Finite Group Databasebeta

← #456×D37 #18003935

Subgroups — #456×D37 #18003935

Order 4736.

708 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 135 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
12−1+4×D37C211yesmaximal, characteristic
2#137×D37C212yesmaximal, complemented
3C4⋊Q8×D37C211yesmaximal, characteristic, complemented
4#14557718C211yesmaximal, characteristic, complemented
5#14557717C212yesmaximal, complemented
6#14557716C211yesmaximal, characteristic, complemented
7#14557715C211yesmaximal, characteristic, complemented
8#14557714C212yesmaximal, complemented
9#104×D37C212yesmaximal, complemented
10#101×D37C211yesmaximal, characteristic, complemented
11C37×#456C211yesmaximal, characteristic, complemented
12#11924584—22no—
13#11924584C2211yescharacteristic
14#11924583—22no—
15#11924583—22no—
16#11924583C2212yes—
17#41×D37—22no—
18#41×D37—22no—
19#41×D37C2212yes—
20C2×Q8×D37—22no—
21C2×Q8×D37C2211yescharacteristic
22C2×Q8×D37C2211yescharacteristic
23#11924569C2212yescomplemented
24#11924568—22no—
25#11924567C2212yescomplemented
26#11924566C2212yescomplemented
27#11924565—22no—
28Q16×D37—22no—
29SD16×D37—22no—
30M16×D37C2212yes—
31#11924536C2211yescharacteristic, complemented
32#11924535—22no—
33#11924534C2211yescharacteristic, complemented
34#11924534C2211yescharacteristic, complemented
35C4⋊C4×D37—22no—
36C42×D37C2211yescharacteristic
37#11924464C2212yescomplemented
38#11924463C2212yescomplemented
39#11924462C2212yescomplemented
40#11924455C2211yescharacteristic, complemented
41#11924454C2212yescomplemented
422−1+4×C37C2211yescharacteristic
43#137×C37C2212yescomplemented
44C4⋊Q8×C37C2211yescharacteristic, complemented
45#104×C37C2212yescomplemented
46#101×C37C2211yescharacteristic, complemented
47#41×C37—22no—
48#41×C37—22no—
49#41×C37C2312yes—
50C2×Q8×C37—22no—
51C2×Q8×C37C2311yescharacteristic
52C2×Q8×C37C2311yescharacteristic
53#10596966—22no—
54#10596966—22no—
55#10596966D412yes—
56Q8×D37—22no—
57Q8×D37—24no—
58Q8×D37—22no—
59Q8×D37—22no—
60Q8×D37—22no—
61Q8×D37—22no—
62Q8×D37—22no—
63Q8×D37—22no—
64Q8×D37D412yes—
65#10596964—22no—
66#10596964—24no—
67#10596964—22no—
68#10596964—24no—
69#10596964—22no—
70#10596964D412yes—
71D4×D37—22no—
72D4×D37—22no—
73D4×D37D412yes—
74#10596962—24no—
75#10596962—24no—
76#10596962—22no—
77#10596962—22no—
78#10596962—22no—
79#10596962C2312yes—
80C2×C4×D37—24no—
81C2×C4×D37—21no—
82C2×C4×D37—22no—
83C2×C4×D37—21no—
84C2×C4×D37D412yes—
85C2×C4×D37C2311yescharacteristic
86C2×Dic74—24no—
87C2×Dic74—22no—
88C2×Dic74—21no—
89C2×Dic74D412yes—
90C2×Dic74C2311yescharacteristic
91C2×Dic74C2311yescharacteristic
92Q16×C37—22no—
93SD16×C37—22no—
94M16×C37C2312yes—
95C4⋊C4×C37—22no—
96C42×C37C2311yescharacteristic
97#10596942—22no—
98#10596942—22no—
99#10596941—22no—
100#10596940—22no—
101C4⋊Dic37—22no—
102#10596936—22no—
103#10596936—22no—
104C4×Dic37—21no—
105C4×Dic37C2311yescharacteristic
106#10596934C2312yes—
