Finite Group Databasebeta

← #17995636

Subgroups — #17995636

Order 4736.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#11924562C211yesmaximal, characteristic, complemented
2#14558111C211yesmaximal, characteristic, complemented
3C2×#11924463C211yesmaximal, characteristic, complemented
4C2×#11924462C211yesmaximal, characteristic, complemented
5#14557436C211yesmaximal, characteristic, complemented
6#14557301C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
8C2×#10596962C2211yescharacteristic
9#11924562—22no—
10#11924562—22no—
11C2×#10596940—21no—
12C2×#10596939—21no—
13C2×#10596934C2211yescharacteristic
14#11924515—21no—
15#11924512—21no—
16#11924510—21no—
17#11924508—22no—
18#11924507—21no—
19C2×C4⋊Dic37C2211yescharacteristic
20C2×C22⋊Dic37—22no—
21C2×C4×Dic37C2211yescharacteristic
22#11924463—22no—
23#11924462—22no—
24C2×#41×C37C2211yescharacteristic
25C22×D4×C37C2211yescharacteristic
26AGL(1,ℤ/8)×C37—22no—
27AGL(1,ℤ/8)×C37—22no—
28C2×SD16×C37—21no—
29C2×D8×C37—21no—
30C2×M16×C37C2211yescharacteristic
31C24×C37—21no—
32#41×C37—21no—
33#41×C37—21no—
34#41×C37—21no—
35#41×C37—21no—
36C2×Q8×C37D411yescharacteristic
37C2×D4×C37—21no—
38C2×D4×C37—21no—
39C2×D4×C37—21no—
40C2×D4×C37—21no—
41C2×D4×C37—21no—
42C2×D4×C37—21no—
43C2×D4×C37D411yescharacteristic
44C2×D4×C37D412yescomplemented
45C22×C4×C37—21no—
46C22×C4×C37C2311yescharacteristic
47C2×C22⋊D37—21no—
48C2×C22⋊D37—22no—
49C22×Dic37—21no—
50C22×Dic37D412yes—
51#10596962—21no—
52#10596962—21no—
53#10596962—21no—
54#10596962—21no—
55C2×D148D411yescharacteristic
56C2×C4×D37—21no—
57C2×Dic74D411yescharacteristic
58SD16×C37—24no—
59D8×C37—24no—
60M16×C37—21no—
61M16×C37D412yescomplemented
62C2×C8×C37—21no—
63C22⋊Dic37—22no—
64C22⋊Dic37—44no—
65#10596940—24no—
66#10596939—24no—
67#10596938—22no—
68C4⋊Dic37—21no—
69C4⋊Dic37—21no—
70#10596936—22no—
71C4×Dic37—21no—
72C4×Dic37—21no—
73#10596934—21no—
74#10596934D412yescomplemented
75C2×#64436—21no—
76C23×C37—21no—
77C23×C37—22no—
78C23×C37—42no—
79C23×C37—42no—
80C23×C37—41no—
81C23×C37—41no—
82C23×C37C2×D411yescharacteristic
83C22×D37—21no—
84Q8×C37—21no—
85Q8×C37—21no—
86D4×C37—22no—
87D4×C37—21no—
88D4×C37—22no—
89D4×C37—21no—
90D4×C37—41no—
91D4×C37—41no—
92D4×C37—41no—
93D4×C37—41no—
94C2×C4×C37—21no—
95C2×C4×C37—41no—
96C2×C4×C37—41no—
97C2×C4×C37C2×D411yescharacteristic
98C2×C4×C37C2×D411yescharacteristic
99C2×C4×C37C2×D411yescharacteristic
100C2×C4×C37C2×D411yescharacteristic
101C2×C4×C37C2×D411yescharacteristic
102C2×C4×C37#4111yescharacteristic
103C22⋊D37—41no—
104C22⋊D37—41no—
105C22⋊D37—82no—
106C2×Dic37—22no—
107C2×Dic37—21no—
108C2×Dic37—22no—
109C2×Dic37—41no—
110C2×Dic37—41no—
111C2×Dic37—42no—
112D148—21no—
113D148—21no—
114C4×D37—41no—
115C4×D37—41no—
116Dic74—21no—
117Dic74—21no—
118C8×C37—22no—
119#64436—22no—
120C22×C37—21no—
121C22×C37—21no—
122C22×C37—44no—
123C22×C37—41no—
124C22×C37—42no—
125C22×C37—41no—
126C22×C37—41no—
127C22×C37—82no—
128C22×C37—81no—
129C22×C37—81no—
130C22×C37#12011yescharacteristic
131C22×C37#12111yescharacteristic
132C22×C37#12711yescharacteristic
133C2×D37—41no—
134C2×D37—41no—
135C2×D37—41no—
136C4×C37—41no—
137C4×C37#12011yescharacteristic
138C4×C37#12011yescharacteristic
139C4×C37#12111yescharacteristic
140C4×C37#12111yescharacteristic
141Dic37—42no—
142Dic37—41no—
143Dic37—42no—
144#2006—371nomaximal, Sylow
145C2×C37—21no—
146C2×C37—21no—
147C2×C37—81no—
148C2×C37—82no—
149C2×C37#39211yescharacteristic
150C2×C37#45311yescharacteristic
151C2×C37#45411yescharacteristic
