Finite Group Databasebeta

← #49172438

Subgroups — #49172438

Order 37504.

1536 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
1C2×#22627671C211yesmaximal, characteristic, complemented
2C2×#41×D293C211yesmaximal, characteristic, complemented
3#34036799C213yesmaximal, complemented
4#34036798C211yesmaximal, characteristic, complemented
5#34036797C213yesmaximal, complemented
6#34036791C211yesmaximal, characteristic, complemented
7#34036788C213yesmaximal, complemented
8#34036786C211yesmaximal, characteristic, complemented
9#34036785C213yesmaximal, complemented
10#34036695C213yesmaximal, complemented
11#34036692C213yesmaximal, complemented
12#34036639C213yesmaximal, complemented
13#34036638C213yesmaximal, complemented
14#34036605C211yesmaximal, characteristic, complemented
15#550×C293C211yesmaximal, characteristic, complemented
16#22627671—28no—
17C2×#17989803C2211yescharacteristic
18C2×#17989802C2213yes—
19C2×#17989802C2213yes—
20C2×#17989801C2213yes—
21C2×#17989801C2213yes—
22C2×#17989801C2213yescomplemented
23#41×D293—28no—
24C22×Dic586C2213yescomplemented
25C2×Q8×D293C2211yescharacteristic
26C2×Q8×D293C2211yescharacteristic, complemented
27C2×D4×D293C2213yes—
28C22×C4×D293C2213yes—
29#22627620C2213yescomplemented
30#22627620C2216yescomplemented
31#22627612C2213yescomplemented
32#22627611C2213yescomplemented
33#22627610C2216yescomplemented
34#22627609C2211yescharacteristic
35#22627606C2213yes—
36#22627604C2213yescomplemented
37#22627603C2213yescomplemented
38#22627602C2213yescomplemented
39#22627601C2213yescomplemented
40#22627599C2213yescomplemented
41#22627598C2213yescomplemented
42#22627597C2213yescomplemented
43#22627596C2216yescomplemented
44#22627596C2216yescomplemented
45#22627596C2213yescomplemented
46#22627595C2213yescomplemented
47#22627595C2213yescomplemented
48#22627594C2213yes—
49#22627592C2213yes—
50#22627592C2216yes—
51Q8×Dic293C2211yescharacteristic, complemented
52C4×Dic586C2213yescomplemented
53#22627589C2213yescomplemented
54C4×D1172C2213yescomplemented
55D4×Dic293C2216yescomplemented
56C4×C22⋊D293C2213yescomplemented
57C4×C22⋊D293C2213yescomplemented
58#22627584C2216yescomplemented
59#22627582C2213yes—
60#22627581C2213yes—
61C2×#41×C293C2211yescharacteristic
62C2×#41×C293C2211yescharacteristic, complemented
63#127×C293C2213yescomplemented
64#121×C293C2213yescomplemented
65#121×C293C2213yescomplemented
66C4×Q8×C293C2211yescharacteristic, complemented
67C4×D4×C293C2213yescomplemented
68C23×D293C2313yes—
69#17989803—24no—
70#17989802—212no—
71#17989802—212no—
72#17989801—212no—
73#17989801—212no—
74#17989801—212no—
75C2×Dic586—212no—
76C2×Dic586C2313yes—
77C2×Dic586C2313yes—
78C2×Dic586C2313yes—
79Q8×D293—24no—
80Q8×D293D418yescomplemented
81C2×D1172C2311yescharacteristic
82C2×D1172C2313yes—
83C2×D1172C2313yes—
84C2×C22⋊D293C2313yes—
85C2×C22⋊D293C2316yes—
