Finite Group Databasebeta

← #33069383

Subgroups — #33069383

Order 18688.

584 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621268C212yesmaximal
2#22620517C211yesmaximal, characteristic, complemented
3#22609485C212yesmaximal, complemented
4#22609478C212yesmaximal, complemented
5#22609325C211yesmaximal, characteristic, complemented
6#22609320C211yesmaximal, characteristic, complemented
7#22609301C211yesmaximal, characteristic, complemented
8#22609296C211yesmaximal, characteristic, complemented
9C2×#17988340C211yesmaximal, characteristic, complemented
10#22608632C212yesmaximal, complemented
11C73×#2966C211yesmaximal, characteristic
12#17989144C2212yes—
13#17988992C2211yescharacteristic
14#17988967C2211yescharacteristic
15#17988938C2212yes—
16#17988891—22no—
17#17988872C2212yes—
18#17988867C2212yes—
19#17988828—21no—
20#17988828—21no—
21#17988826—22no—
22#17988814—22no—
23#17988806—22no—
24#17988797C2212yes—
25#17988792C2212yes—
26C4×Dic292—21no—
27C4×Dic292—21no—
28C2×C4⋊#10596831C2212yes—
29#17988340C2212yes—
30#17988340C2211yescharacteristic
31#17988340C2211yescharacteristic
32#17988334C2212yes—
33#17988334C2212yes—
34#17988332C2212yes—
35#17988332C2212yes—
36#529×C73C2211yescharacteristic
37#446×C73C2211yescharacteristic
38C8×Q8×C73C2211yescharacteristic
39#436×C73C2211yescharacteristic
40#435×C73—21no—
41C8×D4×C73—21no—
42C8×D4×C73C2211yescharacteristic
43#433×C73—21no—
44#432×C73C2211yescharacteristic
45#405×C73—21no—
46C2×C4×C8×C73C2211yescharacteristic
47C2×Dic292—21no—
48C2×Dic292—21no—
49C2×#11924018—22no—
50C2×#11924016—22no—
51C22×#10596831—22no—
52#14556591C2312yes—
53#14556591C2312yes—
54#14556583—22no—
55#14556570—22no—
56#14556570—22no—
57C4×Dic146—22no—
58C4×Dic146—22no—
59#14556550—22no—
60#14556525—21no—
61#14556525—21no—
62#14556524—22no—
63#14556522—22no—
64C4⋊#10596831—22no—
65C4⋊#10596831C2312yes—
66C4⋊#10596831C2312yes—
67#14556516—22no—
68#14556516—22no—
69#14556515—22no—
70#14556514—22no—
71#14556514—22no—
72#14556512—22no—
73C2×M16×C73—21no—
74C22×C8×C73—21no—
75C22×C8×C73D411yescharacteristic
76C22×C8×C73D411yescharacteristic
77#126×C73—21no—
78#125×C73D411yescharacteristic
79#124×C73D411yescharacteristic
80#123×C73—21no—
81C22⋊Q8×C73D411yescharacteristic
82#121×C73D411yescharacteristic
83C4×Q8×C73C2311yescharacteristic
84C4×D4×C73—21no—
85C4×D4×C73C2311yescharacteristic
86#117×C73—21no—
87C2×C42×C73C2311yescharacteristic
88C4⋊C8×C73—21no—
89C4⋊C8×C73—21no—
90C4⋊C8×C73C2311yescharacteristic
91C4⋊C8×C73C2311yescharacteristic
92C22⋊C8×C73—21no—
93C22⋊C8×C73—21no—
94C22⋊C8×C73—21no—
95#97×C73—21no—
96#97×C73—21no—
97C4×C8×C73—21no—
98C4×C8×C73D411yescharacteristic
99C4×C8×C73D411yescharacteristic
100C4×C8×C73C2311yescharacteristic
101C4×C8×C73C2311yescharacteristic
102C2×Dic146—22no—
103C2×Dic146—22no—
104Dic292—42no—
105Dic292—42no—
106#11924018—82no—
107#11924016—82no—
108C2×#10596831—22no—
109C2×#10596831—22no—
110C2×#10596831—82no—
111C4⋊Dic73—22no—
112C4⋊Dic73—22no—
113C4⋊Dic73—22no—
114C4⋊Dic73—22no—
115#11924007—42no—
116#11924007—42no—
117C22⋊Dic73—42no—
118C4×Dic73—42no—
119C2×Q8×C73C2×D411yescharacteristic
120C2×D4×C73—21no—
121C2×D4×C73C2×D411yescharacteristic
122C22×C4×C73—21no—
123C22×C4×C73C2×D411yescharacteristic
124C22×C4×C73C2×D411yescharacteristic
125C22×C4×C73#4111yescharacteristic
126M16×C73—42no—
127C2×C8×C73—22no—
128C2×C8×C73—21no—
129C2×C8×C73—21no—
