Finite Group Databasebeta

← #17997217

Subgroups — #17997217

Order 4736.

600 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 229 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×#11924494C211yesmaximal, characteristic
2C2×#11924493C211yesmaximal, characteristic
3C2×C4⋊C4×D37C211yesmaximal, characteristic
4#14557428C214yesmaximal, complemented
5#14557404C214yesmaximal
6#14557402C211yesmaximal, characteristic, complemented
7#14557399C211yesmaximal, characteristic, complemented
8#14557398C211yesmaximal, characteristic, complemented
9C37×#384C211yesmaximal, characteristic, complemented
10C22×C4×D37C2212yes—
11C22×C4×D37C2211yescharacteristic
12#11924494—24no—
13#11924493C418yescomplemented
14C2×C4⋊Dic37C2211yescharacteristic
15C2×C4⋊Dic37C2211yescharacteristic
16C2×#10596936C2212yes—
17C2×#10596936C2214yes—
18C2×#10596936C2212yes—
19C4⋊C4×D37—24no—
20C2×#10596938C2212yes—
21C2×#10596938C2212yes—
22C2×C4×Dic37C2212yes—
23C2×C4×Dic37C2211yescharacteristic
24#11924443C2214yes—
25#11924443C2214yes—
26C2×C4⋊C4×C37C2211yescharacteristic
27C2×C4⋊C4×C37C2211yescharacteristic
28C2×C4⋊C4×C37C2214yes—
29C2×C42×C37C2211yescharacteristic
30C23×D37C2311yescharacteristic
31C22×C4×C37C2312yes—
32C22×C4×C37C2312yes—
33C22×C4×C37C2312yes—
34C22×C4×C37C2311yescharacteristic
35C22×Dic37C2312yes—
36C22×Dic37C2312yes—
37C22×Dic37C2312yes—
38C22×Dic37C2311yescharacteristic
39C2×C4×D37—24no—
40C2×C4×D37—44no—
41C2×C4×D37C2×C414yes—
42C2×C4×D37D414yes—
43C2×C4×D37Q814yes—
44C4⋊C4×C37—22no—
45C4⋊C4×C37—22no—
46C4⋊C4×C37—44no—
47C42×C37C2×C414yescomplemented
48#10596938—24no—
49#10596938C2×C418yes—
50C4⋊Dic37—22no—
51C4⋊Dic37—22no—
52#10596936—24no—
53#10596936—44no—
54#10596936C2×C418yescomplemented
55C4×Dic37—24no—
56C4×Dic37C2×C414yescomplemented
57C23×C37C2411yescharacteristic
58C22×D37—22no—
59C22×D37—22no—
60C22×D37C22×C412yes—
61C22×D37C2×D412yes—
62C22×D37C2×Q812yes—
63C2×C4×C37—22no—
64C2×C4×C37—22no—
65C2×C4×C37—22no—
66C2×C4×C37—22no—
67C2×C4×C37—24no—
68C2×C4×C37—24no—
69C2×C4×C37C22×C414yes—
70C2×C4×C37C22×C412yes—
71C2×C4×C37C2×D412yes—
72C2×C4×C37C2×Q812yes—
73C2×Dic37—22no—
74C2×Dic37—22no—
75C2×Dic37—22no—
76C2×Dic37—22no—
77C2×Dic37—24no—
78C2×Dic37—24no—
79C2×Dic37C22×C414yes—
80C2×Dic37C22×C412yes—
81C2×Dic37C2×D412yes—
82C2×Dic37C2×Q812yes—
83C4×D37—48no—
84C4×D37C4⋊C4116yescomplemented
85C22×C37C23×C411yescharacteristic
86C22×C37C22×D411yescharacteristic
87C22×C37C22×Q811yescharacteristic
88C22×C37H(5,𝔽2)12yes—
89C22×C372−1+412yes—
90C2×D37—24no—
91C2×D37—22no—
92C2×D37—22no—
93C2×D37—24no—
94C2×D37C2×C4⋊C414yes—
95C4×C37—24no—
96C4×C37—42no—
97C4×C37—42no—
98C4×C37C2×C4⋊C414yes—
99Dic37—24no—
100Dic37—42no—
101Dic37—42no—
102Dic37C2×C4⋊C414yes—
103#2317—371nomaximal, Sylow
104C2×C37C22×C4⋊C411yescharacteristic
105C2×C37#52012yes—
106C2×C37#53511yescharacteristic
