Finite Group Databasebeta

← #34101181

Subgroups — #34101181

Order 20096.

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

normal subgroups only

This group has 370 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×#19097802C211yesmaximal, characteristic, complemented
2#23757851C211yesmaximal, characteristic
3C2×#19097698C211yesmaximal, characteristic, complemented
4C2×#19097697C211yesmaximal, characteristic, complemented
5#23757180C211yesmaximal, characteristic, complemented
6#23757041C211yesmaximal, characteristic, complemented
7C2×#137×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802—22no—
10#19097802—22no—
11C2×#14578944—21no—
12C2×#14578941—21no—
13C2×#14578938C2211yescharacteristic
14#19097750—21no—
15#19097747—21no—
16#19097745—21no—
17#19097743—22no—
18#19097742—21no—
19C2×C4⋊Dic157C2211yescharacteristic
20C2×C4×Dic157C2211yescharacteristic
21#19097698—22no—
22#19097697—22no—
23#19097678—22no—
24C2×#41×C157C2211yescharacteristic
25C22×Q8×C157C2211yescharacteristic
26#137×C157—22no—
27#137×C157—22no—
28C2×Q16×C157—21no—
29C2×SD16×C157—21no—
30C2×M16×C157C2211yescharacteristic
31#14578954—21no—
32#14578954—21no—
33#14578954—21no—
34#14578954—21no—
35C2×Dic314D411yescharacteristic
36C2×D628D411yescharacteristic
37C2×C22⋊D157—22no—
38C2×C22⋊D157—21no—
39C22×Dic157—21no—
40C22×Dic157D412yes—
41C2×C4×D157—21no—
42#14578944—24no—
43#14578941—24no—
44#14578938—21no—
45#14578938D412yescomplemented
46C2×#11929434—21no—
47C4⋊Dic157—21no—
48C4⋊Dic157—21no—
49#14578933—22no—
50C22⋊Dic157—22no—
51#14578931—22no—
52C4×Dic157—21no—
53C4×Dic157—21no—
54#41×C157—21no—
55#41×C157—21no—
56#41×C157—21no—
57#41×C157—21no—
58C2×Q8×C157—21no—
59C2×Q8×C157—21no—
60C2×Q8×C157—21no—
61C2×Q8×C157—21no—
62C2×Q8×C157—21no—
63C2×Q8×C157D411yescharacteristic
64C2×Q8×C157D412yescomplemented
65C2×D4×C157—21no—
66C2×D4×C157D411yescharacteristic
67C22×C4×C157—21no—
68C22×C4×C157—21no—
69C22×C4×C157C2311yescharacteristic
70Q16×C157—24no—
71SD16×C157—24no—
72M16×C157—21no—
73M16×C157D412yescomplemented
74C2×C8×C157—21no—
75C22×D157—21no—
76Dic314—21no—
77Dic314—21no—
78D628—21no—
79D628—21no—
80C22⋊D157—41no—
81C22⋊D157—41no—
82C22⋊D157—82no—
83C2×Dic157—22no—
84C2×Dic157—21no—
85C2×Dic157—22no—
86C2×Dic157—41no—
87C2×Dic157—41no—
88C2×Dic157—42no—
89C4×D157—41no—
90C4×D157—41no—
91#11929434—22no—
92C23×C157—21no—
93C23×C157C2×D411yescharacteristic
94Q8×C157—22no—
95Q8×C157—22no—
96Q8×C157—21no—
97Q8×C157—21no—
98Q8×C157—41no—
99Q8×C157—41no—
100D4×C157—21no—
101D4×C157—21no—
102D4×C157—41no—
103D4×C157—41no—
104C2×C4×C157—21no—
105C2×C4×C157—22no—
106C2×C4×C157—41no—
107C2×C4×C157—41no—
108C2×C4×C157—41no—
109C2×C4×C157—41no—
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2×D411yescharacteristic
112C2×C4×C157C2×D411yescharacteristic
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157#4111yescharacteristic
116C8×C157—22no—
117C22×C157—21no—
118C22×C157—21no—
119C22×C157—41no—
120C22×C157—41no—
121C22×C157—41no—
122C22×C157#12011yescharacteristic
123C22×C157#12111yescharacteristic
124C22×C157#12711yescharacteristic
125C2×D157—41no—
126C2×D157—41no—
127C2×D157—41no—
128C4×C157—41no—
129C4×C157—42no—
130C4×C157#12011yescharacteristic
131C4×C157#12011yescharacteristic
132C4×C157#12111yescharacteristic
133C4×C157#12111yescharacteristic
134Dic157—42no—
135Dic157—41no—
136Dic157—42no—
137C2×C157—21no—
138C2×C157—21no—
139C2×C157—81no—
140C2×C157#39211yescharacteristic
