Finite Group Databasebeta

← C7×#87524 #72276770

Subgroups — C7×#87524 #72276770

Order 2688.

1652 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 1652 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C22×C7×C22⋊A4C211yesmaximal, characteristic, complemented
2C2×C7×#1020C211yesmaximal, characteristic, complemented
3C7×#5936C211yesmaximal, characteristic, complemented
4C7×#2846—31nomaximal, Hall
5C2×C7×C22⋊A4—21no—
6C2×C7×C22⋊A4C2211yescharacteristic
7C7×#1020—21no—
8C7×GL(2,ℤ/4)—43nomaximal
9C26×C7D311yescharacteristic, complemented
10C2×C7×#120—33no—
11C2×C7×#120—33no—
12C2×C7×#120—31no—
13C7×#379—33no—
14C7×#379—33no—
15C7×#379—31no—
16#87524C711yesmaximal, characteristic, direct product factor, Hall
17C7×C22⋊A4D411yescharacteristic, complemented
18C22×C7×A4—43no—
19C22×C7×A4—81no—
20C2×C7×S4—43no—
21C7×SL(2,ℤ/4)—43no—
22C25×C7—21no—
23C25×C7—33no—
24C25×C7—33no—
25C25×C7—61no—
26C25×C7—63no—
27C25×C7—61no—
28C25×C7—62no—
29C25×C7C2×D311yescharacteristic
30C22×C7×D4—33no—
31C22×C7×D4—31no—
32C22×C7×D4—33no—
33C7×#120—63no—
34C7×#120—63no—
35C7×#120—66no—
36C7×#120—66no—
37C7×#120—63no—
38C7×#120—63no—
39C7×#120—61no—
40C7×#120—63no—
41C2×C7×C22⋊C4—33no—
42C2×C7×C22⋊C4—36no—
43C2×C7×C22⋊C4—33no—
44C2×C7×C22⋊C4—33no—
45C2×C7×C22⋊C4—33no—
46C2×C7×C22⋊C4—33no—
47C22×C22⋊A4C2×C711yescharacteristic, complemented
48C2×#1020C2×C711yescharacteristic, complemented
49#5936C2×C711yescharacteristic, complemented
50C2×C7×A4—43no—
51C2×C7×A4—81no—
52C2×C7×A4—83no—
53C2×C7×A4—82no—
54C7×S4—83no—
55C7×#67—161no—
56#2846—31noSylow
57C24×C7—21no—
58C24×C7—33no—
59C24×C7—31no—
60C24×C7—33no—
61C24×C7—33no—
62C24×C7—33no—
63C24×C7—36no—
64C24×C7—36no—
65C24×C7—33no—
66C24×C7—33no—
67C24×C7—31no—
68C24×C7—61no—
69C24×C7—63no—
70C24×C7—63no—
71C24×C7—62no—
72C24×C7—62no—
73C24×C7—61no—
74C24×C7—61no—
75C24×C7—66no—
76C24×C7—63no—
77C24×C7—63no—
78C24×C7—612no—
79C24×C7—63no—
80C24×C7—63no—
81C24×C7—63no—
82C24×C7—612no—
83C24×C7—63no—
84C24×C7—63no—
85C24×C7—62no—
86C24×C7—66no—
87C24×C7—66no—
88C24×C7—66no—
89C24×C7—66no—
90C24×C7—62no—
91C24×C7#6711yescharacteristic, complemented
92C24×C7S413yescomplemented
93C2×C7×D4—63no—
94C2×C7×D4—63no—
95C2×C7×D4—63no—
96C2×C7×D4—612no—
97C2×C7×D4—63no—
98C2×C7×D4—63no—
99C2×C7×D4—63no—
100C2×C7×D4—63no—
101C2×C7×D4—63no—
102C2×C7×D4—63no—
103C2×C7×D4—63no—
104C22×C4×C7—33no—
