Finite Group Databasebeta

← #34101188

Subgroups — #34101188

Order 20096.

380 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 69 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#23758071C211yesmaximal, characteristic, complemented
2C2×#19097796C211yesmaximal, characteristic, complemented
3#23757842C211yesmaximal, characteristic, complemented
4#23757291C211yesmaximal, characteristic, complemented
5#23757277C211yesmaximal, characteristic, complemented
6#23757043C211yesmaximal, characteristic, complemented
7#410×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×D628C2211yescharacteristic
10#19097802—21no—
11#19097801—21no—
12#19097797—21no—
13#19097796—22no—
14#19097796—22no—
15#19097796—21no—
16#19097791—22no—
17C2×#14578941—21no—
18C2×#14578939—21no—
19C2×#14578938C2211yescharacteristic
20C2×#14578938C2211yescharacteristic
21#19097768—21no—
22#19097759—21no—
23#19097746—22no—
24#19097741—22no—
25#19097699—22no—
26#19097691—22no—
27C2×#41×C157C2211yescharacteristic
28#117×C157C2211yescharacteristic
29#117×C157C2211yescharacteristic
30#99×C157—22no—
31C23×D157—21no—
32#14578954—21no—
33#14578954—21no—
34#14578954—21no—
35#14578954—21no—
36C2×Dic314D411yescharacteristic
37C2×D628—24no—
38C2×D628—21no—
39C2×D628—21no—
40C2×D628—21no—
41C2×D628—21no—
42C2×D628—21no—
43C2×D628D412yes—
44C2×D628D411yescharacteristic
45C2×C22⋊D157—21no—
46C2×C4×D157—21no—
47#14578944—22no—
48#14578942—22no—
49#14578941—24no—
50#14578941—22no—
51#14578939—24no—
52#14578939—22no—
53#14578938—21no—
54#14578938—21no—
55#14578938—21no—
56#14578938D412yescomplemented
57C2×#11929434—21no—
58C2×#11929434D412yescomplemented
59#14578931—24no—
60#41×C157—21no—
61#41×C157—21no—
62#41×C157—21no—
63#41×C157—21no—
64C2×Q8×C157D411yescharacteristic, complemented
65C2×D4×C157—21no—
66C2×D4×C157D411yescharacteristic, complemented
67C22×C4×C157—21no—
68C22×C4×C157C2311yescharacteristic
69C4⋊C4×C157—21no—
70C4⋊C4×C157—21no—
71C22⋊C4×C157—21no—
72C22⋊C4×C157D412yes—
73C42×C157—21no—
74C42×C157—21no—
75C22×D157—21no—
76C22×D157—22no—
77C22×D157—41no—
78C22×D157—41no—
79C22×D157—82no—
80Dic314—21no—
81Dic314—21no—
82D628—22no—
83D628—21no—
84D628—22no—
85D628—21no—
86D628—41no—
87D628—41no—
88D628—84no—
89C22⋊D157—41no—
90C22⋊D157—41no—
91C2×Dic157—21no—
92C4×D157—41no—
93C4×D157—41no—
94#11929434—22no—
95#11929434—22no—
96C23×C157—21no—
97C23×C157C2×D411yescharacteristic
98Q8×C157—21no—
99Q8×C157—21no—
100D4×C157—21no—
101D4×C157—21no—
102D4×C157—41no—
103D4×C157—41no—
104C2×C4×C157—22no—
105C2×C4×C157—21no—
106C2×C4×C157—41no—
107C2×C4×C157—41no—
108C2×C4×C157—41no—
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2×D411yescharacteristic
112C2×C4×C157C2×D411yescharacteristic
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157#4111yescharacteristic
115C22×C157—41no—
116C22×C157—41no—
117C22×C157—81no—
118C22×C157#12011yescharacteristic
119C22×C157#12111yescharacteristic
120C22×C157#12711yescharacteristic
121C2×D157—42no—
122C2×D157—44no—
123C2×D157—41no—
124C2×D157—42no—
125C2×D157—82no—
126C2×D157—82no—
127C4×C157—42no—
128C4×C157—41no—
129C4×C157—81no—
130C4×C157#12011yescharacteristic
131C4×C157#12011yescharacteristic
132C4×C157#12111yescharacteristic
133C4×C157#12111yescharacteristic
134Dic157—41no—
135C2×C157—21no—
136C2×C157—21no—
137C2×C157—21no—
138C2×C157—81no—
139C2×C157#39211yescharacteristic
140D157—81no—
141D157—82no—
142C157#200811yescharacteristic, complemented, Sylow
