Finite Group Databasebeta

← #33069954

Subgroups — #33069954

Order 18688.

754 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988933C211yesmaximal, characteristic, complemented
2#22619418C211yesmaximal, characteristic, complemented
3#41×#10596831C211yesmaximal, characteristic, complemented
4#22609373C214yesmaximal, complemented
5#22609366C214yesmaximal, complemented
6#22608689C211yesmaximal, characteristic, complemented
7#22608604C211yesmaximal, characteristic, complemented
8#22608599C211yesmaximal, characteristic, complemented
9C73×#3038C211yesmaximal, characteristic
10C2×#14556621—21no—
11C2×#14556591C2211yescharacteristic
12#17988933—21no—
13#17988933—21no—
14#17988933C2214yes—
15#17988797—21no—
16#17988797C2212yescomplemented
17#17988792—21no—
18#17988792C2212yescomplemented
19Q8×#10596831C2212yescomplemented
20D4×#10596831—22no—
21D4×#10596831C2212yescomplemented
22#17988640—21no—
23#17988640C2211yescharacteristic
24#17988611C2211yescharacteristic
25#17988595—21no—
26C2×#14556512—22no—
27C2×C4×#10596831—21no—
28C2×C4×#10596831C2211yescharacteristic
29#17988339C2214yescomplemented
30#17988334C2214yescomplemented
31#17988331C2214yes—
32C2×AGL(1,ℤ/8)×C73—21no—
33C2×#127×C73C2211yescharacteristic
34C4×#41×C73C2211yescharacteristic
35#460×C73—21no—
36#460×C73C2212yes—
37#459×C73—21no—
38#459×C73C2212yes—
39#423×C73C2211yescharacteristic
40#418×C73—21no—
41C22×Dic146—21no—
42#14556621—44no—
43C2×#11924018—22no—
44C2×#11924016—22no—
45C2×#11924012—21no—
46C22×#10596831—22no—
47C22×#10596831D412yes—
48#14556591—21no—
49#14556591—21no—
50#14556591—21no—
51#14556591C2312yes—
52C2×C4⋊Dic73—21no—
53C2×C4⋊Dic73—21no—
54C4⋊#10596831—22no—
55C4⋊#10596831—21no—
56C4⋊#10596831C2312yes—
57C4⋊#10596831C2312yes—
58#14556514—22no—
59#14556514—22no—
60#14556512—24no—
61#14556512—24no—
62C22⋊#10596831—22no—
63C22⋊#10596831—21no—
64C4×#10596831—22no—
65C4×#10596831—21no—
66C4×#10596831D411yescharacteristic
67C4×#10596831D411yescharacteristic
68C4×#10596831C2312yes—
69C2×#41×C73D411yescharacteristic
70C22×D4×C73—21no—
71C22×D4×C73—21no—
72C22×D4×C73—21no—
73AGL(1,ℤ/8)×C73—44no—
74C2×SD16×C73—22no—
75C2×D8×C73—22no—
76C2×M16×C73—21no—
77#127×C73—21no—
78#127×C73—21no—
79#127×C73—21no—
80#127×C73C2312yes—
81C4×Q8×C73D411yescharacteristic
82C4×Q8×C73C2311yescharacteristic
83C4×D4×C73—21no—
84C4×D4×C73—21no—
85C4×D4×C73D411yescharacteristic
86C4×D4×C73C2311yescharacteristic
87#117×C73—21no—
88#117×C73D411yescharacteristic
89C2×C42×C73—21no—
90C2×C42×C73C2311yescharacteristic
91C4⋊C8×C73—21no—
92C4⋊C8×C73C2312yes—
93#102×C73—22no—
94#102×C73—22no—
95C2×Dic146—22no—
96C2×Dic146—42no—
97C2×Dic146—42no—
98C2×Dic146—41no—
99C22×Dic73—41no—
100#11924018—44no—
101#11924016—44no—
102#11924012—42no—
103C2×#10596831—22no—
104C2×#10596831—22no—
105C2×#10596831—42no—
106C2×#10596831—44no—
107C2×#10596831C2×D412yes—
108C2×#10596831C2×D412yes—
109C4⋊Dic73—22no—
110C4⋊Dic73—22no—
111C4⋊Dic73—41no—
112C4⋊Dic73—41no—
113C24×C73—41no—
114#41×C73—22no—
115#41×C73D814yes—
116C2×Q8×C73C2×D411yescharacteristic
117C2×D4×C73—21no—
118C2×D4×C73—22no—
119C2×D4×C73—22no—
120C2×D4×C73—22no—
121C2×D4×C73—44no—
122C2×D4×C73—41no—
123C2×D4×C73—41no—
124C2×D4×C73—41no—
125C2×D4×C73—42no—
126C2×D4×C73—42no—
127C2×D4×C73—82no—
128C2×D4×C73C2×D411yescharacteristic
129C22×C4×C73—21no—
130C22×C4×C73—21no—
131C22×C4×C73—21no—
132C22×C4×C73C2×D411yescharacteristic
133C22×C4×C73C2×D411yescharacteristic
