Finite Group Databasebeta

← #17994563

Subgroups — #17994563

Order 4736.

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

normal subgroups only

This group has 588 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×#11924581C211yesmaximal, characteristic
2#14557551C211yesmaximal, characteristic, complemented
3#14557549C211yesmaximal, characteristic, complemented
4#14557538C211yesmaximal, characteristic, complemented
5#14557515C211yesmaximal, characteristic, complemented
6#14557506C211yesmaximal, characteristic, complemented
7C37×#418C211yesmaximal, characteristic, complemented
8#11924581—22no—
9#11924581—22no—
10#11924581—22no—
11#11924581—22no—
12C2×#10596962C2211yescharacteristic
13C2×#10596962C2211yescharacteristic
14C2×#10596964—21no—
15C2×#10596964C412yescomplemented
16C22×C22⋊D37—22no—
17C2×D4×D37—21no—
18C2×D4×D37C412yescomplemented
19C2×#10596934C2211yescharacteristic, complemented
20#11924495C2211yescharacteristic, complemented
21#11924464—22no—
22#11924462—22no—
23#11924457—22no—
24#11924454—21no—
25#11924452—21no—
26#11924450—22no—
27C22×D4×C37C2211yescharacteristic
28C2×M16×C37C2211yescharacteristic, complemented
29#117×C37C2211yescharacteristic, complemented
30#102×C37—22no—
31C24×C37—21no—
32C23×D37—22no—
33C2×D4×C37—21no—
34C2×D4×C37—21no—
35C2×D4×C37—21no—
36C2×D4×C37C2×C412yes—
37C2×D4×C37D412yes—
38C2×D4×C37D412yes—
39C22×C4×C37C2311yescharacteristic
40C2×C22⋊D37—28no—
41C2×C22⋊D37—28no—
42C2×C22⋊D37—21no—
43C2×C22⋊D37—22no—
44C2×C22⋊D37—22no—
45C2×C22⋊D37D412yes—
46C22×Dic37—22no—
47#10596964—24no—
48#10596964—24no—
49#10596964—24no—
50#10596964—24no—
51D4×D37—24no—
52D4×D37—24no—
53D4×D37—24no—
54D4×D37—24no—
55#10596962—21no—
56#10596962—21no—
57#10596962—21no—
58#10596962—21no—
59#10596962—21no—
60#10596962—21no—
61#10596962D412yes—
62#10596962D412yes—
63C2×D148C2×C411yescharacteristic
64C2×D148D411yescharacteristic
65C2×C4×D37—21no—
66C2×C4×D37C2×C412yes—
67C2×Dic74C2×C411yescharacteristic
68C2×Dic74D411yescharacteristic
69M16×C37—21no—
70M16×C37—21no—
71C2×C8×C37—21no—
72C4⋊C4×C37—21no—
73C22⋊C4×C37—21no—
74C42×C37—21no—
75C22⋊Dic37—21no—
76C4⋊Dic37—21no—
77C4×Dic37—21no—
78#10596934—21no—
79#10596934—21no—
80C2×#64436—21no—
81C23×C37—22no—
82C23×C37—22no—
83C23×C37—22no—
84C23×C37—82no—
85C23×C37C2×D411yescharacteristic
86C22×D37—21no—
87C22×D37—28no—
88C22×D37—84no—
89C22×D37C22⋊C412yes—
90D4×C37—22no—
91D4×C37—22no—
92D4×C37—22no—
93D4×C37C22⋊C414yes—
94C2×C4×C37—41no—
95C2×C4×C37C22×C411yescharacteristic
96C2×C4×C37C2×D411yescharacteristic
97C2×C4×C37C2×D411yescharacteristic
98C2×C4×C37C2×D411yescharacteristic
99C2×C4×C37C2×D411yescharacteristic
100C2×C4×C37C2×D411yescharacteristic
101C22⋊D37—28no—
102C22⋊D37—28no—
103C22⋊D37—22no—
104C22⋊D37—22no—
105C22⋊D37—22no—
106C22⋊D37—22no—
107C2×Dic37—28no—
108C2×Dic37—21no—
109C2×Dic37—41no—
110C2×Dic37C22⋊C412yes—
111D148—21no—
112D148—21no—
113D148—21no—
114D148C22⋊C412yes—
115C4×D37—22no—
116C4×D37—22no—
117C4×D37—22no—
118C4×D37—22no—
119Dic74—21no—
120Dic74—21no—
121Dic74—21no—
122Dic74C22⋊C412yes—
123C8×C37—41no—
124#64436—41no—
125C22×C37—24no—
126C22×C37—41no—
127C22×C37—44no—
128C22×C37—82no—
129C22×C37—82no—
130C22×C37C2×C22⋊C411yescharacteristic
131C22×C37#12011yescharacteristic
132C22×C37#12011yescharacteristic
133C2×D37—22no—
134C2×D37—22no—
135C2×D37—42no—
136C2×D37—42no—
137C2×D37—88no—
138C4×C37—81no—
139C4×C37C2×C22⋊C411yescharacteristic
140C4×C37C2×C22⋊C411yescharacteristic
141C4×C37#12011yescharacteristic
142C4×C37#12011yescharacteristic
143Dic37—22no—
144Dic37—22no—
145Dic37—81no—
146#1795—371nomaximal, Sylow
147C2×C37—21no—
148C2×C37—21no—
