Finite Group Databasebeta

← #37582255

Subgroups — #37582255

Order 24448.

446 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133462C211yesmaximal, characteristic
2C2×#20259354C211yesmaximal, characteristic, complemented
3#137×D191C211yesmaximal, characteristic, complemented
4#27133428C211yesmaximal, characteristic, complemented
5#27132937C211yesmaximal, characteristic, complemented
6#27132936C211yesmaximal, characteristic, complemented
7#27132935C211yesmaximal, characteristic, complemented
8#27132929C211yesmaximal, characteristic, complemented
9#27132928C211yesmaximal, characteristic, complemented
10#27132927C211yesmaximal, characteristic, complemented
11#27132926C211yesmaximal, characteristic, complemented
12#27132710C211yesmaximal, characteristic, complemented
13#27132646C211yesmaximal, characteristic, complemented
14#101×D191C211yesmaximal, characteristic, complemented
15#473×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17#41×D191C2211yescharacteristic
18C2×Q8×D191C2211yescharacteristic
19#20259355C2211yescharacteristic
20#20259354—21no—
21#20259354—21no—
22#20259354—21no—
23#20259354—21no—
24#20259354—21no—
25#20259354—21no—
26#20259353C2211yescharacteristic
27#20259353C2211yescharacteristic
28#20259352C2211yescharacteristic
29#20259351—21no—
30#20259350C2211yescharacteristic
31#20259349—21no—
32#20259347C2211yescharacteristic
33#20259345—21no—
34#20259343—21no—
35C2×#15690277C2211yescharacteristic, complemented
36Q16×D191—21no—
37C2×#15690276C2211yescharacteristic, complemented
38C2×#15690274C2211yescharacteristic, complemented
39SD16×D191—21no—
40#20259330—21no—
41#20259330C2211yescharacteristic
42#20259329C2211yescharacteristic
43#20259328—21no—
44#20259328C2211yescharacteristic
45C2×#15690270—21no—
46C2×#15690270C2211yescharacteristic
47M16×D191—21no—
48M16×D191C2211yescharacteristic
49M16×D191C2211yescharacteristic
50M16×D191C2211yescharacteristic
51#20259258C2211yescharacteristic, complemented
52#20259257C2211yescharacteristic, complemented
53#20259257C2211yescharacteristic, complemented
54#20259241C2211yescharacteristic, complemented
55#20259241C2211yescharacteristic, complemented
56#20259240C2211yescharacteristic, complemented
57#20259240C2211yescharacteristic, complemented
58#20259238C2211yescharacteristic, complemented
59#20259238C2211yescharacteristic, complemented
60#137×C191C2211yescharacteristic
61#137×C191C2211yescharacteristic
62C2×Q16×C191C2211yescharacteristic, complemented
63#131×C191C2211yescharacteristic
64#108×C191C2211yescharacteristic, complemented
65#101×C191C2211yescharacteristic, complemented
66#101×C191C2211yescharacteristic, complemented
67#15690288—21no—
68#15690288—21no—
69#15690288—21no—
70#15690288—21no—
71#15690288D411yescharacteristic
72#15690287—21no—
73#15690287C2311yescharacteristic
74#15690286—21no—
75#15690286D411yescharacteristic
76C2×Dic382—21no—
77C2×Dic382C2311yescharacteristic
