Finite Group Databasebeta

← C2×#22586668 #31963291

Subgroups — C2×#22586668 #31963291

Order 18176.

1718 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D71C211yesmaximal, characteristic
2C2×#17985961C212yesmaximal
3#22586668C218yesmaximal, direct product factor
4C2×#17985725C211yesmaximal, characteristic, complemented
5C2×#17985724C212yesmaximal, complemented
6C2×#324×C71C211yesmaximal, characteristic, complemented
7C23×C4×D71C411yescharacteristic
8C23×C4×D71C411yescharacteristic
9C23×C4×D71C2211yescharacteristic
10C22×#11765848—22no—
11C22×C4⋊Dic71C411yescharacteristic
12C22×Dic71⋊C4C411yescharacteristic
13C22×Dic71⋊C4C2211yescharacteristic
14C2×C4⋊C4×D71—22no—
15C2×C4⋊C4×D71—22no—
16C2×C4⋊C4×D71—24no—
17C2×C4⋊C4×D71C414yes—
18C2×C4⋊C4×D71C2214yes—
19#17985961—24no—
20#17985961C2218yes—
21C2×C22⋊(C71⋊C8)C2212yes—
22C2×(C2×D71)⋊C8—22no—
23#17985725C2214yescomplemented
24#17985724C2218yescomplemented
25C22×C4⋊C4×C71C2211yescharacteristic
26C2×C22⋊C8×C71C2212yes—
27#324×C71C2214yescomplemented
28C24×D71C2×C411yescharacteristic
29C23×Dic71C2×C411yescharacteristic
30C23×Dic71C2×C411yescharacteristic
31C23×Dic71C2×C411yescharacteristic
32C22×C4×D71—22no—
33C22×C4×D71—21no—
34C22×C4×D71—21no—
35C22×C4×D71—24no—
36C22×C4×D71—22no—
37C22×C4×D71—21no—
38C22×C4×D71—21no—
39C22×C4×D71—24no—
40C22×C4×D71—44no—
41C22×C4×D71—44no—
42C22×C4×D71C2×C414yes—
43C22×C4×D71C2×C414yes—
44C22×C4×D71C2×C412yes—
45C22×C4×D71D412yes—
46C22×C4×D71D412yes—
47C22×C4×D71D414yes—
48C22×C4×D71C2312yes—
49C2×#11765848—28no—
50C2×#11765848—48no—
51C22×C71⋊C8—22no—
52C2×C4⋊Dic71—22no—
53C2×C4⋊Dic71—22no—
54C2×C4⋊Dic71C2×C414yes—
55C2×Dic71⋊C4—22no—
56C2×Dic71⋊C4—22no—
57C2×Dic71⋊C4—22no—
58C2×Dic71⋊C4—22no—
59C2×Dic71⋊C4C2×C412yes—
60C2×Dic71⋊C4C2×C414yes—
61C2×Dic71⋊C4C2312yes—
62C4⋊C4×D71—28no—
63C4⋊C4×D71—28no—
64C4⋊C4×D71—28no—
65C4⋊C4×D71—28no—
66C22⋊(C71⋊C8)—22no—
67C22⋊(C71⋊C8)C2314yes—
68(C2×D71)⋊C8—28no—
69C23×C4×C71C2×C411yescharacteristic
70C23×C4×C71C2×C411yescharacteristic
71C23×C4×C71C2311yescharacteristic
72C22×C8×C71—22no—
73C2×C4⋊C4×C71—22no—
74C2×C4⋊C4×C71—22no—
75C2×C4⋊C4×C71C2×C412yes—
76C2×C4⋊C4×C71C2312yes—
77C22⋊C8×C71—22no—
78C22⋊C8×C71C2314yes—
79C23×D71—22no—
80C23×D71—22no—
81C23×D71—24no—
82C23×D71—24no—
83C23×D71C22⋊C412yes—
84C23×D71C22⋊C411yescharacteristic
85C23×D71C22⋊C411yescharacteristic
86C23×D71C22×C412yes—
87C22×Dic71—21no—
88C22×Dic71—21no—
89C22×Dic71—22no—
90C22×Dic71—21no—
91C22×Dic71—21no—
92C22×Dic71—42no—
93C22×Dic71—42no—
94C22×Dic71C22⋊C412yes—
95C22×Dic71C22⋊C412yes—
96C22×Dic71M1614yescomplemented
97C22×Dic71C22×C412yes—
98C22×Dic71C22×C412yes—
99C22×Dic71C22×C412yes—
100C2×C4×D71—216no—
101C2×C4×D71—28no—
102C2×C4×D71—28no—
103C2×C4×D71—28no—
104C2×C4×D71—24no—
105C2×C4×D71—24no—
106C2×C4×D71—28no—
107C2×C4×D71—28no—
108C2×C4×D71—24no—
