Finite Group Databasebeta

← #34110302

Subgroups — #34110302

Order 20096.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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#23757804C211yesmaximal, characteristic
2C2×Q8×Dic157C211yesmaximal, characteristic, complemented
3#23757609C211yesmaximal, characteristic, complemented
4#23757589C211yesmaximal, characteristic
5#23757582C212yesmaximal, complemented
6#23757493C212yesmaximal, complemented
7#23757429C211yesmaximal, characteristic, complemented
8#23757377C212yesmaximal, complemented
9#23757374C212yesmaximal, complemented
10#23757327C211yesmaximal, characteristic, complemented
11#448×C157C211yesmaximal, characteristic
12C2×#14578944C2212yes—
13#19097764—21no—
14#19097762C2211yescharacteristic
15#19097758C2211yescharacteristic
16#19097752—21no—
17#19097750C2211yescharacteristic
18Q8×Dic157—21no—
19Q8×Dic157—21no—
20Q8×Dic157C2212yes—
21Q8×Dic157C2212yes—
22#19097736C2211yescharacteristic
23D4×Dic157—21no—
24D4×Dic157C2211yescharacteristic
25#19097727—21no—
26C2×C4⋊Dic157—21no—
27C2×C4⋊Dic157C2211yescharacteristic
28C2×C4×Dic157—21no—
29C2×C4×Dic157C2211yescharacteristic
30#19097705C2212yes—
31#19097702C2212yes—
32#19097700C2212yes—
33#19097696C2212yes—
34#19097695C2212yes—
35#19097692C2212yes—
36C22⋊#11929434C2211yescharacteristic
37#19097681C2212yes—
38C22×Q8×C157C2211yescharacteristic
39C2×SD16×C157C2212yes—
40#121×C157C2211yescharacteristic
41#103×C157C2212yes—
42C22⋊C8×C157C2211yescharacteristic
43C2×Dic314C2311yescharacteristic
44C22×Dic157—21no—
45C22×Dic157—21no—
46C22×Dic157D411yescharacteristic
47C22×Dic157D411yescharacteristic
48#14578944—24no—
49C2×#11929434C2312yes—
50C4⋊Dic157—22no—
51C4⋊Dic157—22no—
52C4⋊Dic157—21no—
53C4⋊Dic157—21no—
54C4⋊Dic157C2312yes—
55C4⋊Dic157C2311yescharacteristic
56#14578933—21no—
57#14578933—21no—
58#14578933—21no—
59C22⋊Dic157—21no—
60C22⋊Dic157—21no—
61C22⋊Dic157—21no—
62C22⋊Dic157—21no—
63C4×Dic157—22no—
64C4×Dic157—22no—
65C4×Dic157—21no—
66C4×Dic157D411yescharacteristic
67C4×Dic157D411yescharacteristic
68C4×Dic157C2311yescharacteristic
69C4×Dic157C2311yescharacteristic
70C2×Q8×C157—24no—
71C2×Q8×C157—21no—
72C2×Q8×C157C2312yes—
73C2×D4×C157—21no—
74C2×D4×C157C2311yescharacteristic
75C22×C4×C157—21no—
76C22×C4×C157C2311yescharacteristic
77SD16×C157—24no—
78C2×C8×C157C2312yes—
79C4⋊C4×C157C2311yescharacteristic
80C22⋊C4×C157—21no—
81Dic314—22no—
82C2×Dic157—22no—
83C2×Dic157—22no—
84C2×Dic157—21no—
85C2×Dic157—21no—
86C2×Dic157—41no—
87C2×Dic157—42no—
88C2×Dic157—42no—
89C2×Dic157C2×D411yescharacteristic
90C2×Dic157C2×D411yescharacteristic
91C2×Dic157C2×D411yescharacteristic
92C2×Dic157C2×D411yescharacteristic
93#11929434—22no—
94C23×C157—21no—
95C23×C157C2×D411yescharacteristic
96Q8×C157—24no—
97Q8×C157—22no—
98Q8×C157#4114yes—
99D4×C157—22no—
100D4×C157—42no—
101C2×C4×C157—24no—
102C2×C4×C157—22no—
103C2×C4×C157—22no—
104C2×C4×C157—21no—
105C2×C4×C157C2×D411yescharacteristic
106C2×C4×C157C2411yescharacteristic
107C8×C157—22no—
108C22×C157—21no—
109C22×C157—21no—
110C22×C157—41no—
111C22×C157—41no—
112C22×C157—41no—
113C22×C157#13712yes—
114C22×C157C22×D411yescharacteristic
115C4×C157—24no—
116C4×C157—21no—
117C4×C157—41no—
118C4×C157C2×#4112yes—
119Dic157—21no—
120Dic157—21no—
121Dic157—21no—
122Dic157—21no—