130C2×C8×C73—41no—
131C2×C8×C73—42no—
132C2×C8×C73C2×D411yescharacteristic
133C2×C8×C73C2×D411yescharacteristic
134C2×C8×C73C2×D411yescharacteristic
135C2×C8×C73C2×D411yescharacteristic
136C4⋊C4×C73—21no—
137C4⋊C4×C73—21no—
138C4⋊C4×C73—21no—
139C4⋊C4×C73—21no—
140C4⋊C4×C73C2×D411yescharacteristic
141C4⋊C4×C73C2×D411yescharacteristic
142C22⋊C4×C73—21no—
143C22⋊C4×C73—21no—
144C22⋊C4×C73—21no—
145C22⋊C4×C73—21no—
146C22⋊C4×C73—21no—
147C42×C73—21no—
148C42×C73C2×D411yescharacteristic
149C42×C73C2×D411yescharacteristic
150C42×C73#4111yescharacteristic
151C42×C73C2411yescharacteristic
152C23×C73—21no—
153C23×C73#12111yescharacteristic
154Q8×C73—21no—
155Q8×C73—21no—
156D4×C73—21no—
157D4×C73—21no—
158D4×C73—42no—
159C2×C4×C73—22no—
160C2×C4×C73—21no—
161C2×C4×C73—21no—
162C2×C4×C73—21no—
163C2×C4×C73—21no—
164C2×C4×C73—21no—
165C2×C4×C73—21no—
166C2×C4×C73—21no—
167C2×C4×C73—41no—
168C2×C4×C73—41no—
169C2×C4×C73#12111yescharacteristic
170C2×C4×C73#12111yescharacteristic
171C2×C4×C73#12111yescharacteristic
172C2×C4×C73C22×D411yescharacteristic
173C2×C4×C73C22×D411yescharacteristic
174C2×C4×C73C2×#4111yescharacteristic
175C2×Dic73—42no—
176C2×Dic73—42no—
177Dic146—42no—
178Dic146—42no—
179Dic146—42no—
180Dic146—42no—
181C8×C73—21no—
182C8×C73—21no—
183C8×C73—21no—
184C8×C73—41no—
185C8×C73—41no—
186C8×C73#13512yes—
187#10596831—42no—
188#10596831—42no—
189C22×C73—21no—
190C22×C73—21no—
191C22×C73—41no—
192C22×C73—41no—
193C22×C73—41no—
194C22×C73#47312yes—
195C22×C73C2×#12111yescharacteristic
196C4×C73—21no—
197C4×C73—21no—
198C4×C73—21no—
199C4×C73—21no—
200C4×C73—41no—
201C4×C73—41no—
202C4×C73—41no—
203C4×C73C2×#13511yescharacteristic
204C4×C73#57511yescharacteristic
205C4×C73#57611yescharacteristic
206C4×C73#57811yescharacteristic
207Dic73—82no—
208Dic73—82no—
209#19223—731nomaximal, Sylow
210C2×C73—22no—
211C2×C73—81no—
212C2×C73#303511yescharacteristic
213C2×C73#306111yescharacteristic
214C2×C73C2×#47311yescharacteristic
215#3248—732no—
216#3170—731no—
217#2966D7311yescharacteristic
218#1700—732no—
219#1699—732no—
220#1674—731no—
221#1673—731no—
222#1670—731no—
223#1669—731no—
224C2×#335—731no—
225#1552—732no—
226C73#1922311yescharacteristic, complemented, Sylow
227#533—732no—
228#529C2×D7311yescharacteristic
229#500—731no—
230#494—731no—
231#487—732no—
232#479—1462no—
233#475—732no—
234#474—732no—
235#467—1461no—
236#467—1461no—
237#467—1462no—
238#465—1462no—
239#464—1462no—
240#462—732no—
241#461—732no—
242#446C2×D7311yescharacteristic
243C8×Q8C2×D7311yescharacteristic
244C4×Q16—1461no—
245C4×Q16—1461no—
246#436C2×D7311yescharacteristic
247#435—21no—
248C8×D4—21no—
249C8×D4C2×D7311yescharacteristic
250#433—21no—
251#432C2×D7311yescharacteristic
252C2×C4⋊C8—732no—
253#405—21no—
254C2×C4×C8C2×D7311yescharacteristic
255#335—731no—
256#335—731no—
257#335—732no—
258#332—732no—
259#332—732no—
260#332—732no—
261#332—732no—
262C2×Q16—1462no—
263C2×Q16—1461no—
264C2×Q16—1461no—
265C2×SD16—1462no—
266C2×M16—21no—
267C22×C8—21no—
268C22×C8—1462no—
269C22×C8C22⋊D7311yescharacteristic
270C22×C8C22⋊D7311yescharacteristic
271C4⋊Q8—732no—
272C4⋊Q8—732no—
273#126—21no—
274#125—1462no—
275#125D29211yescharacteristic
276#124D29211yescharacteristic
277#123—21no—
278C22⋊Q8—1462no—
279C22⋊Q8—1462no—
280C22⋊Q8D29211yescharacteristic
281#121D29211yescharacteristic
282C4×Q8—1462no—
283C4×Q8—1462no—
284C4×Q8C22×D7311yescharacteristic
285C4×D4—21no—
