Finite Group Databasebeta

← #34100518

Subgroups — #34100518

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#23758039C211yesmaximal, characteristic, complemented
2#23757823C211yesmaximal, characteristic
3#23757291C211yesmaximal, characteristic, complemented
4#23757289C211yesmaximal, characteristic, complemented
5#23757246C211yesmaximal, characteristic, complemented
6#23757033C211yesmaximal, characteristic, complemented
7#339×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C2×#14578954C2211yescharacteristic
10#19097779—22no—
11#19097777—22no—
12C2×#14578938C2211yescharacteristic
13M16×D157—22no—
14#19097771C411yescharacteristic, complemented
15#19097768C411yescharacteristic, complemented
16#19097759C411yescharacteristic, complemented
17#19097759C411yescharacteristic, complemented
18#19097745—22no—
19C4×D628—22no—
20#19097730C2211yescharacteristic
21#19097699—22no—
22#19097697—22no—
23#19097685—22no—
24C2×M16×C157C2211yescharacteristic
25#117×C157C2211yescharacteristic
26C2×C42×C157C2211yescharacteristic
27#14578954—21no—
28#14578954—21no—
29#14578954—21no—
30#14578954—21no—
31#14578954—21no—
32#14578954—21no—
33#14578954—21no—
34#14578954—21no—
35C2×Dic314C2×C411yescharacteristic
36C2×Dic314C2×C411yescharacteristic
37C2×D628—22no—
38C2×D628C2×C411yescharacteristic
39C2×D628C2×C411yescharacteristic
40C2×C22⋊D157—21no—
41C2×C22⋊D157D412yes—
42C2×C4×D157—21no—
43C2×C4×D157D412yes—
44#14578938—21no—
45#14578938D412yescomplemented
46#14578937—24no—
47C2×#11929434—21no—
48C8×D157—24no—
49C4⋊Dic157—22no—
50C4⋊Dic157—21no—
51C22⋊Dic157—21no—
52#14578931—22no—
53#14578931—22no—
54C4×Dic157—21no—
55C22×C4×C157—21no—
56C22×C4×C157C2311yescharacteristic
57M16×C157—21no—
58M16×C157D412yescomplemented
59C2×C8×C157—21no—
60C4⋊C4×C157—21no—
61C22⋊C4×C157—21no—
62C42×C157—21no—
63C42×C157—21no—
64C42×C157C2×C411yescharacteristic
65C42×C157C2×C411yescharacteristic
66C22×D157—21no—
67C22×D157C22⋊C412yes—
68Dic314—21no—
69Dic314—21no—
70Dic314—21no—
71Dic314—21no—
72D628—21no—
73D628—21no—
74D628—21no—
75D628—21no—
76D628—44no—
77C22⋊D157—22no—
78C22⋊D157—22no—
79C22⋊D157—41no—
80C22⋊D157—41no—
81C2×Dic157—21no—
82C2×Dic157—41no—
83C2×Dic157C22⋊C412yes—
84C4×D157—22no—
85C4×D157—22no—
86C4×D157—41no—
87C4×D157—41no—
88#11929434—22no—
89C23×C157C2×D411yescharacteristic
90C2×C4×C157—21no—
91C2×C4×C157—21no—
92C2×C4×C157—42no—
93C2×C4×C157—41no—
94C2×C4×C157C22×C411yescharacteristic
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157#4111yescharacteristic
99C2×C4×C157#4111yescharacteristic
100C8×C157—22no—
101C22×C157—41no—
102C22×C157C2×C22⋊C411yescharacteristic
103C22×C157#12111yescharacteristic
104C22×C157#12111yescharacteristic
105C2×D157—22no—
106C2×D157—42no—
107C2×D157—42no—
108C2×D157—41no—
109C4×C157—22no—
110C4×C157—41no—
111C4×C157—81no—
112C4×C157C4×D411yescharacteristic
113C4×C157C4×D411yescharacteristic
114C4×C157#12011yescharacteristic
115C4×C157#12311yescharacteristic
116Dic157—22no—
117Dic157—41no—
118Dic157—81no—
119C2×C157—21no—
120C2×C157—21no—
121C2×C157—21no—
122C2×C157#38611yescharacteristic
123D157—42no—
124D157—81no—
125C157#188411yescharacteristic, complemented, Sylow
