Finite Group Databasebeta

← #17997310

Subgroups — #17997310

Order 4736.

612 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C2×#11924525C211yesmaximal, characteristic
2C2×Q8×Dic37C211yesmaximal, characteristic, complemented
3C2×#11924498C211yesmaximal, characteristic
4#14557435C211yesmaximal, characteristic
5#14557422C214yesmaximal, complemented
6#14557407C214yesmaximal, complemented
7#14557402C211yesmaximal, characteristic, complemented
8#14557379C211yesmaximal, characteristic, complemented
9C37×#391C211yesmaximal, characteristic, complemented
10C22×D148C2211yescharacteristic
11C22×C4×D37C2212yes—
12#11924525C418yescomplemented
13Q8×Dic37—24no—
14#11924498—24no—
15C2×C4⋊Dic37C2211yescharacteristic
16C2×C4⋊Dic37C2212yes—
17C2×#10596936C2214yes—
18C2×#10596938C2212yes—
19C2×#10596938C2214yes—
20C2×#10596938C2214yes—
21C2×C4×Dic37C2212yes—
22C2×C4×Dic37C2211yescharacteristic
23C2×C4×Dic37C2211yescharacteristic
24#11924443C2212yes—
25#11924443C2212yes—
26C22×Q8×C37C2211yescharacteristic
27C2×C4⋊C4×C37C2211yescharacteristic
28C2×C42×C37C2211yescharacteristic
29#95×C37C2214yes—
30C23×D37C2312yes—
31C2×Q8×C37—22no—
32C2×Q8×C37—22no—
33C2×Q8×C37C2×C414yescomplemented
34C22×C4×C37C2312yes—
35C22×C4×C37C2312yes—
36C22×C4×C37C2312yes—
37C22×C4×C37C2311yescharacteristic
38C22×Dic37C2312yes—
39C22×Dic37C2312yes—
40C22×Dic37C2311yescharacteristic
41C22×Dic37C2311yescharacteristic
42C2×D148—22no—
43C2×D148—22no—
44C2×D148C2×C414yes—
45C2×C4×D37—24no—
46C2×C4×D37—44no—
47C4⋊C4×C37—22no—
48C42×C37—22no—
49#10596938—24no—
50#10596938—44no—
51#10596938C2×C4116yescomplemented
52C4⋊Dic37—22no—
53C4⋊Dic37—24no—
54#10596936—44no—
55C4×Dic37—24no—
56C4×Dic37C2×C414yescomplemented
57C4×Dic37D414yes—
58C23×C37C2411yescharacteristic
59C22×D37—22no—
60C22×D37—22no—
61C22×D37—24no—
62C22×D37—24no—
63C22×D37C22×C414yes—
64Q8×C37—28no—
65C2×C4×C37—22no—
66C2×C4×C37—22no—
67C2×C4×C37—22no—
68C2×C4×C37—22no—
69C2×C4×C37—24no—
70C2×C4×C37—22no—
71C2×C4×C37—22no—
72C2×C4×C37C22×C414yes—
73C2×C4×C37C22×C412yes—
74C2×C4×C37C2×D412yes—
75C2×C4×C37#4112yes—
76C2×Dic37—22no—
77C2×Dic37—22no—
78C2×Dic37—22no—
79C2×Dic37—22no—
80C2×Dic37—22no—
81C2×Dic37—24no—
82C2×Dic37C22×C414yes—
83C2×Dic37C2×D412yes—
84C2×Dic37C2×D412yes—
85D148—28no—
86C4×D37—48no—
87C22×C37C23×C411yescharacteristic
88C22×C37C22×D411yescharacteristic
89C22×C37C2×#4111yescharacteristic
90C22×C37H(5,𝔽2)12yes—
91C22×C372−1+412yes—
92C2×D37—24no—
93C2×D37—28no—
94C2×D37—42no—
95C2×D37—42no—
96C2×D37—44no—
97C4×C37—24no—
98C4×C37—42no—
99C4×C37—42no—
100C4×C37C4×D414yes—
101Dic37—24no—
102Dic37—22no—
103Dic37—22no—
104Dic37—42no—
105#2326—371nomaximal, Sylow
106C2×C37C2×C4×D411yescharacteristic
107C2×C37#52012yes—
108C2×C37#53511yescharacteristic
109C2×C37#53611yescharacteristic
110C2×C37#54212yes—
