Finite Group Databasebeta

← #34109543

Subgroups — #34109543

Order 20096.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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#23758109C211yesmaximal, characteristic
2#23757849C212yesmaximal, complemented
3#23757844C211yesmaximal, characteristic, complemented
4#23757464C218yesmaximal, complemented
5#23757246C212yesmaximal, complemented
6#410×C157C211yesmaximal, characteristic, complemented
7#19097819—21no—
8#19097818—22no—
9#19097817—22no—
10#19097816—21no—
11#19097816C2214yes—
12C2×#14578954—22no—
13C2×#14578954C2212yes—
14#41×D157—24no—
15C22×Dic314C2211yescharacteristic, complemented
16#19097771—22no—
17#19097769—21no—
18#19097759—22no—
19#19097758—21no—
20#19097755C2218yescomplemented
21#19097752C2214yescomplemented
22#19097751—24no—
23#19097750—22no—
24#19097730C2211yescharacteristic, complemented
25#19097685C2218yescomplemented
26C2×#41×C157C2211yescharacteristic
27#117×C157C2212yescomplemented
28#99×C157C2214yescomplemented
29#14578955—24no—
30#14578954—24no—
31#14578954—24no—
32#14578954—22no—
33#14578954—22no—
34#14578954—24no—
35#14578954—22no—
36#14578954—22no—
37#14578953—24no—
38#14578953—28no—
39C2×Dic314—22no—
40C2×Dic314—21no—
41C2×Dic314—22no—
42C2×Dic314—21no—
43C2×Dic314—22no—
44C2×Dic314D412yes—
45Q8×D157—24no—
46C2×D628—22no—
47C2×D628D412yes—
48C2×C22⋊D157—24no—
49C2×C22⋊D157C2314yes—
50D4×D157—24no—
51D4×D157—28no—
52C22×Dic157C2312yes—
53C2×C4×D157—24no—
54C2×C4×D157D414yes—
55C4⋊Dic157—28no—
56C4⋊Dic157—21no—
57#14578933—24no—
58C22⋊Dic157—24no—
59C22⋊Dic157C2312yes—
60#14578931—28no—
61C4×Dic157—21no—
62#41×C157—22no—
63#41×C157—22no—
64C2×Q8×C157D411yescharacteristic
65C2×D4×C157D411yescharacteristic
66C2×D4×C157C2312yes—
67C22×C4×C157D412yes—
68C22×C4×C157C2311yescharacteristic
69C4⋊C4×C157—22no—
70C22⋊C4×C157C2314yes—
71C42×C157—22no—
72C22×D157—24no—
73C22×D157C2×D414yes—
74Dic314—24no—
75Dic314—22no—
76Dic314—22no—
77Dic314—41no—
78Dic314—41no—
79Dic314—42no—
80D628—24no—
81D628—22no—
82D628—22no—
83C22⋊D157—28no—
84C22⋊D157—24no—
85C22⋊D157—24no—
86C2×Dic157—24no—
87C2×Dic157—24no—
88C2×Dic157—22no—
89C2×Dic157—22no—
90C2×Dic157C2×D414yes—
91C4×D157—28no—
92C4×D157—24no—
93C4×D157—24no—
94C23×C157C2×D412yes—
95C23×C157C2411yescharacteristic
96Q8×C157—22no—
97D4×C157—24no—
98D4×C157—22no—
99C2×C4×C157—22no—
100C2×C4×C157—22no—
101C2×C4×C157—24no—
102C2×C4×C157C2×D412yes—
103C2×C4×C157C2×D412yes—
104C2×C4×C157C2×D412yes—
105C2×C4×C157C2×D411yescharacteristic
106C2×C4×C157C2×D411yescharacteristic
107C22×C157—22no—
108C22×C157—41no—
109C22×C157—42no—
110C22×C157C22×D412yes—
111C22×C157C22×D411yescharacteristic
112C2×D157—24no—
113C2×D157—44no—
114C2×D157—84no—
115C4×C157—22no—
116C4×C157—44no—
117C4×C157#12011yescharacteristic
118C4×C157#12011yescharacteristic
119C4×C157#12012yes—
120Dic157—24no—
121Dic157—42no—
122Dic157—42no—
123C2×C157—21no—
124C2×C157—22no—
125C2×C157—42no—
126C2×C157C2×#12011yescharacteristic
127D157—44no—
128C157#302811yescharacteristic, complemented, Sylow
129#3028—1571nomaximal, Sylow
130#585—1571no—
