Finite Group Databasebeta

← #17997165

Subgroups — #17997165

Order 4736.

588 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 237 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×#11924534C211yesmaximal, characteristic
2C2×Q8×Dic37C211yesmaximal, characteristic
3C2×#11924501C211yesmaximal, characteristic
4#14557439C211yesmaximal, characteristic
5#14557430C214yesmaximal
6#14557402C211yesmaximal, characteristic
7#14557384C214yesmaximal
8C42×Dic37C211yesmaximal, characteristic
9C37×#391C211yesmaximal, characteristic
10C22×Dic74C2211yescharacteristic
11#11924534C418yescomplemented
12Q8×Dic37—24no—
13#11924501—24no—
14C2×C4⋊Dic37C2211yescharacteristic
15C2×C4⋊Dic37C2212yes—
16C2×#10596936C2212yes—
17C2×#10596936C2214yes—
18C2×#10596936C2214yes—
19C2×C4×Dic37C2212yes—
20C2×C4×Dic37C2212yes—
21C2×C4×Dic37C2212yes—
22C2×C4×Dic37C2211yescharacteristic
23C2×C4×Dic37C2211yescharacteristic
24C2×C4×Dic37C2212yes—
25#11924443C2214yes—
26C22×Q8×C37C2211yescharacteristic
27C2×C4⋊C4×C37C2211yescharacteristic
28C2×C42×C37C2211yescharacteristic
29#95×C37C2214yes—
30C2×Q8×C37—22no—
31C2×Q8×C37—22no—
32C2×Q8×C37C2×C414yes—
33C22×C4×C37C2312yes—
34C22×C4×C37C2312yes—
35C22×C4×C37C2312yes—
36C22×C4×C37C2311yescharacteristic
37C22×Dic37C2312yes—
38C22×Dic37C2312yes—
39C22×Dic37C2311yescharacteristic
40C22×Dic37C2312yes—
41C22×Dic37C2311yescharacteristic
42C2×Dic74—22no—
43C2×Dic74—22no—
44C2×Dic74C2×C414yes—
45C4⋊C4×C37—22no—
46C42×C37—22no—
47C4⋊Dic37—22no—
48C4⋊Dic37—24no—
49#10596936—24no—
50#10596936—28no—
51#10596936C2×C4116yes—
52C4×Dic37—24no—
53C4×Dic37—24no—
54C4×Dic37—24no—
55C4×Dic37C2×C414yes—
56C4×Dic37D414yes—
57C4×Dic37Q818yescomplemented
58C23×C37C2411yescharacteristic
59Q8×C37—28no—
60C2×C4×C37—22no—
61C2×C4×C37—22no—
62C2×C4×C37—22no—
63C2×C4×C37—22no—
64C2×C4×C37—22no—
65C2×C4×C37—24no—
66C2×C4×C37C22×C414yes—
67C2×C4×C37C22×C412yes—
68C2×C4×C37C2×D412yes—
69C2×C4×C37C2×Q814yes—
70C2×C4×C37#4112yes—
71C2×Dic37—22no—
72C2×Dic37—22no—
73C2×Dic37—22no—
74C2×Dic37—22no—
75C2×Dic37—24no—
76C2×Dic37C22×C414yes—
77C2×Dic37C22×C414yes—
78C2×Dic37C2×D412yes—
79C2×Dic37C2×D412yes—
80C2×Dic37C2×Q814yes—
81C2×Dic37C2×Q814yes—
82C2×Dic37#4114yes—
83Dic74—28no—
84C22×C37C23×C411yescharacteristic
85C22×C37C22×D411yescharacteristic
86C22×C37C22×Q812yes—
87C22×C37C2×#4112yes—
88C22×C37C2×#4111yescharacteristic
89C4×C37—24no—
90C4×C37—24no—
91C4×C37—42no—
92C4×C37C4×D414yes—
93Dic37—24no—
94Dic37—22no—
95Dic37—42no—
96Dic37C4×Q818yescomplemented
97Dic37C4⋊Q814yescomplemented
98C4×C4⋊Q8—371nomaximal, Sylow
