Finite Group Databasebeta

← #18010583

Subgroups — #18010583

Order 4736.

456 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 88 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×C4⋊#3866C211yesmaximal, characteristic, complemented
2C2×#11924598C212yesmaximal, complemented
3D37⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
4#14558406C212yesmaximal, complemented
5#121×D37C211yesmaximal, characteristic
6C23×#3866—21no—
7C2×D37⋊C8C2212yes—
8C2×C4⋊#3866—21no—
9C2×C4⋊#3866—24no—
10C2×C4⋊#3866C2212yes—
11#11924598—24no—
12#11924593—21no—
13C22⋊#64438—21no—
14C2×D4×D37—21no—
15C2×D4×D37C2211yescharacteristic
16C22×C4×D37C2211yescharacteristic
17#11924511C411yescharacteristic, complemented
18#11924510C411yescharacteristic, complemented
19#11924509—21no—
20#11924508—21no—
21#11924507C411yescharacteristic, complemented
22C4⋊C4×D37C2211yescharacteristic
23C22⋊C4×D37—21no—
24#121×C37C411yescharacteristic, complemented
25C23×D37—21no—
26C23×D37D411yescharacteristic
27C22×#3866—24no—
28C22×#3866—22no—
29C22×#3866—42no—
30C22×#3866—42no—
31C2×D4×C37—21no—
32C2×D4×C37C2×C411yescharacteristic, complemented
33C22×C4×C37C2×C411yescharacteristic
34C2×C22⋊D37—21no—
35C2×C22⋊D37—21no—
36C2×C22⋊D37—21no—
37C22×Dic37D411yescharacteristic
38D4×D37—22no—
39D4×D37—22no—
40D4×D37—42no—
41D4×D37—42no—
42C2×D148—21no—
43C2×D148C2×C411yescharacteristic, complemented
44C2×C4×D37—22no—
45C2×C4×D37—22no—
46C2×C4×D37—21no—
47C2×C4×D37C2×C411yescharacteristic
48C2×C4×D37D411yescharacteristic
49C2×C4×D37D411yescharacteristic
50C2×C4×D37C2311yescharacteristic
51C2×#64438—22no—
52C4⋊#3866—24no—
53C4⋊#3866—22no—
54C4⋊#3866D414yescomplemented
55D37⋊C8—22no—
56D37⋊C8—22no—
57C4⋊C4×C37C2×C411yescharacteristic, complemented
58C22⋊C4×C37—21no—
59C22⋊Dic37—21no—
60#10596938—21no—
61#10596938—21no—
62C4⋊Dic37C2×C411yescharacteristic, complemented
63#10596936—21no—
64C23×C37—21no—
65C23×C37C22⋊C411yescharacteristic
66C22×D37—21no—
67C22×D37—21no—
68C22×D37—21no—
69C22×D37—21no—
70C22×D37—21no—
71C22×D37—41no—
72C22×D37—41no—
73C22×D37—41no—
74C22×D37—41no—
75C22×D37—41no—
76C22×D37—41no—
77C22×D37C22⋊C411yescharacteristic
78C22×D37D812yes—
79C22×D37SD1612yes—
80C22×D37C2×D411yescharacteristic
81C2×#3866—24no—
82C2×#3866—48no—
83C2×#3866—42no—
84C2×#3866—42no—
85D4×C37—22no—
86D4×C37—42no—
87C2×C4×C37—22no—
88C2×C4×C37—21no—
89C2×C4×C37C22⋊C411yescharacteristic
90C2×C4×C37C22×C411yescharacteristic
91C22⋊D37—42no—
92C22⋊D37—42no—
93C22⋊D37—42no—
94C2×Dic37—22no—
95C2×Dic37—21no—
96C2×Dic37C22⋊C411yescharacteristic
97C2×Dic37C2×D411yescharacteristic
98D148—22no—
99D148—42no—
100C4×D37—21no—
101C4×D37—21no—
102C4×D37—22no—
103C4×D37—22no—
104C4×D37—41no—
105C4×D37—41no—
106C4×D37C2×D412yes—
107C4×D37#4112yes—
108#64438—42no—
109C22×C37—21no—
110C22×C37—21no—
111C22×C37—41no—
112C22×C37—41no—
113C22×C37—41no—
114C22×C37#10212yescomplemented
115C22×C37C2×C22⋊C411yescharacteristic
116C2×D37—21no—
117C2×D37—21no—
118C2×D37—22no—
119C2×D37—22no—
120C2×D37—22no—
121C2×D37—22no—
122C2×D37—41no—
123C2×D37—41no—
124C2×D37—41no—
125C2×D37—81no—
126C2×D37—81no—
127C2×D37—81no—
128C2×D37—81no—
129C2×D37#10212yescomplemented
130C2×D37#12011yescharacteristic
131C2×D37#12311yescharacteristic
132C2×D37C2×D811yescharacteristic
133C2×D37C2×SD1611yescharacteristic
134C2×D37AGL(1,ℤ/8)11yescharacteristic
135C2×D37#13711yescharacteristic
136#3866—44no—
137C4×C37—21no—
138C4×C37—41no—
139C4×C37C4×D412yes—
140Dic37—21no—
141Dic37—41no—
142Dic37#12112yes—
143#1874—371nomaximal, Sylow
144C2×C37—22no—
145C2×C37—81no—
146C2×C37#38611yescharacteristic
147C2×C37C2×#10211yescharacteristic
148C2×C37#41711yescharacteristic
149D37—22no—
150D37—81no—
151D37#44711yescharacteristic