105C22×C4×C7—33no—
106C22×C4×C7—31no—
107C7×C22⋊C4—63no—
108C7×C22⋊C4—63no—
109C7×C22⋊C4—63no—
110C7×C22⋊C4—612no—
111C7×C22⋊C4—63no—
112C7×C22⋊C4—63no—
113C7×C22⋊C4—63no—
114C7×C22⋊C4—63no—
115C7×C22⋊C4—63no—
116C7×C22⋊C4—63no—
117C7×C22⋊C4—63no—
118C2×C22⋊A4—21no—
119C2×C22⋊A4C22×C711yescharacteristic
120#1020—21no—
121GL(2,ℤ/4)—43no—
122C22×C3×C7—161no—
123C2×D3×C7—161no—
124C7×A4—43no—
125C7×A4—81no—
126C7×Dic3—161no—
127C26D3×C711yescharacteristic, complemented
128C2×#120—33no—
129C2×#120—33no—
130C2×#120—31no—
131#379—33no—
132#379—33no—
133#379—31no—
134C23×C7—22no—
135C23×C7—21no—
136C23×C7—23no—
137C23×C7—31no—
138C23×C7—33no—
139C23×C7—33no—
140C23×C7—33no—
141C23×C7—312no—
142C23×C7—33no—
143C23×C7—33no—
144C23×C7—33no—
145C23×C7—33no—
146C23×C7—33no—
147C23×C7—33no—
148C23×C7—61no—
149C23×C7—61no—
150C23×C7—62no—
151C23×C7—63no—
152C23×C7—66no—
153C23×C7—66no—
154C23×C7—63no—
155C23×C7—66no—
156C23×C7—63no—
157C23×C7—66no—
158C23×C7—63no—
159C23×C7—63no—
160C23×C7—61no—
161C23×C7—63no—
162C23×C7—63no—
163C23×C7—63no—
164C23×C7—63no—
165C23×C7—63no—
166C23×C7—66no—
167C23×C7—63no—
168C23×C7—612no—
169C23×C7—612no—
170C23×C7—612no—
171C23×C7—63no—
172C23×C7—63no—
173C23×C7—66no—
174C23×C7—66no—
175C23×C7—66no—
176C23×C7—63no—
177C23×C7—63no—
178C23×C7—61no—
179C23×C7—63no—
180C23×C7—62no—
181C23×C7—61no—
182C23×C7—66no—
183C23×C7—612no—
184C23×C7—66no—
185C23×C7—612no—
186C23×C7—612no—
187C23×C7—66no—
188C23×C7—66no—
189C23×C7—66no—
190C23×C7—66no—
191C23×C7—63no—
192C23×C7—61no—
193C23×C7C2×S413yes—
194C7×D4—123no—
195C7×D4—126no—
196C7×D4—126no—
197C7×D4—123no—
198C7×D4—123no—
199C7×D4—123no—
200C7×D4—123no—
201C7×D4—121no—
202C2×C4×C7—66no—
203C2×C4×C7—63no—
204C2×C4×C7—63no—
205C2×C4×C7—63no—
206C2×C4×C7—63no—
207C2×C4×C7—63no—
208C22⋊A4C7×D411yescharacteristic, complemented
209C22×A4—43no—
210C22×A4—81no—
211C2×S4—43no—
212SL(2,ℤ/4)—43no—
213C2×C3×C7—161no—
214C2×C3×C7—321no—
215D3×C7—321no—
216C25—21no—
217C25—33no—
218C25—33no—
219C25—61no—
220C25—63no—
221C25—61no—
222C25—62no—
223C25C2×D3×C711yescharacteristic
224C22×D4—33no—
225C22×D4—31no—
226C22×D4—33no—
227#120—63no—
228#120—63no—
229#120—66no—
230#120—66no—
231#120—63no—
232#120—63no—
233#120—61no—
234#120—63no—
235C2×C22⋊C4—33no—
236C2×C22⋊C4—36no—
237C2×C22⋊C4—33no—
238C2×C22⋊C4—33no—
239C2×C22⋊C4—33no—
240C2×C22⋊C4—33no—
241C22×C7—21no—
