Finite Group Databasebeta

← #34101151

Subgroups — #34101151

Order 20096.

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

normal subgroups only

This group has 478 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×#19097797C211yesmaximal, characteristic, complemented
2#23757840C211yesmaximal, characteristic, complemented
3C2×#19097699C211yesmaximal, characteristic, complemented
4#23757278C211yesmaximal, characteristic, complemented
5C2×#19097688C211yesmaximal, characteristic, complemented
6#23757033C211yesmaximal, characteristic, complemented
7C2×#127×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097797—22no—
10#19097797—22no—
11C2×#14578942—21no—
12C2×#14578939—21no—
13C2×#14578938C2211yescharacteristic
14C2×#14578938C2211yescharacteristic
15#19097767—21no—
16#19097758—21no—
17#19097742—22no—
18#19097740—22no—
19#19097730C2211yescharacteristic
20#19097699—22no—
21#19097692—22no—
22#19097688—22no—
23C22×D4×C157—22no—
24C22×D4×C157C2211yescharacteristic
25C22×D4×C157C2211yescharacteristic
26#127×C157—22no—
27#127×C157—22no—
28C2×C42×C157C2211yescharacteristic
29#14578954—21no—
30#14578954—21no—
31#14578954—21no—
32#14578954—21no—
33C2×Dic314D411yescharacteristic, complemented
34C2×D628D411yescharacteristic, complemented
35C2×C22⋊D157—24no—
36C2×C22⋊D157—21no—
37C2×C4×D157—21no—
38#14578942—24no—
39#14578939—24no—
40#14578938—21no—
41#14578938—21no—
42#14578938—21no—
43#14578938D412yescomplemented
44C2×#11929434—21no—
45C2×#11929434—21no—
46C4⋊Dic157D412yescomplemented
47C22⋊Dic157—24no—
48C22⋊Dic157—21no—
49C4×Dic157—21no—
50C24×C157—21no—
51C24×C157D412yes—
52C2×D4×C157—28no—
53C2×D4×C157—21no—
54C2×D4×C157—21no—
55C2×D4×C157—21no—
56C2×D4×C157—21no—
57C2×D4×C157—21no—
58C2×D4×C157—21no—
59C2×D4×C157—21no—
60C2×D4×C157—21no—
61C2×D4×C157—21no—
62C2×D4×C157—21no—
63C2×D4×C157—21no—
64C2×D4×C157—48no—
65C2×D4×C157D412yescomplemented
66C22×C4×C157—21no—
67C22×C4×C157C2311yescharacteristic
68C42×C157—21no—
69C42×C157D412yescomplemented
70C22×D157—21no—
71Dic314—21no—
72Dic314—21no—
73D628—21no—
74D628—21no—
75C22⋊D157—41no—
76C22⋊D157—41no—
77C22⋊D157—84no—
78C2×Dic157—21no—
79C2×Dic157—22no—
80C4×D157—41no—
81C4×D157—41no—
82#11929434—22no—
83#11929434—41no—
84C23×C157—22no—
85C23×C157—22no—
86C23×C157—22no—
87C23×C157—22no—
88C23×C157—44no—
89C23×C157—44no—
90C23×C157—41no—
91C23×C157—41no—
92C23×C157C2×D411yescharacteristic
93D4×C157—22no—
94D4×C157—22no—
95D4×C157—416no—
96D4×C157—41no—
97D4×C157—41no—
98D4×C157—41no—
99D4×C157—41no—
100D4×C157—41no—
101D4×C157—41no—
102C2×C4×C157—24no—
103C2×C4×C157—22no—
104C2×C4×C157C2×D411yescharacteristic
105C2×C4×C157C2×D411yescharacteristic
106C2×C4×C157C2×D411yescharacteristic
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157#4111yescharacteristic
110C22×C157—22no—
111C22×C157—44no—
112C22×C157—44no—
113C22×C157—42no—
114C22×C157—42no—
115C22×C157—44no—
116C22×C157—82no—
117C22×C157—82no—
118C22×C157—82no—
119C22×C157#12011yescharacteristic
120C22×C157#12011yescharacteristic
121C22×C157#12111yescharacteristic
122C2×D157—41no—
123C2×D157—42no—
124C4×C157—24no—
125C4×C157#12011yescharacteristic
126C4×C157#12111yescharacteristic
127C4×C157#12111yescharacteristic
128C4×C157#12711yescharacteristic
129Dic157—41no—
130Dic157—42no—
131C2×C157—21no—
132C2×C157—21no—
133C2×C157—82no—
