Finite Group Databasebeta

← #34054048

Subgroups — #34054048

Order 19072.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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
1#23750423C212yesmaximal, complemented
2#23750192C211yesmaximal, characteristic, complemented
3#23749803C214yesmaximal, complemented
4#23749793C214yesmaximal, complemented
5#23749642C212yesmaximal, complemented
6C2×#18011828C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C149C211yesmaximal, characteristic, complemented
8C2×#14558736C2211yescharacteristic, complemented
9#18011942—22no—
10#18011941C2214yescomplemented
11#18011937—22no—
12#18011936C2214yescomplemented
13#18011931—24no—
14C2×#14558720C2212yescomplemented
15#18011907C2212yescomplemented
16#18011898C2212yescomplemented
17#18011882—22no—
18#18011880—22no—
19#18011870C2211yescharacteristic, complemented
20#18011838C2218yescomplemented
21#18011828C2214yescomplemented
22H(5,𝔽2)×C149—24no—
23H(5,𝔽2)×C149—22no—
24H(5,𝔽2)×C149C2214yes—
25C2×#41×C149—22no—
26C2×#41×C149C2212yes—
27C22×D4×C149—22no—
28C22×D4×C149—22no—
29C22×D4×C149C2211yescharacteristic
30#14558736—22no—
31#14558736—22no—
32C2×Dic298C2311yescharacteristic, complemented
33C2×D596C2311yescharacteristic, complemented
34C2×C22⋊D149—21no—
35C2×C22⋊D149—24no—
36C2×C4×D149—21no—
37#14558726—24no—
38#14558724—24no—
39#14558723—24no—
40#14558721—24no—
41#14558720—22no—
42#14558720C2314yescomplemented
43C2×#11924746—22no—
44C4⋊Dic149—21no—
45C22⋊Dic149—21no—
46C22⋊Dic149—24no—
47C4×Dic149C2312yescomplemented
48C24×C149—22no—
49C24×C149D412yes—
50#41×C149—28no—
51#41×C149—28no—
52#41×C149—22no—
53#41×C149—22no—
54#41×C149D414yescomplemented
55#41×C149C2314yes—
56C2×Q8×C149D412yescomplemented
57C2×D4×C149—24no—
58C2×D4×C149—28no—
59C2×D4×C149—28no—
60C2×D4×C149—22no—
61C2×D4×C149—28no—
62C2×D4×C149—28no—
63C2×D4×C149—22no—
64C2×D4×C149—22no—
65C2×D4×C149D412yes—
66C2×D4×C149D412yescomplemented
67C2×D4×C149C2312yes—
68C22×C4×C149—22no—
69C22×C4×C149—22no—
70C22×C4×C149C2311yescharacteristic
71C22×D149—21no—
72Dic298—22no—
73D596—22no—
74C22⋊D149—41no—
75C22⋊D149—41no—
76C22⋊D149—84no—
77C2×Dic149—21no—
78C2×Dic149—22no—
79C4×D149—42no—
80#11924746—24no—
81C23×C149—28no—
82C23×C149—22no—
83C23×C149—22no—
84C23×C149—22no—
85C23×C149—82no—
86C23×C149—84no—
87C23×C149C2×D414yes—
88C23×C149C2×D411yescharacteristic
89Q8×C149—22no—
90Q8×C149C2×D414yes—
91D4×C149—216no—
92D4×C149—24no—
93D4×C149—24no—
94D4×C149—24no—
95D4×C149—22no—
96D4×C149—24no—
97D4×C149C2×D414yes—
98C2×C4×C149—28no—
99C2×C4×C149—22no—
100C2×C4×C149—24no—
101C2×C4×C149—24no—
102C2×C4×C149C2×D411yescharacteristic
103C2×C4×C149C2×D412yes—
104C2×C4×C149C2×D412yes—
105C2×C4×C149C2411yescharacteristic
106C22×C149—24no—
107C22×C149—24no—
108C22×C149—41no—
109C22×C149—44no—
110C22×C149—44no—
111C22×C149—44no—
112C22×C149—82no—
113C22×C149—88no—
114C22×C149#12011yescharacteristic
115C22×C149#12011yescharacteristic
116C22×C149C22×D411yescharacteristic
117C2×D149—42no—
118C2×D149—41no—
119C4×C149—24no—
120C4×C149#12012yes—
121C4×C149C22×D412yes—
122Dic149—42no—
123Dic149—41no—
124C2×C149—21no—
125C2×C149—21no—
126C2×C149—21no—
127C2×C149—44no—
128C2×C149—44no—
129C2×C149C2×#12011yescharacteristic
130D149—81no—
131C149#301911yescharacteristic, complemented, Sylow
