Finite Group Databasebeta

← #33114532

Subgroups — #33114532

Order 18688.

994 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×#14556526C211yesmaximal, characteristic, complemented
2C2×#17988614C211yesmaximal, characteristic, complemented
3#22618448C211yesmaximal, characteristic
4C2×#17988581C211yesmaximal, characteristic
5C2×#17988569C211yesmaximal, characteristic
6#22611295C214yesmaximal, complemented
7#22611288C214yesmaximal, complemented
8#22609670C211yesmaximal, characteristic, complemented
9C73×#2911C211yesmaximal, characteristic, complemented
10C22×#11924028C2211yescharacteristic
11C22×#11924012C2211yescharacteristic
12C22×#11924012C2211yescharacteristic
13C23×#10596831C2211yescharacteristic
14C2×#14556526—24no—
15C2×#14556526C2214yes—
16#17988614—22no—
17#17988614C2214yes—
18C2×C4⋊#10596831—22no—
19#17988595C2212yes—
20#17988591C2212yes—
21#17988586—22no—
22#17988581—22no—
23#17988581—22no—
24#17988581—22no—
25C2×#14556514C2212yes—
26C2×#14556511C2212yes—
27#17988569—24no—
28C2×#14556509C2212yes—
29C2×#14556507C2212yes—
30#17988356C2214yes—
31C22×M16×C73C2211yescharacteristic
32C2×#117×C73C2211yescharacteristic
33C2×#117×C73C2211yescharacteristic
34#428×C73—22no—
35#428×C73C2214yes—
36#427×C73—22no—
37C2×#11924028—22no—
38C2×#11924028—21no—
39C2×#11924028—21no—
40C2×#11924028—21no—
41C2×#11924028—21no—
42C2×#11924028—22no—
43C2×#11924028—22no—
44C2×#11924028Q814yes—
45C22×Dic146C2311yescharacteristic
46C22×D292C2311yescharacteristic
47C22×C22⋊D73—21no—
48C22×C4×D73—21no—
49C2×#11924012—21no—
50C2×#11924012—21no—
51C2×#11924012—22no—
52C2×#11924012—22no—
53C2×#11924012—22no—
54C2×#11924012—22no—
55C2×#11924012C2312yes—
56C2×#11924012C2312yes—
57C22×#10596831—22no—
58C22×#10596831—21no—
59C22×#10596831—21no—
60C22×#10596831—22no—
61C22×#10596831D412yes—
62C22×#10596831D412yes—
63C22×#10596831C2312yes—
64#14556526—28no—
65C4⋊#10596831—28no—
66#14556514—24no—
67#14556514—24no—
68#14556511—24no—
69#14556511—24no—
70#14556509—24no—
71#14556507—24no—
72C23×C4×C73C2311yescharacteristic
73C2×M16×C73—21no—
74C2×M16×C73—22no—
75C2×M16×C73D414yes—
76C2×M16×C73C2312yes—
77C22×C8×C73—21no—
78#117×C73—21no—
79#117×C73—21no—
80#117×C73—21no—
81#117×C73—21no—
82#117×C73—21no—
83#117×C73—21no—
84#117×C73D412yes—
85#117×C73C2312yes—
86C2×C4⋊C4×C73D412yes—
87C2×C4⋊C4×C73C2312yes—
88C2×C22⋊C4×C73—21no—
89C2×C22⋊C4×C73—21no—
90C2×C42×C73—21no—
91C2×C42×C73—21no—
92#107×C73—24no—
93#106×C73—24no—
94C23×D73—21no—
95#11924028—24no—
96#11924028—24no—
97#11924028—24no—
98#11924028—24no—
99#11924028—42no—
100#11924028—42no—
101#11924028—42no—
102#11924028—42no—
103C2×Dic146—21no—
104C2×Dic146—21no—
105C2×Dic146—21no—
106C2×Dic146—21no—
107C2×Dic146—21no—
108C2×Dic146C2×Q812yes—
109C2×D292—21no—
110C2×D292—21no—
111C2×D292—21no—
112C2×D292—21no—
113C2×D292—21no—
114C2×D292C2×Q812yes—
115C2×C22⋊D73—24no—
116C2×C22⋊D73—41no—
117C2×C22⋊D73—41no—
