Finite Group Databasebeta

← #17993168

Subgroups — #17993168

Order 4736.

288 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 97 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×C22⋊#64436C211yesmaximal, characteristic
2#14557393C211yesmaximal, characteristic
3#14557270C214yesmaximal, complemented
4C2×C22⋊Q8×C37C211yesmaximal, characteristic
5C22×#64436—21no—
6C2×C4⋊Dic37—21no—
7C2×C22⋊Dic37—21no—
8C22⋊#64436—21no—
9C22⋊#64436C2212yes—
10#11924443C2212yes—
11C22×Q8×C37C411yescharacteristic, complemented
12C23×C4×C37C2211yescharacteristic
13C22⋊Q8×C37—21no—
14C22⋊Q8×C37—21no—
15C22⋊Q8×C37C412yescomplemented
16C22⋊Q8×C37C2212yes—
17C2×C4⋊C4×C37—21no—
18C2×C4⋊C4×C37C411yescharacteristic, complemented
19C2×C22⋊C4×C37—21no—
20C24×C37D411yescharacteristic
21C2×Q8×C37—21no—
22C2×Q8×C37—42no—
23C2×Q8×C37C2×C412yes—
24C22×C4×C37—21no—
25C22×C4×C37—21no—
26C22×C4×C37—42no—
27C22×C4×C37C2×C411yescharacteristic
28C22×C4×C37C2×C411yescharacteristic
29C22×C4×C37D411yescharacteristic
30C22×C4×C37D411yescharacteristic
31C22×C4×C37D411yescharacteristic
32C22×C4×C37C2311yescharacteristic
33C22×Dic37—22no—
34C4⋊C4×C37—22no—
35C4⋊C4×C37—21no—
36C4⋊C4×C37—22no—
37C4⋊C4×C37C2×C412yes—
38C22⋊C4×C37—22no—
39C22⋊C4×C37—22no—
40C22⋊Dic37—42no—
41C4⋊Dic37—42no—
42C2×#64436—22no—
43C2×#64436—42no—
44C23×C37—41no—
45C23×C37—41no—
46C23×C37C22⋊C411yescharacteristic
47C23×C37C22⋊C411yescharacteristic
48C23×C37SD1612yes—
49C23×C37Q1612yes—
50C23×C37C2×D411yescharacteristic
51Q8×C37—42no—
52Q8×C37—42no—
53C2×C4×C37—22no—
54C2×C4×C37—22no—
55C2×C4×C37—22no—
56C2×C4×C37—22no—
57C2×C4×C37—22no—
58C2×C4×C37—41no—
59C2×C4×C37—41no—
60C2×C4×C37—41no—
61C2×C4×C37—42no—
62C2×C4×C37—42no—
63C2×C4×C37C22⋊C411yescharacteristic
64C2×C4×C37C22⋊C411yescharacteristic
65C2×C4×C37C22×C411yescharacteristic
66C2×C4×C37C2×D411yescharacteristic
67C2×Dic37—42no—
68C2×Dic37—42no—
69C2×Dic37—42no—
70#64436—42no—
71C22×C37—21no—
72C22×C37—21no—
73C22×C37—21no—
74C22×C37—21no—
75C22×C37—21no—
76C22×C37—21no—
77C22×C37—42no—
78C22×C37—42no—
79C22×C37#10312yes—
80C22×C37#10312yes—
81C22×C37C2×C22⋊C411yescharacteristic
82C22×C37C2×SD1611yescharacteristic
83C22×C37C2×Q1611yescharacteristic
84C4×C37—21no—
85C4×C37—21no—
86C4×C37—21no—
87C4×C37—42no—
88C4×C37—42no—
89C4×C37#9912yes—
90Dic37—82no—
91#1520—371nomaximal, Sylow
92C2×C37—21no—
93C2×C37—22no—
94C2×C37—22no—
95C2×C37—21no—
96C2×C37C2×#9911yescharacteristic
97C2×C37C2×#10311yescharacteristic
98C2×C37#42111yescharacteristic
99C2×C22⋊Q8D3711yescharacteristic
100C2×C22⋊C8—371no—
101#380—371no—
102#328—374no—
103C37#152011yescharacteristic, complemented, Sylow
104C22×Q8Dic3711yescharacteristic, complemented
105C23×C4C2×D3711yescharacteristic
106C22×C8—741no—
107C22⋊Q8—21no—
108C22⋊Q8—21no—
109C22⋊Q8Dic3712yescomplemented
110C22⋊Q8C2×D3712yes—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—741no—
113C2×C4⋊C4Dic3711yescharacteristic, complemented
114C2×C22⋊C4—21no—
115C2×C22⋊C4—741no—
116C22⋊C8—372no—
117C22⋊C8—741no—
118#95—372no—
119C24C22⋊D3711yescharacteristic
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8C2×Dic3712yes—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—42no—
126C22×C4—742no—
127C22×C4C2×Dic3711yescharacteristic
128C22×C4C2×Dic3711yescharacteristic
129C22×C4C22⋊D3711yescharacteristic
130C22×C4C22⋊D3711yescharacteristic
131C22×C4C22⋊D3711yescharacteristic
132C22×C4C22×D3711yescharacteristic
133C2×C8—742no—
134C2×C8—1482no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—1482no—
139C4⋊C4C2×Dic3712yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—1482no—
143C23—41no—
144C23—41no—
145C23#1059694112yes—
146C23#1059694212yes—
147C23C22⋊Dic3711yescharacteristic
148C23C22⋊Dic3711yescharacteristic
149C23C2×C22⋊D3711yescharacteristic
150Q8—42no—
151Q8—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—1482no—
163C2×C4—1482no—
164C2×C4—1482no—
165C2×C4C22⋊Dic3711yescharacteristic
166C2×C4C22⋊Dic3711yescharacteristic
167C2×C4C22×Dic3711yescharacteristic
168C2×C4C2×C22⋊D3711yescharacteristic
169C8—1482no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22#1192446012yes—
179C22#1192446012yes—
180C22C2×C22⋊Dic3711yescharacteristic
181C22C2×#1059694111yescharacteristic
182C22C2×#1059694211yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—42no—
187C4—42no—
188C4—2962no—
189C4#1192445112yes—
190C2—21no—
191C2—21no—
192C2—22no—
193C2—22no—
194C2C2×#1192445111yescharacteristic
195C2C2×#1192446011yescharacteristic
196C2#1455755011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic37
commutatorC2×C4×C37C22×C4
FrattiniC22×C4C22×D37
FittingC2×C22⋊Q8×C37C2
socleC22×C37C2×C22⋊C4
prime radicalC24×C37D4
cosocleC22×C4×C37C23
hypercenterC2×C22⋊Q8D37
hypocenterC37#1520
LCSG ▷ C2×C4×C37 ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14557270C2semidirect
11. C22×Q8×C37C4semidirect
15. C22⋊Q8×C37C4semidirect
18. C2×C4⋊C4×C37C4semidirect
103. C37#1520semidirect
104. C22×Q8Dic37semidirect
109. C22⋊Q8Dic37semidirect
113. C2×C4⋊C4Dic37semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#644362
#145573932
#145572702
C2×C22⋊Q8×C372
#152037