Finite Group Databasebeta

← #34054039

Subgroups — #34054039

Order 19072.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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#23750461C211yesmaximal, characteristic, complemented
2#23750423C212yesmaximal, complemented
3#23749807C212yesmaximal, complemented
4#23749802C212yesmaximal, complemented
5#23749797C212yesmaximal, complemented
6#23749792C212yesmaximal, complemented
7C2×#18011839C212yesmaximal, complemented
8#23749609C211yesmaximal, characteristic, complemented
9#532×C149C211yesmaximal, characteristic, complemented
10#18011959C2212yes—
11#18011958—22no—
12#18011957—22no—
13#18011956—22no—
14#18011956C2212yes—
15C2×#14558736—22no—
16C2×#14558736C2212yes—
17#41×D149—24no—
18#18011942C2214yescomplemented
19#18011941—22no—
20#18011937C2214yescomplemented
21#18011936—22no—
22#18011931—24no—
23C2×#14558720C2212yescomplemented
24#18011839C2214yescomplemented
25#18011839C2214yescomplemented
26#18011830C2212yescomplemented
27#18011828C2212yescomplemented
28C2×#41×C149C2211yescharacteristic
29C2×#41×C149C2211yescharacteristic, complemented
30C4⋊Q8×C149C2211yescharacteristic, complemented
31#127×C149C2211yescharacteristic, complemented
32#124×C149C2211yescharacteristic, complemented
33#124×C149C2211yescharacteristic, complemented
34#121×C149—22no—
35C4×D4×C149—22no—
36C2×C42×C149C2211yescharacteristic, complemented
37#14558737—24no—
38#14558736—24no—
39#14558736—24no—
40#14558736—22no—
41#14558736—24no—
42#14558736—22no—
43#14558736D414yescomplemented
44#14558736C2314yes—
45#14558735—28no—
46#14558735—24no—
47C2×Dic298—22no—
48C2×Dic298D412yescomplemented
49Q8×D149—24no—
50C2×D596—22no—
51C2×D596D412yescomplemented
52C2×C22⋊D149—24no—
53C2×C22⋊D149—22no—
54D4×D149—28no—
55D4×D149—24no—
56C2×C4×D149—24no—
57C2×C4×D149—22no—
58#14558726—24no—
59#14558724—24no—
60#14558723—24no—
61#14558721—24no—
62#14558720—22no—
63#14558720C2314yes—
64C2×#11924746—22no—
65#41×C149—22no—
66#41×C149—22no—
67#41×C149—22no—
68#41×C149—22no—
69C2×Q8×C149C2311yescharacteristic
70C2×Q8×C149C2311yescharacteristic
71C2×D4×C149—22no—
72C2×D4×C149—21no—
73C2×D4×C149D412yes—
74C2×D4×C149C2311yescharacteristic
75C2×D4×C149C2311yescharacteristic
76C22×C4×C149—21no—
77C22×C4×C149—21no—
78C22×C4×C149D412yes—
79C22×C4×C149C2311yescharacteristic
80C4⋊C4×C149—22no—
81C22⋊C4×C149—22no—
82C22⋊C4×C149—22no—
83C42×C149—21no—
84C42×C149C2311yescharacteristic
85C42×C149C2311yescharacteristic
86C22×D149—24no—
87C22×D149—22no—
88Dic298—24no—
89Dic298—22no—
90Dic298C2×D414yes—
91D596—24no—
92D596—22no—
93D596C2×D414yes—
94C22⋊D149—28no—
95C22⋊D149—24no—
96C22⋊D149—24no—
97C2×Dic149—24no—
98C2×Dic149—22no—
99C4×D149—28no—
100C4×D149—24no—
101C4×D149—24no—
102#11924746—24no—
103C23×C149—21no—
104C23×C149C2×D412yes—
105C23×C149C2×D411yescharacteristic
106Q8×C149—22no—
107Q8×C149—22no—
108D4×C149—24no—
109D4×C149—22no—
110D4×C149—22no—
111D4×C149—44no—
112D4×C149—42no—
113C2×C4×C149—21no—
114C2×C4×C149—22no—
115C2×C4×C149—22no—
116C2×C4×C149—21no—
117C2×C4×C149—21no—
118C2×C4×C149—42no—
119C2×C4×C149—41no—
120C2×C4×C149—41no—
121C2×C4×C149C2×D412yes—
122C2×C4×C149C2×D412yes—
123C2×C4×C149C2×D412yes—
124C2×C4×C149C2×D411yescharacteristic
125C2×C4×C149C2411yescharacteristic
126C22×C149—22no—
127C22×C149—41no—
128C22×C149—42no—
129C22×C149—42no—
130C22×C149—41no—
131C22×C149#12012yes—
132C22×C149C22×D411yescharacteristic
133C2×D149—24no—
134C2×D149—82no—
135C2×D149—84no—
136C4×C149—22no—
137C4×C149—22no—
138C4×C149—41no—
139C4×C149—41no—
140C4×C149#12011yescharacteristic
141C4×C149#12011yescharacteristic
142C4×C149C22×D412yes—
143Dic149—24no—
144C2×C149—21no—
145C2×C149—22no—
146C2×C149—42no—
147C2×C149—81no—
148C2×C149C2×#12011yescharacteristic
149D149—44no—
150C149#301811yescharacteristic, complemented, Sylow
151#3018—1491nomaximal, Sylow
152#585—1491no—
153#575—1492no—
154#532D14911yescharacteristic, complemented
