Finite Group Databasebeta

← #33102133

Subgroups — #33102133

Order 18688.

1132 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989277C211yesmaximal, characteristic
2#22612802C212yesmaximal, complemented
3#22610915C212yesmaximal, complemented
4#22610386C214yesmaximal
5#22610007C212yesmaximal, complemented
6C4×#14556506C211yesmaximal, characteristic
7#22609662C212yesmaximal, complemented
8C73×#1799C211yesmaximal, characteristic
9#17989277—22no—
10#17989277—24no—
11#17989277—22no—
12C2×#14556584C412yescomplemented
13C2×#14556583C411yescharacteristic, complemented
14C2×#14556581C412yescomplemented
15C2×#14556572C2212yes—
16C2×#14556570C411yescharacteristic, complemented
17C2×#14556570C2211yescharacteristic
18C2×C4×Dic146C411yescharacteristic, complemented
19C2×#14556550C411yescharacteristic, complemented
20C2×#14556547C2214yes—
21C2×#14556547C2212yes—
22C22×C4⋊Dic73C2211yescharacteristic
23C2×#14556506C2212yes—
24#17988483—22no—
25#17988470—24no—
26#17988448C2214yes—
27C4⋊C4×Dic73—22no—
28C4×C22⋊Dic73C2212yes—
29#17988354C2211yescharacteristic
30#17988354C2211yescharacteristic
31#17988353C2214yes—
32#17988353C2214yes—
33C2×C4×D4×C73C2211yescharacteristic
34C2×#99×C73C2212yes—
35#380×C73C2212yes—
36#342×C73C2212yes—
37C22×Dic146C2×C411yescharacteristic
38C23×Dic73C2312yes—
39#14556584—24no—
40#14556584—24no—
41#14556583—22no—
42#14556583—22no—
43#14556581—24no—
44#14556581—24no—
45#14556572—24no—
46#14556572—24no—
47#14556570—22no—
48#14556570—22no—
49#14556570—22no—
50#14556570—22no—
51C4×Dic146—22no—
52C4×Dic146—21no—
53C4×Dic146—21no—
54#14556550—22no—
55#14556550—21no—
56#14556550—21no—
57#14556547—24no—
58#14556547—24no—
59#14556547—28no—
60#14556547—44no—
61C2×C4⋊Dic73C2×C411yescharacteristic
62C2×C4⋊Dic73C2×C411yescharacteristic
63C2×C4⋊Dic73C2×C411yescharacteristic
64C2×C4⋊Dic73C2×C412yes—
65C2×C4⋊Dic73D412yes—
66C2×C4⋊Dic73D414yes—
67C2×C4⋊Dic73D412yes—
68C2×#11924007—24no—
69C2×#11924007C2×C412yes—
70C2×#11924007C2×C412yes—
71C2×#11924007C2×C412yes—
72C2×#11924007C2×C412yes—
73C2×#11924007C2×C412yes—
74C2×C22⋊Dic73C2×C412yes—
75C2×C22⋊Dic73C2×C412yes—
76C2×C22⋊Dic73C2312yes—
77C2×C22⋊Dic73C2312yes—
78C2×C22⋊Dic73C2312yes—
79C2×C4×Dic73—24no—
80C2×C4×Dic73—22no—
81C2×C4×Dic73C2×C412yes—
82C2×C4×Dic73C2×C412yes—
83#14556506—24no—
84#14556498—24no—
85#14556498—24no—
86#14556498—22no—
87C22×D4×C73C2311yescharacteristic
88C23×C4×C73C2×C411yescharacteristic
89C23×C4×C73C2311yescharacteristic
90C4×D4×C73—22no—
91C4×D4×C73—22no—
92C2×C4⋊C4×C73—22no—
93C2×C4⋊C4×C73C2×C411yescharacteristic
94C2×C22⋊C4×C73C2×C411yescharacteristic
95C2×C22⋊C4×C73C2314yes—
96C2×C22⋊C4×C73C2311yescharacteristic
97C2×C42×C73C2×C411yescharacteristic
98#99×C73—42no—
99#95×C73—22no—
100C2×Dic146—21no—
101C2×Dic146—21no—
102C2×Dic146—42no—
103C22×Dic73—24no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73C22×C412yes—
109C22×Dic73C22×C412yes—