109C2×C4×D71—24no—
110C2×C4×D71—44no—
111C2×C4×D71—48no—
112C2×C4×D71—44no—
113C2×C4×D71—44no—
114C2×C4×D71—44no—
115C2×C4×D71—48no—
116C2×C4×D71—44no—
117C2×C4×D71—48no—
118C2×C4×D71C22⋊C414yes—
119C2×C4×D71C22⋊C414yes—
120C2×C4×D71C2×D414yes—
121C2×C4×D71C2×D414yes—
122#11765848—416no—
123C2×C71⋊C8—24no—
124C2×C71⋊C8—44no—
125C4⋊Dic71—28no—
126Dic71⋊C4—24no—
127Dic71⋊C4—24no—
128Dic71⋊C4—28no—
129C24×C71C22×C411yescharacteristic
130C22×C4×C71—21no—
131C22×C4×C71—21no—
132C22×C4×C71—22no—
133C22×C4×C71—21no—
134C22×C4×C71—21no—
135C22×C4×C71—42no—
136C22×C4×C71—42no—
137C22×C4×C71M1614yescomplemented
138C22×C4×C71C22×C412yes—
139C22×C4×C71C22×C412yes—
140C22×C4×C71C22×C411yescharacteristic
141C22×C4×C71C2×D412yes—
142C22×C4×C71C2×D412yes—
143C22×C4×C71C2411yescharacteristic
144C2×C8×C71—24no—
145C2×C8×C71—44no—
146C4⋊C4×C71—24no—
147C4⋊C4×C71—24no—
148C23×C71—22no—
149C23×C71—22no—
150C23×C71C2×C22⋊C411yescharacteristic
151C23×C71C2×C22⋊C411yescharacteristic
152C23×C71C2×M1612yes—
153C23×C71C2×M1612yes—
154C23×C71C23×C411yescharacteristic
155C22×D71—216no—
156C22×D71—216no—
157C22×D71—22no—
158C22×D71—22no—
159C22×D71—24no—
160C22×D71—22no—
161C22×D71—22no—
162C22×D71—22no—
163C22×D71—22no—
164C22×D71—24no—
165C22×D71—24no—
166C22×D71—24no—
167C22×D71—24no—
168C22×D71#9914yes—
169C22×D71#10114yes—
170C22×D71C2×C22⋊C412yes—
171C22×D71C2×C22⋊C412yes—
172C2×C4×C71—22no—
173C2×C4×C71—22no—
174C2×C4×C71—24no—
175C2×C4×C71—24no—
176C2×C4×C71—24no—
177C2×C4×C71—22no—
178C2×C4×C71—41no—
179C2×C4×C71—42no—
180C2×C4×C71—41no—
181C2×C4×C71—42no—
182C2×C4×C71—42no—
183C2×C4×C71—42no—
184C2×C4×C71C2×C22⋊C412yes—
185C2×C4×C71C2×M1614yescomplemented
186C2×C4×C71C22×D412yes—
187C2×Dic71—22no—
188C2×Dic71—22no—
189C2×Dic71—24no—
190C2×Dic71—24no—
191C2×Dic71—24no—
192C2×Dic71—22no—
193C2×Dic71—42no—
194C2×Dic71—42no—
195C2×Dic71—42no—
196C2×Dic71—42no—
197C2×Dic71—42no—
198C2×Dic71C2×C22⋊C414yes—
199C2×Dic71C2×M1614yescomplemented
200C4×D71—216no—
201C4×D71—216no—
202C4×D71—28no—
203C4×D71—28no—
204C4×D71—44no—
205C4×D71—44no—
206C4×D71—48no—
207C4×D71—48no—
208C8×C71—44no—
209C71⋊C8—44no—
210C22×C71—21no—
211C22×C71—21no—
212C22×C71—21no—
213C22×C71—21no—
214C22×C71—24no—
215C22×C71—24no—
216C22×C71#40712yes—
217C22×C71#40712yes—
218C22×C71C2×#9912yes—
219C22×C71C2×#10112yes—
220C22×C71C22×C22⋊C411yescharacteristic
221C22×C71C22×M1611yescharacteristic
222C22×C71C22×M1611yescharacteristic
223C2×D71—28no—
224C2×D71—28no—
225C2×D71—216no—
226C2×D71—22no—
227C2×D71—22no—
228C2×D71—22no—
229C2×D71—22no—
230C2×D71—28no—
231C2×D71—28no—
232C2×D71C2×#9914yes—
233C2×D71C2×#10114yes—
234C4×C71—24no—
235C4×C71—22no—
236C4×C71—22no—
237C4×C71—42no—
238Dic71—24no—
239Dic71—24no—
240Dic71—42no—
241C2×#1463—711nomaximal, Sylow
242C2×C71—21no—
243C2×C71—21no—
244C2×C71—22no—
