Finite Group Databasebeta

← #32982468

Subgroups — #32982468

Order 18688.

504 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618987C211yesmaximal, characteristic, complemented
2#22608435C211yesmaximal, characteristic, complemented
3C2×#17988321C211yesmaximal, characteristic, complemented
4#22608291C211yesmaximal, characteristic, complemented
5#22607819C211yesmaximal, characteristic, complemented
6#22607773C211yesmaximal, characteristic, complemented
7C73×#3188C211yesmaximal, characteristic
8#17988659—21no—
9#17988659—21no—
10D4×#10596831—21no—
11D4×#10596831—21no—
12#17988648—21no—
13#17988554—21no—
14#17988551C2211yescharacteristic
15C2×C22⋊#10596831C2211yescharacteristic
16C2×C22⋊#10596831C2211yescharacteristic
17C2×#14556498C2211yescharacteristic
18#17988324—21no—
19#17988321—21no—
20#17988321—21no—
21#17988321—21no—
22#17988311—21no—
23#17988311—21no—
24#525×C73C2211yescharacteristic
25C2×#121×C73C2211yescharacteristic
26#484×C73C411yescharacteristic, complemented
27#482×C73—21no—
28#481×C73C411yescharacteristic, complemented
29#466×C73—21no—
30#465×C73—21no—
31#450×C73C411yescharacteristic, complemented
32#449×C73—21no—
33#447×C73C411yescharacteristic, complemented
34C2×C22⋊C8×C73C2211yescharacteristic
35C23×Dic73—21no—
36C2×#11924012—21no—
37C2×#11924012—21no—
38C22×#10596831—21no—
39C22×#10596831—21no—
40C22×#10596831—21no—
41C4⋊#10596831—21no—
42C4⋊#10596831—21no—
43C22⋊#10596831—21no—
44C22⋊#10596831—21no—
45C22⋊#10596831—21no—
46C22⋊#10596831—21no—
47C22⋊#10596831—21no—
48C22⋊#10596831—21no—
49C22⋊#10596831D411yescharacteristic
50C22⋊#10596831D411yescharacteristic
51C22⋊#10596831D411yescharacteristic
52C22⋊#10596831D411yescharacteristic
53#14556502—21no—
54C4×#10596831—21no—
55#14556498—21no—
56#14556498—21no—
57C2×#41×C73D411yescharacteristic
58C22×D4×C73—21no—
59C22×D4×C73C2×C411yescharacteristic
60C23×C4×C73C2311yescharacteristic
61C2×SD16×C73—21no—
62C2×D8×C73—21no—
63C22×C8×C73—21no—
64#123×C73—21no—
65C22⋊Q8×C73C2×C411yescharacteristic
66C22⋊Q8×C73D411yescharacteristic
67#121×C73—21no—
68#121×C73—21no—
69#121×C73—21no—
70#121×C73—21no—
71#121×C73C2×C411yescharacteristic
72#121×C73D411yescharacteristic
73#120×C73—21no—
74C4×D4×C73—21no—
75C4×D4×C73—21no—
76#117×C73D411yescharacteristic
77C2×C4⋊C4×C73C2×C411yescharacteristic
78C2×C22⋊C4×C73—21no—
79#107×C73—21no—
80#106×C73—21no—
81#103×C73—21no—
82#102×C73—21no—
83#102×C73—21no—
84#102×C73—21no—
85C22⋊C8×C73—21no—
86C22⋊C8×C73C2×C411yescharacteristic
87C22⋊C8×C73C2×C411yescharacteristic
88C22×Dic73—22no—
89C22×Dic73—41no—
90C22×Dic73—41no—
91C22×Dic73—81no—
92C22×Dic73—81no—
93#11924012—81no—
94#11924012—81no—
95C2×#10596831—21no—
96C2×#10596831—21no—
97C2×#10596831—21no—
98C2×#10596831—21no—
99C2×#10596831—41no—
100C2×#10596831—81no—
101C2×#10596831—81no—
102C2×#10596831—81no—
103C24×C73—21no—
104C24×C73#4111yescharacteristic
105#41×C73—81no—
106C2×Q8×C73C22⋊C411yescharacteristic
107C2×D4×C73—21no—
108C2×D4×C73—21no—
109C2×D4×C73—21no—
110C2×D4×C73—21no—
111C2×D4×C73—21no—
112C2×D4×C73—41no—
113C2×D4×C73—41no—
114C2×D4×C73—81no—
115C2×D4×C73—81no—
116C2×D4×C73—81no—
117C2×D4×C73C22⋊C411yescharacteristic
118C22×C4×C73—21no—
119C22×C4×C73—21no—
120C22×C4×C73—81no—
121C22×C4×C73C22×C411yescharacteristic
122C22×C4×C73C2×D411yescharacteristic
123C22×C4×C73C2×D411yescharacteristic
124C22×C4×C73C2×D411yescharacteristic
125C22×C4×C73C2×D411yescharacteristic
126C22×C4×C73#4111yescharacteristic
127SD16×C73—81no—
128D8×C73—81no—
129C2×C8×C73—21no—
130C2×C8×C73—21no—
131C2×C8×C73—81no—
132C4⋊C4×C73—21no—
133C4⋊C4×C73—21no—
134C4⋊C4×C73—21no—
