Finite Group Databasebeta

← #33069121

Subgroups — #33069121

Order 18688.

714 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621056C211yesmaximal, characteristic, complemented
2C2×#17988933C211yesmaximal, characteristic, complemented
3#22618643C211yesmaximal, characteristic, complemented
4#22609462C214yesmaximal, complemented
5#22609338C214yesmaximal, complemented
6#22608677C211yesmaximal, characteristic, complemented
7#22608611C211yesmaximal, characteristic, complemented
8#22608599C211yesmaximal, characteristic, complemented
9C73×#3205C211yesmaximal, characteristic
10C2×#14556598—21no—
11C2×#14556591C2211yescharacteristic
12#17988983—21no—
13#17988933—21no—
14#17988933—21no—
15#17988933—21no—
16#17988933C2212yes—
17#17988933C2212yes—
18#17988903—22no—
19#17988877C2214yescomplemented
20#17988665—21no—
21#17988665C2212yescomplemented
22D4×#10596831—21no—
23D4×#10596831C2212yescomplemented
24#17988640C2211yescharacteristic
25#17988611C2211yescharacteristic
26C2×#14556512—21no—
27C2×#14556512—21no—
28#17988554—21no—
29C2×C4×#10596831C2211yescharacteristic
30#17988339C2212yescomplemented
31#17988337C2212yescomplemented
32#17988335C2214yescomplemented
33#17988331C2214yes—
34#532×C73C2211yescharacteristic
35C2×#127×C73C2211yescharacteristic
36#493×C73—21no—
37#486×C73—21no—
38#459×C73C2214yes—
39#447×C73—22no—
40#431×C73C2211yescharacteristic
41C22×Dic146—21no—
42#14556598—44no—
43C2×#11924012—22no—
44C2×#11924012—21no—
45C22×#10596831—22no—
46C22×#10596831D411yescharacteristic
47C22×#10596831D411yescharacteristic
48#14556591—21no—
49#14556591—21no—
50#14556591—21no—
51#14556591C2312yes—
52C2×C4⋊Dic73—21no—
53C2×C4⋊Dic73—21no—
54#14556522—24no—
55C4⋊#10596831—21no—
56C4⋊#10596831C2312yes—
57C4⋊#10596831C2312yes—
58#14556512—22no—
59#14556512—22no—
60#14556512—22no—
61#14556512—22no—
62#14556512—24no—
63C22⋊#10596831—22no—
64C22⋊#10596831—22no—
65C22⋊#10596831—21no—
66C4×#10596831—21no—
67C4×#10596831D411yescharacteristic
68C4×#10596831D411yescharacteristic
69C4×#10596831C2311yescharacteristic
70C4×#10596831C2311yescharacteristic
71C2×#41×C73—21no—
72C22×D4×C73—21no—
73C22×D4×C73—21no—
74C22×D4×C73—21no—
75C2×D8×C73—24no—
76C4⋊Q8×C73D411yescharacteristic
77#127×C73—21no—
78#127×C73—21no—
79#127×C73—21no—
80#127×C73D411yescharacteristic
81#127×C73C2312yes—
82#124×C73—21no—
83#121×C73—21no—
84#121×C73D412yes—
85C4×D4×C73—21no—
86C4×D4×C73C2312yes—
87C2×C42×C73C2311yescharacteristic
88C4⋊C8×C73C2312yes—
89#102×C73—24no—
90C22⋊C8×C73—21no—
91C4×C8×C73—21no—
92C2×Dic146—22no—
93C2×Dic146—41no—
94C2×Dic146—82no—
95C22×Dic73—41no—
96#11924012—44no—
97#11924012—42no—
98C2×#10596831—22no—
99C2×#10596831—22no—
100C2×#10596831—22no—
101C2×#10596831—22no—
102C2×#10596831—44no—
103C2×#10596831C2×D411yescharacteristic
104C2×#10596831C2×D411yescharacteristic
105C2×#10596831C2×D411yescharacteristic
106C2×#10596831C2×D411yescharacteristic
107C4⋊Dic73—22no—
108C4⋊Dic73—22no—
109C4⋊Dic73—41no—
110C4⋊Dic73—41no—
111C24×C73—41no—
112#41×C73—42no—
113#41×C73—42no—
114C2×Q8×C73—21no—
115C2×D4×C73—22no—
116C2×D4×C73—21no—
117C2×D4×C73—21no—
118C2×D4×C73—22no—
119C2×D4×C73—22no—
120C2×D4×C73—22no—
121C2×D4×C73—41no—
122C2×D4×C73—41no—
123C2×D4×C73—41no—
124C2×D4×C73—42no—