286C4×D4C22×D7311yescharacteristic
287#117—21no—
288#117—1462no—
289C2×C42C22×D7311yescharacteristic
290#107—1461no—
291#107—1461no—
292#107—1462no—
293#106—1462no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—732no—
297C4⋊C8—732no—
298C4⋊C8—1462no—
299C4⋊C8C22×D7311yescharacteristic
300C4⋊C8C22×D7311yescharacteristic
301#103—1462no—
302#103—1462no—
303#103—1462no—
304#103—1462no—
305#103—1462no—
306#102—1462no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310#97—21no—
311#97—21no—
312C4×C8—21no—
313C4×C8C22⋊D7311yescharacteristic
314C4×C8C22⋊D7311yescharacteristic
315C4×C8C22×D7311yescharacteristic
316C4×C8C22×D7311yescharacteristic
317C2×Q8—1462no—
318C2×Q8—1462no—
319C2×Q8C2×D29211yescharacteristic
320C2×D4—21no—
321C2×D4C2×D29211yescharacteristic
322C22×C4—21no—
323C22×C4C2×C22⋊D7311yescharacteristic
324C22×C4C2×D29211yescharacteristic
325C22×C4#1192402811yescharacteristic
326Q16—2922no—
327Q16—2922no—
328Q16—5842no—
329SD16—5842no—
330M16—42no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—5842no—
337C2×C8—1462no—
338C2×C8—1462no—
339C2×C8C2×C22⋊D7311yescharacteristic
340C2×C8C2×C22⋊D7311yescharacteristic
341C2×C8C2×C22⋊D7311yescharacteristic
342C2×C8C2×C22⋊D7311yescharacteristic
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—2922no—
348C4⋊C4—2922no—
349C4⋊C4—1462no—
350C4⋊C4—1462no—
351C4⋊C4—1462no—
352C4⋊C4—1462no—
353C4⋊C4C2×D29211yescharacteristic
354C4⋊C4C2×D29211yescharacteristic
355C22⋊C4—21no—
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—21no—
360C22⋊C4—2922no—
361C42—21no—
362C42—2922no—
363C42C2×C22⋊D7311yescharacteristic
364C42C2×D29211yescharacteristic
365C42#1192402811yescharacteristic
366C42C23×D7311yescharacteristic
367C23—21no—
368C23#1455656511yescharacteristic
369Q8—21no—
370Q8—21no—
371Q8—2922no—
372Q8—2922no—
373Q8—2922no—
374Q8—2922no—
375D4—21no—
376D4—21no—
377D4—42no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—2922no—
389C2×C4—2922no—
390C2×C4#1455656511yescharacteristic
391C2×C4#1455656511yescharacteristic
392C2×C4#1455656511yescharacteristic
393C2×C4C22×C22⋊D7311yescharacteristic
394C2×C4C22×D29211yescharacteristic
395C2×C4C2×#1192402811yescharacteristic
396C8—21no—
397C8—21no—
398C8—21no—
399C8—41no—
400C8—41no—
401C8—2922no—
402C8—2922no—
403C8#1455661112yes—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22#1798886212yes—
410C22C2×#1455656511yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—5842no—
419C4—5842no—
420C4C2×#1455661111yescharacteristic
421C4#1798938511yescharacteristic
422C4#1798939211yescharacteristic
423C4#1798940411yescharacteristic
424C2—22no—
425C2—81no—
426C2#2261938611yescharacteristic
427C2#2261960111yescharacteristic
428C2C2×#1798886211yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#2966C2
socleC22×C73C2×#121
prime radical#121×C73D4
hypercenter#2966D73
hypocenterC73#19223
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
2. #22620517C2semidirect
3. #22609485C2semidirect
4. #22609478C2semidirect
5. #22609325C2semidirect
6. #22609320C2semidirect
7. #22609301C2semidirect
8. #22609296C2semidirect
9. C2×#17988340C2semidirect
10. #22608632C2semidirect
226. C73#19223semidirect

Maximal subgroups

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

subgroupindex
#226212682
#226205172
#226094852
#226094782
#226093252
#226093202
#226093012
#226092962
C2×#179883402
#226086322
C73×#29662
#1922373