155#456—1492no—
156#455—1492no—
157#454—1492no—
158#453—1492no—
159C2×#104—1492no—
160#413—1491no—
1612−1+4—2982no—
1622−1+4—1492no—
163H(5,𝔽2)—2982no—
164H(5,𝔽2)—2982no—
165H(5,𝔽2)—1492no—
166C2×#41—2984no—
167C2×#41—2982no—
168C2×#41—1492no—
169C2×#41C2×D14911yescharacteristic
170C2×#41C2×D14911yescharacteristic, complemented
171#137—2982no—
172#137—1494no—
173AGL(1,ℤ/8)—2982no—
174AGL(1,ℤ/8)—1494no—
175#135—2984no—
176C2×M16—1492no—
177C4⋊Q8C2×D14911yescharacteristic, complemented
178#127C2×D14911yescharacteristic, complemented
179#124C2×D14911yescharacteristic, complemented
180#124C2×D14911yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D14911yescharacteristic, complemented
184#104—1494no—
185#104—1494no—
186#101—1492no—
187#100—1492no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—2984no—
193#41—2984no—
194#41—2988no—
195#41—2984no—
196#41—2984no—
197#41—2982no—
198#41—2982no—
199#41—2984no—
200#41—1494no—
201#41—1494no—
202C2×Q8—2982no—
203C2×Q8—2984no—
204C2×Q8—1492no—
205C2×Q8C22×D14911yescharacteristic
206C2×Q8C22×D14911yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—2982no—
210C2×D4—2988no—
211C2×D4—2984no—
212C2×D4—2984no—
213C2×D4—2982no—
214C2×D4—1492no—
215C2×D4C22⋊D14912yes—
216C2×D4C22×D14911yescharacteristic
217C2×D4C22×D14911yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—2984no—
221C22×C4—2982no—
222C22×C4C22⋊D14912yes—
223C22×C4C22×D14911yescharacteristic
224Q16—2984no—
225SD16—2984no—
226SD16—2984no—
227D8—2984no—
228M16—2982no—
229M16—1494no—
230C2×C8—2982no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D14911yescharacteristic
236C42C22×D14911yescharacteristic
237C23—21no—
238C23—2984no—
239C23—2982no—
240C23C2×C22⋊D14912yes—
241C23C2×C22⋊D14911yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—2984no—
245Q8—2982no—
246Q8—1494no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—2984no—
253D4—2988no—
254D4—2984no—
255D4—2984no—
256D4—2982no—
257D4—1494no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—2988no—
267C2×C4—2984no—
268C2×C4—2982no—
269C2×C4—2984no—
270C2×C4—2984no—
271C2×C4D4×D14912yes—
272C2×C4D4×D14912yes—
273C2×C4C2×C22⋊D14912yes—
274C2×C4C2×C22⋊D14911yescharacteristic
275C2×C4C23×D14911yescharacteristic
276C8—2984no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—2984no—
283C22—11922no—
284C22—11924no—
285C22#1801187912yes—
286C22C22×C22⋊D14911yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—2984no—
292C4#1801187911yescharacteristic
293C4#1801187911yescharacteristic
294C4C2×D4×D14912yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—5964no—
300C2C2×#1801187911yescharacteristic
subgroupG/H
centerC4#18011879
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#532×C149C2
socleC2×C149C2×#120
hypercenter#532D149
hypocenterC149#3018
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #23750461C2semidirect
2. #23750423C2semidirect
3. #23749807C2semidirect
4. #23749802C2semidirect
5. #23749797C2semidirect
6. #23749792C2semidirect
7. C2×#18011839C2semidirect
8. #23749609C2semidirect
9. #532×C149C2semidirect
18. #18011942C22semidirect
20. #18011937C22semidirect
23. C2×#14558720C22semidirect
24. #18011839C22semidirect
25. #18011839C22semidirect
26. #18011830C22semidirect
27. #18011828C22semidirect
29. C2×#41×C149C22semidirect
30. C4⋊Q8×C149C22semidirect
31. #127×C149C22semidirect
32. #124×C149C22semidirect
33. #124×C149C22semidirect
36. C2×C42×C149C22semidirect
43. #14558736D4semidirect
48. C2×Dic298D4semidirect
51. C2×D596D4semidirect
150. C149#3018semidirect
154. #532D149semidirect
170. C2×#41C2×D149semidirect
177. C4⋊Q8C2×D149semidirect
178. #127C2×D149semidirect
179. #124C2×D149semidirect
180. #124C2×D149semidirect
183. C2×C42C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237504612
#237504232
#237498072
#237498022
#237497972
#237497922
C2×#180118392
#237496092
#532×C1492
#3018149