Finite Group Databasebeta

← #33095438

Subgroups — #33095438

Order 18688.

990 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989216C211yesmaximal, characteristic
2#22612533C211yesmaximal, characteristic
3#22612531C211yesmaximal, characteristic
4#22610917C211yesmaximal, characteristic
5#22610916C211yesmaximal, characteristic
6#22610915C211yesmaximal, characteristic
7#22610914C214yesmaximal
8#22610007C214yesmaximal
9C73×#1996C211yesmaximal, characteristic
10#17989216—22no—
11#17989216—22no—
12#17989216—22no—
13#17989216—22no—
14C2×#14556590C412yescomplemented
15C2×#14556589C412yescomplemented
16C2×#14556581C412yescomplemented
17C2×#14556581C412yescomplemented
18C2×#14556570C2211yescharacteristic
19C2×#14556568C2212yes—
20C2×#14556547C2212yes—
21C22×#11924007C2211yescharacteristic
22#17988474—22no—
23#17988471—22no—
24#17988469—22no—
25#17988468—22no—
26#17988448C2212yes—
27#17988448C2212yes—
28#17988447C2211yescharacteristic
29#17988445C2212yes—
30C22⋊C4×Dic73C2212yes—
31C4×C22⋊Dic73C2211yescharacteristic
32#17988354C2214yes—
33#17988353C2214yes—
34#17988353C2214yes—
35C2×C22⋊Q8×C73C2211yescharacteristic
36#380×C73C2211yescharacteristic
37C4×C22⋊C4×C73C2211yescharacteristic
38#342×C73C2214yes—
39C22×Dic146C2×C412yes—
40C22×Dic146C2×C411yescharacteristic
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2311yescharacteristic
43#14556590—24no—
44#14556590—24no—
45#14556589—24no—
46#14556589—24no—
47#14556581—24no—
48#14556581—24no—
49#14556581—24no—
50#14556581—24no—
51#14556570—21no—
52#14556570—21no—
53#14556570—21no—
54#14556570—21no—
55#14556568—24no—
56#14556568—24no—
57#14556547—24no—
58#14556547—24no—
59C2×C4⋊Dic73—21no—
60C2×C4⋊Dic73C2×C412yes—
61C2×C4⋊Dic73C2×C411yescharacteristic
62C2×#11924007—22no—
63C2×#11924007C2×C412yes—
64C2×#11924007C2×C412yes—
65C2×#11924007C2×C412yes—
66C2×#11924007C2×C412yes—
67C2×#11924007C2×C412yes—
68C2×#11924007C2×C412yes—
69C2×#11924007C2×C412yes—
70C2×#11924007D412yes—
71C2×#11924007D412yes—
72C2×#11924007D412yes—
73C2×#11924007D412yes—
74C2×C22⋊Dic73C2312yes—
75C2×C22⋊Dic73C2312yes—
76C2×C22⋊Dic73C2312yes—
77C2×C4×Dic73—22no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73C2×C412yes—
80C2×C4×Dic73C2×C412yes—
81#14556498—22no—
82#14556498—22no—
83#14556498—24no—
84#14556498—24no—
85#14556498—21no—
86#14556498—21no—
87C22×Q8×C73C2×C411yescharacteristic
88C23×C4×C73C2311yescharacteristic
89C22⋊Q8×C73—21no—
90C22⋊Q8×C73—21no—
91C22⋊Q8×C73—21no—
92C22⋊Q8×C73—21no—
93C2×C4⋊C4×C73—21no—
94C2×C4⋊C4×C73C2×C412yes—
95C2×C4⋊C4×C73C2×C411yescharacteristic
96C2×C22⋊C4×C73C2312yes—
97C2×C22⋊C4×C73C2312yes—
98C2×C22⋊C4×C73C2312yes—
99C2×C42×C73—21no—
100#95×C73—21no—
101#95×C73—21no—
102C2×Dic146—21no—
103C2×Dic146—21no—
104C2×Dic146—24no—
105C2×Dic146—82no—