126#1884—1571nomaximal, Sylow
127#568—1571no—
128#532—1571no—
129#421—1571no—
130#421—1571no—
131#410—1571no—
132#339—1571no—
133#339D15711yescharacteristic, complemented
134C2×#41—1571no—
135C2×#41—1571no—
136#131—3142no—
137#131—3142no—
138C2×M16—3142no—
139C2×M16—1571no—
140C2×M16C2×D15711yescharacteristic
141C4⋊Q8—1571no—
142#127—1571no—
143#124—1571no—
144#124—1571no—
145#121—3142no—
146C4×D4—3142no—
147#117—1571no—
148#117C2×D15711yescharacteristic
149C2×C42C2×D15711yescharacteristic
150#104—3142no—
151#104—3142no—
152#99—3142no—
153#41—3141no—
154#41—3141no—
155#41—3141no—
156#41—3141no—
157#41—3141no—
158#41—3141no—
159#41—3141no—
160#41—3141no—
161C2×Q8—1571no—
162C2×Q8—1571no—
163C2×D4—3142no—
164C2×D4—3141no—
165C2×D4—1572no—
166C2×D4—1571no—
167C2×D4—1571no—
168C22×C4—21no—
169C22×C4—3141no—
170C22×C4—1572no—
171C22×C4C22×D15711yescharacteristic
172M16—21no—
173M16—3144no—
174M16—3141no—
175M16—1572no—
176M16D62812yescomplemented
177C2×C8—21no—
178C2×C8—3144no—
179C2×C8—3141no—
180C4⋊C4—21no—
181C4⋊C4—3142no—
182C4⋊C4—3141no—
183C22⋊C4—21no—
184C22⋊C4—3142no—
185C22⋊C4—3142no—
186C22⋊C4—3141no—
187C42—21no—
188C42—21no—
189C42—3141no—
190C42C4×D15711yescharacteristic
191C42C4×D15711yescharacteristic
192C23—3141no—
193C23—1572no—
194C23D4×D15711yescharacteristic
195Q8—3141no—
196Q8—3141no—
197Q8—3141no—
198Q8—3141no—
199D4—3142no—
200D4—3142no—
201D4—3141no—
202D4—3141no—
203D4—3141no—
204D4—3141no—
205D4—6284no—
206D4—6281no—
207D4—6281no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—3142no—
213C2×C4—3142no—
214C2×C4—3141no—
215C2×C4—6281no—
216C2×C4—6281no—
217C2×C4—6281no—
218C2×C4—1572no—
219C2×C4C2×C4×D15711yescharacteristic
220C2×C4D4×D15711yescharacteristic
221C2×C4D4×D15711yescharacteristic
222C2×C4C2×D62811yescharacteristic
223C2×C4#1457895411yescharacteristic
224C2×C4#1457895511yescharacteristic
225C8—22no—
226C8—3142no—
227C22—41no—
228C22—3142no—
229C22—6282no—
230C22—6282no—
231C22—6281no—
232C22C22⋊C4×D15711yescharacteristic
233C22#1909774411yescharacteristic
234C22#1909774611yescharacteristic
235C4—22no—
236C4—41no—
237C4—81no—
238C4—3142no—
239C4—6281no—
240C4—12561no—
241C4#1909773311yescharacteristic
242C4C4×D62811yescharacteristic
243C4#1909774111yescharacteristic
244C4#1909775611yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—6282no—
249C2—12561no—
250C2#2375715111yescharacteristic
subgroupG/H
centerC4#19097741
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#339×C157C2
socleC2×C157#386
prime radical#23757823C2
cosocleC22×C4×C157C23
hypercenter#339D157
hypocenterC157#1884
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #23758039C2semidirect
3. #23757291C2semidirect
4. #23757289C2semidirect
5. #23757246C2semidirect
6. #23757033C2semidirect
7. #339×C157C2semidirect
14. #19097771C4semidirect
15. #19097768C4semidirect
16. #19097759C4semidirect
17. #19097759C4semidirect
45. #14578938D4semidirect
58. M16×C157D4semidirect
125. C157#1884semidirect
133. #339D157semidirect
176. M16D628semidirect

Maximal subgroups

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

subgroupindex
#237580392
#237578232
#237572912
#237572892
#237572462
#237570332
#339×C1572
#1884157