125C2×D4×C73—42no—
126C2×D4×C73—82no—
127C2×D4×C73—82no—
128C2×D4×C73C2×D412yes—
129C22×C4×C73—22no—
130C22×C4×C73C2×D411yescharacteristic
131C22×C4×C73C2×D411yescharacteristic
132C22×C4×C73#4111yescharacteristic
133D8×C73—84no—
134C2×C8×C73—22no—
135C4⋊C4×C73—21no—
136C4⋊C4×C73C2×D412yes—
137C22⋊C4×C73—22no—
138C22⋊C4×C73—22no—
139C42×C73C2×D411yescharacteristic
140C42×C73C2×D411yescharacteristic
141C42×C73#4111yescharacteristic
142C42×C73C2411yescharacteristic
143C23×C73—22no—
144C23×C73—42no—
145C23×C73—82no—
146C23×C73—82no—
147C23×C73—82no—
148C23×C73#12111yescharacteristic
149Q8×C73—42no—
150D4×C73—24no—
151D4×C73—44no—
152D4×C73—42no—
153D4×C73—42no—
154D4×C73—44no—
155D4×C73—42no—
156D4×C73—42no—
157D4×C73—82no—
158D4×C73—82no—
159D4×C73—82no—
160D4×C73—82no—
161C2×C4×C73—22no—
162C2×C4×C73—22no—
163C2×C4×C73—22no—
164C2×C4×C73—42no—
165C2×C4×C73—42no—
166C2×C4×C73#12111yescharacteristic
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73#12111yescharacteristic
169C2×C4×C73#13512yes—
170C2×C4×C73AGL(1,ℤ/8)12yes—
171C2×C4×C73C22×D411yescharacteristic
172C2×C4×C73C22×D411yescharacteristic
173C2×C4×C73C2×#4111yescharacteristic
174C2×Dic73—42no—
175C2×Dic73—82no—
176Dic146—82no—
177Dic146—82no—
178C8×C73—42no—
179#10596831—21no—
180#10596831—21no—
181#10596831—21no—
182#10596831—21no—
183#10596831—42no—
184#10596831—42no—
185C22×C73—21no—
186C22×C73—21no—
187C22×C73—42no—
188C22×C73—42no—
189C22×C73—42no—
190C22×C73—82no—
191C22×C73—82no—
192C22×C73—82no—
193C22×C73—164no—
194C22×C73#46312yes—
195C22×C73C2×#12111yescharacteristic
196C4×C73—21no—
197C4×C73—21no—
198C4×C73—42no—
199C4×C73#46312yes—
200C4×C73#47212yes—
201C4×C73C2×#13511yescharacteristic
202C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
203C4×C73#57712yes—
204Dic73—82no—
205#19196—731nomaximal, Sylow
206C2×C73—22no—
207C2×C73—82no—
208C2×C73—162no—
209C2×C73C2×#46311yescharacteristic
210C2×C73#305711yescharacteristic
211C2×C73C2×#47211yescharacteristic
212#3227—731no—
213#3205D7311yescharacteristic
214C2×#486—731no—
215#2931—731no—
216#1697—734no—
217#1677—734no—
218#1564—731no—
219#1548—731no—
220#1545—731no—
221C73#1919611yescharacteristic, complemented, Sylow
222C2×#131—1461no—
223#532C2×D7311yescharacteristic
224C2×C4⋊Q8—731no—
225C2×#127C2×D7311yescharacteristic
226#498—1461no—
227#493—21no—
228#486—21no—
229#486—732no—
230#486—732no—
231#486—1461no—
232#486—1461no—
233#486—1461no—
234#481—1462no—
235#476—734no—
236#459C2×D7314yes—
237#447—22no—
238#436—732no—
239#436—1461no—
240C8×D4—732no—
241C8×D4—1461no—
242#431—731no—
243#431C2×D7311yescharacteristic
244#423—731no—
245C2×#102—1461no—
246C2×#102—1461no—
247#408—1461no—
248C2×C4×C8—731no—
249#335—732no—
250#334—732no—
251#333—734no—
252#331—734no—
253C2×#41—21no—
254C22×Q8—1461no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—21no—
258C2×D8—24no—
259#131—2924no—
260C2×M16—1462no—
261C2×M16—1461no—
262C22×C8—731no—
263C22×C8—731no—
264C22×C8—1462no—
265C4⋊Q8—732no—
266C4⋊Q8—1461no—
267C4⋊Q8—1461no—
268C4⋊Q8—1461no—
269C4⋊Q8C22⋊D7311yescharacteristic
270#127—21no—
271#127—21no—
272#127—21no—
273#127C22⋊D7311yescharacteristic
274#127C22×D7312yes—
275#124—21no—
276#121—21no—
277#121C22⋊D7312yes—
278C4×D4—21no—