143#2008—1571nomaximal, Sylow
144#575—1571no—
145C2×AGL(1,ℤ/8)—1571no—
146#535—1571no—
147#421—1571no—
148#418—1571no—
149#410D15711yescharacteristic, complemented
150#344—1571no—
151C2×#41—1571no—
152C2×#41C2×D15711yescharacteristic
153C22×D4—1571no—
154#137—3141no—
155#137—3141no—
156AGL(1,ℤ/8)—3142no—
157AGL(1,ℤ/8)—3141no—
158AGL(1,ℤ/8)—3141no—
159AGL(1,ℤ/8)—3142no—
160#135—3142no—
161C2×SD16—3141no—
162C2×D8—3141no—
163C2×M16—1571no—
164C2×M16—1571no—
165#127—3141no—
166#124—3141no—
167#121—3142no—
168#120—3142no—
169#117C2×D15711yescharacteristic
170#117C2×D15711yescharacteristic
171#104—3142no—
172#102—3142no—
173#99—22no—
174C24—3141no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—3141no—
180#41—3141no—
181#41—3141no—
182#41—3141no—
183C2×Q8—1571no—
184C2×Q8D62811yescharacteristic, complemented
185C2×D4—21no—
186C2×D4—3144no—
187C2×D4—3141no—
188C2×D4—3141no—
189C2×D4—3141no—
190C2×D4—3141no—
191C2×D4—3141no—
192C2×D4—3141no—
193C2×D4—1572no—
194C2×D4—1571no—
195C2×D4D62811yescharacteristic, complemented
196C22×C4—21no—
197C22×C4—3141no—
198C22×C4C22×D15711yescharacteristic
199Q16—3142no—
200SD16—3142no—
201SD16—3144no—
202SD16—3142no—
203D8—3144no—
204D8—3142no—
205M16—3141no—
206M16—3141no—
207M16—3141no—
208M16—1572no—
209C2×C8—3141no—
210C2×C8—1572no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—3144no—
215C22⋊C4C22⋊D15712yes—
216C42—21no—
217C42—21no—
218C23—21no—
219C23—3142no—
220C23—3141no—
221C23—6281no—
222C23—6281no—
223C23—12562no—
224C23C2×C22⋊D15711yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—3141no—
228Q8—3141no—
229D4—21no—
230D4—21no—
231D4—41no—
232D4—41no—
233D4—3142no—
234D4—3141no—
235D4—3142no—
236D4—3141no—
237D4—6281no—
238D4—6281no—
239D4—6281no—
240D4—6281no—
241D4—12564no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—3141no—
248C2×C4—6281no—
249C2×C4—6281no—
250C2×C4D4×D15711yescharacteristic
251C2×C4D4×D15711yescharacteristic
252C2×C4D4×D15711yescharacteristic
253C2×C4D4×D15711yescharacteristic
254C2×C4C2×D62811yescharacteristic
255C2×C4#1457895411yescharacteristic
256C8—3142no—
257C8—3142no—
258C22—41no—
259C22—41no—
260C22—81no—
261C22—6282no—
262C22—6284no—
263C22—6281no—
264C22—6282no—
265C22—12562no—
266C22—12562no—
267C22#1909773911yescharacteristic
268C22#1909774511yescharacteristic
269C22#1909776711yescharacteristic
270C4—42no—
271C4—41no—
272C4—81no—
273C4—6281no—
274C4#1909774111yescharacteristic
275C4#1909774111yescharacteristic
276C4#1909774411yescharacteristic
277C4#1909774411yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—81no—
282C2—12562no—
283C2—12561no—
284C2#2375718211yescharacteristic
subgroupG/H
centerC2#23757182
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#410×C157C2
socleC2×C157#392
hypercenter#410D157
hypocenterC157#2008
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #23758071C2semidirect
2. C2×#19097796C2semidirect
3. #23757842C2semidirect
4. #23757291C2semidirect
5. #23757277C2semidirect
6. #23757043C2semidirect
7. #410×C157C2semidirect
56. #14578938D4semidirect
58. C2×#11929434D4semidirect
64. C2×Q8×C157D4semidirect
66. C2×D4×C157D4semidirect
142. C157#2008semidirect
149. #410D157semidirect
184. C2×Q8D628semidirect
195. C2×D4D628semidirect

Maximal subgroups

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

subgroupindex
#237580712
C2×#190977962
#237578422
#237572912
#237572772
#237570432
#410×C1572
#2008157