78Q8×D191—21no—
79Q8×D191—21no—
80Q8×D191—21no—
81Q8×D191—21no—
82Q8×D191D411yescharacteristic
83C2×D764—21no—
84C2×D764C2311yescharacteristic
85D4×D191—21no—
86D4×D191D411yescharacteristic
87C2×C4×D191—21no—
88C2×C4×D191—21no—
89C2×C4×D191—21no—
90C2×C4×D191C2311yescharacteristic
91Dic764—21no—
92#15690277—21no—
93#15690277—21no—
94#15690277—21no—
95#15690277—21no—
96#15690277—21no—
97#15690276—21no—
98#15690276—21no—
99#15690276—21no—
100#15690276—21no—
101#15690275—21no—
102#15690274—21no—
103#15690274—21no—
104#15690274—21no—
105#15690274—21no—
106#15690274—21no—
107D1528—21no—
108#15690272—21no—
109#15690271—21no—
110#15690271C2311yescharacteristic
111#15690271C2311yescharacteristic
112#15690271C2311yescharacteristic
113#15690270—21no—
114#15690270—21no—
115#15690270—21no—
116#15690270—21no—
117#15690270—21no—
118#15690270D411yescharacteristic
119#15690270D411yescharacteristic
120#15690270D411yescharacteristic
121#15690270D411yescharacteristic
122C2×C191⋊C8—21no—
123C2×C191⋊C8C2311yescharacteristic
124C8×D191—21no—
125C8×D191—21no—
126C8×D191—21no—
127C8×D191—21no—
128C8×D191—21no—
129#41×C191C2311yescharacteristic
130C2×Q8×C191C2311yescharacteristic
131C2×Q8×C191C2311yescharacteristic
132Q16×C191—21no—
133Q16×C191—21no—
134Q16×C191—21no—
135Q16×C191—21no—
136SD16×C191—21no—
137SD16×C191—21no—
138M16×C191—21no—
139M16×C191C2311yescharacteristic
140M16×C191C2311yescharacteristic
141M16×C191C2311yescharacteristic
142C2×C8×C191—21no—
143C2×C8×C191C2311yescharacteristic
144C22×D191—21no—
145C22×D191—21no—
146C22×D191#4111yescharacteristic
147Dic382—21no—
148Dic382—21no—
149Dic382—21no—
150Dic382—41no—
151Dic382—41no—
152Dic382C2×D411yescharacteristic
153D764—21no—
154D764—21no—
155D764—21no—
156D764—41no—
157D764—41no—
158D764C2×D411yescharacteristic
159C22⋊D191—21no—
160C22⋊D191—21no—
161C22⋊D191—21no—
162C2×Dic191—21no—
163C2×Dic191—21no—
164C2×Dic191#4111yescharacteristic
165C4×D191—21no—
166C4×D191—21no—
167C4×D191—21no—
168C4×D191—41no—
169C4×D191—41no—
170C4×D191—41no—
171C4×D191—41no—
172C4×D191C2×D411yescharacteristic
173C4×D191C2×D411yescharacteristic
174C4×D191C2×D411yescharacteristic
175C4×D191C2×D411yescharacteristic
176C191⋊C8—21no—
177C191⋊C8—21no—
178C191⋊C8—21no—
179C191⋊C8C2×D411yescharacteristic
180C191⋊C8C2×D411yescharacteristic
181Q8×C191—21no—
182Q8×C191—21no—
183Q8×C191—21no—
184Q8×C191—21no—
185Q8×C191C2×D411yescharacteristic
186D4×C191—21no—
187D4×C191C2×D411yescharacteristic
188C2×C4×C191—21no—
189C2×C4×C191—21no—
190C2×C4×C191—21no—
191C2×C4×C191C2411yescharacteristic
192C8×C191—21no—
193C8×C191—21no—
194C8×C191—21no—
195C8×C191C2×D411yescharacteristic
196C8×C191C2×D411yescharacteristic
197C22×C191—21no—
198C22×C191C2×#4111yescharacteristic
199C2×D191—21no—
200C2×D191—41no—
201C2×D191—41no—
202C2×D191—81no—
203C2×D191—81no—
204C2×D191#12111yescharacteristic
205C2×D191#12111yescharacteristic
206C4×C191—21no—