279C4×D4C22×D7312yes—
280C2×C4⋊C4—1461no—
281C2×C4⋊C4—1461no—
282C2×C42C22×D7311yescharacteristic
283#106—1464no—
284C4⋊C8—732no—
285C4⋊C8—732no—
286C4⋊C8—1461no—
287C4⋊C8C22×D7312yes—
288#102—24no—
289#102—1462no—
290#102—1462no—
291#102—1462no—
292#102—1462no—
293#102—1464no—
294C22⋊C8—21no—
295C22⋊C8—1462no—
296C22⋊C8—1462no—
297C22⋊C8—1461no—
298C4×C8—21no—
299C4×C8—731no—
300C4×C8—731no—
301C4×C8—731no—
302C4×C8—731no—
303C4×C8—1461no—
304C24—41no—
305#41—42no—
306#41—42no—
307C2×Q8—21no—
308C2×Q8—2921no—
309C2×Q8—5842no—
310C2×Q8—1462no—
311C2×D4—22no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4C2×C22⋊D7312yes—
325C22×C4—22no—
326C22×C4—2921no—
327C22×C4D4×D7311yescharacteristic
328C22×C4C2×C22⋊D7311yescharacteristic
329C22×C4#1192402711yescharacteristic
330D8—84no—
331M16—2922no—
332M16—2924no—
333C2×C8—22no—
334C2×C8—2924no—
335C2×C8—731no—
336C2×C8—731no—
337C2×C8—731no—
338C2×C8—731no—
339C2×C8—1462no—
340C2×C8—1462no—
341C2×C8—1462no—
342C2×C8—1462no—
343C4⋊C4—21no—
344C4⋊C4—2921no—
345C4⋊C4—2921no—
346C4⋊C4—1462no—
347C4⋊C4—1462no—
348C4⋊C4C2×C22⋊D7312yes—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C42D4×D7311yescharacteristic
352C42C2×C22⋊D7311yescharacteristic
353C42#1192402711yescharacteristic
354C42C23×D7311yescharacteristic
355C23—22no—
356C23—42no—
357C23—82no—
358C23—82no—
359C23—82no—
360C23#1455656311yescharacteristic
361Q8—42no—
362Q8—5842no—
363Q8—5842no—
364D4—24no—
365D4—44no—
366D4—44no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—82no—
372D4—82no—
373D4—82no—
374D4—82no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—2922no—
381C2×C4—5842no—
382C2×C4#1455656311yescharacteristic
383C2×C4#1455656311yescharacteristic
384C2×C4#1455656311yescharacteristic
385C2×C4#1455661012yes—
386C2×C4#1455661512yes—
387C2×C4C2×D4×D7311yescharacteristic
388C2×C4C22×C22⋊D7311yescharacteristic
389C2×C4C2×#1192402711yescharacteristic
390C8—42no—
391C8—2922no—
392C8—2922no—
393C8—1461no—
394C8—1461no—
395C8—1461no—
396C8—1461no—
397C22—21no—
398C22—21no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—164no—
406C22#1798880012yes—
407C22C2×#1455656311yescharacteristic
408C4—21no—
409C4—21no—
410C4—42no—
411C4—5842no—
412C4#1798880012yes—
413C4#1798885512yes—
414C4C2×#1455661011yescharacteristic
415C4C2×#1455661511yescharacteristic
416C4#1798939812yes—
417C2—22no—
418C2—82no—
419C2—162no—
420C2C2×#1798880011yescharacteristic
421C2#2261956711yescharacteristic
422C2C2×#1798885511yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC73×#3205C2
socleC22×C73C2×#121
hypercenter#3205D73
hypocenterC73#19196
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #22621056C2semidirect
2. C2×#17988933C2semidirect
3. #22618643C2semidirect
4. #22609462C2semidirect
5. #22609338C2semidirect
6. #22608677C2semidirect
7. #22608611C2semidirect
8. #22608599C2semidirect
19. #17988877C22semidirect
21. #17988665C22semidirect
23. D4×#10596831C22semidirect
30. #17988339C22semidirect
31. #17988337C22semidirect
32. #17988335C22semidirect
221. C73#19196semidirect

Maximal subgroups

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

subgroupindex
#226210562
C2×#179889332
#226186432
#226094622
#226093382
#226086772
#226086112
#226085992
C73×#32052
#1919673