106C2×Dic146—81no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—81no—
112C22×Dic73C22×C412yes—
113C22×Dic73C22×C412yes—
114C22×Dic73C22×C412yes—
115C22×Dic73C22×C412yes—
116C22×Dic73C2×D412yes—
117C22×Dic73C2×D412yes—
118C22×Dic73C2×D412yes—
119C22×Dic73C2×D412yes—
120C4⋊Dic73—21no—
121C4⋊Dic73—21no—
122C4⋊Dic73—24no—
123C4⋊Dic73—81no—
124#11924007—22no—
125#11924007—22no—
126#11924007—24no—
127#11924007—24no—
128#11924007—24no—
129#11924007—24no—
130#11924007—82no—
131#11924007C22⋊C414yes—
132#11924007C22⋊C414yes—
133#11924007C22⋊C414yes—
134#11924007C22⋊C414yes—
135C22⋊Dic73—22no—
136C22⋊Dic73—22no—
137C22⋊Dic73—22no—
138C22⋊Dic73—22no—
139C22⋊Dic73—24no—
140C4×Dic73—24no—
141C4×Dic73—24no—
142C4×Dic73—82no—
143C4×Dic73—81no—
144C24×C73C2411yescharacteristic
145C2×Q8×C73—21no—
146C2×Q8×C73—21no—
147C2×Q8×C73—81no—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—81no—
151C22×C4×C73C22×C412yes—
152C22×C4×C73C22×C412yes—
153C22×C4×C73C22×C411yescharacteristic
154C22×C4×C73C22×C411yescharacteristic
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73C2×D411yescharacteristic
157C22×C4×C73C2×D411yescharacteristic
158C22×C4×C73C2×D411yescharacteristic
159C4⋊C4×C73—21no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73—22no—
162C4⋊C4×C73—22no—
163C4⋊C4×C73—81no—
164C22⋊C4×C73—22no—
165C22⋊C4×C73—22no—
166C22⋊C4×C73—22no—
167C22⋊C4×C73—24no—
168C22⋊C4×C73—22no—
169C42×C73—81no—
170C23×C73—41no—
171C23×C73—41no—
172C23×C73C23×C411yescharacteristic
173C23×C73C22×D411yescharacteristic
174C23×C73C22×D411yescharacteristic
175C23×C732−1+412yes—
176C23×C732−1+412yes—
177Q8×C73—81no—
178Q8×C73—81no—
179C2×C4×C73—22no—
180C2×C4×C73—22no—
181C2×C4×C73—41no—
182C2×C4×C73—41no—
183C2×C4×C73—41no—
184C2×C4×C73—41no—
185C2×C4×C73—41no—
186C2×C4×C73—41no—
187C2×C4×C73—42no—
188C2×C4×C73—42no—
189C2×C4×C73—42no—
190C2×C4×C73—42no—
191C2×C4×C73—42no—
192C2×C4×C73—42no—
193C2×C4×C73—42no—
194C2×C4×C73—42no—
195C2×C4×C73C2×C22⋊C411yescharacteristic
196C2×C4×C73C2×C22⋊C411yescharacteristic
197C2×C4×C73C2×C22⋊C411yescharacteristic
198C2×C4×C73C2×C22⋊C411yescharacteristic
199C2×Dic73—24no—
200C2×Dic73—22no—
201C2×Dic73—24no—
202C2×Dic73—22no—
203C2×Dic73—22no—
204C2×Dic73—22no—
205C2×Dic73—42no—
206C2×Dic73—42no—
207C2×Dic73—42no—
208C2×Dic73—42no—
209C2×Dic73—42no—
210C2×Dic73—42no—
211C2×Dic73—42no—
212C2×Dic73—42no—
213C2×Dic73—42no—
214C2×Dic73—42no—
215C2×Dic73—42no—
216C2×Dic73C2×C22⋊C414yes—
217C2×Dic73C2×C22⋊C414yes—
218Dic146—84no—
219Dic146—81no—
220Dic146—81no—
221C22×C73—21no—
222C22×C73—21no—
223C22×C73—21no—
224C22×C73—21no—
225C22×C73—21no—
226C22×C73—21no—
227C22×C73—42no—
228C22×C73—42no—
229C22×C73C22×C22⋊C411yescharacteristic
230C22×C73#51912yes—
231C22×C73#53611yescharacteristic
232C22×C73#53612yes—
233C22×C73#53611yescharacteristic
