Finite Group Databasebeta

← #34116766

Subgroups — #34116766

Order 20096.

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#23758352C211yesmaximal, characteristic, complemented
2#41×#10597845C211yesmaximal, characteristic, complemented
3#23758271C213yesmaximal, complemented
4#23758270C213yesmaximal, complemented
5D157⋊#104C213yesmaximal, complemented
6C2×#19097835C213yesmaximal, complemented
7#23758108C211yesmaximal, characteristic
8#19097859—21no—
9#19097858—23no—
10C2×#14578964C2213yescomplemented
11D157⋊M16C2213yescomplemented
12Q8×#10597845—21no—
13D4×#10597845—23no—
14#19097850C2213yescomplemented
15C2×C4×#10597845C2213yescomplemented
16#19097836C2211yescharacteristic, complemented
17#19097836C2213yescomplemented
18#19097835C2213yescomplemented
19#19097835C2216yescomplemented
20#19097835C2213yescomplemented
21#19097818—21no—
22#19097818C411yescharacteristic, complemented
23#19097818C413yescomplemented
24#19097816—23no—
25#19097816C413yescomplemented
26C2×#14578953—23no—
27C2×#14578953C2213yes—
28#41×D157—21no—
29#41×D157C2213yes—
30#41×D157C2211yescharacteristic
31H(5,𝔽2)×C157C411yescharacteristic, complemented
32C22×#10597845—23no—
33#14578964—23no—
34#14578964C2311yescharacteristic
35#14578964C2313yes—
36#14578963—23no—
37D157⋊C8—23no—
38C2×#11929441—23no—
39C4⋊#10597845—23no—
40C22⋊#10597845—23no—
41C4×#10597845—23no—
42C4×#10597845C2311yescharacteristic
43C4×#10597845C2313yes—
44#14578955D411yescharacteristic
45#14578955C2311yescharacteristic
46#14578954—23no—
47#14578954—23no—
48#14578954C2×C413yes—
49#14578954C2×C413yescomplemented
50#14578953—23no—
51#14578953—26no—
52#14578953—23no—
53#14578953—23no—
54#14578953—23no—
55#14578953—23no—
56#14578953C2×C416yescomplemented
57#14578953D413yes—
58#14578953C2313yes—
59C2×Dic314—23no—
60C2×Dic314C2×C413yescomplemented
61Q8×D157—21no—
62Q8×D157C2×C413yescomplemented
63Q8×D157D411yescharacteristic
64C2×C22⋊D157—23no—
65C2×C22⋊D157—26no—
66D4×D157—23no—
67D4×D157—23no—
68D4×D157C2×C413yescomplemented
69D4×D157D413yes—
70C22×Dic157—23no—
71C2×C4×D157—23no—
72C2×C4×D157C2313yes—
73#41×C157—21no—
74#41×C157C2×C411yescharacteristic
75#41×C157C2×C413yescomplemented
76C2×D4×C157—23no—
77C2×D4×C157C2×C413yescomplemented
78C2×#10597845—21no—
79C2×#10597845—23no—
80C2×#10597845—46no—
81#11929441—23no—
82#11929441—21no—
83C22×D157—23no—
84C22×D157C2×D413yes—
85Dic314—23no—
86Dic314—23no—
87Dic314C22⋊C413yes—
88Dic314C22×C413yes—
89D628—21no—
90D628C22⋊C413yes—
91D628C22×C411yescharacteristic
92C22⋊D157—23no—
93C22⋊D157—26no—
94C22⋊D157—23no—
95C22⋊D157—23no—
96C22⋊D157C22⋊C416yes—
97C2×Dic157—23no—
98C2×Dic157—26no—
99C2×Dic157—23no—
100C2×Dic157C2×D413yes—
101C4×D157—23no—
102C4×D157—23no—
103C4×D157—21no—
104C4×D157C22×C413yes—
105C4×D157C2×D413yes—
106C4×D157C2×D413yes—
107C4×D157C2411yescharacteristic
108C23×C157—23no—
109Q8×C157C22⋊C411yescharacteristic
110Q8×C157C22×C411yescharacteristic
111D4×C157—23no—
112D4×C157—23no—
113D4×C157C22⋊C413yes—
114D4×C157C22×C413yes—
115C2×C4×C157—23no—
116C2×C4×C157C22×C413yes—
117C22×C157—23no—
118C22×C157—83no—
119C22×C157C2×C22⋊C413yes—
120C2×D157—21no—
121C2×D157—43no—
122C2×D157—83no—
123C2×D157C2×C22⋊C413yes—
124C2×D157C22×D411yescharacteristic
125#10597845—22no—
126#10597845—43no—
127C4×C157—21no—
128C4×C157C2×C22⋊C413yes—
129C4×C157C23×C411yescharacteristic
130Dic157—23no—
131Dic157C2×C22⋊C413yes—
132Dic157C22×D411yescharacteristic
133C2×C157—23no—
134C2×C157—43no—
135C2×C157C22×C22⋊C411yescharacteristic
136D157—21no—
137D157—23no—
138D157—41no—
139C157#290111yescharacteristic, complemented, Sylow
140#2901—1571nomaximal, Sylow
141#585—1571no—
142#568—1571no—
143C4×#41—1571no—
144#421—1573no—