86C2×C22⋊D293C2313yes—
87C2×C22⋊D293C2316yes—
88C2×C22⋊D293C2313yes—
89C2×C22⋊D293C2313yes—
90D4×D293—212no—
91C22×Dic293C2316yes—
92C22×Dic293C2313yes—
93C2×C4×D293—212no—
94C2×C4×D293C2313yes—
95C2×C4×D293C2313yes—
96C2×C4×D293C2313yes—
97C2×C4×D293C2311yescharacteristic
98C2×C4×D293C2313yes—
99C4⋊Dic293C2313yes—
100C4⋊Dic293C2313yes—
101#17989781C2313yes—
102#17989781C2311yescharacteristic
103#17989781C2313yes—
104#17989781C2313yes—
105C22⋊Dic293C2313yes—
106C22⋊Dic293C2313yes—
107C22⋊Dic293C2316yes—
108#17989779C2316yes—
109#17989779C2311yescharacteristic
110#17989779C2313yes—
111#17989779C2311yescharacteristic
112#17989779C2313yes—
113#17989779C2311yescharacteristic
114#17989779C2313yes—
115C4×Dic293C2316yes—
116C4×Dic293C2313yes—
117#41×C293—24no—
118#41×C293—24no—
119C2×Q8×C293C2311yescharacteristic
120C2×D4×C293C2313yes—
121C2×D4×C293C2313yes—
122C2×D4×C293C2316yes—
123C2×D4×C293C2313yes—
124C22×C4×C293C2313yes—
125C22×C4×C293C2313yes—
126C4⋊C4×C293C2313yes—
127C4⋊C4×C293C2311yescharacteristic
128C22⋊C4×C293C2313yes—
129C22⋊C4×C293C2313yes—
130C42×C293C2313yes—
131C22×D293—26no—
132C22×D293—43no—
133C22×D293—43no—
134C22×D293C2413yes—
135C22×D293C2411yescharacteristic
136C22×D293C2411yescharacteristic
137Dic586—26no—
138Dic586—26no—
139Dic586C2×D4112yes—
140D1172—22no—
141D1172—26no—
142D1172—43no—
143C22⋊D293—26no—
144C22⋊D293—212no—
145C22⋊D293—26no—
146C22⋊D293—43no—
147C22⋊D293—46no—
148C22⋊D293—43no—
149C2×Dic293—26no—
150C2×Dic293—212no—
151C2×Dic293C2411yescharacteristic
152C2×Dic293C2413yes—
153C2×Dic293C2413yes—
154C2×Dic293C2411yescharacteristic
155C2×Dic293C2413yes—
156C4×D293—22no—
157C4×D293—26no—
158C4×D293—26no—
159C4×D293—26no—
160C4×D293C2×D4112yes—
161C23×C293C2413yes—
162C23×C293C2413yes—
163Q8×C293C2×D414yes—
164D4×C293—26no—
165D4×C293—26no—
166D4×C293—43no—
167D4×C293—46no—
168C2×C4×C293—26no—
169C2×C4×C293—26no—
170C2×C4×C293C2411yescharacteristic
171C2×C4×C293C2413yes—
172C2×C4×C293C2411yescharacteristic
173C2×C4×C293C2411yescharacteristic
174C2×C4×C293C2413yes—
175C22×C293—23no—
176C22×C293—23no—
177C22×C293—23no—
178C22×C293—23no—
179C22×C293—23no—
180C22×C293—23no—
181C22×C293C2511yescharacteristic
182C2×D293—21no—
183C2×D293—21no—
184C2×D293—21no—
185C2×D293—23no—
186C2×D293—21no—
187C2×D293—23no—
188C2×D293—23no—
189C2×D293—21no—
190C2×D293—412no—
191C2×D293C22×D412yes—
192C4×C293—21no—
193C4×C293—21no—
194C4×C293—21no—
195C4×C293—23no—
196C4×C293C22×D416yes—
197Dic293—21no—
198Dic293—23no—
199Dic293—21no—
200Dic293—23no—
201Dic293C22×D416yes—
202C2×C293—43no—
203C2×C293—43no—
204C2×C293C23×D411yescharacteristic
205C2×C293#58511yescharacteristic
206C2×C293#58511yescharacteristic
207D293—22no—
208D293—41no—
209D293—43no—
210C293#348711yescharacteristic, complemented, Sylow
211#3487—2931nomaximal, Sylow
212C2×2−1+4—2931no—
213C22×#41—2931no—
214#550—2931no—
