2×C711yescharacteristic, complemented
84C2×Q8⋊C8C2×C711yescharacteristic, complemented
85C2×D4⋊C8C2×C711yescharacteristic
86#1296C2×C711yescharacteristic, complemented
87C24×C7—21no—
88C2×C7×Q8—21no—
89C2×C7×Q8—21no—
90C2×C7×Q8—22no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—41no—
95C2×C7×Q8—41no—
96C2×C7×Q8—41no—
97C2×C7×Q8—41no—
98C2×C7×Q8—41no—
99C2×C7×Q8SD1612yescomplemented
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4—21no—
103C2×C7×D4—21no—
104C2×C7×D4—21no—
105C2×C7×D4SD1612yes—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—22no—
110C22×C4×C7—41no—
111C22×C4×C7—41no—
112C22×C4×C7—41no—
113C22×C4×C7—41no—
114C22×C4×C7—41no—
115C22×C4×C7—41no—
116C22×C4×C7—41no—
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D411yescharacteristic
120C7×SD16—48no—
121C2×C7×C8—22no—
122C2×C7×C8—22no—
123C2×C7×C8—22no—
124C2×C7×C8—41no—
125C2×C7×C8—41no—
126C2×C7×C8—41no—
127C2×C7×C8—41no—
128C2×C7×C8—41no—
129C2×C7×C8—41no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—21no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—42no—
137C7×C4⋊C4—44no—
138C7×C4⋊C4—41no—
139C7×C4⋊C4—41no—
140C7×C4⋊C4—41no—
141C7×C4⋊C4—41no—
142C7×C4⋊C4—41no—
143C7×C4⋊C4—41no—
144C7×C4⋊C4—41no—
145C7×C4⋊C4—41no—
146C7×C22⋊C4—44no—
147C7×C22⋊C4—41no—
148C7×C22⋊C4—41no—
149C7×C22⋊C4—82no—
150C42×C7—41no—
151C42×C7—41no—
152C42×C7C2×D411yescharacteristic
153C42×C7C2×D411yescharacteristic
154C42×C7C2×D411yescharacteristic
155C42×C7#4111yescharacteristic
156C22×SD16—21no—
157C2×C4⋊Q8C22×C711yescharacteristic
158C2×C22⋊Q8—21no—
159C2×C4×Q8C22×C711yescharacteristic
160C2×C4×D4C22×C711yescharacteristic
161#475—22no—
162#475—22no—
163#461—22no—
164#461—22no—
165C2×#106—21no—
166C2×C4⋊C8C22×C711yescharacteristic
167C2×C4⋊C8C22×C711yescharacteristic
168C2×#103—21no—
169C2×#103—21no—
170C2×C4×C8C22×C711yescharacteristic
171#396—22no—
172#390—21no—
173#389—21no—
174#384C22×C711yescharacteristic
175#380—21no—
176#340—22no—
177Q8⋊C8—22no—
178D4⋊C8—22no—
179C23×C7—22no—
180C23×C7—41no—
181C23×C7—41no—
182C23×C7—41no—
183C23×C7—41no—
184C23×C7—41no—
185C23×C7—41no—
186C23×C7#12011yescharacteristic
187C7×Q8—22no—
188C7×Q8—22no—
189C7×Q8—41no—
190C7×Q8—41no—
191C7×Q8—42no—
192C7×Q8—42no—
193C7×Q8—81no—
194C7×Q8—81no—
195C7×D4—22no—
196C7×D4—22no—
197C7×D4—41no—
198C7×D4—41no—
199C2×C4×C7—21no—
200C2×C4×C7—21no—
201C2×C4×C7—22no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—41no—
206C2×C4×C7—42no—
207C2×C4×C7—42no—
208C2×C4×C7—42no—
209C2×C4×C7—41no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—42no—
213C2×C4×C7—41no—
214C2×C4×C7—81no—
215C2×C4×C7—81no—
216C2×C4×C7—81no—
217C2×C4×C7—81no—
218C2×C4×C7—81no—
219C2×C4×C7—81no—
220C2×C4×C7#12011yescharacteristic
221C2×C4×C7#12011yescharacteristic
222C2×C4×C7#12111yescharacteristic
223C2×C4×C7#12111yescharacteristic
224C2×C4×C7#12111yescharacteristic
225C2×C4×C7#12711yescharacteristic
226C2×C4×C7C2×SD1611yescharacteristic
227C2×C4×C7C2×SD1611yescharacteristic
228C2×C4×C7#13511yescharacteristic
229C2×C4×C7#13511yescharacteristic
230C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
231C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
232C2×C4×C7#13711yescharacteristic
233C2×C4×C7#13711yescharacteristic
234C7×C8—42no—
235C7×C8—42no—
236C7×C8—42no—
237C22×Q8—21no—
238C22×Q8C7×D411yescharacteristic
239C22×D4C7×D411yescharacteristic
240C23×C4—21no—
241C2×SD16—24no—
242C2×SD16—42no—
243C2×SD16—42no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8C7×D412yes—
251C22⋊Q8—42no—
252C22⋊Q8—42no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4C7×D411yescharacteristic
268C2×C4⋊C4C7×D411yescharacteristic
269C2×C4⋊C4C7×D412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C42—21no—
274C2×C42C23×C711yescharacteristic
275#106—24no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8C7×D412yes—
279C4⋊C8C7×D412yes—
280#103—24no—
281#103—24no—
282C4×C8—21no—
283C4×C8—21no—
284#95—22no—
