Finite Group Databasebeta

← #18003954

Subgroups — #18003954

Order 4736.

762 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(5,𝔽2)×D37C211yesmaximal, characteristic, complemented
2#14558332C212yesmaximal, complemented
3#14558101C211yesmaximal, characteristic, complemented
4#14557710C211yesmaximal, characteristic, complemented
5#14557709C212yesmaximal, complemented
6#14557703C211yesmaximal, characteristic, complemented
7#14557702C212yesmaximal, complemented
8#14557701C211yesmaximal, characteristic, complemented
9#14557548C212yesmaximal, complemented
10#100×D37C211yesmaximal, characteristic, complemented
11C37×#455C211yesmaximal, characteristic, complemented
12#11924582—22no—
13#11924582C2212yes—
14#11924581—22no—
15#11924581—22no—
16#11924581C2211yescharacteristic
17C2×#10596966C2211yescharacteristic, complemented
18#41×D37—22no—
19#41×D37C2212yes—
20C2×D4×D37—22no—
21C2×D4×D37—22no—
22C2×D4×D37C2211yescharacteristic
23#11924568—22no—
24#11924567C2212yescomplemented
25#11924564C2212yescomplemented
26#11924563—22no—
27#11924561C2212yescomplemented
28#11924559—22no—
29#11924557—22no—
30M16×D37C2212yescomplemented
31#11924533C2211yescharacteristic, complemented
32#11924532C2211yescharacteristic, complemented
33#11924525C2211yescharacteristic, complemented
34#11924509—22no—
35#11924506—22no—
36#11924493C2211yescharacteristic, complemented
37#11924464C2212yescomplemented
38#11924463C2212yescomplemented
39#11924462C2212yescomplemented
40#11924453C2211yescharacteristic, complemented
41#11924452C2212yescomplemented
42H(5,𝔽2)×C37C2211yescharacteristic
43#137×C37C2212yescomplemented
44#124×C37C2211yescharacteristic, complemented
45#104×C37C2212yescomplemented
46#100×C37C2211yescharacteristic, complemented
47C23×D37—22no—
48C23×D37D412yes—
49#41×C37—22no—
50#41×C37C2312yes—
51C2×Q8×C37C2311yescharacteristic, complemented
52C2×D4×C37—22no—
53C2×D4×C37—22no—
54C2×D4×C37C2311yescharacteristic
55C2×C22⋊D37—22no—
56C2×C22⋊D37—24no—
57C2×C22⋊D37—22no—
58C2×C22⋊D37—22no—
59C2×C22⋊D37D412yes—
60#10596966—22no—
61#10596966—22no—
62#10596966—22no—
63#10596966D412yescomplemented
64Q8×D37D412yescomplemented
65#10596964—24no—
66#10596964—22no—
67#10596964—22no—
68#10596964D412yescomplemented
69D4×D37—24no—
70D4×D37—24no—
71D4×D37—24no—
72D4×D37—22no—
73D4×D37—24no—
74D4×D37—22no—
75D4×D37—22no—
76D4×D37—22no—
77D4×D37—22no—
78D4×D37D412yescomplemented
79#10596962—24no—
80#10596962—22no—
81#10596962—22no—
82#10596962C2312yes—
83C2×D148—22no—
84C2×D148—22no—
85C2×D148—21no—
86C2×D148—22no—
87C2×D148C2311yescharacteristic
88C2×D148C2311yescharacteristic, complemented
89C2×C4×D37—22no—
90C2×C4×D37—22no—
91C2×C4×D37—21no—
92C2×C4×D37C2311yescharacteristic
93Q16×C37—22no—
94SD16×C37—22no—
95M16×C37C2312yescomplemented
96C22⋊C4×C37—22no—
97C42×C37C2311yescharacteristic, complemented
98#10596942—22no—
99#10596941—22no—
100#10596941—22no—
101#10596939—22no—
102#10596938—21no—
103#10596938—22no—
104#10596936—21no—
105C4×Dic37C2311yescharacteristic, complemented
106#10596934C2312yescomplemented
107D296—22no—
108#10596930—22no—
109#10596929—22no—
110C8×D37—22no—
111C23×C37—22no—
112C23×C37C2×D412yes—
113C22×D37—24no—
114C22×D37—22no—
115C22×D37—22no—
116C22×D37—22no—
117C22×D37—22no—
118C22×D37—21no—
