Finite Group Databasebeta

← #17995094

Subgroups — #17995094

Order 4736.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#14558331C211yesmaximal, characteristic
2#14558001C211yesmaximal, characteristic, complemented
3#14557551C211yesmaximal, characteristic, complemented
4C2×#11924463C211yesmaximal, characteristic, complemented
5#14557303C211yesmaximal, characteristic, complemented
6#14557292C211yesmaximal, characteristic, complemented
7C37×#410C211yesmaximal, characteristic, complemented
8C2×#10596962C2211yescharacteristic
9#11924567C412yescomplemented
10#11924566—21no—
11#11924562C412yescomplemented
12#11924561—21no—
13#11924556—22no—
14C2×#10596934C2211yescharacteristic
15C2×#10596934C2211yescharacteristic
16#11924501—21no—
17#11924498—21no—
18#11924496—22no—
19#11924493—21no—
20C2×C4×Dic37C2211yescharacteristic
21C42×D37—21no—
22#11924464—22no—
23#11924463—21no—
24#11924463—21no—
25C2×#41×C37C2211yescharacteristic
26#117×C37C2211yescharacteristic
27#117×C37C2211yescharacteristic
28#99×C37—22no—
29#41×C37—21no—
30#41×C37—21no—
31#41×C37—21no—
32#41×C37—21no—
33C2×Q8×C37C2×C411yescharacteristic, complemented
34C2×D4×C37—21no—
35C2×D4×C37C2×C411yescharacteristic, complemented
36C22×C4×C37—21no—
37C22×C4×C37C2311yescharacteristic
38C2×C22⋊D37—21no—
39C22×Dic37—21no—
40#10596962—21no—
41#10596962—21no—
42#10596962C2×C412yes—
43#10596962D412yes—
44C2×D148D411yescharacteristic
45C2×C4×D37—21no—
46C2×C4×D37—22no—
47C2×Dic74D411yescharacteristic
48C4⋊C4×C37—21no—
49C4⋊C4×C37—21no—
50C22⋊C4×C37—21no—
51C22⋊C4×C37D412yes—
52C42×C37—21no—
53C42×C37—21no—
54#10596942—22no—
55#10596941—22no—
56#10596940—22no—
57#10596939—22no—
58#10596938—22no—
59#10596936—22no—
60C4×Dic37—22no—
61C4×Dic37—21no—
62C4×Dic37D411yescharacteristic
63C4×Dic37D411yescharacteristic
64#10596934—21no—
65#10596934—21no—
66#10596934—21no—
67#10596934C2×C412yescomplemented
68C2×#64436—21no—
69C2×#64436—21no—
70C23×C37—21no—
71C23×C37C2×D411yescharacteristic
72C22×D37—21no—
73Q8×C37—21no—
74Q8×C37—21no—
75D4×C37—21no—
76D4×C37—21no—
77D4×C37—41no—
78D4×C37—41no—
79C2×C4×C37—21no—
80C2×C4×C37—22no—
81C2×C4×C37—41no—
82C2×C4×C37—41no—
83C2×C4×C37—41no—
84C2×C4×C37C22×C411yescharacteristic
85C2×C4×C37C2×D411yescharacteristic
86C2×C4×C37C2×D411yescharacteristic
87C2×C4×C37C2×D411yescharacteristic
88C2×C4×C37#4111yescharacteristic
89C2×C4×C37#4111yescharacteristic
90C22⋊D37—22no—
91C22⋊D37—22no—
92C2×Dic37—21no—
93C2×Dic37—21no—
94C2×Dic37—21no—
95C2×Dic37—42no—
96D148—21no—
97D148C22⋊C412yes—
98C4×D37—22no—
99C4×D37—22no—
100C4×D37—44no—
101Dic74—21no—
102Dic74C22⋊C412yes—
103#64436—22no—
104#64436—41no—
105C22×C37—41no—
106C22×C37—41no—
107C22×C37—81no—
108C22×C37C4×D411yescharacteristic
109C22×C37#12011yescharacteristic
110C22×C37#12111yescharacteristic
111C2×D37—22no—
112C2×D37—42no—
113C4×C37—42no—
114C4×C37—41no—
115C4×C37—81no—
116C4×C37C2×C22⋊C411yescharacteristic
117C4×C37C4×D411yescharacteristic
118C4×C37#12111yescharacteristic
119C4×C37#12311yescharacteristic
120Dic37—22no—
121Dic37—22no—
122Dic37—41no—
123#1906—371nomaximal, Sylow
124C2×C37—21no—
125C2×C37—21no—
126C2×C37—21no—
127C2×C37—81no—
128C2×C37#38611yescharacteristic
129D37—42no—
130#575—371no—