242C22×C7—33no—
243C22×C7—33no—
244C22×C7—31no—
245C22×C7—33no—
246C22×C7—36no—
247C22×C7—36no—
248C22×C7—33no—
249C22×C7—33no—
250C22×C7—33no—
251C22×C7—63no—
252C22×C7—63no—
253C22×C7—62no—
254C22×C7—62no—
255C22×C7—63no—
256C22×C7—66no—
257C22×C7—63no—
258C22×C7—63no—
259C22×C7—63no—
260C22×C7—612no—
261C22×C7—63no—
262C22×C7—612no—
263C22×C7—63no—
264C22×C7—63no—
265C22×C7—62no—
266C22×C7—61no—
267C22×C7—66no—
268C22×C7—66no—
269C22×C7—66no—
270C22×C7—62no—
271C22×C7—66no—
272C22×C7—61no—
273C22×C7—61no—
274C22×C7—123no—
275C22×C7—123no—
276C22×C7—121no—
277C22×C7GL(2,ℤ/4)13yescomplemented
278C22×C7#102011yescharacteristic, complemented
279C4×C7—123no—
280C4×C7—123no—
281C4×C7—121no—
282C2×A4—43no—
283C2×A4—81no—
284C2×A4—83no—
285C2×A4—82no—
286S4—83no—
287#67—161no—
288C3×C7—161noHall
289C24—21no—
290C24—33no—
291C24—31no—
292C24—33no—
293C24—33no—
294C24—33no—
295C24—36no—
296C24—36no—
297C24—33no—
298C24—33no—
299C24—31no—
300C24—61no—
301C24—63no—
302C24—63no—
303C24—62no—
304C24—62no—
305C24—61no—
306C24—61no—
307C24—66no—
308C24—63no—
309C24—63no—
310C24—63no—
311C24—612no—
312C24—63no—
313C24—63no—
314C24—63no—
315C24—612no—
316C24—63no—
317C24—62no—
318C24—62no—
319C24—66no—
320C24—66no—
321C24—66no—
322C24—66no—
323C24C7×#6711yescharacteristic, complemented
324C24C7×S413yescomplemented
325C2×D4—63no—
326C2×D4—63no—
327C2×D4—63no—
328C2×D4—612no—
329C2×D4—63no—
330C2×D4—63no—
331C2×D4—63no—
332C2×D4—63no—
333C2×D4—63no—
334C2×D4—63no—
335C2×D4—63no—
336C22×C4—33no—
337C22×C4—33no—
338C22×C4—31no—
339C22⋊C4—63no—
340C22⋊C4—63no—
341C22⋊C4—63no—
342C22⋊C4—612no—
343C22⋊C4—63no—
344C22⋊C4—63no—
345C22⋊C4—63no—
346C22⋊C4—63no—
347C22⋊C4—63no—
348C22⋊C4—63no—
349C22⋊C4—63no—
350C2×C7—21no—
351C2×C7—33no—
352C2×C7—33no—
353C2×C7—61no—
354C2×C7—62no—
355C2×C7—61no—
356C2×C7—63no—
357C2×C7—241no—
358C2×C7C2×#102011yescharacteristic
359C22×C3—161no—
360C2×D3—161no—
361A4—43no—
362A4—81no—
363Dic3—161no—
364C23—22no—
365C23—21no—
366C23—23no—
367C23—31no—
368C23—33no—
369C23—33no—
370C23—33no—
371C23—312no—
372C23—33no—
373C23—33no—
374C23—33no—
375C23—33no—
376C23—33no—
377C23—33no—
378C23—61no—
379C23—61no—
380C23—62no—
381C23—63no—
382C23—66no—
383C23—66no—
384C23—63no—
385C23—66no—
386C23—63no—
387C23—66no—
388C23—63no—
389C23—63no—
390C23—61no—
391C23—63no—
392C23—63no—
393C23—63no—
394C23—63no—
395C23—63no—