119C22×D37—82no—
120C22×D37—84no—
121C22×D37C2×D412yes—
122C22×D37C2×D411yescharacteristic
123Q8×C37—22no—
124Q8×C37C2×D412yes—
125D4×C37—24no—
126D4×C37—22no—
127D4×C37—22no—
128D4×C37C2×D412yes—
129C2×C4×C37—22no—
130C2×C4×C37—22no—
131C2×C4×C37—21no—
132C2×C4×C37—21no—
133C2×C4×C37C2411yescharacteristic
134C22⋊D37—24no—
135C22⋊D37—24no—
136C22⋊D37—22no—
137C22⋊D37—22no—
138C22⋊D37—24no—
139C22⋊D37—22no—
140C22⋊D37—82no—
141C2×Dic37—22no—
142C2×Dic37—22no—
143C2×Dic37—21no—
144C2×Dic37C2×D411yescharacteristic
145D148—24no—
146D148—22no—
147D148—22no—
148D148—22no—
149D148—41no—
150D148—41no—
151D148—82no—
152D148C2×D412yes—
153C4×D37—24no—
154C4×D37—22no—
155C4×D37—22no—
156C4×D37—22no—
157C4×D37—42no—
158C4×D37C2×D412yes—
159C4×D37C2×D412yes—
160Dic74—22no—
161Dic74C2×D412yes—
162C8×C37—22no—
163#64436—22no—
164C22×C37—22no—
165C22×C37—22no—
166C22×C37—42no—
167C22×C37—82no—
168C22×C37C22×D411yescharacteristic
169C2×D37—22no—
170C2×D37—22no—
171C2×D37—41no—
172C2×D37—44no—
173C2×D37—44no—
174C2×D37—42no—
175C2×D37—42no—
176C2×D37—41no—
177C2×D37—82no—
178C2×D37—84no—
179C2×D37—82no—
180C2×D37#12011yescharacteristic
181C2×D37#12011yescharacteristic
182C4×C37—22no—
183C4×C37—41no—
184C4×C37—41no—
185C4×C37C22×D412yes—
186Dic37—22no—
187Dic37—41no—
188Dic37#12012yes—
189#3019—371nomaximal, Sylow
190C2×C37—21no—
191C2×C37—42no—
192C2×C37—42no—
193C2×C37C2×#12011yescharacteristic
194D37—21no—
195D37—21no—
196D37—42no—
197D37—42no—
198D37—81no—
199C2×H(5,𝔽2)—371no—
200#575—372no—
201#535—371no—
202#455—372no—
203#455—371no—
204#455D3711yescharacteristic, complemented
205#453—371no—
206#453—372no—
207#453—371no—
208#421—372no—
209C2×#100—371no—
210C37#301911yescharacteristic, complemented, Sylow
211H(5,𝔽2)—372no—
212H(5,𝔽2)—371no—
213H(5,𝔽2)—742no—
214H(5,𝔽2)—742no—
215H(5,𝔽2)—742no—
216H(5,𝔽2)C2×D3711yescharacteristic
217C2×#41—372no—
218C2×#41—371no—
219C2×#41—742no—
220C22×D4—371no—
221C22×D4—742no—
222C22×D4—742no—
223#137—372no—
224#137—742no—
225#137C2×D3712yescomplemented
226AGL(1,ℤ/8)—372no—
227AGL(1,ℤ/8)—372no—
228AGL(1,ℤ/8)—742no—
229#135—742no—
230#135—742no—
231C2×M16—372no—
232#127—371no—
233#127—371no—
234#124—371no—
235#124C2×D3711yescharacteristic, complemented
236#121—742no—
237#120—742no—
238#117—371no—
239#104—372no—
240#104—372no—
241#104—372no—
242#104C2×D3712yescomplemented
243#100—371no—
244#100—372no—
245#100C2×D3711yescharacteristic, complemented
246C24—372no—
247C24—742no—
248#41—22no—
249#41—372no—
250#41—372no—
251#41—372no—
252#41—742no—
253#41—742no—
254#41—744no—
255#41—742no—
256#41—742no—
257#41—742no—
258#41—742no—
259#41—744no—
260#41—742no—
261#41C22×D3712yes—
262C2×Q8—372no—
263C2×Q8C22×D3711yescharacteristic, complemented
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—372no—
267C2×D4—372no—
268C2×D4—371no—
269C2×D4—371no—
270C2×D4—742no—
271C2×D4—742no—
272C2×D4—742no—
273C2×D4—741no—
274C2×D4—742no—
275C2×D4—742no—
276C2×D4—742no—
277C2×D4—742no—
278C2×D4—744no—
279C2×D4—744no—
280C2×D4—744no—
281C2×D4—744no—
282C2×D4—742no—
283C2×D4—744no—
284C2×D4—742no—
285C2×D4—742no—
286C2×D4—742no—
287C2×D4C22×D3711yescharacteristic
288C22×C4—371no—
289C22×C4—742no—
290C22×C4—741no—
291C22×C4—742no—
292Q16—22no—
293Q16—742no—