215#550—2933no—
216#550—2933no—
217#550D29311yescharacteristic, complemented
218#548—2931no—
219#548—2933no—
220#548—2931no—
221#548—2933no—
222#535—2933no—
223#535—2933no—
224#525—2933no—
225#525—2933no—
226#519—2931no—
2272−1+4—5868no—
228C2×#41—2931no—
229C2×#41—2933no—
230C2×#41—2933no—
231C2×#41—2933no—
232C2×#41—2933no—
233C2×#41—2933no—
234C2×#41—5868no—
235C2×#41C2×D29311yescharacteristic
236C2×#41C2×D29311yescharacteristic, complemented
237C22×Q8—2931no—
238C22×Q8—2931no—
239C22×Q8—2933no—
240C22×D4—2933no—
241C23×C4—2933no—
242#127—2936no—
243#127—2933no—
244#127C2×D29313yescomplemented
245#124—2933no—
246#124—2936no—
247#124—2933no—
248#123—2931no—
249#123—2933no—
250C22⋊Q8—2933no—
251C22⋊Q8—2933no—
252C22⋊Q8—2933no—
253C22⋊Q8—2933no—
254#121—2933no—
255#121—2933no—
256#121—2933no—
257#121—2936no—
258#121—2936no—
259#121—2933no—
260#121—2933no—
261#121—2933no—
262#121C2×D29313yescomplemented
263#121C2×D29313yescomplemented
264#120—2936no—
265#120—2933no—
266#120—2933no—
267C4×Q8—2931no—
268C4×Q8—2933no—
269C4×Q8—2933no—
270C4×Q8C2×D29311yescharacteristic, complemented
271C4×D4—2936no—
272C4×D4—2933no—
273C4×D4—2933no—
274C4×D4—2936no—
275C4×D4—2933no—
276C4×D4C2×D29313yescomplemented
277#117—2933no—
278#117—2933no—
279C24—2933no—
280#41—24no—
281#41—24no—
282#41—58612no—
283#41—58612no—
284#41—58612no—
285#41—58612no—
286#41—58612no—
287#41—5864no—
288C2×Q8—2938no—
289C2×Q8—2933no—
290C2×Q8—2933no—
291C2×Q8—2933no—
292C2×Q8—58612no—
293C2×Q8—5864no—
294C2×Q8C22×D29311yescharacteristic
295C2×D4—2933no—
296C2×D4—2936no—
297C2×D4—2933no—
298C2×D4—2933no—
299C2×D4—2936no—
300C2×D4—2933no—
301C2×D4—2931no—
302C2×D4—2933no—
303C2×D4—2933no—
304C2×D4—58612no—
305C2×D4C22×D29313yes—
306C2×D4C22×D29313yes—
307C2×D4C22×D29316yes—
308C2×D4C22×D29313yes—
309C22×C4—2933no—
310C22×C4—2936no—
311C22×C4—2933no—
312C22×C4—2933no—
313C22×C4—2933no—
314C22×C4—2933no—
315C22×C4—2931no—
316C22×C4—58612no—
317C22×C4C22×D29313yes—
318C22×C4C22×D29313yes—
319C4⋊C4—2933no—
320C4⋊C4—2933no—
321C4⋊C4—2933no—
322C4⋊C4—2933no—
323C4⋊C4—2931no—
324C4⋊C4—2933no—
325C4⋊C4C22×D29313yes—
326C4⋊C4C22×D29311yescharacteristic
327C22⋊C4—2933no—
328C22⋊C4—2936no—
329C22⋊C4—2933no—
330C22⋊C4—2933no—
331C22⋊C4—2931no—
332C22⋊C4—2931no—
333C22⋊C4—2931no—
334C22⋊C4—2933no—
335C22⋊C4—2936no—
336C22⋊C4—2933no—
337C22⋊C4C22×D29313yes—
338C22⋊C4C22×D29313yes—
339C42—2936no—
340C42—2933no—
341C42C22×D29313yes—
342C23—2931no—
343C23—2933no—
344C23—2931no—
345C23—5866no—
346C23—11723no—
347C23—11723no—
348C23C23×D29313yes—
349C23C23×D29313yes—
350Q8—29312no—
351Q8—5866no—
352Q8—5866no—
353Q8D4×D29314yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—5862no—
359D4—58612no—
360D4—5866no—
361D4—5866no—
362D4—5866no—
363D4—11723no—
364D4—11723no—
365D4—11726no—
366D4—11723no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—2933no—
