Finite Group Databasebeta

← #17993166

Subgroups — #17993166

Order 4736.

288 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22⋊Q8×D37C211yesmaximal, characteristic
2C22⋊C8×D37C211yesmaximal, characteristic
3#14557392C211yesmaximal, characteristic
4#14557271C211yesmaximal, characteristic, complemented
5#14557270C211yesmaximal, characteristic, complemented
6#14557269C211yesmaximal, characteristic, complemented
7C37×#328C211yesmaximal, characteristic, complemented
8C2×Q8×D37C411yescharacteristic, complemented
9C22×C4×D37C2211yescharacteristic
10C2×C8×D37—21no—
11#11924516C411yescharacteristic, complemented
12#11924515C2211yescharacteristic
13#11924514—21no—
14#11924513—21no—
15#11924512C411yescharacteristic, complemented
16C2×#10596936—21no—
17C4⋊C4×D37—21no—
18C4⋊C4×D37C411yescharacteristic, complemented
19C2×#10596938—21no—
20C22⋊C4×D37—21no—
21C22⋊#64436C2211yescharacteristic
22#11924448—21no—
23#11924443C2211yescharacteristic
24C22⋊Q8×C37C2211yescharacteristic
25C22⋊C8×C37C2211yescharacteristic
26#95×C37C2211yescharacteristic
27C23×D37D411yescharacteristic
28C2×Q8×C37C2×C411yescharacteristic
29C22×C4×C37—21no—
30C22×C4×C37C2311yescharacteristic
31C22×Dic37—21no—
32C22×Dic37D411yescharacteristic
33Q8×D37—42no—
34C2×C4×D37—21no—
35C2×C4×D37—21no—
36C2×C4×D37—42no—
37C2×C4×D37C2×C411yescharacteristic
38C2×C4×D37C2×C411yescharacteristic
39C2×C4×D37D411yescharacteristic
40C2×C4×D37D411yescharacteristic
41C2×Dic74—21no—
42C2×Dic74C2×C411yescharacteristic
43C2×C8×C37—21no—
44C4⋊C4×C37—21no—
45C4⋊C4×C37C2×C411yescharacteristic
46C22⋊C4×C37—21no—
47C22⋊Dic37—21no—
48#10596938—21no—
49#10596938—21no—
50#10596938—42no—
51C4⋊Dic37—21no—
52C4⋊Dic37C2×C411yescharacteristic
53#10596936—21no—
54#10596936—21no—
55#10596936—21no—
56#10596936—42no—
57C2×#64436—21no—
58C8×D37—42no—
59C23×C37C2×D411yescharacteristic
60C22×D37—41no—
61C22×D37—41no—
62C22×D37C22⋊C411yescharacteristic
63C22×D37C22⋊C411yescharacteristic
64C22×D37SD1612yes—
65C22×D37Q1612yes—
66Q8×C37—41no—
67C2×C4×C37—21no—
68C2×C4×C37—21no—
69C2×C4×C37—41no—
70C2×C4×C37—41no—
71C2×C4×C37—41no—
72C2×C4×C37—41no—
73C2×C4×C37C22×C411yescharacteristic
74C2×C4×C37C2×D411yescharacteristic
75C2×Dic37—22no—
76C2×Dic37—21no—
77C2×Dic37—21no—
78C2×Dic37—41no—
79C2×Dic37—41no—
80C2×Dic37—41no—
81C2×Dic37C22⋊C411yescharacteristic
82C2×Dic37C22⋊C411yescharacteristic
83C4×D37—22no—
84C4×D37—22no—
85C4×D37—41no—
86C4×D37—41no—
87C4×D37—42no—
88C4×D37—42no—
89Dic74—42no—
90Dic74—41no—
91C8×C37—41no—
92#64436—41no—
93C22×C37—21no—
94C22×C37—21no—
95C22×C37C2×C22⋊C411yescharacteristic
96C22×C37C2×SD1611yescharacteristic
97C22×C37C2×Q1611yescharacteristic
98C2×D37—21no—
99C2×D37—21no—
100C2×D37—21no—
101C2×D37—21no—
102C2×D37—42no—
103C2×D37—42no—
104C2×D37#10312yes—
105C2×D37#10312yes—
106C4×C37—21no—
107C4×C37—21no—
108C4×C37—41no—
109C4×C37—41no—
110C4×C37—81no—
111Dic37—21no—
112Dic37—41no—
113Dic37—41no—
114Dic37—81no—
115Dic37#9912yes—
116#1520—371nomaximal, Sylow
117C2×C37—21no—