118C2×C22⋊D73—42no—
119C22×Dic73—21no—
120C2×C4×D73—24no—
121C2×C4×D73—41no—
122C2×C4×D73—41no—
123C2×C4×D73—42no—
124#11924012—22no—
125#11924012—22no—
126#11924012—22no—
127#11924012—22no—
128#11924012—42no—
129#11924012—42no—
130C2×#10596831—24no—
131C2×#10596831—24no—
132C2×#10596831—22no—
133C2×#10596831—22no—
134C2×#10596831—22no—
135C2×#10596831—22no—
136C2×#10596831—42no—
137C2×#10596831—42no—
138C2×#10596831C2×D414yes—
139C24×C73#4111yescharacteristic
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73C2×D411yescharacteristic
143C22×C4×C73C2×D411yescharacteristic
144C22×C4×C73C2×D411yescharacteristic
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2×D412yes—
147C22×C4×C73C2×Q812yes—
148C22×C4×C73#4111yescharacteristic
149C22×C4×C73#4112yes—
150C22×C4×C73#4112yes—
151C22×C4×C73C2411yescharacteristic
152M16×C73—24no—
153M16×C73—22no—
154M16×C73C2×D414yes—
155C2×C8×C73—24no—
156C2×C8×C73—22no—
157C4⋊C4×C73—22no—
158C4⋊C4×C73—22no—
159C4⋊C4×C73—22no—
160C4⋊C4×C73C2×D414yes—
161C22⋊C4×C73—22no—
162C22⋊C4×C73—22no—
163C22⋊C4×C73—41no—
164C22⋊C4×C73—41no—
165C42×C73—22no—
166C42×C73—22no—
167C42×C73—41no—
168C42×C73—41no—
169C23×C73—42no—
170C23×C73#12111yescharacteristic
171C23×C73#12111yescharacteristic
172C23×C73C22⋊Q812yes—
173C23×C73#12612yes—
174C23×C73C2×#4111yescharacteristic
175C22×D73—22no—
176C22×D73—41no—
177C22×D73—41no—
178C22×D73—82no—
179C2×C4×C73—24no—
180C2×C4×C73—44no—
181C2×C4×C73—42no—
182C2×C4×C73—82no—
183C2×C4×C73#12111yescharacteristic
184C2×C4×C73#12111yescharacteristic
185C2×C4×C73#12112yes—
186C2×C4×C73#12112yes—
187C2×C4×C73#12112yes—
188C2×C4×C73#12112yes—
189C2×C4×C73C22⋊Q812yes—
190C2×C4×C73C22⋊Q812yes—
191C2×C4×C73C22⋊Q812yes—
192C2×C4×C73C22⋊Q812yes—
193C2×C4×C73C22⋊Q812yes—
194C2×C4×C73#12612yes—
195C2×C4×C73C22×D411yescharacteristic
196C2×C4×C73C22×D411yescharacteristic
197C2×C4×C73C22×D411yescharacteristic
198C2×C4×C73C22×Q811yescharacteristic
199C2×C4×C73C2×#4111yescharacteristic
200C2×C4×C73C2×#4111yescharacteristic
201C22⋊D73—44no—
202C22⋊D73—44no—
203C2×Dic73—22no—
204C2×Dic73—41no—
205C2×Dic73—41no—
206D292—22no—
207D292—22no—
208D292—41no—
209D292—41no—
210C4×D73—44no—
211C4×D73—44no—
212Dic146—22no—
213Dic146—22no—
214Dic146—41no—
215Dic146—41no—
216C8×C73—24no—
217#10596831—24no—
218#10596831—42no—
219#10596831—42no—
220C22×C73—22no—
221C22×C73—22no—
222C22×C73—22no—
223C22×C73—42no—
224C22×C73—42no—
225C22×C73#39612yes—
226C22×C73#39612yes—
227C22×C73C2×#12111yescharacteristic
228C22×C73C2×C22⋊Q811yescharacteristic
229C22×C73C2×#12611yescharacteristic
230C2×D73—42no—
231C2×D73—82no—
232C2×D73—82no—
233C2×D73—82no—
234C4×C73—44no—
235C4×C73—82no—
236C4×C73#39612yes—
237C4×C73#39612yes—
238C4×C73C2×#12111yescharacteristic
239C4×C73C2×#12111yescharacteristic
240C4×C73C2×C22⋊Q811yescharacteristic
241C4×C73C2×C22⋊Q811yescharacteristic
242Dic73—42no—
243#23601—731nomaximal, Sylow