123Dic157—41no—
124Dic157—42no—
125Dic157—41no—
126C2×C157—22no—
127C2×C157—81no—
128C2×C157#52511yescharacteristic
129C2×C157C2×#13711yescharacteristic
130C2×C157#57511yescharacteristic
131C157#311711yescharacteristic, complemented, Sylow
132#3117—1571nomaximal, Sylow
133#529—1571no—
134C2×C4×Q8—1571no—
135#484—1571no—
136#481—1571no—
137#479—1572no—
138#464—1572no—
139#451—1571no—
140#448D15711yescharacteristic
141#441—1572no—
142#440—1572no—
143#432—1571no—
144C22×Q8C2×D15711yescharacteristic
145C2×Q16—1572no—
146C2×SD16C2×D15712yes—
147#126—3141no—
148#125—1571no—
149#124—1571no—
150#123—3141no—
151C22⋊Q8—1571no—
152#121C2×D15711yescharacteristic
153C4×Q8—3141no—
154C4×Q8—3141no—
155C4×Q8—1572no—
156C4×Q8—1572no—
157C4×Q8—1571no—
158C4×D4—3141no—
159C4×D4—1571no—
160#117—3141no—
161C2×C4⋊C4—3141no—
162C2×C4⋊C4—1571no—
163C2×C42—3141no—
164C2×C42—1571no—
165#107—1572no—
166#106—1572no—
167C4⋊C8—1572no—
168#103—1572no—
169#103—1572no—
170#103C2×D15712yes—
171#102—1572no—
172C22⋊C8—1571no—
173C22⋊C8C2×D15711yescharacteristic
174#97—1572no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—1571no—
178C2×Q8C22×D15712yes—
179C2×D4—21no—
180C2×D4C22×D15711yescharacteristic
181C22×C4—21no—
182C22×C4—3141no—
183C22×C4—3141no—
184C22×C4—1571no—
185C22×C4—1571no—
186C22×C4C22×D15711yescharacteristic
187Q16—3144no—
188SD16—24no—
189C2×C8—1572no—
190C2×C8C22×D15712yes—
191C4⋊C4—3142no—
192C4⋊C4—3142no—
193C4⋊C4—3141no—
194C4⋊C4—3141no—
195C4⋊C4—3141no—
196C4⋊C4—3141no—
197C4⋊C4—3141no—
198C4⋊C4—1572no—
199C4⋊C4—1571no—
200C4⋊C4C22×D15711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—3141no—
203C22⋊C4—3141no—
204C22⋊C4—3141no—
205C22⋊C4—3141no—
206C42—3142no—
207C42—3142no—
208C42—3141no—
209C42—1571no—
210C42—1571no—
211C42—1571no—
212C42—1571no—
213C23—21no—
214C23D4×D15711yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—3142no—
218Q8#1457895314yes—
219D4—22no—
220D4—42no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—3142no—
226C2×C4—3142no—
227C2×C4—3141no—
228C2×C4—3141no—
229C2×C4—6281no—
230C2×C4—6282no—
231C2×C4—6282no—
232C2×C4—1571no—
233C2×C4—1571no—
234C2×C4—1571no—
235C2×C4—1571no—
236C2×C4D4×D15711yescharacteristic
237C2×C4C23×D15711yescharacteristic
238C8—22no—
239C8—3142no—
240C22—21no—
241C22—21no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22#1909780012yes—
246C22C2×D4×D15711yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—3141no—
251C4—3141no—
252C4—3141no—
253C4—3141no—
254C4—6281no—
255C4—6282no—
256C4—6281no—
257C4C2×#1457895312yes—
258C2—22no—
259C2—81no—
260C2#2375777911yescharacteristic
261C2C2×#1909780011yescharacteristic
262C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#448×C157C2
socleC22×C157C22×D4
prime radical#121×C157C22
hypercenter#448D157
hypocenterC157#3117
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic157C2semidirect
3. #23757609C2semidirect
5. #23757582C2semidirect
6. #23757493C2semidirect
7. #23757429C2semidirect
8. #23757377C2semidirect
9. #23757374C2semidirect
10. #23757327C2semidirect
131. C157#3117semidirect

Maximal subgroups

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

subgroupindex
#237578042
C2×Q8×Dic1572
#237576092
#237575892
#237575822
#237574932
#237574292
#237573772
#237573742
#237573272
#448×C1572
#3117157