294SD16—22no—
295SD16—742no—
296SD16—742no—
297SD16—742no—
298D8—742no—
299D8—742no—
300M16—372no—
301M16—742no—
302M16C22×D3712yescomplemented
303C2×C8—742no—
304C4⋊C4—741no—
305C22⋊C4—22no—
306C22⋊C4—741no—
307C22⋊C4—742no—
308C42—371no—
309C42C22×D3711yescharacteristic, complemented
310C23—22no—
311C23—371no—
312C23—372no—
313C23—2962no—
314C23—2964no—
315C23—742no—
316C23—742no—
317C23—742no—
318C23—742no—
319C23—744no—
320C23—741no—
321C23D4×D3712yes—
322Q8—22no—
323Q8—372no—
324Q8—742no—
325Q8D4×D3712yes—
326D4—24no—
327D4—22no—
328D4—22no—
329D4—372no—
330D4—2962no—
331D4—2962no—
332D4—742no—
333D4—744no—
334D4—744no—
335D4—742no—
336D4—742no—
337D4—744no—
338D4—744no—
339D4—742no—
340D4—742no—
341D4—742no—
342D4—1481no—
343D4—1481no—
344D4D4×D3712yes—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—371no—
350C2×C4—372no—
351C2×C4—372no—
352C2×C4—742no—
353C2×C4—742no—
354C2×C4—744no—
355C2×C4—742no—
356C2×C4—742no—
357C2×C4—742no—
358C2×C4—741no—
359C2×C4—1482no—
360C2×C4C23×D3711yescharacteristic
361C8—22no—
362C8—742no—
363C22—22no—
364C22—22no—
365C22—42no—
366C22—82no—
367C22—371no—
368C22—371no—
369C22—2962no—
370C22—2962no—
371C22—2964no—
372C22—742no—
373C22—742no—
374C22—1482no—
375C22—1481no—
376C22—1482no—
377C22—1484no—
378C22—1484no—
379C22—1481no—
380C22C2×D4×D3711yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—372no—
385C4—742no—
386C4—1481no—
387C4C2×D4×D3712yes—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—2961no—
392C2—741no—
393C2—741no—
394C2—1482no—
395C2—1482no—
396C2#120×D3711yescharacteristic
subgroupG/H
centerC2#120×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#455C2
socleC2×C37C2×#120
hypercenter#455D37
hypocenterC37#3019
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. H(5,𝔽2)×D37C2semidirect
2. #14558332C2semidirect
3. #14558101C2semidirect
4. #14557710C2semidirect
5. #14557709C2semidirect
6. #14557703C2semidirect
7. #14557702C2semidirect
8. #14557701C2semidirect
9. #14557548C2semidirect
10. #100×D37C2semidirect
11. C37×#455C2semidirect
17. C2×#10596966C22semidirect
24. #11924567C22semidirect
25. #11924564C22semidirect
27. #11924561C22semidirect
30. M16×D37C22semidirect
31. #11924533C22semidirect
32. #11924532C22semidirect
33. #11924525C22semidirect
36. #11924493C22semidirect
37. #11924464C22semidirect
38. #11924463C22semidirect
39. #11924462C22semidirect
40. #11924453C22semidirect
41. #11924452C22semidirect
43. #137×C37C22semidirect
44. #124×C37C22semidirect
45. #104×C37C22semidirect
46. #100×C37C22semidirect
51. C2×Q8×C37C23semidirect
63. #10596966D4semidirect
64. Q8×D37D4semidirect
68. #10596964D4semidirect
78. D4×D37D4semidirect
88. C2×D148C23semidirect
95. M16×C37C23semidirect
97. C42×C37C23semidirect
105. C4×Dic37C23semidirect
106. #10596934C23semidirect
204. #455D37semidirect
210. C37#3019semidirect
225. #137C2×D37semidirect
235. #124C2×D37semidirect
242. #104C2×D37semidirect
245. #100C2×D37semidirect
263. C2×Q8C22×D37semidirect
302. M16C22×D37semidirect
309. C42C22×D37semidirect

Maximal subgroups

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

subgroupindex
H(5,𝔽2)×D372
#145583322
#145581012
#145577102
#145577092
#145577032
#145577022
#145577012
#145575482
#100×D372
C37×#4552
#301937