134C22×C4×C73#4111yescharacteristic
135SD16×C73—44no—
136D8×C73—44no—
137M16×C73—42no—
138C2×C8×C73—22no—
139C4⋊C4×C73—21no—
140C4⋊C4×C73—21no—
141C4⋊C4×C73C2×D411yescharacteristic
142C4⋊C4×C73C2×D411yescharacteristic
143C22⋊C4×C73—21no—
144C22⋊C4×C73—21no—
145C22⋊C4×C73—21no—
146C42×C73—21no—
147C42×C73—21no—
148C42×C73—21no—
149C42×C73C2×D411yescharacteristic
150C42×C73C2×D411yescharacteristic
151C42×C73#4111yescharacteristic
152C42×C73C2411yescharacteristic
153C23×C73—21no—
154C23×C73—42no—
155C23×C73—82no—
156C23×C73—82no—
157C23×C73—82no—
158C23×C73#12111yescharacteristic
159Q8×C73—21no—
160Q8×C73C2×D812yes—
161D4×C73—24no—
162D4×C73—21no—
163D4×C73—42no—
164D4×C73—42no—
165D4×C73—44no—
166D4×C73—82no—
167D4×C73—82no—
168D4×C73—82no—
169D4×C73—82no—
170D4×C73C2×D812yes—
171C2×C4×C73—21no—
172C2×C4×C73—22no—
173C2×C4×C73—24no—
174C2×C4×C73—22no—
175C2×C4×C73—21no—
176C2×C4×C73—21no—
177C2×C4×C73—42no—
178C2×C4×C73—42no—
179C2×C4×C73#12111yescharacteristic
180C2×C4×C73#12111yescharacteristic
181C2×C4×C73#12111yescharacteristic
182C2×C4×C73C2×D812yes—
183C2×C4×C73#13512yes—
184C2×C4×C73C22×D411yescharacteristic
185C2×C4×C73C22×D411yescharacteristic
186C2×C4×C73C2×#4111yescharacteristic
187C2×Dic73—42no—
188C2×Dic73—82no—
189Dic146—42no—
190Dic146—42no—
191Dic146—82no—
192C8×C73—42no—
193#10596831—22no—
194#10596831—22no—
195#10596831—42no—
196#10596831—42no—
197C22×C73—21no—
198C22×C73—21no—
199C22×C73—22no—
200C22×C73—41no—
201C22×C73—41no—
202C22×C73—82no—
203C22×C73—82no—
204C22×C73—82no—
205C22×C73—164no—
206C22×C73#46612yes—
207C22×C73C2×#12111yescharacteristic
208C4×C73—21no—
209C4×C73—22no—
210C4×C73—21no—
211C4×C73—41no—
212C4×C73—41no—
213C4×C73#46612yes—
214C4×C73#47112yes—
215C4×C73C22×D811yescharacteristic
216C4×C73C2×#13511yescharacteristic
217C4×C73#57512yes—
218Dic73—82no—
219#19276—731nomaximal, Sylow
220C2×C73—22no—
221C2×C73—42no—
222C2×C73—162no—
223C2×C73#304011yescharacteristic
224C2×C73C2×#46611yescharacteristic
225C2×C73C2×#47111yescharacteristic
226C2×#486—731no—
227#3039—731no—
228#3038D7311yescharacteristic
229C8×#41—731no—
230#1682—734no—
231#1681—734no—
232#1567—731no—
233#1546—731no—
234#1545—731no—
235C73#1927611yescharacteristic, complemented, Sylow
236C2×#137—1461no—
237C2×AGL(1,ℤ/8)—21no—
238C2×C4⋊Q8—731no—
239C2×#127C2×D7311yescharacteristic
240C4×#41C2×D7311yescharacteristic
241#486—734no—
242#486—1461no—
243#486—1461no—
244#462—732no—
245#462—1461no—
246#461—732no—
247#461—1461no—
248#460—21no—
249#460C2×D7312yes—
250#459—21no—
251#459C2×D7312yes—
252C8×Q8—732no—
253C8×D4—732no—
254C8×D4—1462no—
255#431—731no—
256#431—1461no—
257#423—731no—
258#423C2×D7311yescharacteristic
259#419—1461no—
260#418—21no—
261C2×#102—1462no—
262C2×C4×C8—731no—
263C2×C4×C8—1461no—
264#335—734no—
265#332—734no—
266#331—734no—
267C2×#41C22⋊D7311yescharacteristic
268C22×Q8—1461no—
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—21no—
272#137—2924no—
273AGL(1,ℤ/8)—44no—
274C2×Q16—1462no—
275C2×SD16—22no—
276C2×SD16—1462no—
277C2×D8—22no—
278C2×M16—21no—
279C2×M16—1461no—
280C22×C8—732no—
281C22×C8—1462no—
282C4⋊Q8—732no—
283C4⋊Q8—1461no—
284C4⋊Q8—1461no—
285C4⋊Q8—1461no—
286#127—21no—
287#127—21no—
288#127—21no—
289#127C22×D7312yes—
290C4×Q8C22⋊D7311yescharacteristic
291C4×Q8C22×D7311yescharacteristic
292C4×D4—21no—
293C4×D4—21no—
294C4×D4C22⋊D7311yescharacteristic
295C4×D4C22×D7311yescharacteristic
296#117—21no—
297#117C22⋊D7311yescharacteristic
