Finite Group Databasebeta

← #18010851

Subgroups — #18010851

Order 4736.

280 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 54 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×#11924596C211yesmaximal, characteristic, complemented
2C2×#11924594C211yesmaximal, characteristic, complemented
3#14558435C212yesmaximal, complemented
4#14558433C212yesmaximal, complemented
5#14558092C211yesmaximal, characteristic
6C2×#10596957—21no—
7C2×C22⋊#3866—21no—
8#11924596—21no—
9#11924596C2212yes—
10#11924594—21no—
11#11924594C2212yes—
12C22×C22⋊D37C2211yescharacteristic
13#11924519C411yescharacteristic, complemented
14#11924519C2211yescharacteristic
15#11924517—21no—
16#11924508—21no—
17#11924508—21no—
18#11924505C411yescharacteristic, complemented
19#11924505C2211yescharacteristic
20#11924504—21no—
21C2×C22⋊Dic37C411yescharacteristic, complemented
22C22×D4×C37C411yescharacteristic, complemented
23C24×C37C2×C411yescharacteristic
24C24×C37C2×C411yescharacteristic, complemented
25C23×D37D411yescharacteristic
26C22×#3866—41no—
27C2×D4×C37—21no—
28C2×D4×C37—21no—
29C2×D4×C37—41no—
30C2×D4×C37—41no—
31C22×C4×C37C2×C411yescharacteristic, complemented
32C2×C22⋊D37—21no—
33C2×C22⋊D37—21no—
34C2×C22⋊D37—41no—
35C2×C22⋊D37—41no—
36C2×C22⋊D37C2×C411yescharacteristic
37C2×C22⋊D37D411yescharacteristic
38C2×C22⋊D37D411yescharacteristic
39C2×C22⋊D37C2311yescharacteristic
40C22×Dic37—21no—
41C22×Dic37D411yescharacteristic
42C22⋊#3866—22no—
43C22⋊#3866—41no—
44#10596957—22no—
45#10596957—41no—
46C2×#64438—41no—
47C22⋊Dic37—21no—
48C22⋊Dic37—21no—
49C22⋊Dic37—21no—
50C22⋊Dic37C2×C411yescharacteristic, complemented
51C22⋊Dic37C2×C411yescharacteristic, complemented
52#10596938—21no—
53#10596938—21no—
54#10596936—21no—
55#10596936—21no—
56C23×C37—21no—
57C23×C37—21no—
58C23×C37—21no—
59C23×C37—41no—
60C23×C37—81no—
61C23×C37—81no—
62C23×C37C22⋊C411yescharacteristic
63C23×C37C22⋊C411yescharacteristic
64C23×C37C22×C411yescharacteristic
65C22×D37—21no—
66C22×D37—21no—
67C22×D37—81no—
68C22×D37C22⋊C411yescharacteristic
69C22×D37C2×D411yescharacteristic
70C2×#3866—42no—
71C2×#3866—82no—
72D4×C37—42no—
73D4×C37—41no—
74D4×C37—41no—
75C2×C4×C37—21no—
76C2×C4×C37—41no—
77C22⋊D37—41no—
78C22⋊D37—41no—
79C22⋊D37—41no—
80C22⋊D37—41no—
81C22⋊D37—81no—
82C22⋊D37—81no—
83C2×Dic37—21no—
84C2×Dic37—21no—
85C2×Dic37—21no—
86C2×Dic37—21no—
87C2×Dic37—81no—
88C2×Dic37C22⋊C411yescharacteristic
89C2×Dic37C2×D411yescharacteristic
90#64438—42no—
91C22×C37—21no—
92C22×C37—21no—
93C22×C37—41no—
94C22×C37—41no—
95C22×C37—41no—
96C22×C37—41no—
97C22×C37—41no—
98C22×C37—41no—
99C22×C37—81no—
100C22×C37—81no—
101C22×C37—82no—
102C22×C37#9911yescharacteristic
103C22×C37#9911yescharacteristic
104C22×C37C2×C22⋊C411yescharacteristic
105C2×D37—21no—
106C2×D37—21no—
107C2×D37—42no—
108C2×D37—41no—
109C2×D37—41no—
110C2×D37—81no—
111#3866—82no—
112C4×C37—41no—
113Dic37—21no—
114Dic37—21no—
115Dic37—41no—
116Dic37—41no—
117#2121—371nomaximal, Sylow
118C2×C37—21no—
119C2×C37—21no—
120C2×C37—21no—
121C2×C37—41no—
122C2×C37—41no—