110C22×Dic73C22×C412yes—
111C22×Dic73C22×C412yes—
112C22×Dic73C2×D414yes—
113C22×Dic73C2×D414yes—
114C4⋊Dic73—21no—
115C4⋊Dic73—21no—
116C4⋊Dic73C22⋊C412yes—
117C4⋊Dic73C22⋊C411yescharacteristic
118C4⋊Dic73C22⋊C411yescharacteristic
119C4⋊Dic73C22⋊C412yes—
120C4⋊Dic73C22⋊C412yes—
121C4⋊Dic73C22⋊C414yes—
122C4⋊Dic73C22⋊C414yes—
123#11924007—24no—
124#11924007—24no—
125#11924007—24no—
126#11924007—22no—
127#11924007—22no—
128#11924007—24no—
129#11924007—48no—
130C22⋊Dic73—24no—
131C22⋊Dic73—22no—
132C22⋊Dic73—22no—
133C22⋊Dic73—22no—
134C22⋊Dic73—24no—
135C22⋊Dic73—22no—
136C22⋊Dic73—22no—
137C22⋊Dic73#4114yes—
138C4×Dic73—22no—
139C4×Dic73—22no—
140C4×Dic73—22no—
141C4×Dic73—22no—
142C4×Dic73—48no—
143C4×Dic73—44no—
144C24×C73C22×C411yescharacteristic
145C24×C73C2411yescharacteristic
146C2×D4×C73—22no—
147C2×D4×C73—22no—
148C2×D4×C73—21no—
149C2×D4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—21no—
152C22×C4×C73—24no—
153C22×C4×C73—42no—
154C22×C4×C73—42no—
155C22×C4×C73C22×C411yescharacteristic
156C22×C4×C73C22×C411yescharacteristic
157C22×C4×C73C22×C411yescharacteristic
158C22×C4×C73C22×C411yescharacteristic
159C22×C4×C73C22×C411yescharacteristic
160C22×C4×C73C2×D412yes—
161C22×C4×C73C2×D412yes—
162C4⋊C4×C73—22no—
163C4⋊C4×C73—44no—
164C22⋊C4×C73—22no—
165C22⋊C4×C73—22no—
166C22⋊C4×C73—24no—
167C22⋊C4×C73—24no—
168C42×C73—21no—
169C42×C73—21no—
170C23×C73—21no—
171C23×C73—21no—
172C23×C73—21no—
173C23×C73—41no—
174C23×C73—41no—
175C23×C73—41no—
176C23×C73—41no—
177C23×C73C23×C411yescharacteristic
178C23×C73C22×D412yes—
179C23×C73C2×#4112yes—
180C23×C73H(5,𝔽2)11yescharacteristic
181C23×C732−1+411yescharacteristic
182D4×C73—42no—
183D4×C73—42no—
184C2×C4×C73—24no—
185C2×C4×C73—22no—
186C2×C4×C73—21no—
187C2×C4×C73—21no—
188C2×C4×C73—21no—
189C2×C4×C73—21no—
190C2×C4×C73—21no—
191C2×C4×C73—41no—
192C2×C4×C73—42no—
193C2×C4×C73—42no—
194C2×C4×C73—42no—
195C2×C4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73—44no—
198C2×C4×C73—44no—
199C2×C4×C73—44no—
200C2×C4×C73C2×C22⋊C411yescharacteristic
201C2×C4×C73C2×C22⋊C412yes—
202C2×C4×C73C2×C22⋊C411yescharacteristic
203C2×Dic73—22no—
204C2×Dic73—22no—
205C2×Dic73—22no—
206C2×Dic73—22no—
207C2×Dic73—24no—
208C2×Dic73—24no—
209C2×Dic73—24no—
210C2×Dic73—24no—
211C2×Dic73—22no—
212C2×Dic73—22no—
213C2×Dic73—24no—
214C2×Dic73—42no—
215C2×Dic73—42no—
216C2×Dic73—42no—
217C2×Dic73—42no—
218C2×Dic73—42no—
219C2×Dic73—42no—
220C2×Dic73—42no—
221C2×Dic73C2×C22⋊C412yes—
222C2×Dic73C2×C22⋊C412yes—
223C2×Dic73C2×C22⋊C414yes—
224Dic146—42no—
225Dic146—42no—
226C22×C73—22no—
227C22×C73—21no—
228C22×C73—22no—
229C22×C73—21no—
230C22×C73—42no—
231C22×C73—41no—
232C22×C73—41no—
233C22×C73—41no—
234C22×C73—41no—
235C22×C73—41no—
236C22×C73—81no—
237C22×C73—81no—
238C22×C73C22×C22⋊C411yescharacteristic
239C22×C73C4×#4111yescharacteristic
240C22×C73#52011yescharacteristic
241C22×C73#54612yes—