145#421—1573no—
146C2×#104—1573no—
147C2×#104—1573no—
1482−1+4—3141no—
1492−1+4—1571no—
1502−1+4—1573no—
151H(5,𝔽2)—3143no—
152H(5,𝔽2)—1573no—
153H(5,𝔽2)#1059784511yescharacteristic, complemented
154C2×#41—3143no—
155C2×#41—3141no—
156C2×#41—1573no—
157C2×#41—1573no—
158C2×#41—1571no—
159#131—3143no—
160#131—3141no—
161C2×M16—1573no—
162C2×M16—1573no—
163C4×Q8—3141no—
164C4×D4—3143no—
165#117—1573no—
166C2×C42—1573no—
167#104—1573no—
168#104—1571no—
169#104—1576no—
170#104—1573no—
171#104—1573no—
172#41—21no—
173#41—3143no—
174#41—3143no—
175#41—3143no—
176#41—3146no—
177#41—3143no—
178#41—3143no—
179#41—3143no—
180#41—3143no—
181#41—1571no—
182#41—1576no—
183#41—1573no—
184#41—1573no—
185#41—1571no—
186#41—1573no—
187#41—1573no—
188#41C2×#1059784511yescharacteristic
189#41C2×#1059784513yescomplemented
190C2×Q8—3143no—
191C2×Q8—3141no—
192C2×Q8—1573no—
193C2×Q8—1573no—
194C2×Q8—1571no—
195C2×D4—23no—
196C2×D4—3143no—
197C2×D4—3143no—
198C2×D4—3143no—
199C2×D4—3146no—
200C2×D4—1573no—
201C2×D4—1573no—
202C2×D4C2×#1059784513yescomplemented
203C22×C4—3143no—
204C22×C4—3143no—
205C22×C4—3143no—
206C22×C4—1573no—
207M16—3143no—
208M16—3143no—
209M16—1571no—
210M16—1573no—
211C2×C8—3143no—
212C2×C8—3143no—
213C4⋊C4—3143no—
214C22⋊C4—3143no—
215C42—3143no—
216C42—1571no—
217C42—1573no—
218C23—23no—
219C23—3143no—
220C23—1573no—
221Q8—3143no—
222Q8—3143no—
223Q8—1573no—
224Q8—1573no—
225Q8C22⋊#1059784511yescharacteristic
226Q8C22×#1059784511yescharacteristic
227D4—23no—
228D4—23no—
229D4—3141no—
230D4—3143no—
231D4—3146no—
232D4—3143no—
233D4—3143no—
234D4—1576no—
235D4—1571no—
236D4—1573no—
237D4C22⋊#1059784513yes—
238D4C22×#1059784513yes—
239C2×C4—23no—
240C2×C4—3143no—
241C2×C4—3141no—
242C2×C4—3141no—
243C2×C4—3143no—
244C2×C4—3143no—
245C2×C4—3146no—
246C2×C4—3143no—
247C2×C4—3143no—
248C2×C4—6286no—
249C2×C4—1573no—
250C2×C4—1571no—
251C2×C4—1573no—
252C2×C4—1573no—
253C2×C4—1573no—
254C2×C4C22×#1059784513yes—
255C8—3143no—
256C8—3141no—
257C22—23no—
258C22—83no—
259C22—3141no—
260C22—6283no—
261C22—1571no—
262C22—1573no—
263C22—12563no—
264C22C2×C22⋊#1059784513yes—
265C4—21no—
266C4—3142no—
267C4—3143no—
268C4—6283no—
269C4—1571no—
270C4—1573no—
271C4C2×C22⋊#1059784513yes—
272C4C23×#1059784511yescharacteristic
273C2—23no—
274C2—43no—
275C2—3141no—
276C2—3143no—
277C2—6281no—
278C2C22×C22⋊#1059784511yescharacteristic
subgroupG/H
centerC2C22×C22⋊#10597845
commutatorC4×C157C23×C4
FrattiniC4C23×#10597845
FittingH(5,𝔽2)×C157C4
socleC2×C157C22×C22⋊C4
prime radical#23758108C2
cosocleC4×D157C24
hypercenterH(5,𝔽2)#10597845
hypocenterC157#2901
LCSG ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #23758352C2semidirect
2. #41×#10597845C2semidirect
3. #23758271C2semidirect
4. #23758270C2semidirect
5. D157⋊#104C2semidirect
6. C2×#19097835C2semidirect
10. C2×#14578964C22semidirect
11. D157⋊M16C22semidirect
14. #19097850C22semidirect
15. C2×C4×#10597845C22semidirect
16. #19097836C22semidirect
17. #19097836C22semidirect
18. #19097835C22semidirect
19. #19097835C22semidirect
20. #19097835C22semidirect
22. #19097818C4semidirect
23. #19097818C4semidirect
25. #19097816C4semidirect
31. H(5,𝔽2)×C157C4semidirect
49. #14578954C2×C4semidirect
56. #14578953C2×C4semidirect
60. C2×Dic314C2×C4semidirect
62. Q8×D157C2×C4semidirect
68. D4×D157C2×C4semidirect
75. #41×C157C2×C4semidirect
77. C2×D4×C157C2×C4semidirect
139. C157#2901semidirect
153. H(5,𝔽2)#10597845semidirect
189. #41C2×#10597845semidirect
202. C2×D4C2×#10597845semidirect

Maximal subgroups

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

subgroupindex
#237583522
#41×#105978452
#237582712
#237582702
D157⋊#1042
C2×#190978352
#237581082
#2901157