234C4×C73—21no—
235C4×C73—21no—
236C4×C73—21no—
237C4×C73—21no—
238C4×C73—42no—
239C4×C73—42no—
240C4×C73—82no—
241C4×C73—82no—
242Dic73—24no—
243Dic73—24no—
244Dic73—42no—
245Dic73—42no—
246Dic73—82no—
247Dic73—82no—
248#21828—731nomaximal, Sylow
249C2×C73—21no—
250C2×C73—22no—
251C2×C73—21no—
252C2×C73—22no—
253C2×C73#231011yescharacteristic
254C2×C73#288311yescharacteristic
255C2×C73#288411yescharacteristic
256C2×#544—731no—
257#2313—731no—
258#2313—731no—
259#1996—731no—
260#1996—731no—
261#1996—734no—
262#1996—731no—
263#1996D7311yescharacteristic
264#1820—734no—
265C73#2182811yescharacteristic, complemented, Sylow
266#544—1462no—
267#544—1462no—
268#544—1462no—
269#544—1462no—
270C2×C4⋊Q8—732no—
271C2×C4⋊Q8—732no—
272C2×#125—732no—
273C2×#125—732no—
274C2×C22⋊Q8—731no—
275C2×C22⋊Q8—732no—
276C2×C22⋊Q8C2×D7311yescharacteristic
277C2×#117—732no—
278C22×C4⋊C4—731no—
279#387—1462no—
280#387—1462no—
281#387—1462no—
282#387—1462no—
283#381—732no—
284#381—732no—
285#380—731no—
286#380—732no—
287#380C2×D7311yescharacteristic
288C4×C22⋊C4—731no—
289C4×C22⋊C4—732no—
290C4×C22⋊C4C2×D7311yescharacteristic
291#342—734no—
292#342—734no—
293#342—734no—
294#342C2×D7314yes—
295C22×Q8—732no—
296C22×Q8—731no—
297C22×Q8C4×D7311yescharacteristic
298C23×C4—731no—
299C23×C4—731no—
300C23×C4C22×D7311yescharacteristic
301C4⋊Q8—1464no—
302C4⋊Q8—1464no—
303C4⋊Q8—1464no—
304C4⋊Q8—1464no—
305#125—1464no—
306#125—1464no—
307#125—1464no—
308#125—1464no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—1464no—
314C22⋊Q8—1464no—
315C22⋊Q8—1461no—
316C22⋊Q8—1461no—
317C22⋊Q8—1461no—
318C22⋊Q8—1461no—
319#117—1464no—
320#117—1464no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—732no—
323C2×C4⋊C4—732no—
324C2×C4⋊C4—732no—
325C2×C4⋊C4—732no—
326C2×C4⋊C4—732no—
327C2×C4⋊C4—732no—
328C2×C4⋊C4—732no—
329C2×C4⋊C4—732no—
330C2×C4⋊C4—732no—
331C2×C4⋊C4—731no—
332C2×C4⋊C4—732no—
333C2×C4⋊C4—732no—
334C2×C4⋊C4—732no—
335C2×C4⋊C4—1462no—
336C2×C4⋊C4—1461no—
337C2×C4⋊C4C4×D7311yescharacteristic
338C2×C4⋊C4C4×D7312yes—
339C2×C22⋊C4—732no—
340C2×C22⋊C4—732no—
341C2×C22⋊C4—732no—
342C2×C22⋊C4C22×D7312yes—
343C2×C22⋊C4C22×D7312yes—
344C2×C22⋊C4C22×D7312yes—
345C2×C42—21no—
346C2×C42—732no—
347C2×C42—732no—
348C2×C42—1462no—
349C2×C42—1461no—
350#95—21no—
351#95—21no—
352#95—1462no—
353#95—1462no—
354#95—1464no—
355#95—1464no—
356#95—1461no—
357#95—1461no—
358C24C23×D7311yescharacteristic
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—81no—
362C2×Q8—5841no—
363C2×Q8—5842no—
364C2×Q8—1464no—
365C2×Q8—1461no—
366C2×Q8—1461no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—81no—
370C22×C4—5841no—
371C22×C4—732no—
372C22×C4—732no—
373C22×C4—732no—
374C22×C4—732no—
375C22×C4—732no—
376C22×C4—732no—
377C22×C4—732no—
378C22×C4—732no—