245C2×C71#146314yescomplemented
246C2×C71C2×#40711yescharacteristic
247C2×C71C22×#9911yescharacteristic
248C2×C71C22×#10111yescharacteristic
249D71—24no—
250D71—24no—
251C23×C4⋊C4—711no—
252C2×#407—712no—
253#1463—718no—
254C2×#324—712no—
255C2×#324—711no—
256C2×#324D7111yescharacteristic, complemented
257C71C2×#146311yescharacteristic, complemented, Sylow
258C24×C4—711no—
259C24×C4—711no—
260C24×C4—711no—
261C22×M16—1422no—
262C22×C4⋊C4—711no—
263C22×C4⋊C4—711no—
264C22×C4⋊C4—711no—
265C22×C4⋊C4—714no—
266C22×C4⋊C4—714no—
267C22×C4⋊C4—1422no—
268C22×C4⋊C4—1424no—
269C22×C4⋊C4—1422no—
270C22×C4⋊C4C2×D7111yescharacteristic
271#407—718no—
272#407—1424no—
273C2×C22⋊C8—712no—
274C2×C22⋊C8—1422no—
275C2×C22⋊C8C2×D7112yes—
276#324—714no—
277#324—718no—
278#324C2×D7114yescomplemented
279C25—711no—
280C23×C4—2844no—
281C23×C4—2844no—
282C23×C4—711no—
283C23×C4—712no—
284C23×C4—712no—
285C23×C4—714no—
286C23×C4—712no—
287C23×C4—711no—
288C23×C4—714no—
289C23×C4—711no—
290C23×C4—712no—
291C23×C4—714no—
292C23×C4—1422no—
293C23×C4—1421no—
294C23×C4—1421no—
295C23×C4—1424no—
296C23×C4—1422no—
297C23×C4—1421no—
298C23×C4—1421no—
299C23×C4—1424no—
300C23×C4C4×D7111yescharacteristic
301C23×C4C4×D7111yescharacteristic
302C23×C4C22×D7111yescharacteristic
303C2×M16—2848no—
304C2×M16—1428no—
305C22×C8—22no—
306C22×C8—1422no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—714no—
310C2×C4⋊C4—712no—
311C2×C4⋊C4—714no—
312C2×C4⋊C4—712no—
313C2×C4⋊C4—1428no—
314C2×C4⋊C4—1428no—
315C2×C4⋊C4—1428no—
316C2×C4⋊C4—1428no—
317C2×C4⋊C4—1422no—
318C2×C4⋊C4—1422no—
319C2×C4⋊C4—1422no—
320C2×C4⋊C4—1422no—
321C2×C4⋊C4—1422no—
322C2×C4⋊C4—1422no—
323C2×C4⋊C4C4×D7112yes—
324C2×C4⋊C4C22×D7112yes—
325C22⋊C8—22no—
326C22⋊C8—714no—
327C22⋊C8—1428no—
328C22⋊C8—1422no—
329C22⋊C8C22×D7114yes—
330C24—711no—
331C24—711no—
332C24—712no—
333C24—712no—
334C24—1424no—
335C24—1424no—
336C24—1422no—
337C24—1422no—
338C24C2×C4×D7111yescharacteristic
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—2844no—
347C22×C4—2844no—
348C22×C4—2848no—
349C22×C4—2844no—
350C22×C4—2848no—
351C22×C4—2844no—
352C22×C4—2848no—
353C22×C4—2844no—
354C22×C4—2842no—
355C22×C4—2842no—
356C22×C4—712no—
357C22×C4—712no—
358C22×C4—714no—
359C22×C4—714no—
360C22×C4—712no—
361C22×C4—712no—
362C22×C4—712no—
363C22×C4—714no—
364C22×C4—714no—
365C22×C4—714no—
366C22×C4—1424no—
367C22×C4—1424no—
368C22×C4—1428no—
369C22×C4—1424no—
370C22×C4—1424no—
371C22×C4—1428no—
372C22×C4—14216no—
373C22×C4—1428no—
374C22×C4—1428no—
375C22×C4—1421no—
376C22×C4—1421no—
377C22×C4—1421no—
378C22×C4—1421no—
379C22×C4—1422no—
380C22×C4—1428no—
381C22×C4#1176584814yescomplemented
382C22×C4C2×C4×D7112yes—
383C22×C4C2×C4×D7112yes—
384C22×C4C2×C4×D7111yescharacteristic
385C22×C4D4×D7112yes—
386C22×C4D4×D7112yes—
387C22×C4C23×D7111yescharacteristic