141C2×C157#45511yescharacteristic
142C2×C157#45611yescharacteristic
143D157—81no—
144C157#200711yescharacteristic, complemented, Sylow
145#2007—1571nomaximal, Sylow
146C2×#137—1571no—
147C2×#137D15711yescharacteristic, complemented
148#537—1571no—
149C2×#104—1571no—
150C2×#104—1571no—
151#391—1571no—
152#343—1571no—
153C2×#41—1571no—
154C2×#41C2×D15711yescharacteristic
155C22×Q8C2×D15711yescharacteristic
156#137—22no—
157#137—22no—
158#137—3142no—
159#137—3142no—
160C2×Q16—21no—
161C2×Q16—3141no—
162C2×SD16—21no—
163C2×SD16—3141no—
164C2×M16—1571no—
165C2×M16C2×D15711yescharacteristic
166C22⋊Q8—3141no—
167C22⋊Q8—3141no—
168#121—3141no—
169#121—3141no—
170#121—3142no—
171C2×C4⋊C4—1571no—
172C2×C42—1571no—
173#104—3142no—
174#104—3142no—
175#95—3142no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—3141no—
181#41—3141no—
182#41—3141no—
183#41—3141no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—1571no—
190C2×Q8C22⋊D15711yescharacteristic
191C2×Q8C22⋊D15712yescomplemented
192C2×D4—21no—
193C2×D4—3141no—
194C2×D4—3142no—
195C2×D4—1571no—
196C2×D4C22⋊D15711yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—3141no—
200C22×C4—3141no—
201C22×C4—1572no—
202C22×C4C22×D15711yescharacteristic
203Q16—24no—
204Q16—3144no—
205SD16—24no—
206SD16—3144no—
207M16—21no—
208M16—3141no—
209M16—1572no—
210M16C22⋊D15712yescomplemented
211C2×C8—21no—
212C2×C8—3141no—
213C4⋊C4—3142no—
214C4⋊C4—3141no—
215C4⋊C4—3141no—
216C22⋊C4—3142no—
217C22⋊C4—3142no—
218C42—3141no—
219C42—3141no—
220C23—21no—
221C23—3141no—
222C23D4×D15711yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—21no—
226Q8—21no—
227Q8—41no—
228Q8—41no—
229Q8—3141no—
230Q8—3141no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—3141no—
236D4—3141no—
237D4—6281no—
238D4—6281no—
239D4—12562no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—3142no—
247C2×C4—3142no—
248C2×C4—3141no—
249C2×C4—6281no—
250C2×C4—6281no—
251C2×C4—6281no—
252C2×C4—6281no—
253C2×C4—6282no—
254C2×C4D4×D15711yescharacteristic
255C2×C4D4×D15711yescharacteristic
256C2×C4C2×C22⋊D15711yescharacteristic
257C2×C4C2×C22⋊D15711yescharacteristic
258C2×C4C2×C22⋊D15711yescharacteristic
259C2×C4#1457895311yescharacteristic
260C8—22no—
261C8—3142no—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—6281no—
268C22—6281no—
269C22—6281no—
270C22#1909773911yescharacteristic
271C22#1909774311yescharacteristic
272C22#1909776711yescharacteristic
273C4—41no—
274C4—42no—
275C4—6282no—
276C4—6281no—
277C4—6282no—
278C4#1909773911yescharacteristic
279C4#1909774011yescharacteristic
280C4#1909774211yescharacteristic
281C4#1909774311yescharacteristic
282C2—21no—
283C2—21no—
284C2—81no—
285C2—12561no—
286C2#2375718111yescharacteristic
287C2#2375744911yescharacteristic
288C2#2375745411yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×#137×C157C2
socleC22×C157#120
prime radical#23757851C2
hypercenterC2×#137D157
hypocenterC157#2007
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#19097802C2semidirect
3. C2×#19097698C2semidirect
4. C2×#19097697C2semidirect
5. #23757180C2semidirect
6. #23757041C2semidirect
7. C2×#137×C157C2semidirect
45. #14578938D4semidirect
64. C2×Q8×C157D4semidirect
73. M16×C157D4semidirect
144. C157#2007semidirect
147. C2×#137D157semidirect
191. C2×Q8C22⋊D157semidirect
210. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978022
#237578512
C2×#190976982
C2×#190976972
#237571802
#237570412
C2×#137×C1572
#2007157