135C4⋊C4×C73—21no—
136C4⋊C4×C73C22⋊C411yescharacteristic
137C4⋊C4×C73C22⋊C411yescharacteristic
138C22⋊C4×C73—21no—
139C22⋊C4×C73—21no—
140C22⋊C4×C73—21no—
141C22⋊C4×C73—21no—
142C22⋊C4×C73—21no—
143C22⋊C4×C73—41no—
144C22⋊C4×C73—41no—
145C42×C73—21no—
146C23×C73—21no—
147C23×C73—21no—
148C23×C73—21no—
149C23×C73—21no—
150C23×C73—21no—
151C23×C73—41no—
152C23×C73—41no—
153C23×C73—41no—
154C23×C73—41no—
155C23×C73—81no—
156C23×C73—81no—
157C23×C73C4×D411yescharacteristic
158C23×C73#12111yescharacteristic
159C23×C73#12311yescharacteristic
160Q8×C73—41no—
161D4×C73—41no—
162D4×C73—41no—
163D4×C73—41no—
164D4×C73—41no—
165D4×C73—41no—
166D4×C73—41no—
167D4×C73—41no—
168D4×C73—81no—
169D4×C73—81no—
170D4×C73—82no—
171D4×C73—81no—
172D4×C73—81no—
173C2×C4×C73—21no—
174C2×C4×C73—21no—
175C2×C4×C73—21no—
176C2×C4×C73—41no—
177C2×C4×C73—41no—
178C2×C4×C73—41no—
179C2×C4×C73—41no—
180C2×C4×C73—41no—
181C2×C4×C73—41no—
182C2×C4×C73—41no—
183C2×C4×C73—41no—
184C2×C4×C73—41no—
185C2×C4×C73—81no—
186C2×C4×C73C2×C22⋊C411yescharacteristic
187C2×C4×C73C4×D411yescharacteristic
188C2×C4×C73#12011yescharacteristic
189C2×C4×C73#12111yescharacteristic
190C2×Dic73—42no—
191C2×Dic73—81no—
192C2×Dic73—82no—
193C2×Dic73—81no—
194C2×Dic73—82no—
195C8×C73—41no—
196C8×C73—41no—
197#10596831—41no—
198#10596831—41no—
199#10596831—41no—
200#10596831—41no—
201#10596831—81no—
202C22×C73—21no—
203C22×C73—21no—
204C22×C73—21no—
205C22×C73—21no—
206C22×C73—21no—
207C22×C73—21no—
208C22×C73—21no—
209C22×C73—41no—
210C22×C73—41no—
211C22×C73—41no—
212C22×C73—41no—
213C22×C73—41no—
214C22×C73—41no—
215C22×C73—41no—
216C22×C73—81no—
217C22×C73—81no—
218C22×C73—81no—
219C22×C73—161no—
220C22×C73—161no—
221C22×C73#38611yescharacteristic
222C22×C73#44311yescharacteristic
223C22×C73#44411yescharacteristic
224C22×C73#47111yescharacteristic
225C22×C73#47211yescharacteristic
226C4×C73—21no—
227C4×C73—21no—
228C4×C73—21no—
229C4×C73—21no—
230C4×C73—41no—
231C4×C73—41no—
232C4×C73—41no—
233C4×C73—81no—
234Dic73—82no—
235#9264—731nomaximal, Sylow
236C2×C73—21no—
237C2×C73—21no—
238C2×C73—21no—
239C2×C73—21no—
240C2×C73—41no—
241C2×C73—81no—
242C2×C73—161no—
243C2×C73#187611yescharacteristic
244C2×C73#190711yescharacteristic
245C2×C73#191311yescharacteristic
246#3188D7311yescharacteristic
247#2977—731no—
248#1504—731no—
249C2×#327—731no—
250#1471—731no—
251#1298—731no—
252#1280—731no—
253C73#926411yescharacteristic, complemented, Sylow
254#525C2×D7311yescharacteristic
255C2×#121C2×D7311yescharacteristic
256#484Dic7311yescharacteristic, complemented
257#482—21no—
258#481Dic7311yescharacteristic, complemented
259#466—21no—
260#465—21no—
261#450Dic7311yescharacteristic, complemented
262#449—21no—
263#447Dic7311yescharacteristic, complemented
264#435—1461no—
265#435—1461no—
266C8×D4—1461no—
267C8×D4—1461no—
268#433—1461no—
269#408—1461no—
270#407—731no—
271C2×C22⋊C8—731no—
272C2×C22⋊C8—731no—
273C2×C22⋊C8C2×D7311yescharacteristic
274C2×#95—731no—
275#328—1461no—
276#327—1461no—
277#327—1461no—
278#327—1461no—
279#323—1461no—
280#323—1461no—
281C2×#41C22⋊D7311yescharacteristic
282C22×D4—21no—
283C22×D4C2×Dic7311yescharacteristic
284C23×C4—1461no—
285C23×C4C22×D7311yescharacteristic
286C2×SD16—21no—
287C2×D8—21no—
288C2×M16—1461no—
289C2×M16—1461no—
290C22×C8—21no—
291C22×C8—1461no—
292C22×C8—1461no—
293C22×C8—1461no—
294#123—21no—
295C22⋊Q8C2×Dic7311yescharacteristic
296C22⋊Q8C22⋊D7311yescharacteristic
297#121—21no—
298#121—21no—
299#121—21no—
300#121—21no—
301#121C2×Dic7311yescharacteristic
302#121C22⋊D7311yescharacteristic
303#120—21no—
304C4×D4—21no—
305C4×D4—21no—
