Finite Group Databasebeta

← #34100640

Subgroups — #34100640

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
2C4×#14578954C211yesmaximal, characteristic, complemented
3#23757289C211yesmaximal, characteristic, complemented
4#23757247C211yesmaximal, characteristic, complemented
5#23757044C211yesmaximal, characteristic, complemented
6#23757033C211yesmaximal, characteristic, complemented
7C2×#104×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802C412yescomplemented
10#19097801—21no—
11#19097797C412yescomplemented
12#19097796—21no—
13#19097791—22no—
14C2×#14578938C2211yescharacteristic
15C4×Dic314—21no—
16C4×D628—21no—
17C4×C22⋊D157—22no—
18#19097730C2211yescharacteristic
19#19097730C2211yescharacteristic
20#19097728—21no—
21C42×D157—21no—
22#19097697—22no—
23#19097686—22no—
24C2×#41×C157C2211yescharacteristic
25C2×M16×C157C2211yescharacteristic
26C2×C42×C157C2211yescharacteristic
27#104×C157—21no—
28#104×C157—21no—
29#14578954—21no—
30#14578954—21no—
31#14578954C2×C412yes—
32#14578954D412yes—
33C2×Dic314D411yescharacteristic
34C2×D628D411yescharacteristic
35C2×C22⋊D157—21no—
36C2×C4×D157—22no—
37C2×C4×D157—21no—
38#14578944—22no—
39#14578942—22no—
40#14578941—22no—
41#14578939—22no—
42#14578938—21no—
43#14578938C2×C412yescomplemented
44C2×#11929434—21no—
45C4⋊Dic157—21no—
46C4⋊Dic157—21no—
47#14578933—22no—
48C22⋊Dic157—21no—
49C22⋊Dic157D412yes—
50#14578931—22no—
51C4×Dic157—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×C157C2×C411yescharacteristic, complemented
59C2×D4×C157—21no—
60C2×D4×C157C2×C411yescharacteristic, complemented
61C22×C4×C157—21no—
62C22×C4×C157—21no—
63C22×C4×C157C2311yescharacteristic
64M16×C157—21no—
65M16×C157—21no—
66C2×C8×C157—21no—
67C42×C157—21no—
68C42×C157D411yescharacteristic
69C42×C157D411yescharacteristic
70C22×D157—21no—
71Dic314—21no—
72Dic314C22⋊C412yes—
73D628—21no—
74D628C22⋊C412yes—
75C22⋊D157—22no—
76C22⋊D157—22no—
77C2×Dic157—21no—
78C2×Dic157—22no—
79C2×Dic157—41no—
80C4×D157—22no—
81C4×D157—22no—
82C4×D157—44no—
83#11929434—22no—
84C23×C157—21no—
85C23×C157C2×D411yescharacteristic
86Q8×C157—21no—
87Q8×C157—21no—
88D4×C157—21no—
89D4×C157—21no—
90D4×C157—41no—
91D4×C157—41no—
92C2×C4×C157—21no—
93C2×C4×C157—21no—
94C2×C4×C157—21no—
95C2×C4×C157—42no—
96C2×C4×C157—41no—
97C2×C4×C157—41no—
98C2×C4×C157C22×C411yescharacteristic
99C2×C4×C157C2×D411yescharacteristic
100C2×C4×C157C2×D411yescharacteristic
101C2×C4×C157C2×D411yescharacteristic
102C2×C4×C157#4111yescharacteristic
103C2×C4×C157#4111yescharacteristic
104C8×C157—41no—
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—22no—
114C4×C157—41no—
115C4×C157—41no—
116C4×C157C2×C22⋊C411yescharacteristic
117C4×C157C4×D411yescharacteristic
118C4×C157#12111yescharacteristic
119C4×C157#12311yescharacteristic
120Dic157—22no—
121Dic157—42no—
122Dic157—81no—
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×#104D15711yescharacteristic, complemented
135#410—1571no—
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×M16C2×D15711yescharacteristic
147C4×Q8—3141no—
148C4×D4—3141no—
149C4×D4—3142no—
150#117—3141no—
151#117—1571no—
152#117—1571no—
153C2×C42—3141no—
154C2×C42C2×D15711yescharacteristic
155#104—21no—
156#104—21no—
157#104—3142no—
158#99—3142no—
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—21no—
175C22×C4—3142no—
176C22×C4—3141no—
177C22×C4C22×D15711yescharacteristic
178Q16—3142no—
179SD16—3142no—
180SD16—3142no—
181D8—3142no—
182M16—21no—
183M16—21no—
184M16—3141no—
185M16—1572no—
186C2×C8—21no—
187C2×C8—3141no—
188C4⋊C4—3142no—
189C4⋊C4—3141no—
190C4⋊C4—3141no—
191C22⋊C4—3142no—
192C22⋊C4—3141no—
193C22⋊C4—1572no—
194C42—21no—
195C42—3142no—
196C42—3141no—
197C42—3141no—
198C42C22⋊D15711yescharacteristic
199C42C22⋊D15711yescharacteristic
200C23—21no—
201C23—3141no—
202C23D4×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—21no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—3141no—
222C2×C4—3142no—
223C2×C4—3142no—
224C2×C4—3142no—
225C2×C4—6284no—
226C2×C4—6281no—
227C2×C4C2×C4×D15711yescharacteristic
228C2×C4D4×D15711yescharacteristic
229C2×C4D4×D15711yescharacteristic
230C2×C4C2×C22⋊D15711yescharacteristic
231C2×C4#1457895311yescharacteristic
232C2×C4#1457895411yescharacteristic
233C8—41no—
234C8—3142no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—3142no—
239C22—6282no—
240C22#1909773111yescharacteristic
241C22#1909773911yescharacteristic
242C22#1909774411yescharacteristic
243C4—22no—
244C4—41no—
245C4—41no—
246C4—3142no—
247C4—6282no—
248C4—12561no—
249C4C22⋊C4×D15711yescharacteristic
250C4C4×C22⋊D15711yescharacteristic
251C4#1909774211yescharacteristic
252C4#1909775311yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—81no—
257C2—6282no—
258C2#2375714911yescharacteristic
subgroupG/H
centerC4#19097742
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×#104×C157C2
socleC2×C157#386
prime radical#23758071C2
cosocleC22×C4×C157C23
hypercenterC2×#104D157
hypocenterC157#1906
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C4×#14578954C2semidirect
3. #23757289C2semidirect
4. #23757247C2semidirect
5. #23757044C2semidirect
6. #23757033C2semidirect
7. C2×#104×C157C2semidirect
9. #19097802C4semidirect
11. #19097797C4semidirect
43. #14578938C2×C4semidirect
58. C2×Q8×C157C2×C4semidirect
60. C2×D4×C157C2×C4semidirect
129. C157#1906semidirect
134. C2×#104D157semidirect
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
C4×#145789542
#237572892
#237572472
#237570442
#237570332
C2×#104×C1572
#1906157