Finite Group Databasebeta

← C2×#22608259 #33051823

Subgroups — C2×#22608259 #33051823

Order 18688.

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×D73C211yesmaximal, characteristic
2C2×#17988549C212yesmaximal
3#22608259C218yesmaximal, direct product factor
4C2×#17988313C211yesmaximal, characteristic, complemented
5C2×#17988312C212yesmaximal, complemented
6C2×#324×C73C211yesmaximal, characteristic, complemented
7C23×C4×D73C411yescharacteristic
8C23×C4×D73C411yescharacteristic
9C23×C4×D73C2211yescharacteristic
10C22×#11924011—22no—
11C22×C4⋊Dic73C411yescharacteristic
12C22×#11924007C411yescharacteristic
13C22×#11924007C2211yescharacteristic
14C2×C4⋊C4×D73—22no—
15C2×C4⋊C4×D73—22no—
16C2×C4⋊C4×D73—24no—
17C2×C4⋊C4×D73C414yes—
18C2×C4⋊C4×D73C2214yes—
19#17988549—24no—
20#17988549C2218yes—
21C2×C22⋊#10596831C2212yes—
22C2×#14556503—22no—
23#17988313C2214yescomplemented
24#17988312C2218yescomplemented
25C22×C4⋊C4×C73C2211yescharacteristic
26C2×C22⋊C8×C73C2212yes—
27#324×C73C2214yescomplemented
28C24×D73C2×C411yescharacteristic
29C23×Dic73C2×C411yescharacteristic
30C23×Dic73C2×C411yescharacteristic
31C23×Dic73C2×C411yescharacteristic
32C22×C4×D73—22no—
33C22×C4×D73—21no—
34C22×C4×D73—21no—
35C22×C4×D73—24no—
36C22×C4×D73—22no—
37C22×C4×D73—21no—
38C22×C4×D73—21no—
39C22×C4×D73—24no—
40C22×C4×D73—44no—
41C22×C4×D73—44no—
42C22×C4×D73C2×C414yes—
43C22×C4×D73C2×C414yes—
44C22×C4×D73C2×C412yes—
45C22×C4×D73D412yes—
46C22×C4×D73D412yes—
47C22×C4×D73D414yes—
48C22×C4×D73C2312yes—
49C2×#11924011—28no—
50C2×#11924011—48no—
51C22×#10596831—22no—
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—22no—
54C2×C4⋊Dic73C2×C414yes—
55C2×#11924007—22no—
56C2×#11924007—22no—
57C2×#11924007—22no—
58C2×#11924007—22no—
59C2×#11924007C2×C412yes—
60C2×#11924007C2×C414yes—
61C2×#11924007C2312yes—
62C4⋊C4×D73—28no—
63C4⋊C4×D73—28no—
64C4⋊C4×D73—28no—
65C4⋊C4×D73—28no—
66C22⋊#10596831—22no—
67C22⋊#10596831C2314yes—
68#14556503—28no—
69C23×C4×C73C2×C411yescharacteristic
70C23×C4×C73C2×C411yescharacteristic
71C23×C4×C73C2311yescharacteristic
72C22×C8×C73—22no—
73C2×C4⋊C4×C73—22no—
74C2×C4⋊C4×C73—22no—
75C2×C4⋊C4×C73C2×C412yes—
76C2×C4⋊C4×C73C2312yes—
77C22⋊C8×C73—22no—
78C22⋊C8×C73C2314yes—
79C23×D73—22no—
80C23×D73—22no—
81C23×D73—24no—
82C23×D73—24no—
83C23×D73C22⋊C412yes—
84C23×D73C22⋊C411yescharacteristic
85C23×D73C22⋊C411yescharacteristic
86C23×D73C22×C412yes—
87C22×Dic73—21no—
88C22×Dic73—21no—
89C22×Dic73—22no—
90C22×Dic73—21no—
91C22×Dic73—21no—
92C22×Dic73—42no—
93C22×Dic73—42no—
94C22×Dic73C22⋊C412yes—
95C22×Dic73C22⋊C412yes—
96C22×Dic73M1614yescomplemented
97C22×Dic73C22×C412yes—
98C22×Dic73C22×C412yes—
99C22×Dic73C22×C412yes—
100C2×C4×D73—216no—
101C2×C4×D73—28no—
102C2×C4×D73—28no—
103C2×C4×D73—28no—
104C2×C4×D73—24no—
105C2×C4×D73—24no—
106C2×C4×D73—28no—
107C2×C4×D73—28no—
