Finite Group Databasebeta

← #34061321

Subgroups — #34061321

Order 19072.

738 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 198 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#23750704C211yesmaximal, characteristic, complemented
2#41×#10597010C211yesmaximal, characteristic, complemented
3#23750623C213yesmaximal, complemented
4#23750622C213yesmaximal, complemented
5D149⋊#104C213yesmaximal, complemented
6C2×#18011975C213yesmaximal, complemented
7#23750460C211yesmaximal, characteristic
8#18011999—21no—
9#18011998—23no—
10C2×#14558746C2213yescomplemented
11D149⋊M16C2213yescomplemented
12Q8×#10597010—21no—
13D4×#10597010—23no—
14#18011990C2213yescomplemented
15C2×C4×#10597010C2213yescomplemented
16#18011976C2211yescharacteristic, complemented
17#18011976C2213yescomplemented
18#18011975C2213yescomplemented
19#18011975C2216yescomplemented
20#18011975C2213yescomplemented
21#18011958—21no—
22#18011958C411yescharacteristic, complemented
23#18011958C413yescomplemented
24#18011956—23no—
25#18011956C413yescomplemented
26C2×#14558735—23no—
27C2×#14558735C2213yes—
28#41×D149—21no—
29#41×D149C2213yes—
30#41×D149C2211yescharacteristic
31H(5,𝔽2)×C149C411yescharacteristic, complemented
32C22×#10597010—23no—
33#14558746—23no—
34#14558746C2311yescharacteristic
35#14558746C2313yes—
36#14558745—23no—
37D149⋊C8—23no—
38C2×#11924753—23no—
39C4⋊#10597010—23no—
40C22⋊#10597010—23no—
41C4×#10597010—23no—
42C4×#10597010C2311yescharacteristic
43C4×#10597010C2313yes—
44#14558737D411yescharacteristic
45#14558737C2311yescharacteristic
46#14558736—23no—
47#14558736—23no—
48#14558736C2×C413yes—
49#14558736C2×C413yescomplemented
50#14558735—23no—
51#14558735—26no—
52#14558735—23no—
53#14558735—23no—
54#14558735—23no—
55#14558735—23no—
56#14558735C2×C416yescomplemented
57#14558735D413yes—
58#14558735C2313yes—
59C2×Dic298—23no—
60C2×Dic298C2×C413yescomplemented
61Q8×D149—21no—
62Q8×D149C2×C413yescomplemented
63Q8×D149D411yescharacteristic
64C2×C22⋊D149—23no—
65C2×C22⋊D149—26no—
66D4×D149—23no—
67D4×D149—23no—
68D4×D149C2×C413yescomplemented
69D4×D149D413yes—
70C22×Dic149—23no—
71C2×C4×D149—23no—
72C2×C4×D149C2313yes—
73#41×C149—21no—
74#41×C149C2×C411yescharacteristic
75#41×C149C2×C413yescomplemented
76C2×D4×C149—23no—
77C2×D4×C149C2×C413yescomplemented
78C2×#10597010—21no—
79C2×#10597010—23no—
80C2×#10597010—46no—
81#11924753—23no—
82#11924753—21no—
83C22×D149—23no—
84C22×D149C2×D413yes—
85Dic298—23no—
86Dic298—23no—
87Dic298C22⋊C413yes—
88Dic298C22×C413yes—
89D596—21no—
90D596C22⋊C413yes—
91D596C22×C411yescharacteristic
92C22⋊D149—23no—
93C22⋊D149—26no—
94C22⋊D149—23no—
95C22⋊D149—23no—
96C22⋊D149C22⋊C416yes—
97C2×Dic149—23no—
98C2×Dic149—26no—
99C2×Dic149—23no—
100C2×Dic149C2×D413yes—
101C4×D149—23no—
102C4×D149—23no—
103C4×D149—21no—
104C4×D149C22×C413yes—
105C4×D149C2×D413yes—
106C4×D149C2×D413yes—
107C4×D149C2411yescharacteristic
108C23×C149—23no—
109Q8×C149C22⋊C411yescharacteristic
110Q8×C149C22×C411yescharacteristic
111D4×C149—23no—
112D4×C149—23no—
113D4×C149C22⋊C413yes—
114D4×C149C22×C413yes—
115C2×C4×C149—23no—
116C2×C4×C149C22×C413yes—
117C22×C149—23no—
118C22×C149—83no—
119C22×C149C2×C22⋊C413yes—
120C2×D149—21no—
121C2×D149—43no—
122C2×D149—83no—
123C2×D149C2×C22⋊C413yes—
124C2×D149C22×D411yescharacteristic
125#10597010—22no—
126#10597010—43no—
127C4×C149—21no—
128C4×C149C2×C22⋊C413yes—
129C4×C149C23×C411yescharacteristic
130Dic149—23no—
131Dic149C2×C22⋊C413yes—
132Dic149C22×D411yescharacteristic
133C2×C149—23no—
134C2×C149—43no—
135C2×C149C22×C22⋊C411yescharacteristic
136D149—21no—
137D149—23no—
138D149—41no—
139C149#290111yescharacteristic, complemented, Sylow
140#2901—1491nomaximal, Sylow
141#585—1491no—
142#568—1491no—
