Finite Group Databasebeta

← #353×D37 #17996252

Subgroups — #353×D37 #17996252

Order 4736.

368 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 57 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#127×D37C211yesmaximal, characteristic
2#99×D37C212yesmaximal, complemented
3#14557326C211yesmaximal, characteristic, complemented
4#14557325C212yesmaximal, complemented
5C37×#353C211yesmaximal, characteristic, complemented
6C2×D4×D37—22no—
7C2×D4×D37C411yescharacteristic, complemented
8C2×D4×D37C2211yescharacteristic
9#11924533C411yescharacteristic, complemented
10#11924532—22no—
11#11924532C411yescharacteristic, complemented
12#11924532C2211yescharacteristic
13C22⋊C4×D37—22no—
14C42×D37C411yescharacteristic, complemented
15#11924451C2212yescomplemented
16#11924450—22no—
17#127×C37C2211yescharacteristic
18#99×C37C2212yescomplemented
19C23×D37—21no—
20C23×D37D412yes—
21C2×D4×C37—22no—
22C2×D4×C37C2×C411yescharacteristic, complemented
23C2×D4×C37C2311yescharacteristic
24C2×C22⋊D37—22no—
25C2×C22⋊D37—22no—
26C2×C22⋊D37—21no—
27C2×C22⋊D37D412yes—
28D4×D37—41no—
29D4×D37—41no—
30D4×D37—41no—
31D4×D37—41no—
32D4×D37—48no—
33C2×D148—22no—
34C2×D148C2×C411yescharacteristic
35C2×D148C2×C411yescharacteristic, complemented
36C2×C4×D37—21no—
37C2×C4×D37—42no—
38C2×C4×D37C2×C411yescharacteristic
39C22⋊C4×C37—22no—
40C42×C37C2×C411yescharacteristic, complemented
41C22⋊Dic37—22no—
42#10596938—42no—
43C4×Dic37—21no—
44C4×Dic37C2×C411yescharacteristic, complemented
45C23×C37—21no—
46C23×C37C2×D412yes—
47C22×D37—22no—
48C22×D37—22no—
49C22×D37—21no—
50C22×D37—44no—
51C22×D37—44no—
52C22×D37—41no—
53C22×D37—41no—
54C22×D37C22⋊C411yescharacteristic
55C22×D37C22⋊C412yes—
56D4×C37—41no—
57D4×C37—41no—
58D4×C37—44no—
59C2×C4×C37—21no—
60C2×C4×C37—42no—
61C2×C4×C37C22×C411yescharacteristic
62C22⋊D37—42no—
63C22⋊D37—44no—
64C22⋊D37—44no—
65C22⋊D37—42no—
66C2×Dic37—21no—
67C2×Dic37—42no—
68C2×Dic37C22⋊C411yescharacteristic
69D148—41no—
70D148—41no—
71D148—44no—
72C4×D37—24no—
73C4×D37—21no—
74C4×D37—21no—
75C4×D37—84no—
76C22×C37—22no—
77C22×C37—42no—
78C22×C37—42no—
79C22×C37—41no—
80C22×C37C2×C22⋊C411yescharacteristic
81C2×D37—22no—
82C2×D37—22no—
83C2×D37—44no—
84C2×D37—42no—
85C2×D37—42no—
86C2×D37—41no—
87C2×D37—82no—
88C2×D37—82no—
89C2×D37—82no—
90C2×D37#9911yescharacteristic
91C2×D37#9911yescharacteristic
92C4×C37—22no—
93C4×C37—21no—
94C4×C37—82no—
95Dic37—22no—
96Dic37—21no—
97Dic37—82no—
98C2×#353—371nomaximal, Sylow
99C2×C37—21no—
100C2×C37—42no—
101C2×C37—81no—
102C2×C37C2×#9911yescharacteristic
103D37—21no—
104D37—42no—
105D37—81no—
106D37#35312yesdirect product factor
107C2×#127—371no—
108C2×#99—372no—
109#353—371no—
110#353—372no—