207C4×C191—41no—
208C4×C191—41no—
209C4×C191C22×D411yescharacteristic
210C4×C191C22×D411yescharacteristic
211Dic191—21no—
212Dic191—41no—
213Dic191#12111yescharacteristic
214Dic191#12111yescharacteristic
215C2×C191—21no—
216C2×C191—41no—
217C2×C191C2×#12111yescharacteristic
218D191—21no—
219D191—21no—
220D191—41no—
221D191—81no—
222C191#306911yescharacteristic, complemented, Sylow
223#3069—1911nomaximal, Sylow
224#575—1911no—
225C2×#137—1911no—
226C2×#137—1911no—
227#568—1911no—
228#473—1911no—
229#473—1911no—
230#473—1911no—
231#473D19111yescharacteristic, complemented
232#471—1911no—
233#471—1911no—
234#471—1911no—
235#471—1911no—
236#430—1911no—
237#413—1911no—
238C2×#101—1911no—
239C2×#41—1911no—
240C2×#41—1911no—
241C22×Q8—1911no—
242#137—3821no—
243#137—3821no—
244#137—3821no—
245#137—3821no—
246#137—3821no—
247#137—3821no—
248#137—3821no—
249#137—1911no—
250#137—1911no—
251#137—1911no—
252#137—1911no—
253#137C2×D19111yescharacteristic
254#137C2×D19111yescharacteristic
255AGL(1,ℤ/8)—3821no—
256AGL(1,ℤ/8)—1911no—
257AGL(1,ℤ/8)—1911no—
258#135—3821no—
259#135—3821no—
260C2×Q16—3821no—
261C2×Q16—1911no—
262C2×Q16C2×D19111yescharacteristic, complemented
263C2×SD16—3821no—
264C2×SD16—1911no—
265C2×SD16—1911no—
266#131—3821no—
267#131—3821no—
268#131—1911no—
269#131—1911no—
270#131—1911no—
271#131C2×D19111yescharacteristic
272C2×M16—3821no—
273C2×M16—3821no—
274C2×M16—1911no—
275C2×M16—1911no—
276C2×M16—1911no—
277C2×M16—1911no—
278#108—1911no—
279#108—1911no—
280#108—1911no—
281#108C2×D19111yescharacteristic, complemented
282#101—1911no—
283#101—1911no—
284#101—1911no—
285#101—1911no—
286#101C2×D19111yescharacteristic, complemented
287#101C2×D19111yescharacteristic, complemented
288#100—1911no—
289#100—1911no—
290#41—3821no—
291#41—3821no—
292#41—3821no—
293#41—3821no—
294#41—3821no—
295#41—3821no—
296#41—1911no—
297#41—1911no—
298#41—1911no—
299#41C22×D19111yescharacteristic
300C2×Q8—3821no—
301C2×Q8—3821no—
302C2×Q8—3821no—
303C2×Q8—3821no—
304C2×Q8—3821no—
305C2×Q8—1911no—
306C2×Q8—1911no—
307C2×Q8C22×D19111yescharacteristic
308C2×Q8C22×D19111yescharacteristic
309C2×D4—3821no—
310C2×D4—3821no—
311C2×D4—1911no—
312C2×D4—1911no—
313C22×C4—3821no—
314C22×C4—3821no—
315C22×C4—3821no—
316C22×C4—1911no—
317Q16—21no—
318Q16—21no—
319Q16—21no—
320Q16—21no—
321Q16—3821no—
322Q16—3821no—
323Q16—3821no—
324Q16—3821no—
325Q16—3821no—
326Q16—3821no—
327SD16—21no—
328SD16—21no—
329SD16—3821no—
330SD16—3821no—
331SD16—3821no—
332SD16—3821no—
333SD16—3821no—
334SD16—3821no—
335SD16—3821no—
336SD16—3821no—
337SD16—3821no—
338SD16—3821no—
339D8—3821no—
340D8—3821no—
341M16—21no—
342M16—3821no—
343M16—3821no—
344M16—3821no—
345M16—3821no—
346M16—3821no—
347M16—3821no—
348M16—1911no—
349M16—1911no—
350M16—1911no—
351M16—1911no—
352M16—1911no—
353M16—1911no—
354M16—1911no—
355M16C22×D19111yescharacteristic
356M16C22×D19111yescharacteristic