388M16—28416no—
389C2×C8—24no—
390C2×C8—44no—
391C2×C8—2844no—
392C2×C8—1424no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—1428no—
396C4⋊C4—1428no—
397C4⋊C4—1424no—
398C4⋊C4—1424no—
399C23—22no—
400C23—22no—
401C23—712no—
402C23—712no—
403C23—714no—
404C23—714no—
405C23—1424no—
406C23—1424no—
407C23—1424no—
408C23—1424no—
409C23—14216no—
410C23—14216no—
411C23—1424no—
412C23—1422no—
413C23—1422no—
414C23—1422no—
415C23—1422no—
416C23—1422no—
417C23—1422no—
418C23C22⋊C4×D7111yescharacteristic
419C23C22⋊C4×D7111yescharacteristic
420C23M16×D7112yes—
421C23C2×#1176584812yes—
422C23C22×C4×D7111yescharacteristic
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—2844no—
436C2×C4—2844no—
437C2×C4—2848no—
438C2×C4—2848no—
439C2×C4—2842no—
440C2×C4—2842no—
441C2×C4—2842no—
442C2×C4—2842no—
443C2×C4—2842no—
444C2×C4—714no—
445C2×C4—714no—
446C2×C4—1428no—
447C2×C4—1428no—
448C2×C4—14216no—
449C2×C4—14216no—
450C2×C4—1422no—
451C2×C4—1422no—
452C2×C4—1424no—
453C2×C4—1424no—
454C2×C4—1424no—
455C2×C4—1422no—
456C2×C4C22⋊C4×D7112yes—
457C2×C4C2×#1176584814yescomplemented
458C2×C4C2×D4×D7112yes—
459C8—44no—
460C8—2844no—
461C22—24no—
462C22—24no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—714no—
468C22—714no—
469C22—14216no—
470C22—1428no—
471C22—1428no—
472C22—1428no—
473C22—1428no—
474C22—1422no—
475C22—1422no—
476C22—1422no—
477C22—1422no—
478C22#1798596112yes—
479C22#1798596112yes—
480C22#99×D7112yes—
481C22#101×D7112yes—
482C22C2×C22⋊C4×D7111yescharacteristic
483C22C2×M16×D7111yescharacteristic
484C22C22×#1176584811yescharacteristic
485C4—24no—
486C4—22no—
487C4—22no—
488C4—42no—
489C4—2842no—
490C4—1424no—
491C4—1424no—
492C2—22no—
493C2—21no—
494C2—21no—
495C2—1424no—
496C2—1424no—
497C2#2258666814yesdirect product factor
498C2C2×#1798596111yescharacteristic
499C2C2×#99×D7111yescharacteristic
500C2C2×#101×D7111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D71
commutatorC23×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#324×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×D71C2×C4
cosocleC22×C4×C71C24
hypercenterC2×#324D71
hypocenterC71C2×#1463
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #22586668C2direct (N × H)
4. C2×#17985725C2semidirect
5. C2×#17985724C2semidirect
6. C2×#324×C71C2semidirect
23. #17985725C22semidirect
24. #17985724C22semidirect
27. #324×C71C22semidirect
96. C22×Dic71M16semidirect
137. C22×C4×C71M16semidirect
185. C2×C4×C71C2×M16semidirect
199. C2×Dic71C2×M16semidirect
245. C2×C71#1463semidirect
256. C2×#324D71semidirect
257. C71C2×#1463semidirect
278. #324C2×D71semidirect
381. C22×C4#11765848semidirect
457. C2×C4C2×#11765848semidirect
497. C2#22586668direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D712
C2×#179859612
#225866682
C2×#179857252
C2×#179857242
C2×#324×C712
C2×#146371