108C2×C4×D73—24no—
109C2×C4×D73—24no—
110C2×C4×D73—44no—
111C2×C4×D73—48no—
112C2×C4×D73—44no—
113C2×C4×D73—44no—
114C2×C4×D73—44no—
115C2×C4×D73—48no—
116C2×C4×D73—44no—
117C2×C4×D73—48no—
118C2×C4×D73C22⋊C414yes—
119C2×C4×D73C22⋊C414yes—
120C2×C4×D73C2×D414yes—
121C2×C4×D73C2×D414yes—
122#11924011—416no—
123C2×#10596831—24no—
124C2×#10596831—44no—
125C4⋊Dic73—28no—
126#11924007—24no—
127#11924007—24no—
128#11924007—28no—
129C24×C73C22×C411yescharacteristic
130C22×C4×C73—21no—
131C22×C4×C73—21no—
132C22×C4×C73—22no—
133C22×C4×C73—21no—
134C22×C4×C73—21no—
135C22×C4×C73—42no—
136C22×C4×C73—42no—
137C22×C4×C73M1614yescomplemented
138C22×C4×C73C22×C412yes—
139C22×C4×C73C22×C412yes—
140C22×C4×C73C22×C411yescharacteristic
141C22×C4×C73C2×D412yes—
142C22×C4×C73C2×D412yes—
143C22×C4×C73C2411yescharacteristic
144C2×C8×C73—24no—
145C2×C8×C73—44no—
146C4⋊C4×C73—24no—
147C4⋊C4×C73—24no—
148C23×C73—22no—
149C23×C73—22no—
150C23×C73C2×C22⋊C411yescharacteristic
151C23×C73C2×C22⋊C411yescharacteristic
152C23×C73C2×M1612yes—
153C23×C73C2×M1612yes—
154C23×C73C23×C411yescharacteristic
155C22×D73—216no—
156C22×D73—216no—
157C22×D73—22no—
158C22×D73—22no—
159C22×D73—24no—
160C22×D73—22no—
161C22×D73—22no—
162C22×D73—22no—
163C22×D73—22no—
164C22×D73—24no—
165C22×D73—24no—
166C22×D73—24no—
167C22×D73—24no—
168C22×D73#9914yes—
169C22×D73#10114yes—
170C22×D73C2×C22⋊C412yes—
171C22×D73C2×C22⋊C412yes—
172C2×C4×C73—22no—
173C2×C4×C73—22no—
174C2×C4×C73—24no—
175C2×C4×C73—24no—
176C2×C4×C73—24no—
177C2×C4×C73—22no—
178C2×C4×C73—41no—
179C2×C4×C73—42no—
180C2×C4×C73—41no—
181C2×C4×C73—42no—
182C2×C4×C73—42no—
183C2×C4×C73—42no—
184C2×C4×C73C2×C22⋊C412yes—
185C2×C4×C73C2×M1614yescomplemented
186C2×C4×C73C22×D412yes—
187C2×Dic73—22no—
188C2×Dic73—22no—
189C2×Dic73—24no—
190C2×Dic73—24no—
191C2×Dic73—24no—
192C2×Dic73—22no—
193C2×Dic73—42no—
194C2×Dic73—42no—
195C2×Dic73—42no—
196C2×Dic73—42no—
197C2×Dic73—42no—
198C2×Dic73C2×C22⋊C414yes—
199C2×Dic73C2×M1614yescomplemented
200C4×D73—216no—
201C4×D73—216no—
202C4×D73—28no—
203C4×D73—28no—
204C4×D73—44no—
205C4×D73—44no—
206C4×D73—48no—
207C4×D73—48no—
208C8×C73—44no—
209#10596831—44no—
210C22×C73—21no—
211C22×C73—21no—
212C22×C73—21no—
213C22×C73—21no—
214C22×C73—24no—
215C22×C73—24no—
216C22×C73#40712yes—
217C22×C73#40712yes—
218C22×C73C2×#9912yes—
219C22×C73C2×#10112yes—
220C22×C73C22×C22⋊C411yescharacteristic
221C22×C73C22×M1611yescharacteristic
222C22×C73C22×M1611yescharacteristic
223C2×D73—28no—
224C2×D73—28no—
225C2×D73—216no—
226C2×D73—22no—
227C2×D73—22no—
228C2×D73—22no—
229C2×D73—22no—
230C2×D73—28no—
231C2×D73—28no—
232C2×D73C2×#9914yes—
233C2×D73C2×#10114yes—
234C4×C73—24no—
235C4×C73—22no—
236C4×C73—22no—
237C4×C73—42no—
238Dic73—24no—
239Dic73—24no—
240Dic73—42no—
241C2×#1463—731nomaximal, Sylow
242C2×C73—21no—
243C2×C73—21no—