111#353D3711yescharacteristic, direct product factor
112C37C2×#35311yescharacteristic, complemented, Sylow
113C22×D4—371no—
114C22×D4—371no—
115C22×D4—742no—
116#127—371no—
117#127—371no—
118#127—371no—
119#127—742no—
120#127C2×D3711yescharacteristic
121C2×C22⋊C4—742no—
122C2×C42—371no—
123#99—372no—
124#99—742no—
125#99C2×D3712yescomplemented
126C24—372no—
127C24—741no—
128C2×D4—22no—
129C2×D4—372no—
130C2×D4—371no—
131C2×D4—371no—
132C2×D4—742no—
133C2×D4—741no—
134C2×D4—742no—
135C2×D4—742no—
136C2×D4—1481no—
137C2×D4—1481no—
138C2×D4—1481no—
139C2×D4—1481no—
140C2×D4—1488no—
141C2×D4C4×D3711yescharacteristic, complemented
142C2×D4C22×D3711yescharacteristic
143C22×C4—371no—
144C22×C4—741no—
145C22×C4—1482no—
146C22⋊C4—22no—
147C22⋊C4—742no—
148C22⋊C4—1482no—
149C42—371no—
150C42—741no—
151C42C4×D3711yescharacteristic, complemented
152C23—21no—
153C23—371no—
154C23—372no—
155C23—742no—
156C23—742no—
157C23—741no—
158C23—1481no—
159C23—1481no—
160C23—1484no—
161C23—1484no—
162C23D4×D3712yes—
163D4—41no—
164D4—41no—
165D4—44no—
166D4—1481no—
167D4—1481no—
168D4—1484no—
169D4—1482no—
170D4—1484no—
171D4—1484no—
172D4—1482no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—371no—
176C2×C4—2964no—
177C2×C4—741no—
178C2×C4—741no—
179C2×C4—744no—
180C2×C4—741no—
181C2×C4—1482no—
182C2×C4C2×C4×D3711yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—371no—
188C22—371no—
189C22—2962no—
190C22—2962no—
191C22—2962no—
192C22—742no—
193C22—742no—
194C22—1482no—
195C22—1482no—
196C22—1484no—
197C22—1481no—
198C22C22⋊C4×D3711yescharacteristic
199C4—22no—
200C4—21no—
201C4—82no—
202C4—2962no—
203C4—742no—
204C4—741no—
205C2—21no—
206C2—42no—
207C2—81no—
208C2—372no—
209C2—2961no—
210C2—741no—
211C2—1482no—
212C2#99×D3711yescharacteristic
subgroupG/H
centerC2#99×D37
commutatorC2×C4×C37C22×C4
FrattiniC2×D4C22×D37
FittingC37×#353C2
socleC2×C37C2×#99
prime radical#127×D37C2
cosocleC2×D4×C37C23
hypercenter#353D37
hypocenterC37C2×#353
LCSG ▷ C2×C4×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D37C2semidirect
3. #14557326C2semidirect
4. #14557325C2semidirect
5. C37×#353C2semidirect
7. C2×D4×D37C4semidirect
9. #11924533C4semidirect
11. #11924532C4semidirect
14. C42×D37C4semidirect
15. #11924451C22semidirect
18. #99×C37C22semidirect
22. C2×D4×C37C2×C4semidirect
35. C2×D148C2×C4semidirect
40. C42×C37C2×C4semidirect
44. C4×Dic37C2×C4semidirect
106. D37#353direct (N × H)
111. #353D37direct (N × H)
112. C37C2×#353semidirect
125. #99C2×D37semidirect
141. C2×D4C4×D37semidirect
151. C42C4×D37semidirect

Maximal subgroups

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

subgroupindex
#127×D372
#99×D372
#145573262
#145573252
C37×#3532
C2×#35337