118C2×C37—21no—
119C2×C37C2×#9911yescharacteristic
120C2×C37C2×#10311yescharacteristic
121C2×C37#42111yescharacteristic
122D37—22no—
123D37—22no—
124C2×C22⋊Q8—371no—
125C2×C22⋊C8—371no—
126#380—371no—
127#328—371no—
128#328—371no—
129#328—371no—
130#328D3711yescharacteristic, complemented
131C37#152011yescharacteristic, complemented, Sylow
132C22×Q8—371no—
133C23×C4—371no—
134C22×C8—741no—
135C22⋊Q8—371no—
136C22⋊Q8—371no—
137C22⋊Q8—371no—
138C22⋊Q8—741no—
139C22⋊Q8—741no—
140C22⋊Q8C2×D3711yescharacteristic
141C2×C4⋊C4—371no—
142C2×C4⋊C4—741no—
143C2×C4⋊C4—741no—
144C2×C22⋊C4—741no—
145C2×C22⋊C4—741no—
146C22⋊C8—371no—
147C22⋊C8—741no—
148C22⋊C8C2×D3711yescharacteristic
149#95—371no—
150#95C2×D3711yescharacteristic
151C24—371no—
152C2×Q8—371no—
153C2×Q8—741no—
154C2×Q8—1482no—
155C2×Q8C4×D3711yescharacteristic
156C22×C4—21no—
157C22×C4—371no—
158C22×C4—371no—
159C22×C4—371no—
160C22×C4—371no—
161C22×C4—371no—
162C22×C4—741no—
163C22×C4—741no—
164C22×C4—741no—
165C22×C4—1482no—
166C22×C4C22×D3711yescharacteristic
167C2×C8—21no—
168C2×C8—741no—
169C2×C8—1482no—
170C4⋊C4—21no—
171C4⋊C4—371no—
172C4⋊C4—741no—
173C4⋊C4—741no—
174C4⋊C4—741no—
175C4⋊C4—741no—
176C4⋊C4—1482no—
177C4⋊C4C4×D3711yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—741no—
180C22⋊C4—741no—
181C22⋊C4—741no—
182C22⋊C4—1482no—
183C23—371no—
184C23—371no—
185C23—372no—
186C23—372no—
187C23—1481no—
188C23—1481no—
189C23D4×D3711yescharacteristic
190Q8—41no—
191Q8—1481no—
192Q8—1482no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—371no—
200C2×C4—371no—
201C2×C4—742no—
202C2×C4—742no—
203C2×C4—742no—
204C2×C4—741no—
205C2×C4—741no—
206C2×C4—1481no—
207C2×C4—1481no—
208C2×C4—1482no—
209C2×C4—1482no—
210C2×C4—1481no—
211C2×C4—1481no—
212C2×C4—1481no—
213C2×C4C2×C4×D3711yescharacteristic
214C2×C4D4×D3711yescharacteristic
215C8—41no—
216C8—1481no—
217C22—21no—
218C22—21no—
219C22—372no—
220C22—372no—
221C22—741no—
222C22—741no—
223C22—741no—
224C22—741no—
225C22—1482no—
226C22—1482no—
227C22C22⋊C4×D3711yescharacteristic
228C22SD16×D3711yescharacteristic
229C22Q16×D3711yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—372no—
236C4—2961no—
237C4—741no—
238C4—1481no—
239C4—1481no—
240C2—21no—
241C2—21no—
242C2—742no—
243C2—742no—
244C2#99×D3711yescharacteristic
245C2#103×D3711yescharacteristic
246C2#1455754811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D37
commutatorC2×C4×C37C22×C4
FrattiniC22×C4C22×D37
FittingC37×#328C2
socleC22×C37C2×C22⋊C4
prime radicalC23×D37D4
cosocleC22×C4×C37C23
hypercenter#328D37
hypocenterC37#1520
LCSG ▷ C2×C4×C37 ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #14557271C2semidirect
5. #14557270C2semidirect
6. #14557269C2semidirect
7. C37×#328C2semidirect
8. C2×Q8×D37C4semidirect
11. #11924516C4semidirect
15. #11924512C4semidirect
18. C4⋊C4×D37C4semidirect
130. #328D37semidirect
131. C37#1520semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D372
C22⋊C8×D372
#145573922
#145572712
#145572702
#145572692
C37×#3282
#152037