244C2×C73—21no—
245C2×C73—21no—
246C2×C73—22no—
247C2×C73—22no—
248C2×C73C2×#39611yescharacteristic
249C2×C73#306712yes—
250D73—82no—
251C22×#108—731no—
252#2911D7311yescharacteristic, complemented
253C2×#424—731no—
254#2896—731no—
255C2×#416—731no—
256C2×#413—731no—
257#2064—734no—
258#2063—734no—
259#1742—731no—
260C73#2360111yescharacteristic, complemented, Sylow
261C22×#41—731no—
262C22×M16—731no—
263C22×M16—731no—
264C22×M16C2×D7311yescharacteristic
265C23×C8—731no—
266C2×#117C2×D7311yescharacteristic
267C2×#117C2×D7311yescharacteristic
268C2×#108—734no—
269C2×#108—1464no—
270#428—22no—
271#428C2×D7314yes—
272#427—22no—
273#424—734no—
274#424—1462no—
275C2×C4⋊C8—1462no—
276#419—732no—
277#418—732no—
278#417—1462no—
279#416—1462no—
280#416—1462no—
281#416—1462no—
282C2×#103—732no—
283C2×#102—732no—
284#413—1464no—
285C2×#101—732no—
286C2×#100—732no—
287#343—734no—
288C2×#41—734no—
289C2×#41—1461no—
290C2×#41—1461no—
291C2×#41—1461no—
292C2×#41—1461no—
293C2×#41—1462no—
294C2×#41—1462no—
295C2×#41—1462no—
296C22×Q8—731no—
297C22×D4—731no—
298C22×D4—1461no—
299C23×C4—1461no—
300C23×C4C22×D7311yescharacteristic
301C2×M16—21no—
302C2×M16—22no—
303C2×M16—732no—
304C2×M16—732no—
305C2×M16—1462no—
306C2×M16—1462no—
307C2×M16—1462no—
308C2×M16—1462no—
309C2×M16—1461no—
310C2×M16—1461no—
311C2×M16D29214yes—
312C2×M16C22×D7312yes—
313C22×C8—21no—
314C22×C8—732no—
315C22×C8—732no—
316C22×C8—732no—
317C22×C8—1461no—
318C22×C8—1461no—
319C22×C8—1462no—
320C22×C8—1462no—
321#117—21no—
322#117—21no—
323#117—21no—
324#117—21no—
325#117—21no—
326#117—21no—
327#117C22⋊D7312yes—
328#117C22×D7312yes—
329C2×C4⋊C4C22⋊D7312yes—
330C2×C4⋊C4C22×D7312yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C42—21no—
334C2×C42—21no—
335#108—1468no—
336#107—24no—
337#106—24no—
338C4⋊C8—1468no—
339#103—1464no—
340#103—1464no—
341#102—1464no—
342#102—1464no—
343#101—1464no—
344#100—1464no—
345C24—1461no—
346C24#1192402711yescharacteristic
347#41—2922no—
348#41—2922no—
349#41—2922no—
350#41—2922no—
351#41—1464no—
352#41—1464no—
353#41—1464no—
354#41—1464no—
355C2×Q8—732no—
356C2×Q8—1461no—
357C2×Q8—1461no—
358C2×Q8—1461no—
359C2×Q8—1461no—
360C2×Q8—1461no—
361C2×D4—2921no—
362C2×D4—2921no—
363C2×D4—2922no—
364C2×D4—732no—
365C2×D4—1464no—
366C2×D4—1461no—
367C2×D4—1461no—
368C2×D4—1461no—
369C2×D4—1461no—
370C2×D4—1461no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—2921no—
374C22×C4—2921no—
375C22×C4—2922no—
376C22×C4—1461no—
377C22×C4—1464no—
378C22×C4D4×D7311yescharacteristic
379C22×C4D4×D7311yescharacteristic
380C22×C4C2×C22⋊D7311yescharacteristic
381C22×C4C2×C22⋊D7311yescharacteristic
382C22×C4C2×D29212yes—
383C22×C4Q8×D7312yes—
384C22×C4#1192402711yescharacteristic
385C22×C4#1192402812yes—
386C22×C4#1192402912yes—
387C22×C4C23×D7311yescharacteristic
388M16—24no—
389M16—22no—
390M16—2922no—
391M16—2922no—
392M16—1462no—
393M16—1462no—
394M16—1462no—
395M16—1462no—