379C22×C4—1462no—
380C22×C4—1462no—
381C22×C4—1462no—
382C22×C4—1462no—
383C22×C4C2×C4×D7312yes—
384C22×C4C2×C4×D7312yes—
385C22×C4C2×C4×D7311yescharacteristic
386C22×C4C2×C4×D7311yescharacteristic
387C22×C4D4×D7311yescharacteristic
388C22×C4D4×D7311yescharacteristic
389C22×C4D4×D7311yescharacteristic
390C22×C4D4×D7311yescharacteristic
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—81no—
396C4⋊C4—5842no—
397C4⋊C4—5841no—
398C4⋊C4—734no—
399C4⋊C4—734no—
400C4⋊C4—734no—
401C4⋊C4—734no—
402C4⋊C4—1464no—
403C4⋊C4—1464no—
404C4⋊C4—1464no—
405C4⋊C4—1464no—
406C4⋊C4—1464no—
407C4⋊C4—1462no—
408C4⋊C4—1462no—
409C4⋊C4—1461no—
410C4⋊C4—1461no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—24no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—1462no—
417C22⋊C4—1462no—
418C22⋊C4—1464no—
419C22⋊C4—1462no—
420C22⋊C4—1462no—
421C42—81no—
422C42—5841no—
423C42—5842no—
424C42—1464no—
425C42—1464no—
426C23—41no—
427C23—41no—
428C23C22×C4×D7311yescharacteristic
429C23C2×D4×D7311yescharacteristic
430C23C2×D4×D7311yescharacteristic
431C23#1455663812yes—
432C23#1455663912yes—
433Q8—81no—
434Q8—81no—
435Q8—5841no—
436Q8—5841no—
437Q8—5844no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—2922no—
455C2×C4—2922no—
456C2×C4—2922no—
457C2×C4—2922no—
458C2×C4—2922no—
459C2×C4—2922no—
460C2×C4—2922no—
461C2×C4—2922no—
462C2×C4—2922no—
463C2×C4—2922no—
464C2×C4—2922no—
465C2×C4—734no—
466C2×C4—734no—
467C2×C4—1462no—
468C2×C4—1462no—
469C2×C4—1462no—
470C2×C4—1464no—
471C2×C4—1462no—
472C2×C4—1464no—
473C2×C4C22⋊C4×D7311yescharacteristic
474C2×C4C22⋊C4×D7311yescharacteristic
475C2×C4C22⋊C4×D7311yescharacteristic
476C2×C4C22⋊C4×D7311yescharacteristic
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—42no—
484C22—42no—
485C22C2×C22⋊C4×D7311yescharacteristic
486C22#1798906312yes—
487C22#1798915711yescharacteristic
488C22#1798916012yes—
489C22#1798916211yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—82no—
498C4—2922no—
499C4—2922no—
500C4—5842no—
501C4—5842no—
502C4—1464no—
503C4—1464no—
504C2—21no—
505C2—22no—
506C2—21no—
507C2—22no—
508C2#2261251411yescharacteristic
509C2#2261838911yescharacteristic
510C2#2261839311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D73
commutatorC23×C73C23×C4
FrattiniC24C23×D73
FittingC73×#1996C2
socleC22×C73C22×C22⋊C4
prime radical, cosocleC24×C73C24
hypercenter#1996D73
hypocenterC73#21828
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
14. C2×#14556590C4semidirect
15. C2×#14556589C4semidirect
16. C2×#14556581C4semidirect
17. C2×#14556581C4semidirect
265. C73#21828semidirect

Maximal subgroups

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

subgroupindex
C2×#179892162
#226125332
#226125312
#226109172
#226109162
#226109152
#226109142
#226100072
C73×#19962
#2182873