298C2×C4⋊C4—1461no—
299C2×C4⋊C4—1461no—
300C2×C42—21no—
301C2×C42C22×D7311yescharacteristic
302C4⋊C8—21no—
303C4⋊C8—732no—
304C4⋊C8—732no—
305C4⋊C8—1462no—
306C4⋊C8—1461no—
307C4⋊C8C22×D7312yes—
308#103—1462no—
309#103—1462no—
310#102—22no—
311#102—22no—
312#102—1464no—
313#102—1464no—
314C22⋊C8—1462no—
315C22⋊C8—1461no—
316C4×C8—731no—
317C4×C8—731no—
318C4×C8—732no—
319C4×C8—1462no—
320C4×C8—1461no—
321C24—41no—
322#41—22no—
323#41#1192401314yes—
324C2×Q8—2921no—
325C2×Q8—2922no—
326C2×Q8—2922no—
327C2×Q8—1462no—
328C2×Q8C2×C22⋊D7311yescharacteristic
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—44no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—82no—
340C2×D4C2×C22⋊D7311yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—2921no—
345C22×C4D4×D7311yescharacteristic
346C22×C4C2×C22⋊D7311yescharacteristic
347C22×C4#1192402711yescharacteristic
348Q16—2924no—
349SD16—44no—
350SD16—2924no—
351D8—44no—
352M16—42no—
353M16—2922no—
354C2×C8—22no—
355C2×C8—2924no—
356C2×C8—2922no—
357C2×C8—732no—
358C2×C8—732no—
359C2×C8—1462no—
360C2×C8—1462no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—2921no—
364C4⋊C4—2921no—
365C4⋊C4—1462no—
366C4⋊C4—1462no—
367C4⋊C4C2×C22⋊D7311yescharacteristic
368C4⋊C4C2×C22⋊D7311yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42D4×D7311yescharacteristic
376C42C2×C22⋊D7311yescharacteristic
377C42#1192402711yescharacteristic
378C42C23×D7311yescharacteristic
379C23—21no—
380C23—42no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23#1455656311yescharacteristic
385Q8—21no—
386Q8—2922no—
387Q8—2922no—
388Q8—5842no—
389Q8C2×#1192401312yes—
390D4—24no—
391D4—21no—
392D4—44no—
393D4—42no—
394D4—42no—
395D4—82no—
396D4—82no—
397D4—82no—
398D4—82no—
399D4C2×#1192401312yes—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—2922no—
409C2×C4—5842no—
410C2×C4#1455656311yescharacteristic
411C2×C4#1455656311yescharacteristic
412C2×C4#1455656311yescharacteristic
413C2×C4C2×#1192401312yes—
414C2×C4#1455661112yes—
415C2×C4C2×D4×D7311yescharacteristic
416C2×C4C22×C22⋊D7311yescharacteristic
417C2×C4C2×#1192402711yescharacteristic
418C8—42no—
419C8—2922no—
420C8—2922no—
421C8—1462no—
422C8—1462no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—164no—
432C22#1798882112yes—
433C22C2×#1455656311yescharacteristic
434C4—21no—
435C4—22no—
436C4—21no—
437C4—41no—
438C4—41no—
439C4—5842no—
440C4#1798882112yes—
441C4#1798884812yes—
442C4C22×#1192401311yescharacteristic
443C4C2×#1455661111yescharacteristic
444C4#1798938312yes—
445C2—22no—
446C2—42no—
447C2—162no—
448C2#2261942311yescharacteristic
449C2C2×#1798882111yescharacteristic
450C2C2×#1798884811yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC73×#3038C2
socleC22×C73C2×#121
hypercenter#3038D73
hypocenterC73#19276
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. C2×#17988933C2semidirect
2. #22619418C2semidirect
3. #41×#10596831C2semidirect
4. #22609373C2semidirect
5. #22609366C2semidirect
6. #22608689C2semidirect
7. #22608604C2semidirect
8. #22608599C2semidirect
16. #17988797C22semidirect
18. #17988792C22semidirect
19. Q8×#10596831C22semidirect
21. D4×#10596831C22semidirect
29. #17988339C22semidirect
30. #17988334C22semidirect
235. C73#19276semidirect

Maximal subgroups

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

subgroupindex
C2×#179889332
#226194182
#41×#105968312
#226093732
#226093662
#226086892
#226086042
#226085992
C73×#30382
#1927673