111D37—44no—
112C2×#124—371no—
113C2×C4×Q8—371no—
114C2×C4×D4—371no—
115#391D3711yescharacteristic, complemented
116#390—371no—
117#388—374no—
118#385—374no—
119#384—371no—
120#376—371no—
121C37#232611yescharacteristic, complemented, Sylow
122C22×Q8C2×D3711yescharacteristic
123C22×D4—371no—
124C23×C4—372no—
125#124—378no—
126C4×Q8—744no—
127C4×D4—744no—
128C2×C4⋊C4—371no—
129C2×C4⋊C4—372no—
130C2×C4⋊C4—374no—
131C2×C4⋊C4C2×D3711yescharacteristic
132C2×C22⋊C4—374no—
133C2×C22⋊C4—372no—
134C2×C22⋊C4—374no—
135C2×C42—371no—
136C2×C42—371no—
137C2×C42—372no—
138C2×C42C2×D3711yescharacteristic
139#95—372no—
140#95—372no—
141#95C2×D3714yes—
142C24—372no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8C4×D3714yescomplemented
146C2×D4—374no—
147C2×D4—742no—
148C2×D4—742no—
149C22×C4—371no—
150C22×C4—372no—
151C22×C4—371no—
152C22×C4—372no—
153C22×C4—744no—
154C22×C4—1484no—
155C22×C4C22×D3712yes—
156C22×C4C22×D3712yes—
157C22×C4C22×D3712yes—
158C22×C4C22×D3711yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—744no—
161C4⋊C4—742no—
162C4⋊C4—1484no—
163C22⋊C4—3716no—
164C22⋊C4—744no—
165C22⋊C4—1484no—
166C42—22no—
167C42—374no—
168C42—374no—
169C42—744no—
170C23—374no—
171C23—742no—
172C23—742no—
173C23—744no—
174C23—744no—
175C23C23×D3711yescharacteristic
176Q8—28no—
177D4—748no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—374no—
186C2×C4—372no—
187C2×C4—372no—
188C2×C4—742no—
189C2×C4—742no—
190C2×C4—742no—
191C2×C4—742no—
192C2×C4—742no—
193C2×C4—744no—
194C2×C4—1488no—
195C2×C4C2×C4×D3714yes—
196C2×C4C2×C4×D3712yes—
197C2×C4D4×D3712yes—
198C2×C4#1059696412yes—
199C22—744no—
200C22—748no—
201C22—1482no—
202C22—1482no—
203C22—1484no—
204C22C22×C4×D3711yescharacteristic
205C22C2×D4×D3711yescharacteristic
206C22C2×#1059696411yescharacteristic
207C22#1192458212yes—
208C22#1192458412yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—744no—
213C4—742no—
214C4—742no—
215C4—1482no—
216C4#1192449614yes—
217C2—1484no—
218C2C2×#1192449611yescharacteristic
219C2#1455801512yes—
220C2#1455810111yescharacteristic
221C2#1455810811yescharacteristic
222C2#1455815612yes—
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#391C2
socle, cosocleC23×C37C24
prime radicalC22×D148C22
hypercenter#391D37
hypocenterC37#2326
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic37C2semidirect
5. #14557422C2semidirect
6. #14557407C2semidirect
7. #14557402C2semidirect
8. #14557379C2semidirect
9. C37×#391C2semidirect
12. #11924525C4semidirect
33. C2×Q8×C37C2×C4semidirect
51. #10596938C2×C4semidirect
56. C4×Dic37C2×C4semidirect
115. #391D37semidirect
121. C37#2326semidirect
145. C2×Q8C4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245252
C2×Q8×Dic372
C2×#119244982
#145574352
#145574222
#145574072
#145574022
#145573792
C37×#3912
#232637