396M16C2×D29214yes—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—2922no—
400C2×C8—2922no—
401C2×C8—734no—
402C2×C8—1462no—
403C2×C8—1462no—
404C2×C8—1464no—
405C2×C8—1464no—
406C2×C8—1462no—
407C2×C8—1462no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4C2×C22⋊D7314yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C42—22no—
417C42—22no—
418C42—41no—
419C42—41no—
420C23—42no—
421C23—2921no—
422C23—2921no—
423C23—5842no—
424C23—1462no—
425C23#1455656311yescharacteristic
426C23#1455656311yescharacteristic
427C23#1455657112yes—
428C23#1455658612yes—
429C23C2×#1192402711yescharacteristic
430Q8—2921no—
431Q8—2921no—
432Q8—1462no—
433Q8—1462no—
434D4—2924no—
435D4—2924no—
436D4—2921no—
437D4—2921no—
438D4—1462no—
439D4—1462no—
440C2×C4—24no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—82no—
444C2×C4—2921no—
445C2×C4—2921no—
446C2×C4—2924no—
447C2×C4—2924no—
448C2×C4—1462no—
449C2×C4#1455656311yescharacteristic
450C2×C4#1455656311yescharacteristic
451C2×C4#1455656512yes—
452C2×C4#1455656512yes—
453C2×C4#1455656612yes—
454C2×C4#1455656612yes—
455C2×C4#1455656712yes—
456C2×C4#1455656712yes—
457C2×C4#1455656912yes—
458C2×C4#1455656912yes—
459C2×C4#1455657112yes—
460C2×C4#1455658612yes—
461C2×C4C2×D4×D7311yescharacteristic
462C2×C4C22×C22⋊D7311yescharacteristic
463C2×C4C22×D29211yescharacteristic
464C2×C4C2×Q8×D7311yescharacteristic
465C2×C4C2×#1192402811yescharacteristic
466C2×C4C2×#1192402911yescharacteristic
467C8—24no—
468C8—2922no—
469C8—2922no—
470C8—1464no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—42no—
475C22—42no—
476C22—2922no—
477C22—5842no—
478C22—5842no—
479C22—5842no—
480C22#1798850912yes—
481C22#1798850912yes—
482C22C2×#1455656311yescharacteristic
483C22C2×#1455657111yescharacteristic
484C22C2×#1455658611yescharacteristic
485C4—44no—
486C4—82no—
487C4—2922no—
488C4#1798850912yes—
489C4#1798850912yes—
490C4C2×#1455656511yescharacteristic
491C4C2×#1455656611yescharacteristic
492C4C2×#1455656711yescharacteristic
493C4C2×#1455656911yescharacteristic
494C2—21no—
495C2—21no—
496C2—22no—
497C2—22no—
498C2—5842no—
499C2C2×#1798850911yescharacteristic
500C2#2261964812yes—
subgroupG/H
centerC2×C4#14556563
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2911C2
socleC22×C73C2×#121
prime radicalC22×#11924028C22
hypercenter#2911D73
hypocenterC73#23601
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2911

Complements

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

NH ≅ G/Ntype
1. C22×#14556526C2semidirect
2. C2×#17988614C2semidirect
6. #22611295C2semidirect
7. #22611288C2semidirect
8. #22609670C2semidirect
9. C73×#2911C2semidirect
252. #2911D73semidirect
260. C73#23601semidirect

Maximal subgroups

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

subgroupindex
C22×#145565262
C2×#179886142
#226184482
C2×#179885812
C2×#179885692
#226112952
#226112882
#226096702
C73×#29112
#2360173