242C22×C73#54712yes—
243C4×C73—21no—
244C4×C73—22no—
245C4×C73—41no—
246C4×C73—41no—
247C4×C73—41no—
248C4×C73—84no—
249C4×C73#41012yes—
250Dic73—22no—
251Dic73—24no—
252Dic73—22no—
253Dic73—44no—
254Dic73—42no—
255Dic73—42no—
256Dic73—82no—
257#22478—731nomaximal, Sylow
258C2×C73—22no—
259C2×C73—22no—
260C2×C73—21no—
261C2×C73—21no—
262C2×C73—81no—
263C2×C73#233911yescharacteristic
264C2×C73C2×#41011yescharacteristic
265C2×C73#288411yescharacteristic
266C2×#552—731no—
267#2344—732no—
268#1996—732no—
269#1899—734no—
270#1820—732no—
271#1799D7311yescharacteristic
272C4×#99—731no—
273#1740—732no—
274C73#2247811yescharacteristic, complemented, Sylow
275#552—1462no—
276#552—1464no—
277#552—1462no—
278C2×#126—732no—
279C2×#125—732no—
280C2×#125—731no—
281C2×#123—732no—
282C2×C22⋊Q8—731no—
283C2×C22⋊Q8—731no—
284C2×C4×Q8—731no—
285C2×C4×D4C2×D7311yescharacteristic
286C2×#117—731no—
287C2×#117—732no—
288C2×#117—734no—
289C22×C4⋊C4—731no—
290C2×#99—732no—
291C2×#99C2×D7312yes—
292#389—1462no—
293#387—1464no—
294#381—734no—
295#380C2×D7312yes—
296C4×C4⋊C4—1462no—
297C4×C22⋊C4—732no—
298#342—734no—
299#342—734no—
300#342—731no—
301#342—731no—
302#342C2×D7312yes—
303C22×Q8—731no—
304C22×D4C22×D7311yescharacteristic
305C23×C4—732no—
306C23×C4C4×D7311yescharacteristic
307C23×C4C22×D7311yescharacteristic
308#126—1464no—
309#126—1464no—
310#125—1462no—
311#125—1462no—
312#125—1464no—
313#125—1464no—
314#123—1464no—
315#123—1464no—
316C22⋊Q8—1462no—
317C22⋊Q8—1462no—
318C22⋊Q8—1462no—
319C22⋊Q8—1462no—
320C4×Q8—1462no—
321C4×Q8—1461no—
322C4×Q8—1461no—
323C4×D4—22no—
324C4×D4—22no—
325#117—2924no—
326#117—1468no—
327#117—1464no—
328#117—1464no—
329#117—1462no—
330#117—1461no—
331#117—1461no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—731no—
334C2×C4⋊C4—731no—
335C2×C4⋊C4—732no—
336C2×C4⋊C4—732no—
337C2×C4⋊C4—734no—
338C2×C4⋊C4—732no—
339C2×C4⋊C4—732no—
340C2×C4⋊C4—731no—
341C2×C4⋊C4—732no—
342C2×C4⋊C4—732no—
343C2×C4⋊C4—732no—
344C2×C4⋊C4—732no—
345C2×C4⋊C4—1464no—
346C2×C4⋊C4C4×D7311yescharacteristic
347C2×C22⋊C4—732no—
348C2×C22⋊C4—732no—
349C2×C22⋊C4—732no—
350C2×C22⋊C4—732no—
351C2×C22⋊C4—732no—
352C2×C22⋊C4C4×D7311yescharacteristic
353C2×C22⋊C4C22×D7314yes—
354C2×C22⋊C4C22×D7311yescharacteristic
355C2×C42—732no—
356C2×C42—732no—
357C2×C42—1464no—
358C2×C42—1462no—
359C2×C42C4×D7311yescharacteristic
360#99—42no—
361#99—1464no—
362#95—22no—
363#95—1464no—
364#95—1464no—
365#95—1462no—
366C24C2×C4×D7311yescharacteristic
367C24C23×D7311yescharacteristic
368C2×Q8—2922no—
369C2×Q8—1461no—
370C2×Q8—1461no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—732no—
381C22×C4—732no—
382C22×C4—732no—
383C22×C4—732no—
384C22×C4—734no—
385C22×C4—734no—
386C22×C4—1462no—
387C22×C4—1462no—
388C22×C4—1464no—
389C22×C4—1462no—
390C22×C4—1462no—
391C22×C4C2×C4×D7311yescharacteristic
392C22×C4C2×C4×D7311yescharacteristic