149C2×C37—21no—
150C2×C37—44no—
151C2×C37#37911yescharacteristic
152D37—42no—
153D37—42no—
154C2×H(5,𝔽2)—371no—
155#421—371no—
156#421—371no—
157#418—371no—
158#418D3711yescharacteristic, complemented
159#413—371no—
160#410—371no—
161C37#179511yescharacteristic, complemented, Sylow
162H(5,𝔽2)—742no—
163H(5,𝔽2)—742no—
164H(5,𝔽2)—742no—
165H(5,𝔽2)—742no—
166C2×#41—371no—
167C2×#41—372no—
168C2×#41—371no—
169C2×#41—741no—
170C22×D4—372no—
171C22×D4—741no—
172C22×D4—742no—
173C22×D4C2×D3711yescharacteristic
174C2×M16—371no—
175C2×M16C2×D3711yescharacteristic, complemented
176#117—371no—
177#117C2×D3711yescharacteristic, complemented
178#104—742no—
179#104—742no—
180#102—22no—
181#102—742no—
182#101—741no—
183#100—741no—
184#99—742no—
185C24—21no—
186C24—742no—
187#41—372no—
188#41—372no—
189#41—744no—
190#41—744no—
191#41—744no—
192#41—744no—
193#41—741no—
194#41—741no—
195#41—741no—
196#41—741no—
197#41—741no—
198#41—741no—
199C2×Q8—371no—
200C2×Q8—371no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—372no—
205C2×D4—371no—
206C2×D4—371no—
207C2×D4—742no—
208C2×D4—744no—
209C2×D4—744no—
210C2×D4—744no—
211C2×D4—742no—
212C2×D4—748no—
213C2×D4—748no—
214C2×D4—741no—
215C2×D4—744no—
216C2×D4C4×D3712yes—
217C2×D4D14812yes—
218C2×D4C22⋊D3712yes—
219C22×C4—372no—
220C22×C4—742no—
221C22×C4—741no—
222C22×C4C22×D3711yescharacteristic
223M16—21no—
224M16—21no—
225M16—741no—
226M16—741no—
227C2×C8—21no—
228C2×C8—741no—
229C4⋊C4—21no—
230C4⋊C4—741no—
231C22⋊C4—21no—
232C22⋊C4—741no—
233C42—21no—
234C42—741no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—82no—
239C23—372no—
240C23—2964no—
241C23—748no—
242C23—741no—
243C23D4×D3711yescharacteristic
244Q8—372no—
245Q8—741no—
246Q8—741no—
247Q8—741no—
248D4—22no—
249D4—22no—
250D4—22no—
251D4—372no—
252D4—748no—
253D4—748no—
254D4—742no—
255D4—742no—
256D4—742no—
257D4—742no—
258D4—741no—
259D4—741no—
260D4—741no—
261D4#1059693814yes—
262C2×C4—41no—
263C2×C4—372no—
264C2×C4—748no—
265C2×C4—741no—
266C2×C4—742no—
267C2×C4—742no—
268C2×C4—742no—
269C2×C4—742no—
270C2×C4—1481no—
271C2×C4C2×C4×D3711yescharacteristic
272C2×C4C2×D14811yescharacteristic
273C2×C4D4×D3711yescharacteristic
274C2×C4D4×D3711yescharacteristic
275C2×C4D4×D3711yescharacteristic
276C2×C4C2×C22⋊D3711yescharacteristic
277C8—41no—
278C8—1481no—
279C22—24no—
280C22—41no—
281C22—44no—
282C22—82no—
283C22—82no—
284C22—2968no—
285C22—742no—
286C22—742no—
287C22—1482no—
288C22—1482no—
289C22C22⋊C4×D3711yescharacteristic
290C22#1192450411yescharacteristic
291C22#1192450611yescharacteristic
292C4—81no—
293C4—2961no—
294C4—742no—
295C4—742no—
296C4C22⋊C4×D3711yescharacteristic
297C4C2×#1059693811yescharacteristic
298C4#1192450411yescharacteristic
299C4#1192450611yescharacteristic
300C2—21no—
301C2—21no—
302C2—21no—
303C2—44no—
304C2—1482no—
305C2—1482no—
306C2#1455738811yescharacteristic
subgroupG/H
centerC2#14557388
commutatorC2×C4×C37C22×C4
FrattiniC22×C4C22×D37
FittingC37×#418C2
socleC2×C37#379
prime radicalC2×#11924581C2
cosocleC22×C4×C37C23
hypercenter#418D37
hypocenterC37#1795
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #14557551C2semidirect
3. #14557549C2semidirect
4. #14557538C2semidirect
5. #14557515C2semidirect
6. #14557506C2semidirect
7. C37×#418C2semidirect
15. C2×#10596964C4semidirect
18. C2×D4×D37C4semidirect
19. C2×#10596934C22semidirect
20. #11924495C22semidirect
28. C2×M16×C37C22semidirect
29. #117×C37C22semidirect
158. #418D37semidirect
161. C37#1795semidirect
175. C2×M16C2×D37semidirect
177. #117C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245812
#145575512
#145575492
#145575382
#145575152
#145575062
C37×#4182
#179537