132#3019—1491nomaximal, Sylow
133C2×H(5,𝔽2)D14911yescharacteristic, complemented
134#575—1492no—
135#535—1491no—
136#455—1494no—
137#453—1494no—
138#421—1492no—
139C2×#100—1491no—
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D14914yes—
143C2×#41—22no—
144C2×#41—1491no—
145C2×#41C2×D14912yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D14911yescharacteristic
149#137—2982no—
150#137—1494no—
151AGL(1,ℤ/8)—2982no—
152AGL(1,ℤ/8)—1494no—
153#135—2984no—
154C2×M16—1492no—
155#127—1492no—
156#124—1492no—
157#121—2982no—
158#120—2982no—
159#117—1491no—
160#104—1498no—
161#100—1494no—
162C24—22no—
163C24C22⋊D14912yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—2982no—
169#41—2982no—
170#41C22⋊D14914yescomplemented
171#41C22×D14914yes—
172C2×Q8—1491no—
173C2×Q8C22⋊D14912yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—2981no—
183C2×D4—2984no—
184C2×D4—1491no—
185C2×D4C22⋊D14912yes—
186C2×D4C22⋊D14912yescomplemented
187C2×D4C22×D14912yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—2981no—
191C22×C4C22×D14911yescharacteristic
192Q16—2984no—
193SD16—2984no—
194SD16—2984no—
195D8—2984no—
196M16—2982no—
197M16—1494no—
198C2×C8—2982no—
199C4⋊C4—2981no—
200C22⋊C4—2981no—
201C22⋊C4—2984no—
202C42—1492no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—2981no—
210C23C2×C22⋊D14914yes—
211C23C2×C22⋊D14911yescharacteristic
212Q8—22no—
213Q8—2982no—
214Q8C2×C22⋊D14914yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—2982no—
222D4—5961no—
223D4—5961no—
224D4—11924no—
225D4C2×C22⋊D14914yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—2982no—
231C2×C4—2981no—
232C2×C4—5962no—
233C2×C4C2×C22⋊D14912yes—
234C2×C4C2×C22⋊D14912yes—
235C2×C4C2×C22⋊D14911yescharacteristic
236C2×C4C23×D14911yescharacteristic
237C8—2984no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—5962no—
247C22—5961no—
248C22#1801188011yescharacteristic
249C22#1801188011yescharacteristic
250C22C22×C22⋊D14911yescharacteristic
251C4—24no—
252C4—5962no—
253C4—5961no—
254C4#1801188012yes—
255C4C22×C22⋊D14912yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—11921no—
262C2C2×#1801188011yescharacteristic
subgroupG/H
centerC2C2×#18011880
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×H(5,𝔽2)×C149C2
socleC2×C149C2×#120
hypercenterC2×H(5,𝔽2)D149
hypocenterC149#3019
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #23750423C2semidirect
2. #23750192C2semidirect
3. #23749803C2semidirect
4. #23749793C2semidirect
5. #23749642C2semidirect
6. C2×#18011828C2semidirect
7. C2×H(5,𝔽2)×C149C2semidirect
8. C2×#14558736C22semidirect
10. #18011941C22semidirect
12. #18011936C22semidirect
14. C2×#14558720C22semidirect
15. #18011907C22semidirect
16. #18011898C22semidirect
19. #18011870C22semidirect
20. #18011838C22semidirect
21. #18011828C22semidirect
32. C2×Dic298C23semidirect
33. C2×D596C23semidirect
42. #14558720C23semidirect
47. C4×Dic149C23semidirect
54. #41×C149D4semidirect
56. C2×Q8×C149D4semidirect
66. C2×D4×C149D4semidirect
131. C149#3019semidirect
133. C2×H(5,𝔽2)D149semidirect
170. #41C22⋊D149semidirect
173. C2×Q8C22⋊D149semidirect
186. C2×D4C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
#237504232
#237501922
#237498032
#237497932
#237496422
C2×#180118282
C2×H(5,𝔽2)×C1492
#3019149