396C23—66no—
397C23—63no—
398C23—612no—
399C23—612no—
400C23—612no—
401C23—63no—
402C23—63no—
403C23—66no—
404C23—66no—
405C23—66no—
406C23—63no—
407C23—63no—
408C23—61no—
409C23—63no—
410C23—62no—
411C23—61no—
412C23—66no—
413C23—612no—
414C23—66no—
415C23—612no—
416C23—612no—
417C23—66no—
418C23—66no—
419C23—66no—
420C23—66no—
421C23—63no—
422C23—61no—
423C23C2×C7×S413yes—
424D4—123no—
425D4—126no—
426D4—126no—
427D4—123no—
428D4—123no—
429D4—123no—
430D4—123no—
431D4—121no—
432C2×C4—66no—
433C2×C4—63no—
434C2×C4—63no—
435C2×C4—63no—
436C2×C4—63no—
437C2×C4—63no—
438C7#8752411yescharacteristic, direct product factor, Sylow
439C2×C3—161no—
440C2×C3—321no—
441D3—321no—
442C22—21no—
443C22—33no—
444C22—33no—
445C22—31no—
446C22—33no—
447C22—36no—
448C22—36no—
449C22—33no—
450C22—33no—
451C22—33no—
452C22—63no—
453C22—63no—
454C22—62no—
455C22—62no—
456C22—63no—
457C22—66no—
458C22—63no—
459C22—63no—
460C22—63no—
461C22—612no—
462C22—63no—
463C22—612no—
464C22—63no—
465C22—63no—
466C22—62no—
467C22—61no—
468C22—66no—
469C22—66no—
470C22—66no—
471C22—62no—
472C22—66no—
473C22—61no—
474C22—61no—
475C22—123no—
476C22—123no—
477C22—121no—
478C22C7×GL(2,ℤ/4)13yescomplemented
479C22C7×#102011yescharacteristic, complemented
480C4—123no—
481C4—123no—
482C4—121no—
483C3—161noSylow
484C2—21no—
485C2—33no—
486C2—33no—
487C2—61no—
488C2—62no—
489C2—61no—
490C2—63no—
491C2—241no—
492C2C2×C7×#102011yescharacteristic
subgroupG/H
centerC2×C7C2×#1020
commutator, cosocleC2×C22⋊A4C22×C7
FrattiniC2C2×C7×#1020
FittingC26×C7D3
socleC25×C7C2×D3
hypercenterC22×C7#1020
hypocenterC22⋊A4C7×D4
LCSG ▷ C2×C22⋊A4 ▷ C22⋊A4
UCSC1 ◁ C2×C7 ◁ C22×C7

Complements

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

NH ≅ G/Ntype
1. C22×C7×C22⋊A4C2semidirect
2. C2×C7×#1020C2semidirect
3. C7×#5936C2semidirect
9. C26×C7D3semidirect
16. #87524C7direct (N × H)
17. C7×C22⋊A4D4semidirect
47. C22×C22⋊A4C2×C7semidirect
48. C2×#1020C2×C7semidirect
49. #5936C2×C7semidirect
91. C24×C7#67semidirect
92. C24×C7S4semidirect
127. C26D3×C7semidirect
208. C22⋊A4C7×D4semidirect
277. C22×C7GL(2,ℤ/4)semidirect
278. C22×C7#1020semidirect
323. C24C7×#67semidirect
324. C24C7×S4semidirect
438. C7#87524direct (N × H)
478. C22C7×GL(2,ℤ/4)semidirect
479. C22C7×#1020semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊A42
C2×C7×#10202
C7×#59362
C7×#28463
C7×GL(2,ℤ/4)4
#875247