Finite Group Databasebeta

← #18005368

Subgroups — #18005368

Order 4736.

416 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 137 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×#11924515C211yesmaximal, characteristic
2#14557868C212yesmaximal, complemented
3#14557866C212yesmaximal, complemented
4#14557772C214yesmaximal, complemented
5#14557689C212yesmaximal, complemented
6#14557688C212yesmaximal, complemented
7C2×C22⋊#64436C211yesmaximal, characteristic
8C2×C22⋊Q8×C37C211yesmaximal, characteristic
9C22×Dic74C2211yescharacteristic
10C2×#10596942C2214yescomplemented
11C22×#64436C2212yes—
12#11924515—21no—
13#11924515—21no—
14#11924515C2212yes—
15#11924515C2212yes—
16C2×C4⋊Dic37C2211yescharacteristic
17C2×#10596936—21no—
18C2×C22⋊Dic37—21no—
19#11924469C2214yescomplemented
20#11924460C2214yescomplemented
21#11924459C2214yescomplemented
22C22⋊#64436C2212yes—
23C22⋊#64436C2212yes—
24C22×Q8×C37C2211yescharacteristic
25C23×C4×C37C2211yescharacteristic
26C22⋊Q8×C37—21no—
27C22⋊Q8×C37—21no—
28C22⋊Q8×C37C2212yes—
29C22⋊Q8×C37C2212yes—
30C2×C4⋊C4×C37—21no—
31C2×C4⋊C4×C37C2211yescharacteristic
32C2×C22⋊C4×C37—21no—
33C24×C37D411yescharacteristic
34C2×Q8×C37—21no—
35C2×Q8×C37—42no—
36C2×Q8×C37C2312yes—
37C22×C4×C37—21no—
38C22×C4×C37—21no—
39C22×C4×C37—42no—
40C22×C4×C37D411yescharacteristic
41C22×C4×C37D411yescharacteristic
42C22×C4×C37D411yescharacteristic
43C22×C4×C37C2311yescharacteristic
44C22×C4×C37C2311yescharacteristic
45C22×C4×C37C2311yescharacteristic
46C22×Dic37—21no—
47C22×Dic37—21no—
48C2×Dic74—21no—
49C2×Dic74—42no—
50C2×Dic74C2312yes—
51C4⋊C4×C37—22no—
52C4⋊C4×C37—22no—
53C4⋊C4×C37—21no—
54C4⋊C4×C37C2312yes—
55C22⋊C4×C37—22no—
56C22⋊C4×C37—22no—
57C22⋊Dic37—22no—
58C22⋊Dic37—22no—
59#10596942—44no—
60C4⋊Dic37—21no—
61C4⋊Dic37C2312yes—
62#10596936—22no—
63#10596936—22no—
64C2×#64436—42no—
65C2×#64436C2314yes—
66C23×C37—41no—
67C23×C37—41no—
68C23×C37Q1614yescomplemented
69C23×C37C2×D411yescharacteristic
70C23×C37C2×D411yescharacteristic
71C23×C37C2×D411yescharacteristic
72Q8×C37—24no—
73Q8×C37—42no—
74C2×C4×C37—22no—
75C2×C4×C37—22no—
76C2×C4×C37—22no—
77C2×C4×C37—22no—
78C2×C4×C37—22no—
79C2×C4×C37—41no—
80C2×C4×C37—41no—
81C2×C4×C37—41no—
82C2×C4×C37—42no—
83C2×C4×C37—42no—
84C2×C4×C37C2×D411yescharacteristic
85C2×C4×C37C2×D411yescharacteristic
86C2×C4×C37C2×D411yescharacteristic
87C2×C4×C37C2411yescharacteristic
88C2×Dic37—22no—
89C2×Dic37—22no—
90C2×Dic37—42no—
91C2×Dic37—42no—
92Dic74—24no—
93Dic74—42no—
94#64436—24no—
95C22×C37—21no—
96C22×C37—21no—
97C22×C37—21no—
98C22×C37—21no—
99C22×C37—21no—
100C22×C37—21no—
101C22×C37—44no—
102C22×C37C2×Q1612yes—
103C22×C37C2×Q1612yes—
104C22×C37C2×Q1612yes—
105C22×C37C22×D411yescharacteristic
106C4×C37—21no—
107C4×C37—21no—
108C4×C37—21no—
109C4×C37—42no—
110C4×C37—42no—
111C4×C37H(5,𝔽2)12yes—
112Dic37—42no—
113Dic37—42no—
114#3189—371nomaximal, Sylow
115C2×C37—22no—
116C2×C37—22no—
117C2×C37—22no—
118C2×C37#53411yescharacteristic
119C2×C37C22×Q1611yescharacteristic
120C2×C37#57511yescharacteristic