134C2×C157—82no—
135C2×C157#39211yescharacteristic
136C2×C157#45312yes—
137D157—81no—
138C157#200211yescharacteristic, complemented, Sylow
139#2002—1571nomaximal, Sylow
140C2×AGL(1,ℤ/8)—1571no—
141#535—1571no—
142C2×#127D15711yescharacteristic, complemented
143C2×#104—1571no—
144#418—1571no—
145C2×#100—1571no—
146#339—1571no—
147C2×#41—1571no—
148C22×D4—22no—
149C22×D4C2×D15711yescharacteristic
150C22×D4C2×D15711yescharacteristic
151AGL(1,ℤ/8)—3142no—
152AGL(1,ℤ/8)—3142no—
153C2×SD16—3141no—
154C2×D8—3141no—
155C2×M16—1571no—
156C2×M16—1571no—
157#127—22no—
158#127—22no—
159#127—3141no—
160#124—3141no—
161#121—3142no—
162#120—3142no—
163#117—1571no—
164C2×C42C2×D15711yescharacteristic
165#104—3142no—
166#102—3142no—
167#100—3142no—
168C24—21no—
169C24C22⋊D15712yes—
170#41—3141no—
171#41—3141no—
172#41—3141no—
173#41—3141no—
174C2×Q8—1571no—
175C2×D4—28no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—48no—
188C2×D4—3141no—
189C2×D4—3144no—
190C2×D4—1571no—
191C2×D4C22⋊D15712yescomplemented
192C22×C4—21no—
193C22×C4—3141no—
194C22×C4C22×D15711yescharacteristic
195SD16—3144no—
196D8—3144no—
197M16—3141no—
198M16—3141no—
199M16—3141no—
200M16—1572no—
201C2×C8—3141no—
202C2×C8—3141no—
203C4⋊C4—1572no—
204C22⋊C4—3144no—
205C22⋊C4—3141no—
206C42—21no—
207C42—3141no—
208C42C22⋊D15712yescomplemented
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—44no—
214C23—44no—
215C23—41no—
216C23—41no—
217C23—3141no—
218C23C2×C22⋊D15711yescharacteristic
219Q8—3141no—
220Q8—3141no—
221D4—22no—
222D4—22no—
223D4—416no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—3141no—
231D4—3141no—
232D4—6281no—
233D4—6281no—
234D4—12564no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—3142no—
238C2×C4—3141no—
239C2×C4—6281no—
240C2×C4—6281no—
241C2×C4D4×D15711yescharacteristic
242C2×C4D4×D15711yescharacteristic
243C2×C4D4×D15711yescharacteristic
244C2×C4C2×C22⋊D15711yescharacteristic
245C2×C4C2×C22⋊D15711yescharacteristic
246C2×C4#1457895311yescharacteristic
247C8—3142no—
248C8—6281no—
249C22—22no—
250C22—44no—
251C22—44no—
252C22—42no—
253C22—42no—
254C22—44no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—6281no—
259C22—6282no—
260C22#1909773911yescharacteristic
261C22#1909774011yescharacteristic
262C22#1909774311yescharacteristic
263C4—24no—
264C4—6281no—
265C4—6282no—
266C4#1909773911yescharacteristic
267C4#1909774211yescharacteristic
268C4#1909774311yescharacteristic
269C4#1909776711yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—82no—
274C2—12561no—
275C2#2375718111yescharacteristic
276C2#2375744012yes—
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×#127×C157C2
socleC22×C157#120
hypercenterC2×#127D157
hypocenterC157#2002
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#19097797C2semidirect
2. #23757840C2semidirect
3. C2×#19097699C2semidirect
4. #23757278C2semidirect
5. C2×#19097688C2semidirect
6. #23757033C2semidirect
7. C2×#127×C157C2semidirect
33. C2×Dic314D4semidirect
34. C2×D628D4semidirect
43. #14578938D4semidirect
46. C4⋊Dic157D4semidirect
65. C2×D4×C157D4semidirect
69. C42×C157D4semidirect
138. C157#2002semidirect
142. C2×#127D157semidirect
191. C2×D4C22⋊D157semidirect
208. C42C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977972
#237578402
C2×#190976992
#237572782
C2×#190976882
#237570332
C2×#127×C1572
#2002157