244C2×C73—22no—
245C2×C73#146314yescomplemented
246C2×C73C2×#40711yescharacteristic
247C2×C73C22×#9911yescharacteristic
248C2×C73C22×#10111yescharacteristic
249D73—24no—
250D73—24no—
251C23×C4⋊C4—731no—
252C2×#407—732no—
253#1463—738no—
254C2×#324—732no—
255C2×#324—731no—
256C2×#324D7311yescharacteristic, complemented
257C73C2×#146311yescharacteristic, complemented, Sylow
258C24×C4—731no—
259C24×C4—731no—
260C24×C4—731no—
261C22×M16—1462no—
262C22×C4⋊C4—731no—
263C22×C4⋊C4—731no—
264C22×C4⋊C4—731no—
265C22×C4⋊C4—734no—
266C22×C4⋊C4—734no—
267C22×C4⋊C4—1462no—
268C22×C4⋊C4—1464no—
269C22×C4⋊C4—1462no—
270C22×C4⋊C4C2×D7311yescharacteristic
271#407—738no—
272#407—1464no—
273C2×C22⋊C8—732no—
274C2×C22⋊C8—1462no—
275C2×C22⋊C8C2×D7312yes—
276#324—734no—
277#324—738no—
278#324C2×D7314yescomplemented
279C25—731no—
280C23×C4—2924no—
281C23×C4—2924no—
282C23×C4—731no—
283C23×C4—732no—
284C23×C4—732no—
285C23×C4—734no—
286C23×C4—732no—
287C23×C4—731no—
288C23×C4—734no—
289C23×C4—731no—
290C23×C4—732no—
291C23×C4—734no—
292C23×C4—1462no—
293C23×C4—1461no—
294C23×C4—1461no—
295C23×C4—1464no—
296C23×C4—1462no—
297C23×C4—1461no—
298C23×C4—1461no—
299C23×C4—1464no—
300C23×C4C4×D7311yescharacteristic
301C23×C4C4×D7311yescharacteristic
302C23×C4C22×D7311yescharacteristic
303C2×M16—2928no—
304C2×M16—1468no—
305C22×C8—22no—
306C22×C8—1462no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—734no—
310C2×C4⋊C4—732no—
311C2×C4⋊C4—734no—
312C2×C4⋊C4—732no—
313C2×C4⋊C4—1468no—
314C2×C4⋊C4—1468no—
315C2×C4⋊C4—1468no—
316C2×C4⋊C4—1468no—
317C2×C4⋊C4—1462no—
318C2×C4⋊C4—1462no—
319C2×C4⋊C4—1462no—
320C2×C4⋊C4—1462no—
321C2×C4⋊C4—1462no—
322C2×C4⋊C4—1462no—
323C2×C4⋊C4C4×D7312yes—
324C2×C4⋊C4C22×D7312yes—
325C22⋊C8—22no—
326C22⋊C8—734no—
327C22⋊C8—1468no—
328C22⋊C8—1462no—
329C22⋊C8C22×D7314yes—
330C24—731no—
331C24—731no—
332C24—732no—
333C24—732no—
334C24—1464no—
335C24—1464no—
336C24—1462no—
337C24—1462no—
338C24C2×C4×D7311yescharacteristic
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—2924no—
347C22×C4—2924no—
348C22×C4—2928no—
349C22×C4—2924no—
350C22×C4—2928no—
351C22×C4—2924no—
352C22×C4—2928no—
353C22×C4—2924no—
354C22×C4—2922no—
355C22×C4—2922no—
356C22×C4—732no—
357C22×C4—732no—
358C22×C4—734no—
359C22×C4—734no—
360C22×C4—732no—
361C22×C4—732no—
362C22×C4—734no—
363C22×C4—732no—
364C22×C4—734no—
365C22×C4—734no—
366C22×C4—1464no—
367C22×C4—1464no—
368C22×C4—1468no—
369C22×C4—1464no—
370C22×C4—1464no—
371C22×C4—1468no—
372C22×C4—14616no—
373C22×C4—1468no—
374C22×C4—1468no—
375C22×C4—1461no—
376C22×C4—1461no—
377C22×C4—1461no—
378C22×C4—1461no—
379C22×C4—1462no—
380C22×C4—1468no—
381C22×C4#1192401114yescomplemented
382C22×C4C2×C4×D7312yes—
383C22×C4C2×C4×D7312yes—
384C22×C4C2×C4×D7311yescharacteristic
385C22×C4D4×D7312yes—
386C22×C4D4×D7312yes—
387C22×C4C23×D7311yescharacteristic