143C4×#41—1491no—
144#421—1493no—
145#421—1493no—
146C2×#104—1493no—
147C2×#104—1493no—
1482−1+4—2981no—
1492−1+4—1491no—
1502−1+4—1493no—
151H(5,𝔽2)—2983no—
152H(5,𝔽2)—1493no—
153H(5,𝔽2)#1059701011yescharacteristic, complemented
154C2×#41—2983no—
155C2×#41—2981no—
156C2×#41—1493no—
157C2×#41—1493no—
158C2×#41—1491no—
159#131—2983no—
160#131—2981no—
161C2×M16—1493no—
162C2×M16—1493no—
163C4×Q8—2981no—
164C4×D4—2983no—
165#117—1493no—
166C2×C42—1493no—
167#104—1493no—
168#104—1496no—
169#104—1493no—
170#104—1491no—
171#104—1493no—
172#41—21no—
173#41—2983no—
174#41—2983no—
175#41—2983no—
176#41—2986no—
177#41—2983no—
178#41—2983no—
179#41—2983no—
180#41—2983no—
181#41—1491no—
182#41—1496no—
183#41—1493no—
184#41—1493no—
185#41—1491no—
186#41—1493no—
187#41—1493no—
188#41C2×#1059701011yescharacteristic
189#41C2×#1059701013yescomplemented
190C2×Q8—2983no—
191C2×Q8—2981no—
192C2×Q8—1493no—
193C2×Q8—1493no—
194C2×Q8—1491no—
195C2×D4—23no—
196C2×D4—2983no—
197C2×D4—2983no—
198C2×D4—2983no—
199C2×D4—2986no—
200C2×D4—1493no—
201C2×D4—1493no—
202C2×D4C2×#1059701013yescomplemented
203C22×C4—2983no—
204C22×C4—2983no—
205C22×C4—2983no—
206C22×C4—1493no—
207M16—2983no—
208M16—2983no—
209M16—1491no—
210M16—1493no—
211C2×C8—2983no—
212C2×C8—2983no—
213C4⋊C4—2983no—
214C22⋊C4—2983no—
215C42—2983no—
216C42—1491no—
217C42—1493no—
218C23—23no—
219C23—2983no—
220C23—1493no—
221Q8—2983no—
222Q8—2983no—
223Q8—1493no—
224Q8—1493no—
225Q8C22⋊#1059701011yescharacteristic
226Q8C22×#1059701011yescharacteristic
227D4—23no—
228D4—23no—
229D4—2981no—
230D4—2983no—
231D4—2986no—
232D4—2983no—
233D4—2983no—
234D4—1496no—
235D4—1493no—
236D4—1491no—
237D4C22⋊#1059701013yes—
238D4C22×#1059701013yes—
239C2×C4—23no—
240C2×C4—2983no—
241C2×C4—2981no—
242C2×C4—2981no—
243C2×C4—2983no—
244C2×C4—2983no—
245C2×C4—2986no—
246C2×C4—2983no—
247C2×C4—2983no—
248C2×C4—5966no—
249C2×C4—1493no—
250C2×C4—1493no—
251C2×C4—1493no—
252C2×C4—1493no—
253C2×C4—1491no—
254C2×C4C22×#1059701013yes—
255C8—2983no—
256C8—2981no—
257C22—23no—
258C22—83no—
259C22—2981no—
260C22—5963no—
261C22—1491no—
262C22—1493no—
263C22—11923no—
264C22C2×C22⋊#1059701013yes—
265C4—21no—
266C4—2982no—
267C4—2983no—
268C4—5963no—
269C4—1491no—
270C4—1493no—
271C4C2×C22⋊#1059701013yes—
272C4C23×#1059701011yescharacteristic
273C2—23no—
274C2—43no—
275C2—2981no—
276C2—2983no—
277C2—5961no—
278C2C22×C22⋊#1059701011yescharacteristic
subgroupG/H
centerC2C22×C22⋊#10597010
commutatorC4×C149C23×C4
FrattiniC4C23×#10597010
FittingH(5,𝔽2)×C149C4
socleC2×C149C22×C22⋊C4
prime radical#23750460C2
cosocleC4×D149C24
hypercenterH(5,𝔽2)#10597010
hypocenterC149#2901
LCSG ▷ C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #23750704C2semidirect
2. #41×#10597010C2semidirect
3. #23750623C2semidirect
4. #23750622C2semidirect
5. D149⋊#104C2semidirect
6. C2×#18011975C2semidirect
10. C2×#14558746C22semidirect
11. D149⋊M16C22semidirect
14. #18011990C22semidirect
15. C2×C4×#10597010C22semidirect
16. #18011976C22semidirect
17. #18011976C22semidirect
18. #18011975C22semidirect
19. #18011975C22semidirect
20. #18011975C22semidirect
22. #18011958C4semidirect
23. #18011958C4semidirect
25. #18011956C4semidirect
31. H(5,𝔽2)×C149C4semidirect
49. #14558736C2×C4semidirect
56. #14558735C2×C4semidirect
60. C2×Dic298C2×C4semidirect
62. Q8×D149C2×C4semidirect
68. D4×D149C2×C4semidirect
75. #41×C149C2×C4semidirect
77. C2×D4×C149C2×C4semidirect
139. C149#2901semidirect
153. H(5,𝔽2)#10597010semidirect
189. #41C2×#10597010semidirect
202. C2×D4C2×#10597010semidirect

Maximal subgroups

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

subgroupindex
#237507042
#41×#105970102
#237506232
#237506222
D149⋊#1042
C2×#180119752
#237504602
#2901149