285C22×C7—42no—
286C22×C7—42no—
287C22×C7—42no—
288C22×C7—81no—
289C22×C7—81no—
290C22×C7—81no—
291C22×C7—81no—
292C22×C7#39211yescharacteristic
293C22×C7#44811yescharacteristic
294C22×C7#44911yescharacteristic
295C22×C7#45011yescharacteristic
296C22×C7#45211yescharacteristic
297C22×C7#45511yescharacteristic
298C22×C7#45611yescharacteristic
299C4×C7—21no—
300C4×C7—21no—
301C4×C7—42no—
302C4×C7—82no—
303C4×C7—81no—
304C4×C7—82no—
305C4×C7—81no—
306C4×C7#44811yescharacteristic
307C4×C7#44911yescharacteristic
308C4×C7#45011yescharacteristic
309C4×C7#45211yescharacteristic
310C4×C7#46511yescharacteristic
311C4×C7#46511yescharacteristic
312C4×C7#46811yescharacteristic
313C4×C7#46811yescharacteristic
314C24—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—22no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8C7×SD1612yescomplemented
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4C7×SD1612yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4C2×C7×D411yescharacteristic
345C22×C4C2×C7×D411yescharacteristic
346C22×C4C2×C7×D411yescharacteristic
347SD16—48no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C22⋊C4—44no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—82no—
377C42—41no—
378C42—41no—
379C42C2×C7×D411yescharacteristic
380C42C2×C7×D411yescharacteristic
381C42C2×C7×D411yescharacteristic
382C42C7×#4111yescharacteristic
383C2×C7—82no—
384C2×C7#164211yescharacteristic
385C2×C7#164311yescharacteristic
386C2×C7#165011yescharacteristic
387C2×C7#165111yescharacteristic
388C2×C7#200011yescharacteristic
389C2×C7#200011yescharacteristic
390C2×C7#200711yescharacteristic
391C23—22no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23C7×#12011yescharacteristic
399Q8—22no—
400Q8—22no—
401Q8—41no—
402Q8—41no—
403Q8—42no—
404Q8—42no—
405Q8—81no—
406Q8—81no—
407D4—22no—
408D4—22no—
409D4—41no—
410D4—41no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4C7×#12011yescharacteristic
433C2×C4C7×#12011yescharacteristic
434C2×C4C7×#12111yescharacteristic
435C2×C4C7×#12111yescharacteristic
436C2×C4C7×#12111yescharacteristic
437C2×C4C7×#12711yescharacteristic
438C2×C4C2×C7×SD1611yescharacteristic
439C2×C4C2×C7×SD1611yescharacteristic
440C2×C4C7×#13511yescharacteristic
441C2×C4C7×#13511yescharacteristic
442C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
443C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
444C2×C4C7×#13711yescharacteristic
445C2×C4C7×#13711yescharacteristic
446C8—42no—
447C8—42no—
448C8—42no—
449C7#1023211yescharacteristic, direct product factor, Sylow
450C22—42no—
451C22—42no—
452C22—42no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22C7×#39211yescharacteristic
458C22C7×#44811yescharacteristic
459C22C7×#44911yescharacteristic
460C22C7×#45011yescharacteristic
461C22C7×#45211yescharacteristic
462C22C7×#45511yescharacteristic
463C22C7×#45611yescharacteristic
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—82no—
469C4—81no—
470C4—82no—
471C4C7×#44811yescharacteristic
472C4C7×#44911yescharacteristic
473C4C7×#45011yescharacteristic
474C4C7×#45211yescharacteristic
475C4C7×#46511yescharacteristic
476C4C7×#46511yescharacteristic
477C4C7×#46811yescharacteristic
478C4C7×#46811yescharacteristic
479C2—82no—
480C2C7×#164211yescharacteristic
481C2C7×#164311yescharacteristic
482C2C7×#165011yescharacteristic
483C2C7×#165111yescharacteristic
484C2C7×#200011yescharacteristic
485C2C7×#200011yescharacteristic
486C2C7×#200711yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#475C2semidirect
4. C7×#1990C2semidirect
5. C2×C7×Q8⋊C8C2semidirect
7. C7×#1296C2semidirect
31. #10232C7direct (N × H)
80. C2×#475C2×C7semidirect
83. #1990C2×C7semidirect
84. C2×Q8⋊C8C2×C7semidirect
86. #1296C2×C7semidirect
99. C2×C7×Q8SD16semidirect
326. C2×Q8C7×SD16semidirect
449. C7#10232direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4752
C2×C7×#4612
C7×#25382
C7×#19902
C2×C7×Q8⋊C82
C2×C7×D4⋊C82
C7×#12962
#102327