107Dic148—22no—
108#10596930—22no—
109#10596929—22no—
110C8×D37—22no—
111C22×D37—22no—
112C22×D37C2×D411yescharacteristic
113Q8×C37—22no—
114Q8×C37—22no—
115Q8×C37—22no—
116Q8×C37C2×D412yes—
117D4×C37—22no—
118D4×C37—22no—
119D4×C37C2×D412yes—
120C2×C4×C37—24no—
121C2×C4×C37—21no—
122C2×C4×C37—22no—
123C2×C4×C37—21no—
124C2×C4×C37C2×D412yes—
125C2×C4×C37C2411yescharacteristic
126C22⋊D37—24no—
127C22⋊D37—22no—
128C22⋊D37—22no—
129C22⋊D37—22no—
130C2×Dic37—24no—
131C2×Dic37—21no—
132C2×Dic37—22no—
133C2×Dic37—21no—
134C2×Dic37C2×D412yes—
135C2×Dic37C2×D411yescharacteristic
136D148—22no—
137D148—22no—
138D148C2×D412yes—
139C4×D37—24no—
140C4×D37—24no—
141C4×D37—24no—
142C4×D37—22no—
143C4×D37—22no—
144C4×D37—22no—
145C4×D37—22no—
146C4×D37—22no—
147C4×D37—41no—
148C4×D37—41no—
149C4×D37C2×D412yes—
150C4×D37C2×D412yes—
151Dic74—24no—
152Dic74—22no—
153Dic74—22no—
154Dic74—22no—
155Dic74—24no—
156Dic74—22no—
157Dic74—42no—
158Dic74C2×D412yes—
159C8×C37—22no—
160#64436—22no—
161C22×C37—22no—
162C22×C37C22×D411yescharacteristic
163C2×D37—22no—
164C2×D37—22no—
165C2×D37—44no—
166C2×D37#12011yescharacteristic
167C2×D37#12011yescharacteristic
168C4×C37—22no—
169C4×C37—22no—
170C4×C37—22no—
171C4×C37—41no—
172C4×C37C22×D412yes—
173Dic37—22no—
174Dic37—22no—
175Dic37—22no—
176Dic37—41no—
177Dic37#12012yes—
178C2×#456—371nomaximal, Sylow
179C2×C37—21no—
180C2×C37—42no—
181C2×C37C2×#12011yescharacteristic
182D37—21no—
183D37—42no—
184D37#45612yesdirect product factor
185C2×2−1+4—371no—
186C2×#137—372no—
187C2×C4⋊Q8—371no—
188#456—371no—
189#456—372no—
190#456—371no—
191#456—372no—
192#456—371no—
193#456D3711yescharacteristic, direct product factor
194C2×#104—372no—
195C2×#101—371no—
196C37C2×#45611yescharacteristic, complemented, Sylow
1972−1+4—372no—
1982−1+4—371no—
1992−1+4—742no—
2002−1+4—742no—
2012−1+4—742no—
2022−1+4C2×D3711yescharacteristic
203C2×#41—372no—
204C2×#41—742no—
205C2×#41—742no—
206C22×Q8—371no—
207C22×Q8—371no—
208C22×Q8—742no—
209#137—372no—
210#137—372no—
211#137—372no—
212#137—742no—
213#137—742no—
214#137C2×D3712yescomplemented
215C2×Q16—742no—
216C2×SD16—742no—
217C2×M16—372no—
218C4⋊Q8—371no—
219C4⋊Q8—371no—
220C4⋊Q8—371no—
221C4⋊Q8—742no—
222C4⋊Q8C2×D3711yescharacteristic, complemented
223C2×C4⋊C4—742no—
224C2×C42—371no—
225#104—372no—
226#104—372no—
227#104—372no—
228#104C2×D3712yescomplemented
229#101—371no—
230#101—372no—
231#101C2×D3711yescharacteristic, complemented
232#41—22no—
233#41—22no—
234#41—372no—
235#41—372no—
236#41—372no—
237#41—742no—
238#41—742no—
239#41—744no—
240#41—744no—
241#41—742no—
242#41—744no—
243#41—742no—
244#41—744no—
245#41—742no—
246#41—742no—
247#41—742no—
248#41—742no—
249#41C22×D3712yes—
250C2×Q8—22no—
251C2×Q8—372no—
252C2×Q8—371no—
253C2×Q8—372no—
254C2×Q8—371no—
255C2×Q8—741no—
256C2×Q8—742no—
257C2×Q8—744no—
258C2×Q8—742no—
259C2×Q8—742no—
260C2×Q8—742no—
261C2×Q8—742no—
262C2×Q8—742no—
263C2×Q8—742no—