152D37—81no—
153C2×AGL(1,ℤ/8)—371no—
154C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
155#537—371no—
156C2×#104—371no—
157C2×#104—371no—
158#390—371no—
159#343—371no—
160C37#200611yescharacteristic, complemented, Sylow
161C2×#41—371no—
162C2×#41C2×D3711yescharacteristic
163C22×D4C2×D3711yescharacteristic
164AGL(1,ℤ/8)—22no—
165AGL(1,ℤ/8)—22no—
166AGL(1,ℤ/8)—742no—
167AGL(1,ℤ/8)—742no—
168C2×SD16—21no—
169C2×SD16—741no—
170C2×D8—21no—
171C2×D8—741no—
172C2×M16—371no—
173C2×M16C2×D3711yescharacteristic
174C22⋊Q8—741no—
175C22⋊Q8—741no—
176#121—741no—
177#121—742no—
178#121—741no—
179C2×C4⋊C4—371no—
180C2×C22⋊C4—742no—
181C2×C42—371no—
182#104—742no—
183#104—742no—
184C24—21no—
185#41—21no—
186#41—21no—
187#41—21no—
188#41—21no—
189#41—741no—
190#41—741no—
191#41—741no—
192#41—741no—
193C2×Q8—371no—
194C2×Q8C22⋊D3711yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—371no—
202C2×D4—742no—
203C2×D4—741no—
204C2×D4C22⋊D3711yescharacteristic
205C2×D4C22⋊D3712yescomplemented
206C22×C4—21no—
207C22×C4—372no—
208C22×C4—741no—
209C22×C4—741no—
210C22×C4C22×D3711yescharacteristic
211SD16—24no—
212SD16—744no—
213D8—24no—
214D8—744no—
215M16—21no—
216M16—372no—
217M16—741no—
218M16C22⋊D3712yescomplemented
219C2×C8—21no—
220C2×C8—741no—
221C4⋊C4—742no—
222C4⋊C4—741no—
223C4⋊C4—741no—
224C22⋊C4—742no—
225C22⋊C4—742no—
226C22⋊C4—1484no—
227C42—741no—
228C42—741no—
229C23—21no—
230C23—22no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—741no—
236C23D4×D3711yescharacteristic
237Q8—21no—
238Q8—21no—
239Q8—741no—
240Q8—741no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—21no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—2962no—
250D4—741no—
251D4—741no—
252D4—1481no—
253D4—1481no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—742no—
258C2×C4—742no—
259C2×C4—741no—
260C2×C4—1481no—
261C2×C4—1481no—
262C2×C4—1481no—
263C2×C4—1481no—
264C2×C4—1482no—
265C2×C4D4×D3711yescharacteristic
266C2×C4D4×D3711yescharacteristic
267C2×C4#1059696411yescharacteristic
268C2×C4C2×C22⋊D3711yescharacteristic
269C2×C4C2×C22⋊D3711yescharacteristic
270C2×C4C2×C22⋊D3711yescharacteristic
271C8—22no—
272C8—742no—
273C22—21no—
274C22—21no—
275C22—44no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22—1481no—
284C22—1481no—
285C22—1481no—
286C22#1192450411yescharacteristic
287C22#1192450811yescharacteristic
288C22#1192453211yescharacteristic
289C4—41no—
290C4—1482no—
291C4—1481no—
292C4—1482no—
293C4#1192450411yescharacteristic
294C4#1192450511yescharacteristic
295C4#1192450711yescharacteristic
296C4#1192450811yescharacteristic
297C2—21no—
298C2—21no—
299C2—82no—
300C2—81no—
301C2—2961no—
302C2#1455744111yescharacteristic
303C2#1455769911yescharacteristic
304C2#1455770411yescharacteristic
subgroupG/H
centerC22#11924504
commutator, cosocleC22×C4×C37C23
FrattiniC22×C4C22×D37
FittingC2×AGL(1,ℤ/8)×C37C2
socleC22×C37#120
hypercenterC2×AGL(1,ℤ/8)D37
hypocenterC37#2006
LCSG ▷ C22×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11924562C2semidirect
2. #14558111C2semidirect
3. C2×#11924463C2semidirect
4. C2×#11924462C2semidirect
5. #14557436C2semidirect
6. #14557301C2semidirect
7. C2×AGL(1,ℤ/8)×C37C2semidirect
44. C2×D4×C37D4semidirect
61. M16×C37D4semidirect
74. #10596934D4semidirect
154. C2×AGL(1,ℤ/8)D37semidirect
160. C37#2006semidirect
205. C2×D4C22⋊D37semidirect
218. M16C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245622
#145581112
C2×#119244632
C2×#119244622
#145574362
#145573012
C2×AGL(1,ℤ/8)×C372
#200637