Finite Group Databasebeta

← C2×#22622958 #33966948

Subgroups — C2×#22622958 #33966948

Order 18688.

1906 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989310C211yesmaximal, characteristic
2#22622958C2116yesmaximal, direct product factor
3C2×#17988979C212yesmaximal
4C2×#17988966C211yesmaximal, characteristic, complemented
5C2×#17988809C212yesmaximal
6C2×#17988797C211yesmaximal, characteristic, complemented
7C2×#17988794C212yesmaximal, complemented
8C2×#17988789C211yesmaximal, characteristic
9C2×#17988718C211yesmaximal, characteristic, complemented
10C2×#17988692C212yesmaximal, complemented
11C2×C4×#11924015C211yesmaximal, characteristic
12C2×#557×C73C211yesmaximal, characteristic, complemented
13C22×#11924018C2212yes—
14C22×#11924018C2212yes—
15C22×#11924018C2212yes—
16C22×#11924015C2211yescharacteristic
17C22×#11924015C2212yes—
18#17989310—24no—
19#17989310C2218yes—
20C2×#14556591C2211yescharacteristic
21C2×#14556586—22no—
22C2×#14556579C2212yes—
23C2×#14556567—21no—
24C2×C4×Dic146C2212yes—
25C2×C4×D292C2211yescharacteristic
26C2×#14556549—21no—
27#17988979C22116yes—
28#17988966C2218yescomplemented
29#17988809C22116yes—
30#17988797C2218yescomplemented
31#17988794C22116yescomplemented
32#17988789C2218yes—
33#17988718C2218yescomplemented
34#17988692C22116yescomplemented
35C4×#11924015C2218yes—
36C2×#14556522C2211yescharacteristic
37C2×C4⋊#10596831C2211yescharacteristic
38C2×C4⋊#10596831C2212yes—
39C2×#14556515C2211yescharacteristic
40C2×#14556515C2212yes—
41C2×#14556514C2212yes—
42C2×#14556511C2211yescharacteristic
43C2×#14556502C2212yes—
44C2×C4×#10596831C2211yescharacteristic
45#557×C73—24no—
46#557×C73C2218yescomplemented
47C2×C4⋊Q8×C73C2211yescharacteristic
48C2×C4⋊Q8×C73C2212yes—
49C2×#125×C73—21no—
50C2×#125×C73—22no—
51C2×C4×Q8×C73—21no—
52C2×C4×Q8×C73C2211yescharacteristic
53C2×C4×Q8×C73C2212yes—
54C2×C4×Q8×C73C2211yescharacteristic
55C22×Dic146C2312yes—
56C22×D292C2311yescharacteristic
57C22×C4×D73—21no—
58C2×#11924018—28no—
59C2×#11924018—44no—
60C2×#11924018—44no—
61C2×#11924018C2318yes—
62C2×#11924018C2318yes—
63C2×#11924018C2318yes—
64C2×#11924015—24no—
65C2×#11924015—44no—
66C2×#11924015C2314yes—
67C2×#11924015C2318yes—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2312yes—
70C22×#10596831C2311yescharacteristic
71#14556591—22no—
72#14556591C2314yes—
73#14556586—216no—
74#14556579—24no—
75#14556579C2318yes—
76#14556567—28no—
77C4×Dic146—24no—
78C4×Dic146C2318yes—
79C4×D292—22no—
80C4×D292C2314yes—
81#14556549—28no—
82C2×C4⋊Dic73—21no—
83C2×C4⋊Dic73—21no—
84C2×C4⋊Dic73C2311yescharacteristic
85C2×#11924007—22no—
86C2×#11924005—22no—
87C2×#11924005—21no—
88C2×#11924005—22no—
89C2×C4×Dic73—22no—
90#14556522C2314yes—
91C4⋊#10596831C2314yes—
92C4⋊#10596831C2318yes—
93#14556515C2314yes—
94#14556515C2318yes—
95#14556514C2318yes—
96#14556511C2314yes—
97#14556502C2318yes—
98C4×#10596831C2314yes—
99C22×Q8×C73D411yescharacteristic
100C22×Q8×C73C2311yescharacteristic
101C22×Q8×C73C2312yes—
102C4⋊Q8×C73—22no—
103C4⋊Q8×C73—24no—
104C4⋊Q8×C73C2314yes—
105C4⋊Q8×C73C2318yes—