393C22×C4C2×C4×D7311yescharacteristic
394C22×C4C2×C4×D7311yescharacteristic
395C22×C4C2×C4×D7311yescharacteristic
396C22×C4D4×D7312yes—
397C22×C4D4×D7312yes—
398C4⋊C4—22no—
399C4⋊C4—44no—
400C4⋊C4—2928no—
401C4⋊C4—732no—
402C4⋊C4—731no—
403C4⋊C4—731no—
404C4⋊C4—732no—
405C4⋊C4—732no—
406C4⋊C4—734no—
407C4⋊C4—734no—
408C4⋊C4—1464no—
409C4⋊C4—1464no—
410C4⋊C4—1462no—
411C4⋊C4—1462no—
412C4⋊C4—1464no—
413C4⋊C4—1464no—
414C4⋊C4—1461no—
415C4⋊C4—1461no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—734no—
421C22⋊C4—1462no—
422C22⋊C4—1462no—
423C22⋊C4—1462no—
424C22⋊C4—1462no—
425C22⋊C4—1464no—
426C22⋊C4—1464no—
427C22⋊C4—1462no—
428C42—21no—
429C42—21no—
430C42—2924no—
431C42—2928no—
432C42—1462no—
433C42—1462no—
434C42—1462no—
435C42—1462no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23C22×C4×D7311yescharacteristic
444C23C2×D4×D7312yes—
445C23#41×D7312yes—
446C23#1455663611yescharacteristic
447C23#1455663911yescharacteristic
448Q8—2922no—
449Q8—2922no—
450D4—42no—
451D4—42no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—2922no—
469C2×C4—2922no—
470C2×C4—2922no—
471C2×C4—2922no—
472C2×C4—2922no—
473C2×C4—2922no—
474C2×C4—2922no—
475C2×C4—732no—
476C2×C4—732no—
477C2×C4—734no—
478C2×C4—1464no—
479C2×C4—1462no—
480C2×C4—1462no—
481C2×C4—1464no—
482C2×C4—1464no—
483C2×C4—1464no—
484C2×C4—1464no—
485C2×C4—1462no—
486C2×C4—1462no—
487C2×C4—1462no—
488C2×C4—1462no—
489C2×C4C22⋊C4×D7311yescharacteristic
490C2×C4C22⋊C4×D7312yes—
491C2×C4C22⋊C4×D7311yescharacteristic
492C22—22no—
493C22—21no—
494C22—22no—
495C22—21no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—81no—
503C22—81no—
504C22C2×C22⋊C4×D7311yescharacteristic
505C22#1798905511yescharacteristic
506C22#1798906811yescharacteristic
507C22#1798923412yes—
508C22#1798923912yes—
509C4—21no—
510C4—22no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—84no—
515C4—2924no—
516C4—2922no—
517C4—2922no—
518C4—5842no—
519C4—1462no—
520C4—1462no—
521C4—1464no—
522C4#1798855912yes—
523C2—22no—
524C2—22no—
525C2—21no—
526C2—21no—
527C2—81no—
528C2#2261275911yescharacteristic
529C2C2×#1798855911yescharacteristic
530C2#2261839311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D73
commutatorC23×C73C23×C4
FrattiniC24C23×D73
FittingC73×#1799C2
socleC22×C73C22×C22⋊C4
prime radicalC22×D4×C73C23
cosocleC24×C73C24
hypercenter#1799D73
hypocenterC73#22478
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #22612802C2semidirect
3. #22610915C2semidirect
5. #22610007C2semidirect
7. #22609662C2semidirect
12. C2×#14556584C4semidirect
13. C2×#14556583C4semidirect
14. C2×#14556581C4semidirect
16. C2×#14556570C4semidirect
18. C2×C4×Dic146C4semidirect
19. C2×#14556550C4semidirect
274. C73#22478semidirect

Maximal subgroups

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

subgroupindex
C2×#179892772
#226128022
#226109152
#226103862
#226100072
C4×#145565062
#226096622
C73×#17992
#2247873