Finite Group Databasebeta

← #33069442

Subgroups — #33069442

Order 18688.

584 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621268C212yesmaximal
2#22620517C211yesmaximal, characteristic, complemented
3#22609449C211yesmaximal, characteristic, complemented
4#22609444C211yesmaximal, characteristic, complemented
5#22609425C211yesmaximal, characteristic, complemented
6#22609420C211yesmaximal, characteristic, complemented
7#22609361C212yesmaximal, complemented
8#22609354C212yesmaximal, complemented
9C2×#17988338C211yesmaximal, 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#17988338C2211yescharacteristic
30#17988338C2211yescharacteristic
31#17988338C2212yes—
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—22no—
155D4×C73—22no—
156D4×C73—42no—
157C2×C4×C73—22no—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—22no—
161C2×C4×C73—21no—
162C2×C4×C73—21no—
163C2×C4×C73—21no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73#12111yescharacteristic
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73#12111yescharacteristic
169C2×C4×C73C22×D411yescharacteristic
170C2×C4×C73C22×D411yescharacteristic
171C2×C4×C73C2×#4111yescharacteristic
172C2×Dic73—42no—
173C2×Dic73—42no—
174Dic146—42no—
175Dic146—42no—
176Dic146—42no—
177Dic146—42no—
178C8×C73—21no—
179C8×C73—21no—
180C8×C73—21no—
181C8×C73—41no—
182C8×C73—41no—
183C8×C73#13512yes—
184#10596831—42no—
185#10596831—42no—
186C22×C73—21no—
187C22×C73—21no—
188C22×C73—41no—
189C22×C73—41no—
190C22×C73—41no—
191C22×C73#47312yes—
192C22×C73C2×#12111yescharacteristic
193C4×C73—21no—
194C4×C73—21no—
195C4×C73—21no—
196C4×C73—21no—
197C4×C73—41no—
198C4×C73—41no—
199C4×C73—41no—
200C4×C73C2×#13511yescharacteristic
201C4×C73#57511yescharacteristic
202C4×C73#57712yes—
203Dic73—82no—
204Dic73—82no—
205#19228—731nomaximal, Sylow
206C2×C73—22no—
207C2×C73—81no—
208C2×C73#303511yescharacteristic
209C2×C73#306011yescharacteristic
210C2×C73C2×#47311yescharacteristic
211#3248—732no—
212#3170—731no—
213#2966D7311yescharacteristic
214#1694—731no—
215#1693—731no—
216#1690—731no—
217#1689—731no—
218#1680—732no—
219#1679—732no—
220C2×#334—731no—
221#1552—732no—
222C73#1922811yescharacteristic, complemented, Sylow
223#533—732no—
224#529C2×D7311yescharacteristic
225#500—731no—
226#494—731no—
227#487—732no—
228#479—1462no—
229#475—732no—
230#474—732no—
231#467—1461no—
232#467—1461no—
233#467—1462no—
234#465—1462no—
235#464—1462no—
236#462—732no—
237#461—732no—
238#446C2×D7311yescharacteristic
239C8×Q8C2×D7311yescharacteristic
240C4×Q16—1461no—
241C4×Q16—1461no—
242#436C2×D7311yescharacteristic
243#435—21no—
244C8×D4—21no—
245C8×D4C2×D7311yescharacteristic
246#433—21no—
247#432C2×D7311yescharacteristic
248C2×C4⋊C8—732no—
249#405—21no—
250C2×C4×C8C2×D7311yescharacteristic
251#334—732no—
252#334—731no—
253#334—731no—
254#332—732no—
255#332—732no—
256#332—732no—
257#332—732no—
258C2×Q16—1462no—
259C2×Q16—1461no—
260C2×Q16—1461no—
261C2×SD16—1462no—
262C2×M16—21no—
263C22×C8—21no—
264C22×C8—1462no—
265C22×C8C22⋊D7311yescharacteristic
266C22×C8C22⋊D7311yescharacteristic
267C4⋊Q8—732no—
268C4⋊Q8—732no—
269#126—21no—
270#125—1462no—
271#125D29211yescharacteristic
272#124D29211yescharacteristic
273#123—21no—
274C22⋊Q8—1462no—
275C22⋊Q8—1462no—
276C22⋊Q8D29211yescharacteristic
277#121D29211yescharacteristic