107C2×C37#53611yescharacteristic
108C2×C37#54412yes—
109D37—24no—
110C2×#117—371no—
111C2×#117—371no—
112C22×C4⋊C4—371no—
113#389—374no—
114#385—374no—
115#384—371no—
116#384D3711yescharacteristic, complemented
117#382—371no—
118#382—371no—
119C37#231711yescharacteristic, complemented, Sylow
120C23×C4—371no—
121C23×C4—372no—
122#117—378no—
123#117—744no—
124C2×C4⋊C4—372no—
125C2×C4⋊C4—372no—
126C2×C4⋊C4—371no—
127C2×C4⋊C4—371no—
128C2×C4⋊C4—374no—
129C2×C4⋊C4—744no—
130C2×C4⋊C4C2×D3711yescharacteristic
131C2×C4⋊C4C2×D3711yescharacteristic
132C2×C4⋊C4C2×D3714yes—
133C2×C22⋊C4—372no—
134C2×C22⋊C4—372no—
135C2×C42—371no—
136C2×C42—372no—
137C2×C42C2×D3711yescharacteristic
138#95—374no—
139#95—374no—
140C24—371no—
141C22×C4—371no—
142C22×C4—374no—
143C22×C4—374no—
144C22×C4—374no—
145C22×C4—372no—
146C22×C4—372no—
147C22×C4—372no—
148C22×C4—744no—
149C22×C4—1484no—
150C22×C4C22×D3712yes—
151C22×C4C22×D3712yes—
152C22×C4C22×D3712yes—
153C22×C4C22×D3711yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—44no—
157C4⋊C4—378no—
158C4⋊C4—744no—
159C4⋊C4—742no—
160C4⋊C4—742no—
161C4⋊C4—1484no—
162C22⋊C4—378no—
163C22⋊C4—744no—
164C42—374no—
165C42—744no—
166C42C4×D3714yescomplemented
167C23—372no—
168C23—372no—
169C23—372no—
170C23—742no—
171C23—742no—
172C23C23×D3711yescharacteristic
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—372no—
180C2×C4—372no—
181C2×C4—372no—
182C2×C4—3716no—
183C2×C4—374no—
184C2×C4—742no—
185C2×C4—742no—
186C2×C4—742no—
187C2×C4—742no—
188C2×C4—744no—
189C2×C4—744no—
190C2×C4—1488no—
191C2×C4C2×C4×D3714yes—
192C2×C4C2×C4×D3712yes—
193C2×C4D4×D3712yes—
194C2×C4Q8×D3712yes—
195C22—374no—
196C22—744no—
197C22—742no—
198C22—742no—
199C22—744no—
200C22C22×C4×D3711yescharacteristic
201C22C2×D4×D3711yescharacteristic
202C22C2×Q8×D3711yescharacteristic
203C22#1192458112yes—
204C22#1192458412yes—
205C4—24no—
206C4—42no—
207C4—42no—
208C4—374no—
209C4—744no—
210C4—1482no—
211C4—1482no—
212C4C4⋊C4×D3714yes—
213C2—744no—
214C2C2×C4⋊C4×D3711yescharacteristic
215C2#1455801412yes—
216C2#1455809611yescharacteristic
217C2#1455810811yescharacteristic
218C2#1455816012yes—
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#384C2
socle, cosocleC23×C37C24
prime radicalC23×D37C23
hypercenter#384D37
hypocenterC37#2317
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
4. #14557428C2semidirect
6. #14557402C2semidirect
7. #14557399C2semidirect
8. #14557398C2semidirect
9. C37×#384C2semidirect
13. #11924493C4semidirect
47. C42×C37C2×C4semidirect
54. #10596936C2×C4semidirect
56. C4×Dic37C2×C4semidirect
84. C4×D37C4⋊C4semidirect
116. #384D37semidirect
119. C37#2317semidirect
166. C42C4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119244942
C2×#119244932
C2×C4⋊C4×D372
#145574282
#145574042
#145574022
#145573992
#145573982
C37×#3842
#231737