106#125×C73—28no—
107#125×C73—216no—
108C4×Q8×C73—28no—
109C4×Q8×C73—22no—
110C4×Q8×C73—24no—
111C4×Q8×C73D414yes—
112C4×Q8×C73C2314yes—
113C4×Q8×C73C2318yes—
114C4×Q8×C73C2314yes—
115C2×C4⋊C4×C73—22no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—22no—
118C2×C4⋊C4×C73—21no—
119C2×C4⋊C4×C73—22no—
120C2×C4⋊C4×C73—21no—
121C2×C4⋊C4×C73D411yescharacteristic
122C2×C4⋊C4×C73D412yes—
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C42×C73—22no—
125C2×C42×C73—21no—
126C2×C42×C73C2311yescharacteristic
127C2×C42×C73C2312yes—
128C23×D73—21no—
129C2×Dic146—24no—
130C2×Dic146—24no—
131C2×Dic146—22no—
132C2×Dic146C2414yes—
133C2×D292—22no—
134C2×D292—22no—
135C2×D292—21no—
136C2×D292C2412yes—
137C22×Dic73—21no—
138C22×Dic73—22no—
139C2×C4×D73—24no—
140C2×C4×D73—44no—
141#11924018—216no—
142#11924018—48no—
143#11924018—48no—
144#11924015—28no—
145#11924015—48no—
146C2×#10596831—24no—
147C2×#10596831—22no—
148C2×#10596831#4114yes—
149C2×#10596831C2412yes—
150C2×#10596831C2414yes—
151C2×#10596831C2412yes—
152C4⋊Dic73—24no—
153C4⋊Dic73—24no—
154C4⋊Dic73—21no—
155C4⋊Dic73C2412yes—
156#11924007—28no—
157#11924005—28no—
158#11924005—24no—
159#11924005—28no—
160C4×Dic73—28no—
161C2×Q8×C73—24no—
162C2×Q8×C73—21no—
163C2×Q8×C73—22no—
164C2×Q8×C73—22no—
165C2×Q8×C73—24no—
166C2×Q8×C73—24no—
167C2×Q8×C73—22no—
168C2×Q8×C73C2×D414yes—
169C2×Q8×C73C2412yes—
170C2×Q8×C73C2414yes—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—22no—
174C22×C4×C73C2×D412yes—
175C22×C4×C73C2×D412yes—
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2411yescharacteristic
179C4⋊C4×C73—28no—
180C4⋊C4×C73—24no—
181C4⋊C4×C73—28no—
182C4⋊C4×C73—24no—
183C4⋊C4×C73—28no—
184C4⋊C4×C73—24no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73C2×D414yes—
187C4⋊C4×C73C2×D418yes—
188C4⋊C4×C73C2412yes—
189C42×C73—28no—
190C42×C73—24no—
191C42×C73C2×D412yes—
192C42×C73C2×D414yes—
193C42×C73C2412yes—
194C42×C73C2414yes—
195C23×C73C22×D411yescharacteristic
196C22×D73—22no—
197C22×D73—42no—
198C22×D73—42no—
199C22×D73—42no—
200Q8×C73—28no—
201Q8×C73—22no—
202Q8×C73—24no—
203Q8×C73—24no—
204Q8×C73—22no—
205Q8×C73—42no—
206Q8×C73—44no—
207C2×C4×C73—24no—
208C2×C4×C73—24no—
209C2×C4×C73—22no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—24no—
213C2×C4×C73—42no—
214C2×C4×C73—44no—
215C2×C4×C73—42no—
216C2×C4×C73#13714yes—
217C2×C4×C73C22×D414yes—
218C2×C4×C73C22×D414yes—
219C2×C4×C73C22×D412yes—
220C2×C4×C73C22×D411yescharacteristic
221C2×C4×C73C22×D412yes—
222C2×C4×C73C2×#4112yes—
223C2×C4×C73H(5,𝔽2)12yes—
224C2×C4×C73C2511yescharacteristic
225C2×Dic73—22no—
226C2×Dic73—24no—
227C2×Dic73—44no—
228C2×Dic73—42no—
229D292—22no—
230D292—22no—
231D292—42no—
232C4×D73—48no—
233Dic146—24no—
234Dic146—24no—
235Dic146—44no—
236#10596831—24no—
237#10596831—22no—
238#10596831C2×#4114yes—
239C22×C73#55012yes—
240C22×C73C2×#13712yes—
241C22×C73#57812yes—
242C22×C73C23×D411yescharacteristic
243C2×D73—42no—
244C2×D73—42no—
245C2×D73—42no—
246C2×D73—84no—
247C4×C73—24no—
248C4×C73—24no—
249C4×C73—22no—
250C4×C73—42no—
251C4×C73—44no—
252C4×C73—42no—
