Finite Group Databasebeta

← #34100634

Subgroups — #34100634

Order 20096.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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
2#23757741C211yesmaximal, characteristic, complemented
3#23757291C211yesmaximal, characteristic, complemented
4C2×#19097698C211yesmaximal, characteristic, complemented
5#23757043C211yesmaximal, characteristic, complemented
6#23757032C211yesmaximal, characteristic, complemented
7#410×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802C412yescomplemented
10#19097801—21no—
11#19097797C412yescomplemented
12#19097796—21no—
13#19097791—22no—
14C2×#14578938C2211yescharacteristic
15C2×#14578938C2211yescharacteristic
16#19097736—21no—
17#19097733—21no—
18#19097731—22no—
19#19097728—21no—
20C2×C4×Dic157C2211yescharacteristic
21C42×D157—21no—
22#19097699—22no—
23#19097698—21no—
24#19097698—21no—
25C2×#41×C157C2211yescharacteristic
26#117×C157C2211yescharacteristic
27#117×C157C2211yescharacteristic
28#99×C157—22no—
29#14578954—21no—
30#14578954—21no—
31#14578954C2×C412yes—
32#14578954D412yes—
33C2×Dic314D411yescharacteristic
34C2×D628D411yescharacteristic
35C2×C22⋊D157—21no—
36C22×Dic157—21no—
37C2×C4×D157—22no—
38C2×C4×D157—21no—
39#14578944—22no—
40#14578942—22no—
41#14578941—22no—
42#14578939—22no—
43#14578938—21no—
44#14578938—21no—
45#14578938—21no—
46#14578938C2×C412yescomplemented
47C2×#11929434—21no—
48C2×#11929434—21no—
49#14578933—22no—
50#14578931—22no—
51C4×Dic157—22no—
52C4×Dic157—21no—
53C4×Dic157D411yescharacteristic
54C4×Dic157D411yescharacteristic
55#41×C157—21no—
56#41×C157—21no—
57#41×C157—21no—
58#41×C157—21no—
59C2×Q8×C157C2×C411yescharacteristic, complemented
60C2×D4×C157—21no—
61C2×D4×C157C2×C411yescharacteristic, complemented
62C22×C4×C157—21no—
63C22×C4×C157C2311yescharacteristic
64C4⋊C4×C157—21no—
65C4⋊C4×C157—21no—
66C22⋊C4×C157—21no—
67C22⋊C4×C157D412yes—
68C42×C157—21no—
69C42×C157—21no—
70C22×D157—21no—
71Dic314—21no—
72Dic314C22⋊C412yes—
73D628—21no—
74D628C22⋊C412yes—
75C22⋊D157—22no—
76C22⋊D157—22no—
77C2×Dic157—21no—
78C2×Dic157—21no—
79C2×Dic157—21no—
80C2×Dic157—42no—
81C4×D157—22no—
82C4×D157—22no—
83C4×D157—44no—
84#11929434—22no—
85#11929434—41no—
86C23×C157—21no—
87C23×C157C2×D411yescharacteristic
88Q8×C157—21no—
89Q8×C157—21no—
90D4×C157—21no—
91D4×C157—21no—
92D4×C157—41no—
93D4×C157—41no—
94C2×C4×C157—21no—
95C2×C4×C157—22no—
96C2×C4×C157—41no—
97C2×C4×C157—41no—
98C2×C4×C157—41no—
99C2×C4×C157C22×C411yescharacteristic
100C2×C4×C157C2×D411yescharacteristic
101C2×C4×C157C2×D411yescharacteristic
102C2×C4×C157C2×D411yescharacteristic
103C2×C4×C157#4111yescharacteristic
104C2×C4×C157#4111yescharacteristic
105C22×C157—41no—
106C22×C157—41no—
107C22×C157—81no—
108C22×C157C4×D411yescharacteristic
109C22×C157#12011yescharacteristic
110C22×C157#12111yescharacteristic
111C2×D157—22no—
112C2×D157—42no—
113C4×C157—42no—
114C4×C157—41no—
115C4×C157—81no—
116C4×C157C2×C22⋊C411yescharacteristic
117C4×C157C4×D411yescharacteristic
118C4×C157#12111yescharacteristic
119C4×C157#12311yescharacteristic
120Dic157—22no—
121Dic157—22no—
122Dic157—41no—
123C2×C157—21no—
124C2×C157—21no—
125C2×C157—21no—
126C2×C157—81no—
127C2×C157#38611yescharacteristic
128D157—42no—