278C4×Q8—1462no—
279C4×Q8—1462no—
280C4×Q8C22×D7311yescharacteristic
281C4×D4—21no—
282C4×D4C22×D7311yescharacteristic
283#117—21no—
284#117—1462no—
285C2×C42C22×D7311yescharacteristic
286#107—1461no—
287#107—1461no—
288#107—1462no—
289#106—1462no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—732no—
293C4⋊C8—732no—
294C4⋊C8—1462no—
295C4⋊C8C22×D7311yescharacteristic
296C4⋊C8C22×D7311yescharacteristic
297#103—1462no—
298#103—1462no—
299#103—1462no—
300#103—1462no—
301#103—1462no—
302#102—1462no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306#97—21no—
307#97—21no—
308C4×C8—21no—
309C4×C8C22⋊D7311yescharacteristic
310C4×C8C22⋊D7311yescharacteristic
311C4×C8C22×D7311yescharacteristic
312C4×C8C22×D7311yescharacteristic
313C2×Q8—1462no—
314C2×Q8—1462no—
315C2×Q8C2×D29211yescharacteristic
316C2×D4—21no—
317C2×D4C2×D29211yescharacteristic
318C22×C4—21no—
319C22×C4C2×C22⋊D7311yescharacteristic
320C22×C4C2×D29211yescharacteristic
321C22×C4#1192402811yescharacteristic
322Q16—2922no—
323Q16—2922no—
324Q16—5842no—
325SD16—5842no—
326M16—42no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—5842no—
333C2×C8—1462no—
334C2×C8—1462no—
335C2×C8C2×C22⋊D7311yescharacteristic
336C2×C8C2×C22⋊D7311yescharacteristic
337C2×C8C2×C22⋊D7311yescharacteristic
338C2×C8C2×C22⋊D7311yescharacteristic
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—2922no—
344C4⋊C4—2922no—
345C4⋊C4—1462no—
346C4⋊C4—1462no—
347C4⋊C4—1462no—
348C4⋊C4—1462no—
349C4⋊C4C2×D29211yescharacteristic
350C4⋊C4C2×D29211yescharacteristic
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—2922no—
357C42—21no—
358C42—2922no—
359C42C2×C22⋊D7311yescharacteristic
360C42C2×D29211yescharacteristic
361C42#1192402811yescharacteristic
362C42C23×D7311yescharacteristic
363C23—21no—
364C23#1455656511yescharacteristic
365Q8—22no—
366Q8—2922no—
367Q8—2922no—
368Q8—2922no—
369Q8—2922no—
370D4—22no—
371D4—42no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—2922no—
382C2×C4—2922no—
383C2×C4#1455656511yescharacteristic
384C2×C4#1455656511yescharacteristic
385C2×C4#1455656511yescharacteristic
386C2×C4C22×C22⋊D7311yescharacteristic
387C2×C4C22×D29211yescharacteristic
388C2×C4C2×#1192402811yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—41no—
393C8—41no—
394C8—2922no—
395C8—2922no—
396C8#1455661112yes—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#1798886212yes—
403C22C2×#1455656511yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—5842no—
412C4—5842no—
413C4C2×#1455661111yescharacteristic
414C4#1798938511yescharacteristic
415C4#1798939912yes—
416C2—22no—
417C2—81no—
418C2#2261938611yescharacteristic
419C2#2261959311yescharacteristic
420C2C2×#1798886211yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#2966C2
socleC22×C73C2×#121
prime radical#121×C73D4
hypercenter#2966D73
hypocenterC73#19228
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. #22609449C2semidirect
4. #22609444C2semidirect
5. #22609425C2semidirect
6. #22609420C2semidirect
7. #22609361C2semidirect
8. #22609354C2semidirect
9. C2×#17988338C2semidirect
10. #22608632C2semidirect
222. C73#19228semidirect

Maximal subgroups

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

subgroupindex
#226212682
#226205172
#226094492
#226094442
#226094252
#226094202
#226093612
#226093542
C2×#179883382
#226086322
C73×#29662
#1922873