357M16C22×D19111yescharacteristic
358C2×C8—21no—
359C2×C8—3821no—
360C2×C8—3821no—
361C2×C8—3821no—
362C2×C8—3821no—
363C2×C8—3821no—
364C2×C8—3821no—
365C2×C8—1911no—
366C2×C8C22×D19111yescharacteristic
367C23—3821no—
368C23—3821no—
369C23—1911no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—3821no—
375Q8—3821no—
376Q8—3821no—
377Q8—1911no—
378Q8—7641no—
379Q8—7641no—
380Q8D4×D19111yescharacteristic
381D4—21no—
382D4—3821no—
383D4—3821no—
384D4—3821no—
385D4—3821no—
386D4—3821no—
387D4—3821no—
388D4—1911no—
389D4—7641no—
390D4—7641no—
391D4D4×D19111yescharacteristic
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—3821no—
396C2×C4—3821no—
397C2×C4—3821no—
398C2×C4—3821no—
399C2×C4—3821no—
400C2×C4—1911no—
401C2×C4—1911no—
402C2×C4—1911no—
403C2×C4—1911no—
404C2×C4—1911no—
405C2×C4—7641no—
406C2×C4—7641no—
407C2×C4—7641no—
408C2×C4—7641no—
409C2×C4C23×D19111yescharacteristic
410C8—21no—
411C8—21no—
412C8—21no—
413C8—3821no—
414C8—3821no—
415C8—3821no—
416C8—1911no—
417C8—1911no—
418C8D4×D19111yescharacteristic
419C8D4×D19111yescharacteristic
420C22—21no—
421C22—3821no—
422C22—1911no—
423C22—1911no—
424C22—15281no—
425C22—15281no—
426C22—7641no—
427C22—7641no—
428C22#41×D19111yescharacteristic
429C4—21no—
430C4—41no—
431C4—41no—
432C4—3821no—
433C4—1911no—
434C4—1911no—
435C4—7641no—
436C4C2×D4×D19111yescharacteristic
437C4C2×D4×D19111yescharacteristic
438C2—21no—
439C2—41no—
440C2—3821no—
441C2—3821no—
442C2—15281no—
443C2—7641no—
444C2#121×D19111yescharacteristic
subgroupG/H
centerC2#121×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#473×C191C2
socleC2×C191C2×#121
prime radical#27133462C2
hypercenter#473D191
hypocenterC191#3069
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
2. C2×#20259354C2semidirect
3. #137×D191C2semidirect
4. #27133428C2semidirect
5. #27132937C2semidirect
6. #27132936C2semidirect
7. #27132935C2semidirect
8. #27132929C2semidirect
9. #27132928C2semidirect
10. #27132927C2semidirect
11. #27132926C2semidirect
12. #27132710C2semidirect
13. #27132646C2semidirect
14. #101×D191C2semidirect
15. #473×C191C2semidirect
35. C2×#15690277C22semidirect
37. C2×#15690276C22semidirect
38. C2×#15690274C22semidirect
51. #20259258C22semidirect
52. #20259257C22semidirect
53. #20259257C22semidirect
54. #20259241C22semidirect
55. #20259241C22semidirect
56. #20259240C22semidirect
57. #20259240C22semidirect
58. #20259238C22semidirect
59. #20259238C22semidirect
62. C2×Q16×C191C22semidirect
64. #108×C191C22semidirect
65. #101×C191C22semidirect
66. #101×C191C22semidirect
222. C191#3069semidirect
231. #473D191semidirect
262. C2×Q16C2×D191semidirect
281. #108C2×D191semidirect
286. #101C2×D191semidirect
287. #101C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271334622
C2×#202593542
#137×D1912
#271334282
#271329372
#271329362
#271329352
#271329292
#271329282
#271329272
#271329262
#271327102
#271326462
#101×D1912
#473×C1912
#3069191