129C157#190611yescharacteristic, complemented, Sylow
130#1906—1571nomaximal, Sylow
131#575—1571no—
132C4×#41—1571no—
133#421—1571no—
134C2×#104—1571no—
135#410D15711yescharacteristic, complemented
136#344—1571no—
137#339—1571no—
138C2×#41—1571no—
139C2×#41C2×D15711yescharacteristic
140#137—3141no—
141#137—1572no—
142AGL(1,ℤ/8)—3141no—
143AGL(1,ℤ/8)—1572no—
144#135—3142no—
145C2×M16—1571no—
146C2×M16—1571no—
147C4×Q8—3141no—
148C4×D4—3141no—
149C4×D4—3142no—
150#117—3141no—
151#117C2×D15711yescharacteristic
152#117C2×D15711yescharacteristic
153C2×C42—3141no—
154C2×C42—1571no—
155#104—3142no—
156#104—3141no—
157#104—3141no—
158#99—22no—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—3141no—
164#41—3141no—
165#41—1572no—
166#41—1572no—
167C2×Q8—1571no—
168C2×Q8C4×D15711yescharacteristic, complemented
169C2×D4—21no—
170C2×D4—3141no—
171C2×D4—1571no—
172C2×D4C4×D15711yescharacteristic, complemented
173C22×C4—21no—
174C22×C4—3142no—
175C22×C4—3141no—
176C22×C4—3141no—
177C22×C4C22×D15711yescharacteristic
178Q16—3142no—
179SD16—3142no—
180SD16—3142no—
181D8—3142no—
182M16—3141no—
183M16—3141no—
184M16—3141no—
185M16—1572no—
186C2×C8—3141no—
187C2×C8—3141no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—3142no—
191C22⋊C4—21no—
192C22⋊C4—3142no—
193C22⋊C4C22⋊D15712yes—
194C42—21no—
195C42—21no—
196C42—3142no—
197C42—3141no—
198C42—1571no—
199C42—1571no—
200C23—21no—
201C23—3141no—
202C23C2×C22⋊D15711yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—3141no—
206Q8—1572no—
207D4—21no—
208D4—21no—
209D4—41no—
210D4—41no—
211D4—3142no—
212D4—3142no—
213D4—3141no—
214D4—1572no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—3141no—
221C2×C4—3141no—
222C2×C4—3142no—
223C2×C4—3142no—
224C2×C4—3141no—
225C2×C4—6284no—
226C2×C4—6282no—
227C2×C4C2×C4×D15711yescharacteristic
228C2×C4D4×D15711yescharacteristic
229C2×C4D4×D15711yescharacteristic
230C2×C4D4×D15711yescharacteristic
231C2×C4#1457895311yescharacteristic
232C2×C4#1457895411yescharacteristic
233C8—3142no—
234C8—6281no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—3142no—
239C22—6282no—
240C22C4×C22⋊D15711yescharacteristic
241C22#1909773911yescharacteristic
242C22#1909774211yescharacteristic
243C4—42no—
244C4—41no—
245C4—81no—
246C4—3142no—
247C4—3142no—
248C4—6281no—
249C4C22⋊C4×D15711yescharacteristic
250C4#1909773111yescharacteristic
251C4#1909774411yescharacteristic
252C4#1909775311yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—81no—
257C2—6282no—
258C2#2375714911yescharacteristic
subgroupG/H
centerC4#19097744
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#410×C157C2
socleC2×C157#386
prime radical#23758071C2
cosocleC22×C4×C157C23
hypercenter#410D157
hypocenterC157#1906
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #23757741C2semidirect
3. #23757291C2semidirect
4. C2×#19097698C2semidirect
5. #23757043C2semidirect
6. #23757032C2semidirect
7. #410×C157C2semidirect
9. #19097802C4semidirect
11. #19097797C4semidirect
46. #14578938C2×C4semidirect
59. C2×Q8×C157C2×C4semidirect
61. C2×D4×C157C2×C4semidirect
129. C157#1906semidirect
135. #410D157semidirect
168. C2×Q8C4×D157semidirect
172. C2×D4C4×D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
#237577412
#237572912
C2×#190976982
#237570432
#237570322
#410×C1572
#1906157