370C2×C4—29312no—
371C2×C4—2931no—
372C2×C4—2933no—
373C2×C4—2933no—
374C2×C4—2931no—
375C2×C4—5866no—
376C2×C4—5862no—
377C2×C4—5866no—
378C2×C4—58612no—
379C2×C4—5866no—
380C2×C4—5866no—
381C2×C4C23×D29313yes—
382C2×C4C23×D29311yescharacteristic
383C2×C4C23×D29311yescharacteristic
384C2×C4C23×D29311yescharacteristic
385C2×C4C23×D29313yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—2932no—
393C22—5861no—
394C22—5861no—
395C22—5861no—
396C22—5863no—
397C22—5861no—
398C22—5863no—
399C22—5861no—
400C22—5863no—
401C22—117212no—
402C22C24×D29311yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—2936no—
408C4—5861no—
409C4—5863no—
410C4—5861no—
411C4—5863no—
412C4C2×D4×D29316yes—
413C2—43no—
414C2—43no—
415C2—5862no—
416C2—11723no—
417C2—11721no—
418C2C22×D4×D29311yescharacteristic
419C2#3403696411yescharacteristic
420C2#3403696511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D293
commutator, socle, cosocleC22×C293C25
Fitting#550×C293C2
hypercenter#550D293
hypocenterC293#3487
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#22627671C2semidirect
2. C2×#41×D293C2semidirect
3. #34036799C2semidirect
4. #34036798C2semidirect
5. #34036797C2semidirect
6. #34036791C2semidirect
7. #34036788C2semidirect
8. #34036786C2semidirect
9. #34036785C2semidirect
10. #34036695C2semidirect
11. #34036692C2semidirect
12. #34036639C2semidirect
13. #34036638C2semidirect
14. #34036605C2semidirect
15. #550×C293C2semidirect
22. C2×#17989801C22semidirect
24. C22×Dic586C22semidirect
26. C2×Q8×D293C22semidirect
29. #22627620C22semidirect
30. #22627620C22semidirect
31. #22627612C22semidirect
32. #22627611C22semidirect
33. #22627610C22semidirect
36. #22627604C22semidirect
37. #22627603C22semidirect
38. #22627602C22semidirect
39. #22627601C22semidirect
40. #22627599C22semidirect
41. #22627598C22semidirect
42. #22627597C22semidirect
43. #22627596C22semidirect
44. #22627596C22semidirect
45. #22627596C22semidirect
46. #22627595C22semidirect
47. #22627595C22semidirect
51. Q8×Dic293C22semidirect
52. C4×Dic586C22semidirect
53. #22627589C22semidirect
54. C4×D1172C22semidirect
55. D4×Dic293C22semidirect
56. C4×C22⋊D293C22semidirect
57. C4×C22⋊D293C22semidirect
58. #22627584C22semidirect
62. C2×#41×C293C22semidirect
63. #127×C293C22semidirect
64. #121×C293C22semidirect
65. #121×C293C22semidirect
66. C4×Q8×C293C22semidirect
67. C4×D4×C293C22semidirect
80. Q8×D293D4semidirect
210. C293#3487semidirect
217. #550D293semidirect
236. C2×#41C2×D293semidirect
244. #127C2×D293semidirect
262. #121C2×D293semidirect
263. #121C2×D293semidirect
270. C4×Q8C2×D293semidirect
276. C4×D4C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276712
C2×#41×D2932
#340367992
#340367982
#340367972
#340367912
#340367882
#340367862
#340367852
#340366952
#340366922
#340366392
#340366382
#340366052
#550×C2932
#3487293