253C4×C73#55012yes—
254C4×C73C2×#13714yes—
255C4×C73C22×#4111yescharacteristic
256C4×C73C2×H(5,𝔽2)11yescharacteristic
257Dic73—44no—
258Dic73—42no—
259C2×#3398—731nomaximal, Sylow
260C2×C73#339814yescomplemented
261C2×C73C2×#55011yescharacteristic
262C2×C73C22×#13711yescharacteristic
263C2×C73C2×#57811yescharacteristic
264D73—82no—
265C2×#557D7311yescharacteristic, complemented
266C2×#556—731no—
267#3398—7316no—
268C2×#497—732no—
269C2×#494—731no—
270C2×#464—732no—
271C2×#462—732no—
272C2×#462—731no—
273C2×#461—731no—
274C2×#446—731no—
275C2×#441—732no—
276C2×C4×SD16—731no—
277C73C2×#339811yescharacteristic, complemented, Sylow
278C22×Q16—732no—
279C22×Q16—732no—
280C22×Q16—732no—
281C22×SD16—732no—
282C22×SD16—731no—
283#557—24no—
284#557C2×D7318yescomplemented
285#556—738no—
286#556—1464no—
287C2×C4⋊Q8—731no—
288C2×C4⋊Q8C2×D7311yescharacteristic
289C2×C4⋊Q8C2×D7312yes—
290C2×#126—1462no—
291C2×#125—21no—
292C2×#125—22no—
293C2×#124—732no—
294C2×C22⋊Q8—1461no—
295C2×C4×Q8—21no—
296C2×C4×Q8—732no—
297C2×C4×Q8C2×D7311yescharacteristic
298C2×C4×Q8C2×D7312yes—
299C2×C4×Q8C2×D7311yescharacteristic
300C2×C4×D4—731no—
301C2×#117—1461no—
302#497—7316no—
303#494—738no—
304#464—7316no—
305#462—7316no—
306#462—738no—
307#461—738no—
308#446—738no—
309#441—7316no—
310C4×SD16—738no—
311C2×#106—731no—
312C2×C4⋊C8—731no—
313C2×C4⋊C8—732no—
314C2×#103—731no—
315C2×#103—732no—
316C2×#103—732no—
317C2×#102—731no—
318C2×#97—732no—
319C2×C4×C8—731no—
320C22×Q8—732no—
321C22×Q8C22⋊D7311yescharacteristic
322C22×Q8C22×D7311yescharacteristic
323C22×Q8C22×D7312yes—
324C22×D4—731no—
325C23×C4—1461no—
326C2×Q16—2924no—
327C2×Q16—2924no—
328C2×Q16—738no—
329C2×Q16—738no—
330C2×Q16—738no—
331C2×Q16—1468no—
332C2×SD16—2924no—
333C2×SD16—738no—
334C2×SD16—734no—
335C2×SD16—1464no—
336C22×C8—731no—
337C22×C8—732no—
338C22×C8—731no—
339C4⋊Q8—22no—
340C4⋊Q8—24no—
341C4⋊Q8—734no—
342C4⋊Q8—1462no—
343C4⋊Q8C22×D7314yes—
344C4⋊Q8C22×D7318yes—
345#126—14616no—
346#125—28no—
347#125—216no—
348#124—738no—
349#124—1464no—
350C22⋊Q8—1468no—
351C4×Q8—28no—
352C4×Q8—22no—
353C4×Q8—24no—
354C4×Q8—738no—
355C4×Q8—1464no—
356C4×Q8C22⋊D7314yes—
357C4×Q8C22×D7314yes—
358C4×Q8C22×D7318yes—
359C4×Q8C22×D7314yes—
360C4×D4—734no—
361C4×D4—1462no—
362#117—1468no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—1461no—
371C2×C4⋊C4—1462no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4C22⋊D7311yescharacteristic
374C2×C4⋊C4C22⋊D7312yes—
375C2×C4⋊C4C22×D7311yescharacteristic
376C2×C22⋊C4—1462no—
377C2×C22⋊C4—1461no—
378C2×C22⋊C4—1462no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—1462no—
382C2×C42C22×D7311yescharacteristic
383C2×C42C22×D7312yes—
384#106—734no—
385C4⋊C8—734no—
386C4⋊C8—738no—
387#103—734no—
388#103—738no—
389#103—738no—
390#102—734no—
391#97—738no—
392C4×C8—734no—
393C24—1461no—
394C2×Q8—24no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—734no—
402C2×Q8—1464no—
403C2×Q8—1464no—
404C2×Q8—1462no—
405C2×Q8C2×C22⋊D7314yes—
406C2×Q8C23×D7312yes—
407C2×Q8C23×D7314yes—
408C2×D4—732no—
409C2×D4—1461no—
410C2×D4—1462no—