306#117C22⋊D7311yescharacteristic
307C2×C4⋊C4C2×Dic7311yescharacteristic
308C2×C22⋊C4—21no—
309#107—21no—
310#106—21no—
311C4⋊C8—1461no—
312C4⋊C8—1461no—
313#103—21no—
314#102—21no—
315#102—21no—
316#102—21no—
317C22⋊C8—21no—
318C22⋊C8—731no—
319C22⋊C8—731no—
320C22⋊C8—731no—
321C22⋊C8—731no—
322C22⋊C8—1461no—
323C22⋊C8—1461no—
324C22⋊C8—1461no—
325C22⋊C8—1461no—
326C22⋊C8—1461no—
327C22⋊C8—1461no—
328C22⋊C8C2×Dic7311yescharacteristic
329C22⋊C8C2×Dic7311yescharacteristic
330#97—1461no—
331C4×C8—1461no—
332#95—1461no—
333#95—1461no—
334C24—21no—
335C24#1192402711yescharacteristic
336#41—81no—
337C2×Q8C22⋊Dic7311yescharacteristic
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—81no—
346C2×D4—81no—
347C2×D4—81no—
348C2×D4C22⋊Dic7311yescharacteristic
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—81no—
352C22×C4—2921no—
353C22×C4—2921no—
354C22×C4—5841no—
355C22×C4—5841no—
356C22×C4—1462no—
357C22×C4C22×Dic7311yescharacteristic
358C22×C4D4×D7311yescharacteristic
359C22×C4D4×D7311yescharacteristic
360C22×C4C2×C22⋊D7311yescharacteristic
361C22×C4C2×C22⋊D7311yescharacteristic
362C22×C4#1192402711yescharacteristic
363SD16—81no—
364D8—81no—
365M16—5841no—
366M16—5841no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—81no—
370C2×C8—2921no—
371C2×C8—5841no—
372C2×C8—5841no—
373C2×C8—5841no—
374C2×C8—1461no—
375C2×C8—1461no—
376C2×C8—1461no—
377C2×C8—1461no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4C22⋊Dic7311yescharacteristic
383C4⋊C4C22⋊Dic7311yescharacteristic
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C42—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—81no—
402C23—81no—
403C23D4×Dic7311yescharacteristic
404C23#1455656311yescharacteristic
405C23#1455657211yescharacteristic
406Q8—41no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—81no—
415D4—81no—
416D4—82no—
417D4—81no—
418D4—81no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—81no—
432C2×C4—2922no—
433C2×C4—5841no—
434C2×C4—5842no—
435C2×C4—5841no—
436C2×C4—5842no—
437C2×C4C2×C22⋊Dic7311yescharacteristic
438C2×C4D4×Dic7311yescharacteristic
439C2×C4#1455655911yescharacteristic
440C2×C4#1455656311yescharacteristic
441C8—41no—
442C8—41no—
443C8—2921no—
444C8—2921no—
445C8—2921no—
446C8—2921no—
447C8—5841no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—161no—
466C22—161no—
467C22#1798846711yescharacteristic
468C22#1798870411yescharacteristic
469C22#1798870911yescharacteristic
470C22#1798884811yescharacteristic
471C22#1798885511yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—81no—
480C4—5842no—
481C2—21no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—41no—
486C2—81no—
487C2—161no—
488C2#2261026011yescharacteristic
489C2#2261044111yescharacteristic
490C2#2261047411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
Fitting, prime radicalC73×#3188C2
socleC22×C73#386
cosocleC23×C4×C73C23
hypercenter#3188D73
hypocenterC73#9264
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3188

Complements

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

NH ≅ G/Ntype
1. #22618987C2semidirect
2. #22608435C2semidirect
3. C2×#17988321C2semidirect
4. #22608291C2semidirect
5. #22607819C2semidirect
6. #22607773C2semidirect
26. #484×C73C4semidirect
28. #481×C73C4semidirect
31. #450×C73C4semidirect
33. #447×C73C4semidirect
253. C73#9264semidirect
256. #484Dic73semidirect
258. #481Dic73semidirect
261. #450Dic73semidirect
263. #447Dic73semidirect

Maximal subgroups

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

subgroupindex
#226189872
#226084352
C2×#179883212
#226082912
#226078192
#226077732
C73×#31882
#926473