99C2×C37C2×C4×D411yescharacteristic
100C2×C37C2×C4×Q812yes—
101C2×C37C2×C4⋊Q811yescharacteristic
102C2×C37#53211yescharacteristic
103C2×C37#53312yes—
104C2×C4⋊Q8—371no—
105C2×C4×Q8—371no—
106C2×C4×Q8—371no—
107#391—371no—
108#391D3711yescharacteristic
109#389—374no—
110#384—371no—
111C4×C4⋊C4—374no—
112C43—371no—
113C37C4×C4⋊Q811yescharacteristic, complemented, Sylow
114C22×Q8—371no—
115C22×Q8C2×D3711yescharacteristic
116C4⋊Q8—378no—
117C4×Q8—744no—
118C4×Q8—744no—
119C2×C4⋊C4—374no—
120C2×C4⋊C4—372no—
121C2×C4⋊C4—371no—
122C2×C4⋊C4—372no—
123C2×C4⋊C4—374no—
124C2×C4⋊C4C2×D3711yescharacteristic
125C2×C42—371no—
126C2×C42—371no—
127C2×C42—372no—
128C2×C42—372no—
129C2×C42—372no—
130C2×C42—372no—
131C2×C42C2×D3711yescharacteristic
132#95—374no—
133#95C2×D3714yes—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—374no—
137C2×Q8—742no—
138C2×Q8—742no—
139C2×Q8C4×D3714yes—
140C22×C4—371no—
141C22×C4—372no—
142C22×C4—372no—
143C22×C4—371no—
144C22×C4—372no—
145C22×C4C22×D3712yes—
146C22×C4C22×D3712yes—
147C22×C4C22×D3712yes—
148C22×C4C22×D3711yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—3716no—
151C4⋊C4—748no—
152C4⋊C4—744no—
153C4⋊C4—744no—
154C4⋊C4—742no—
155C42—22no—
156C42—374no—
157C42—374no—
158C42—378no—
159C42—744no—
160C42—744no—
161C42—744no—
162C23C23×D3711yescharacteristic
163Q8—28no—
164Q8—748no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—374no—
172C2×C4—372no—
173C2×C4—374no—
174C2×C4—374no—
175C2×C4—374no—
176C2×C4—374no—
177C2×C4—372no—
178C2×C4—742no—
179C2×C4—742no—
180C2×C4—742no—
181C2×C4—742no—
182C2×C4—744no—
183C2×C4C2×C4×D3714yes—
184C2×C4C2×C4×D3712yes—
185C2×C4D4×D3712yes—
186C2×C4#1059696412yes—
187C2×C4Q8×D3714yes—
188C22C22×C4×D3711yescharacteristic
189C22C2×D4×D3711yescharacteristic
190C22C2×Q8×D3712yes—
191C22#41×D3712yes—
192C22C2×#1059696411yescharacteristic
193C4—24no—
194C4—24no—
195C4—42no—
196C4—378no—
197C4—374no—
198C4—744no—
199C4—742no—
200C4—1482no—
201C4#1192449614yes—
202C2C2×#1192449611yescharacteristic
203C2C4×Q8×D3712yes—
204C2C4⋊Q8×D3711yescharacteristic
205C2#1455808211yescharacteristic
206C2#1455808512yes—
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#391C2
socle, prime radical, cosocleC23×C37C24
hypercenter#391D37
hypocenterC37C4×C4⋊Q8
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #11924534C4semidirect
57. C4×Dic37Q8semidirect
96. Dic37C4×Q8semidirect
97. Dic37C4⋊Q8semidirect
113. C37C4×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#119245342
C2×Q8×Dic372
C2×#119245012
#145574392
#145574302
#145574022
#145573842
C42×Dic372
C37×#3912
C4×C4⋊Q837