411C2×D4—1462no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—2924no—
416C22×C4—1461no—
417C22×C4—1462no—
418C22×C4—1464no—
419C22×C4C2×C22⋊D7312yes—
420C22×C4C2×C22⋊D7312yes—
421C22×C4C2×C22⋊D7311yescharacteristic
422C22×C4C2×C22⋊D7311yescharacteristic
423C22×C4C23×D7311yescharacteristic
424Q16—2928no—
425Q16—2928no—
426Q16—14616no—
427SD16—2928no—
428SD16—1468no—
429C2×C8—734no—
430C2×C8—732no—
431C2×C8—732no—
432C2×C8—734no—
433C2×C8—1464no—
434C2×C8—1462no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—24no—
439C4⋊C4—28no—
440C4⋊C4—24no—
441C4⋊C4—21no—
442C4⋊C4—732no—
443C4⋊C4—1468no—
444C4⋊C4—1461no—
445C4⋊C4—1464no—
446C4⋊C4—1464no—
447C4⋊C4C2×C22⋊D7314yes—
448C4⋊C4C2×C22⋊D7318yes—
449C4⋊C4C23×D7312yes—
450C22⋊C4—1468no—
451C22⋊C4—1464no—
452C22⋊C4—1468no—
453C42—28no—
454C42—24no—
455C42—1468no—
456C42C2×C22⋊D7312yes—
457C42C2×C22⋊D7314yes—
458C42C23×D7312yes—
459C42C23×D7314yes—
460C23—2922no—
461C23—2922no—
462C23—2922no—
463C23—1462no—
464C23C22×C22⋊D7311yescharacteristic
465Q8—28no—
466Q8—22no—
467Q8—24no—
468Q8—24no—
469Q8—22no—
470Q8—42no—
471Q8—44no—
472Q8—2924no—
473Q8—1464no—
474Q8—1464no—
475D4—2922no—
476D4—1462no—
477D4—1462no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—2922no—
488C2×C4—2924no—
489C2×C4—2928no—
490C2×C4—1462no—
491C2×C4—1464no—
492C2×C4#1455662214yes—
493C2×C4C22×C22⋊D7314yes—
494C2×C4C22×C22⋊D7314yes—
495C2×C4C22×C22⋊D7312yes—
496C2×C4C22×C22⋊D7312yes—
497C2×C4C22×C22⋊D7311yescharacteristic
498C2×C4#41×D7312yes—
499C2×C4#1455663712yes—
500C2×C4C24×D7311yescharacteristic
501C8—734no—
502C8—1464no—
503C8—1462no—
504C22—2922no—
505C22—2922no—
506C22—2922no—
507C22—5844no—
508C22#1798925612yes—
509C22C2×#1455662212yes—
510C22#1798940212yes—
511C22C23×C22⋊D7311yescharacteristic
512C4—24no—
513C4—24no—
514C4—22no—
515C4—42no—
516C4—44no—
517C4—42no—
518C4—2922no—
519C4—2924no—
520C4#1798925612yes—
521C4C2×#1455662214yes—
522C4C2×#41×D7311yescharacteristic
523C4C2×#1455663711yescharacteristic
524C2—5842no—
525C2#2262295814yesdirect product factor
526C2C2×#1798925611yescharacteristic
527C2C22×#1455662211yescharacteristic
528C2C2×#1798940211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#557×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
hypercenterC2×#557D73
hypocenterC73C2×#3398
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
2. #22622958C2direct (N × H)
4. C2×#17988966C2semidirect
6. C2×#17988797C2semidirect
7. C2×#17988794C2semidirect
9. C2×#17988718C2semidirect
10. C2×#17988692C2semidirect
12. C2×#557×C73C2semidirect
28. #17988966C22semidirect
30. #17988797C22semidirect
31. #17988794C22semidirect
33. #17988718C22semidirect
34. #17988692C22semidirect
46. #557×C73C22semidirect
260. C2×C73#3398semidirect
265. C2×#557D73semidirect
277. C73C2×#3398semidirect
284. #557C2×D73semidirect
525. C2#22622958direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179893102
#226229582
C2×#179889792
C2×#179889662
C2×#179888092
C2×#179887972
C2×#179887942
C2×#179887892
C2×#179887182
C2×#179886922
C2×C4×#119240152
C2×#557×C732
C2×#339873