123C2×C37—81no—
124C2×C37C2×#9911yescharacteristic
125D37—41no—
126D37—41no—
127#534—371no—
128C2×#100—371no—
129C2×#99—371no—
130#352—372no—
131AΣL(1,𝔽16)—372no—
132C37#212111yescharacteristic, complemented, Sylow
133C22×D4—371no—
134C22×D4#386611yescharacteristic, complemented
135C2×M16—741no—
136#123—371no—
137#123—371no—
138#123—741no—
139#121—741no—
140#121—741no—
141#120—371no—
142#120—371no—
143#120—741no—
144C2×C22⋊C4—371no—
145C2×C22⋊C4—741no—
146#100—372no—
147#100—741no—
148#99—372no—
149#99—741no—
150C24—371no—
151C24C2×#386611yescharacteristic
152C24C2×#386611yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—371no—
158C2×D4—371no—
159C2×D4—371no—
160C2×D4—371no—
161C2×D4—741no—
162C2×D4—741no—
163C2×D4—1481no—
164C2×D4—1481no—
165C22×C4—371no—
166C22×C4—741no—
167C22×C4—1481no—
168C22×C4C2×#386611yescharacteristic, complemented
169M16—742no—
170M16—1481no—
171C2×C8—1481no—
172C4⋊C4—741no—
173C4⋊C4—741no—
174C22⋊C4—371no—
175C22⋊C4—371no—
176C22⋊C4—741no—
177C22⋊C4—741no—
178C22⋊C4—741no—
179C22⋊C4—741no—
180C22⋊C4—741no—
181C22⋊C4—742no—
182C22⋊C4—1481no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—41no—
187C23—81no—
188C23—81no—
189C23—371no—
190C23—371no—
191C23—2961no—
192C23—741no—
193C23—741no—
194C23C22⋊#386611yescharacteristic
195C23C22⋊#386611yescharacteristic
196C23C22×#386611yescharacteristic
197D4—42no—
198D4—41no—
199D4—41no—
200D4—2961no—
201D4—2961no—
202D4—1481no—
203D4—1481no—
204D4—1481no—
205D4—1481no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—371no—
209C2×C4—371no—
210C2×C4—2962no—
211C2×C4—2961no—
212C2×C4—741no—
213C2×C4—741no—
214C2×C4—741no—
215C2×C4—741no—
216C2×C4—1482no—
217C8—1482no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—81no—
227C22—81no—
228C22—82no—
229C22—2961no—
230C22—741no—
231C22—741no—
232C22—1481no—
233C22—1481no—
234C22—1482no—
235C22#1192459411yescharacteristic
236C22#1192459411yescharacteristic
237C22C2×C22⋊#386611yescharacteristic
238C4—41no—
239C4—2962no—
240C4—741no—
241C4—741no—
242C4—1481no—
243C4—1481no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—81no—
250C2—1481no—
251C2—1481no—
252C2C2×#1192459411yescharacteristic
subgroupG/H
centerC2C2×#11924594
commutatorC23×C37C22×C4
FrattiniC23C22×#3866
FittingC22×D4×C37C4
socleC2×C37C2×#99
prime radical#14558092C2
cosocleC2×C22⋊D37C23
hypercenterC22×D4#3866
hypocenterC37#2121
LCSG ▷ C23×C37 ▷ C22×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#11924596C2semidirect
2. C2×#11924594C2semidirect
3. #14558435C2semidirect
4. #14558433C2semidirect
13. #11924519C4semidirect
18. #11924505C4semidirect
21. C2×C22⋊Dic37C4semidirect
22. C22×D4×C37C4semidirect
24. C24×C37C2×C4semidirect
31. C22×C4×C37C2×C4semidirect
50. C22⋊Dic37C2×C4semidirect
51. C22⋊Dic37C2×C4semidirect
132. C37#2121semidirect
134. C22×D4#3866semidirect
152. C24C2×#3866semidirect
168. C22×C4C2×#3866semidirect

Maximal subgroups

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

subgroupindex
C2×#119245962
C2×#119245942
#145584352
#145584332
#145580922
#212137