264C2×Q8—744no—
265C2×Q8—742no—
266C2×Q8C22×D3711yescharacteristic
267C2×Q8C22×D3711yescharacteristic
268C2×D4—372no—
269C2×D4—742no—
270C2×D4—742no—
271C22×C4—371no—
272C22×C4—372no—
273C22×C4—742no—
274C22×C4—741no—
275C22×C4—744no—
276C22×C4—741no—
277Q16—22no—
278Q16—742no—
279Q16—742no—
280Q16—742no—
281SD16—22no—
282SD16—742no—
283SD16—742no—
284SD16—742no—
285M16—372no—
286M16—742no—
287M16C22×D3712yes—
288C2×C8—742no—
289C4⋊C4—22no—
290C4⋊C4—742no—
291C4⋊C4—742no—
292C4⋊C4—742no—
293C42—371no—
294C42—741no—
295C42C22×D3711yescharacteristic
296C23—371no—
297C23—742no—
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8—372no—
302Q8—742no—
303Q8—744no—
304Q8—744no—
305Q8—742no—
306Q8—742no—
307Q8—742no—
308Q8—1482no—
309Q8D4×D3712yes—
310D4—22no—
311D4—22no—
312D4—372no—
313D4—742no—
314D4—742no—
315D4—742no—
316D4—742no—
317D4—744no—
318D4—742no—
319D4D4×D3712yes—
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—371no—
325C2×C4—372no—
326C2×C4—372no—
327C2×C4—372no—
328C2×C4—742no—
329C2×C4—742no—
330C2×C4—742no—
331C2×C4—742no—
332C2×C4—744no—
333C2×C4—744no—
334C2×C4—744no—
335C2×C4—742no—
336C2×C4—744no—
337C2×C4—741no—
338C2×C4—742no—
339C2×C4—741no—
340C2×C4—1481no—
341C2×C4—1481no—
342C2×C4D4×D3712yes—
343C2×C4C23×D3711yescharacteristic
344C8—22no—
345C8—742no—
346C22—22no—
347C22—371no—
348C22—371no—
349C22—742no—
350C22—742no—
351C22—1484no—
352C22C2×D4×D3711yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—372no—
358C4—742no—
359C4—742no—
360C4—742no—
361C4—1481no—
362C4C2×D4×D3712yes—
363C2—21no—
364C2—42no—
365C2—372no—
366C2—741no—
367C2—1482no—
368C2#120×D3711yescharacteristic
subgroupG/H
centerC2#120×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#456C2
socleC2×C37C2×#120
prime radical2−1+4×D37C2
hypercenter#456D37
hypocenterC37C2×#456
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
2. #137×D37C2semidirect
3. C4⋊Q8×D37C2semidirect
4. #14557718C2semidirect
5. #14557717C2semidirect
6. #14557716C2semidirect
7. #14557715C2semidirect
8. #14557714C2semidirect
9. #104×D37C2semidirect
10. #101×D37C2semidirect
11. C37×#456C2semidirect
23. #11924569C22semidirect
25. #11924567C22semidirect
26. #11924566C22semidirect
31. #11924536C22semidirect
33. #11924534C22semidirect
34. #11924534C22semidirect
37. #11924464C22semidirect
38. #11924463C22semidirect
39. #11924462C22semidirect
40. #11924455C22semidirect
41. #11924454C22semidirect
43. #137×C37C22semidirect
44. C4⋊Q8×C37C22semidirect
45. #104×C37C22semidirect
46. #101×C37C22semidirect
184. D37#456direct (N × H)
193. #456D37direct (N × H)
196. C37C2×#456semidirect
214. #137C2×D37semidirect
222. C4⋊Q8C2×D37semidirect
228. #104C2×D37semidirect
231. #101C2×D37semidirect

Maximal subgroups

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

subgroupindex
2−1+4×D372
#137×D372
C4⋊Q8×D372
#145577182
#145577172
#145577162
#145577152
#145577142
#104×D372
#101×D372
C37×#4562
C2×#45637