121C2×C22⋊Q8—371no—
122C2×C22⋊Q8D3711yescharacteristic
123#484—372no—
124#484—372no—
125#467—374no—
126#451—372no—
127#451—372no—
128C2×C22⋊C8—371no—
129C37#318911yescharacteristic, complemented, Sylow
130C22×Q8—371no—
131C22×Q8C2×D3711yescharacteristic
132C23×C4C2×D3711yescharacteristic
133C2×Q16—374no—
134C22×C8—372no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137C22⋊Q8—372no—
138C22⋊Q8—372no—
139C22⋊Q8—741no—
140C22⋊Q8—741no—
141C22⋊Q8C2×D3712yes—
142C22⋊Q8C2×D3712yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—371no—
145C2×C4⋊C4—741no—
146C2×C4⋊C4C2×D3711yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—741no—
149#107—374no—
150#103—374no—
151#103—374no—
152C22⋊C8—372no—
153C22⋊C8—372no—
154C24C22⋊D3711yescharacteristic
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—372no—
158C2×Q8—741no—
159C2×Q8—1482no—
160C2×Q8C22×D3712yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—741no—
165C22×C4—741no—
166C22×C4C22⋊D3711yescharacteristic
167C22×C4C22⋊D3711yescharacteristic
168C22×C4C22⋊D3711yescharacteristic
169C22×C4C22×D3711yescharacteristic
170C22×C4C22×D3711yescharacteristic
171C22×C4C22×D3711yescharacteristic
172Q16—1484no—
173C2×C8—374no—
174C2×C8—1482no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—372no—
179C4⋊C4—742no—
180C4⋊C4—742no—
181C4⋊C4—741no—
182C4⋊C4C22×D3712yes—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—742no—
186C22⋊C4—742no—
187C23—41no—
188C23—41no—
189C23#1059694214yescomplemented
190C23C2×C22⋊D3711yescharacteristic
191C23C2×C22⋊D3711yescharacteristic
192C23C2×C22⋊D3711yescharacteristic
193Q8—24no—
194Q8—42no—
195Q8—744no—
196Q8—1482no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—742no—
208C2×C4—742no—
209C2×C4—1482no—
210C2×C4—1482no—
211C2×C4C2×C22⋊D3711yescharacteristic
212C2×C4C2×C22⋊D3711yescharacteristic
213C2×C4C2×C22⋊D3711yescharacteristic
214C2×C4C23×D3711yescharacteristic
215C8—744no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—44no—
223C22C2×#1059694212yes—
224C22C2×#1059694212yes—
225C22C2×#1059694212yes—
226C22C22×C22⋊D3711yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—42no—
232C4—1482no—
233C4—1482no—
234C4#1192458112yes—
235C2—22no—
236C2—22no—
237C2—22no—
238C2#1455809211yescharacteristic
239C2C22×#1059694211yescharacteristic
240C2#1455833111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×C22⋊Q8×C37C2
socleC22×C37C22×D4
prime radicalC24×C37D4
hypercenterC2×C22⋊Q8D37
hypocenterC37#3189
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #14557868C2semidirect
3. #14557866C2semidirect
4. #14557772C2semidirect
5. #14557689C2semidirect
6. #14557688C2semidirect
10. C2×#10596942C22semidirect
19. #11924469C22semidirect
20. #11924460C22semidirect
21. #11924459C22semidirect
68. C23×C37Q16semidirect
129. C37#3189semidirect
189. C23#10596942semidirect

Maximal subgroups

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

subgroupindex
C2×#119245152
#145578682
#145578662
#145577722
#145576892
#145576882
C2×C22⋊#644362
C2×C22⋊Q8×C372
#318937