388M16—29216no—
389C2×C8—24no—
390C2×C8—44no—
391C2×C8—2924no—
392C2×C8—1464no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—1468no—
396C4⋊C4—1468no—
397C4⋊C4—1464no—
398C4⋊C4—1464no—
399C23—22no—
400C23—22no—
401C23—732no—
402C23—732no—
403C23—734no—
404C23—734no—
405C23—1464no—
406C23—1464no—
407C23—1464no—
408C23—1464no—
409C23—14616no—
410C23—14616no—
411C23—1464no—
412C23—1462no—
413C23—1462no—
414C23—1462no—
415C23—1462no—
416C23—1462no—
417C23—1462no—
418C23C22⋊C4×D7311yescharacteristic
419C23C22⋊C4×D7311yescharacteristic
420C23M16×D7312yes—
421C23C2×#1192401112yes—
422C23C22×C4×D7311yescharacteristic
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—2924no—
436C2×C4—2924no—
437C2×C4—2928no—
438C2×C4—2928no—
439C2×C4—2922no—
440C2×C4—2922no—
441C2×C4—2922no—
442C2×C4—2922no—
443C2×C4—2922no—
444C2×C4—734no—
445C2×C4—734no—
446C2×C4—1468no—
447C2×C4—1468no—
448C2×C4—14616no—
449C2×C4—14616no—
450C2×C4—1462no—
451C2×C4—1462no—
452C2×C4—1464no—
453C2×C4—1464no—
454C2×C4—1464no—
455C2×C4—1462no—
456C2×C4C22⋊C4×D7312yes—
457C2×C4C2×#1192401114yescomplemented
458C2×C4C2×D4×D7312yes—
459C8—44no—
460C8—2924no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—24no—
466C22—24no—
467C22—734no—
468C22—734no—
469C22—14616no—
470C22—1468no—
471C22—1468no—
472C22—1462no—
473C22—1462no—
474C22—1468no—
475C22—1468no—
476C22—1462no—
477C22—1462no—
478C22#1798854912yes—
479C22#1798854912yes—
480C22#99×D7312yes—
481C22#101×D7312yes—
482C22C2×C22⋊C4×D7311yescharacteristic
483C22C2×M16×D7311yescharacteristic
484C22C22×#1192401111yescharacteristic
485C4—22no—
486C4—22no—
487C4—24no—
488C4—42no—
489C4—2922no—
490C4—1464no—
491C4—1464no—
492C2—22no—
493C2—21no—
494C2—21no—
495C2—1464no—
496C2—1464no—
497C2#2260825914yesdirect product factor
498C2C2×#1798854911yescharacteristic
499C2C2×#99×D7311yescharacteristic
500C2C2×#101×D7311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D73
commutatorC23×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#324×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC24×D73C2×C4
cosocleC22×C4×C73C24
hypercenterC2×#324D73
hypocenterC73C2×#1463
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #22608259C2direct (N × H)
4. C2×#17988313C2semidirect
5. C2×#17988312C2semidirect
6. C2×#324×C73C2semidirect
23. #17988313C22semidirect
24. #17988312C22semidirect
27. #324×C73C22semidirect
96. C22×Dic73M16semidirect
137. C22×C4×C73M16semidirect
185. C2×C4×C73C2×M16semidirect
199. C2×Dic73C2×M16semidirect
245. C2×C73#1463semidirect
256. C2×#324D73semidirect
257. C73C2×#1463semidirect
278. #324C2×D73semidirect
381. C22×C4#11924011semidirect
457. C2×C4C2×#11924011semidirect
497. C2#22608259direct (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×D732
C2×#179885492
#226082592
C2×#179883132
C2×#179883122
C2×#324×C732
C2×#146373