131#536—1571no—
132#536—1572no—
133#458—1578no—
134#410—1572no—
135#410D15711yescharacteristic, complemented
1362−1+4—3142no—
1372−1+4—3141no—
138H(5,𝔽2)—3142no—
139H(5,𝔽2)—3141no—
140H(5,𝔽2)—1574no—
141C2×#41—3144no—
142C2×#41—3142no—
143C2×#41—1572no—
144C2×#41C2×D15711yescharacteristic
145C22×Q8—1571no—
146C4⋊Q8—3142no—
147C4⋊Q8—3141no—
148#124—3142no—
149#124—3141no—
150#123—1574no—
151#123—1578no—
152C22⋊Q8—3144no—
153C22⋊Q8—3142no—
154#117—1571no—
155#117C2×D15712yescomplemented
156#99—1578no—
157#99C2×D15714yescomplemented
158#41—22no—
159#41—22no—
160#41—3144no—
161#41—3144no—
162#41—3144no—
163#41—3144no—
164#41—3148no—
165#41—3142no—
166#41—3142no—
167#41—3142no—
168#41—3144no—
169#41—3142no—
170C2×Q8—3142no—
171C2×Q8—3144no—
172C2×Q8—3142no—
173C2×Q8—3142no—
174C2×Q8—3141no—
175C2×Q8—3141no—
176C2×Q8—1572no—
177C2×Q8D62811yescharacteristic
178C2×D4—3142no—
179C2×D4—3144no—
180C2×D4—3148no—
181C2×D4—3144no—
182C2×D4—1574no—
183C2×D4—1572no—
184C2×D4D62811yescharacteristic
185C2×D4C22×D15712yes—
186C22×C4—3144no—
187C22×C4—1572no—
188C22×C4—1574no—
189C22×C4D62812yes—
190C22×C4C22×D15711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—3148no—
193C4⋊C4—3141no—
194C4⋊C4—3144no—
195C22⋊C4—3148no—
196C22⋊C4—3144no—
197C22⋊C4—1572no—
198C22⋊C4C22×D15714yes—
199C42—22no—
200C42—3141no—
201C23—3144no—
202C23—1574no—
203C23C2×D62812yes—
204C23C23×D15711yescharacteristic
205Q8—22no—
206Q8—3144no—
207Q8—3142no—
208Q8—3142no—
209Q8—6281no—
210Q8—6281no—
211Q8—6282no—
212D4—24no—
213D4—22no—
214D4—3144no—
215D4—3148no—
216D4—3144no—
217D4—3144no—
218D4—3142no—
219D4—3142no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—3148no—
224C2×C4—3144no—
225C2×C4—3144no—
226C2×C4—3144no—
227C2×C4—3144no—
228C2×C4—3142no—
229C2×C4—3142no—
230C2×C4—1574no—
231C2×C4D4×D15712yes—
232C2×C4D4×D15712yes—
233C2×C4C2×D62812yes—
234C2×C4C2×D62811yescharacteristic
235C2×C4C2×D62811yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—3144no—
240C22—6284no—
241C22—12564no—
242C22C2×D4×D15712yes—
243C22C22×D62811yescharacteristic
244C4—22no—
245C4—44no—
246C4—3144no—
247C4—6282no—
248C4—6282no—
249C4#1909774111yescharacteristic
250C4#1909774111yescharacteristic
251C4#1909774112yes—
252C2—22no—
253C2—21no—
254C2—42no—
255C2—6284no—
256C2C2×#1909774111yescharacteristic
subgroupG/H
centerC2C2×#19097741
commutator, cosocleC23×C157C24
FrattiniC23C23×D157
Fitting#410×C157C2
socleC2×C157C2×#120
prime radical#23758109C2
hypercenter#410D157
hypocenterC157#3028
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #23757849C2semidirect
3. #23757844C2semidirect
4. #23757464C2semidirect
5. #23757246C2semidirect
6. #410×C157C2semidirect
15. C22×Dic314C22semidirect
20. #19097755C22semidirect
21. #19097752C22semidirect
24. #19097730C22semidirect
25. #19097685C22semidirect
27. #117×C157C22semidirect
28. #99×C157C22semidirect
128. C157#3028semidirect
135. #410D157semidirect
155. #117C2×D157semidirect
157. #99C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581092
#237578492
#237578442
#237574642
#237572462
#410×C1572
#3028157