152D37#44811yescharacteristic
153D37#48011yescharacteristic
154D37#48111yescharacteristic
155C2×#121—371no—
156C22×C4⋊C4—371no—
157C2×#102—372no—
158C2×C22⋊C8—371no—
159#340—372no—
160C37#187411yescharacteristic, complemented, Sylow
161C22×D4—371no—
162C22×D4—741no—
163C23×C4—371no—
164C23×C4—741no—
165C22×C8—372no—
166#121—371no—
167#121—371no—
168#121—371no—
169#121—741no—
170#121—741no—
171#121#386611yescharacteristic, complemented
172C2×C4⋊C4—372no—
173C2×C4⋊C4—371no—
174C2×C4⋊C4—744no—
175C2×C4⋊C4—741no—
176C2×C22⋊C4—741no—
177#102—744no—
178C22⋊C8—741no—
179C22⋊C8—741no—
180C24—371no—
181C24—741no—
182C2×D4—21no—
183C2×D4—371no—
184C2×D4—741no—
185C2×D4—742no—
186C2×D4—742no—
187C2×D4—741no—
188C2×D4—741no—
189C2×D4—741no—
190C2×D4—1482no—
191C2×D4—1482no—
192C2×D4C2×#386611yescharacteristic, complemented
193C22×C4—371no—
194C22×C4—371no—
195C22×C4—371no—
196C22×C4—371no—
197C22×C4—371no—
198C22×C4—742no—
199C22×C4—742no—
200C22×C4—741no—
201C22×C4—742no—
202C22×C4—744no—
203C22×C4—1482no—
204C22×C4—1482no—
205C22×C4C2×#386611yescharacteristic
206C2×C8—742no—
207C2×C8—742no—
208C2×C8—742no—
209C4⋊C4—374no—
210C4⋊C4—371no—
211C4⋊C4—741no—
212C4⋊C4—742no—
213C4⋊C4—744no—
214C4⋊C4C2×#386611yescharacteristic, complemented
215C22⋊C4—21no—
216C22⋊C4—741no—
217C22⋊C4—741no—
218C22⋊C4—741no—
219C23—21no—
220C23—371no—
221C23—372no—
222C23—372no—
223C23—371no—
224C23—741no—
225C23—741no—
226C23—741no—
227C23—741no—
228C23—741no—
229C23—1481no—
230C23—1481no—
231C23—1481no—
232C23—1481no—
233C23—1481no—
234C23—1481no—
235C23C22⋊#386611yescharacteristic
236D4—22no—
237D4—42no—
238D4—742no—
239D4—1482no—
240D4—1482no—
241D4—1482no—
242D4—1482no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—371no—
246C2×C4—372no—
247C2×C4—372no—
248C2×C4—371no—
249C2×C4—742no—
250C2×C4—741no—
251C2×C4—741no—
252C2×C4—742no—
253C2×C4—742no—
254C2×C4—744no—
255C2×C4—741no—
256C2×C4—1481no—
257C2×C4—1481no—
258C2×C4—1482no—
259C2×C4—1482no—
260C2×C4—1488no—
261C2×C4C22⋊#386611yescharacteristic
262C2×C4C22×#386611yescharacteristic
263C8—1482no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—371no—
270C22—371no—
271C22—371no—
272C22—371no—
273C22—371no—
274C22—371no—
275C22—372no—
276C22—2961no—
277C22—2961no—
278C22—2961no—
279C22—2961no—
280C22—742no—
281C22—742no—
282C22—742no—
283C22—742no—
284C22—741no—
285C22—741no—
286C22—1481no—
287C22—1481no—
288C22—1481no—
289C22#1192459812yescomplemented
290C22C2×C22⋊#386611yescharacteristic
291C4—21no—
292C4—41no—
293C4—372no—
294C4—741no—
295C4—1484no—
296C4—1481no—
297C4D4×#386612yes—
298C2—22no—
299C2—81no—
300C2—371no—
301C2—371no—
302C2—371no—
303C2—371no—
304C2—2961no—
305C2—742no—
306C2#1455848111yescharacteristic
307C2C2×#1192459811yescharacteristic
308C2#1455852411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#3866
commutatorC2×C4×C37C22×C4
FrattiniC2×C4C22×#3866
Fitting#121×C37C4
socleC22×C37C2×C22⋊C4
prime radical#121×D37C2
cosocleC2×C4×D37C23
hypercenter#121#3866
hypocenterC37#1874
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊#3866C2semidirect
2. C2×#11924598C2semidirect
3. D37⋊(C22⋊C8)C2semidirect
4. #14558406C2semidirect
17. #11924511C4semidirect
18. #11924510C4semidirect
21. #11924507C4semidirect
24. #121×C37C4semidirect
32. C2×D4×C37C2×C4semidirect
43. C2×D148C2×C4semidirect
54. C4⋊#3866D4semidirect
57. C4⋊C4×C37C2×C4semidirect
62. C4⋊Dic37C2×C4semidirect
114. C22×C37#102semidirect
129. C2×D37#102semidirect
160. C37#1874semidirect
171. #121#3866semidirect
192. C2×D4C2×#3866semidirect
214. C4⋊C4C2×#3866semidirect
289. C22#11924598semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#38662
C2×#119245982
D37⋊(C22⋊C8)2
#145584062
#121×D372
#187437