131C4×#41—371no—
132#421—371no—
133C2×#104—371no—
134#410D3711yescharacteristic, complemented
135#344—371no—
136#339—371no—
137C37#190611yescharacteristic, complemented, Sylow
138C2×#41—371no—
139C2×#41C2×D3711yescharacteristic
140#137—372no—
141#137—741no—
142AGL(1,ℤ/8)—372no—
143AGL(1,ℤ/8)—741no—
144#135—742no—
145C2×M16—371no—
146C2×M16—371no—
147C4×Q8—741no—
148C4×D4—741no—
149C4×D4—742no—
150#117—741no—
151#117C2×D3711yescharacteristic
152#117C2×D3711yescharacteristic
153C2×C42—371no—
154C2×C42—741no—
155#104—742no—
156#104—741no—
157#104—741no—
158#99—22no—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—372no—
164#41—372no—
165#41—741no—
166#41—741no—
167C2×Q8—371no—
168C2×Q8C4×D3711yescharacteristic, complemented
169C2×D4—21no—
170C2×D4—371no—
171C2×D4—741no—
172C2×D4C4×D3711yescharacteristic, complemented
173C22×C4—21no—
174C22×C4—742no—
175C22×C4—741no—
176C22×C4—741no—
177C22×C4C22×D3711yescharacteristic
178Q16—742no—
179SD16—742no—
180SD16—742no—
181D8—742no—
182M16—372no—
183M16—741no—
184M16—741no—
185M16—741no—
186C2×C8—741no—
187C2×C8—741no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—742no—
191C22⋊C4—21no—
192C22⋊C4—742no—
193C22⋊C4C22⋊D3712yes—
194C42—21no—
195C42—21no—
196C42—371no—
197C42—371no—
198C42—742no—
199C42—741no—
200C23—21no—
201C23—741no—
202C23C2×C22⋊D3711yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—372no—
206Q8—741no—
207D4—21no—
208D4—21no—
209D4—41no—
210D4—41no—
211D4—372no—
212D4—742no—
213D4—742no—
214D4—741no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—741no—
221C2×C4—741no—
222C2×C4—742no—
223C2×C4—742no—
224C2×C4—741no—
225C2×C4—1484no—
226C2×C4—1482no—
227C2×C4C2×C4×D3711yescharacteristic
228C2×C4#1059696211yescharacteristic
229C2×C4D4×D3711yescharacteristic
230C2×C4D4×D3711yescharacteristic
231C2×C4D4×D3711yescharacteristic
232C2×C4#1059696411yescharacteristic
233C8—742no—
234C8—1481no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—742no—
239C22—1482no—
240C22C4×C22⋊D3711yescharacteristic
241C22#1192450411yescharacteristic
242C22#1192450711yescharacteristic
243C4—42no—
244C4—41no—
245C4—81no—
246C4—742no—
247C4—742no—
248C4—1481no—
249C4C22⋊C4×D3711yescharacteristic
250C4#1192449611yescharacteristic
251C4#1192450911yescharacteristic
252C4#1192451811yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—81no—
257C2—1482no—
258C2#1455740911yescharacteristic
subgroupG/H
centerC4#11924509
commutatorC2×C4×C37C22×C4
FrattiniC22×C4C22×D37
FittingC37×#410C2
socleC2×C37#386
prime radical#14558331C2
cosocleC22×C4×C37C23
hypercenter#410D37
hypocenterC37#1906
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #14558001C2semidirect
3. #14557551C2semidirect
4. C2×#11924463C2semidirect
5. #14557303C2semidirect
6. #14557292C2semidirect
7. C37×#410C2semidirect
9. #11924567C4semidirect
11. #11924562C4semidirect
33. C2×Q8×C37C2×C4semidirect
35. C2×D4×C37C2×C4semidirect
67. #10596934C2×C4semidirect
134. #410D37semidirect
137. C37#1906semidirect
168. C2×Q8C4×D37semidirect
172. C2×D4C4×D37semidirect

Maximal subgroups

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

subgroupindex
#145583312
#145580012
#145575512
C2×#119244632
#145573032
#145572922
C37×#4102
#190637