Finite Group Databasebeta

← #34110257

Subgroups — #34110257

Order 20096.

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

normal subgroups only

This group has 424 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#23757821C211yesmaximal, characteristic
2C4×Q8×D157C211yesmaximal, characteristic, complemented
3#23757585C212yesmaximal, complemented
4#23757569C211yesmaximal, characteristic, complemented
5#23757568C211yesmaximal, characteristic, complemented
6#23757470C211yesmaximal, characteristic, complemented
7#23757410C212yesmaximal, complemented
8#23757374C212yesmaximal, complemented
9#23757370C212yesmaximal, complemented
10#23757329C211yesmaximal, characteristic, complemented
11#460×C157C211yesmaximal, characteristic, complemented
12C2×#14578953—21no—
13C2×Q8×D157C2211yescharacteristic
14C2×#14578942C2212yes—
15#19097771C2211yescharacteristic
16#19097758—21no—
17#19097742—21no—
18#19097742C2212yes—
19Q8×Dic157C2211yescharacteristic
20C4×Dic314—21no—
21C4×Dic314C2211yescharacteristic
22#19097736—21no—
23#19097736C2211yescharacteristic
24D4×Dic157—21no—
25D4×Dic157C2212yes—
26C4⋊C4×D157—21no—
27C4⋊C4×D157C2211yescharacteristic
28C42×D157—21no—
29C42×D157C2211yescharacteristic
30#19097705C2212yes—
31#19097704C2212yes—
32C4⋊#11929434C2211yescharacteristic
33#19097696C2212yes—
34#19097695C2212yes—
35#19097692C2212yes—
36#19097683C2212yes—
37#19097681C2212yes—
38C2×SD16×C157C2212yes—
39#127×C157C2211yescharacteristic
40C4×Q8×C157C2211yescharacteristic
41C4⋊C8×C157C2211yescharacteristic
42#102×C157C2212yes—
43#14578953—44no—
44C2×Dic314—21no—
45C2×Dic314—21no—
46C2×Dic314C2311yescharacteristic
47Q8×D157—22no—
48Q8×D157—22no—
49C2×C22⋊D157—22no—
50C22×Dic157—22no—
51C2×C4×D157—21no—
52C2×C4×D157—21no—
53C2×C4×D157D411yescharacteristic
54C2×C4×D157D411yescharacteristic
55C2×C4×D157C2311yescharacteristic
56#14578942—24no—
57C2×#11929434C2312yes—
58C4⋊Dic157—21no—
59C4⋊Dic157—21no—
60C4⋊Dic157C2311yescharacteristic
61C4⋊Dic157C2312yes—
62#14578933—21no—
63#14578933—21no—
64#14578933—21no—
65C22⋊Dic157—22no—
66C22⋊Dic157—22no—
67C4×Dic157—21no—
68C4×Dic157—21no—
69C4×Dic157—21no—
70C4×Dic157D411yescharacteristic
71C4×Dic157D411yescharacteristic
72C4×Dic157C2311yescharacteristic
73C2×Q8×C157C2311yescharacteristic
74C2×D4×C157—21no—
75C2×D4×C157—21no—
76C2×D4×C157C2312yes—
77SD16×C157—24no—
78C2×C8×C157C2312yes—
79C4⋊C4×C157—21no—
80C4⋊C4×C157C2311yescharacteristic
81C42×C157—21no—
82C42×C157C2311yescharacteristic
83C22×D157C2×D411yescharacteristic
84Dic314—24no—
85Dic314—21no—
86Dic314—42no—
87Dic314#4112yescomplemented
88C22⋊D157—44no—
89C2×Dic157—21no—
90C2×Dic157—21no—
91C2×Dic157—22no—
92C2×Dic157—44no—
93C2×Dic157C2×D411yescharacteristic
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96C4×D157—24no—
97C4×D157—22no—
98C4×D157—22no—
99C4×D157—21no—
100C4×D157—21no—
101C4×D157—42no—
102#11929434—22no—
103C23×C157—22no—
104Q8×C157—21no—
105Q8×C157#4112yescomplemented
106D4×C157—22no—
107D4×C157—22no—
108D4×C157—42no—
109D4×C157—42no—
110C2×C4×C157—21no—
111C2×C4×C157—21no—
112C2×C4×C157C2×D411yescharacteristic
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157C2411yescharacteristic
115C8×C157—22no—
116C22×C157—42no—
117C22×C157—42no—
118C22×C157—42no—
119C22×C157C22×D411yescharacteristic
120C2×D157—21no—
121C2×D157—21no—
122C2×D157AGL(1,ℤ/8)12yes—
123C4×C157—22no—
124C4×C157—21no—
125C4×C157—41no—
126C4×C157#13712yes—
127C4×C157C2×#4111yescharacteristic
128C4×C157C2×#4111yescharacteristic
129Dic157—22no—
130Dic157—21no—
131Dic157—21no—
132Dic157—21no—
133Dic157—41no—
134Dic157—42no—
135C2×C157—82no—
136C2×C157#52511yescharacteristic
137C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C157C2×#13711yescharacteristic
139D157—22no—
140C157#311211yescharacteristic, complemented, Sylow
141#3112—1571nomaximal, Sylow
142#532—1571no—
143C2×C4×Q8—1571no—
144#480—1572no—
145#477—1571no—
146#477—1571no—
147#460—1571no—
148#460D15711yescharacteristic, complemented
149#448—1572no—
150#440—1572no—
151#440—1572no—
152#432—1571no—
153C2×#41—3141no—
154C22×Q8—1571no—
155C2×SD16—1572no—
156C2×SD16C2×D15712yes—
157C4⋊Q8—1571no—
158#127C2×D15711yescharacteristic
159#124—3141no—
160#121—3141no—
161#121—1572no—
162C4×Q8—3141no—
163C4×Q8—3141no—
164C4×Q8—1571no—
165C4×Q8—1571no—
166C4×Q8—1571no—
167C4×Q8C2×D15711yescharacteristic
168C4×D4—3141no—
169C4×D4—1572no—
170C2×C4⋊C4—3141no—
171C2×C4⋊C4—1571no—
172C2×C42—3141no—
173C2×C42—1571no—
174#107—1572no—
175#107—1572no—
176C4⋊C8—1571no—
177C4⋊C8C2×D15711yescharacteristic
178#103—1572no—
179#103—1572no—
180#102—1572no—
181#102C2×D15712yes—
182C22⋊C8—1572no—
183#97—1572no—
184#41—6284no—
185C2×Q8—3141no—
186C2×Q8—3142no—
187C2×Q8—3142no—
188C2×Q8—3141no—
189C2×Q8—1571no—
190C2×Q8C22×D15711yescharacteristic
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—3142no—
194C2×D4C22×D15712yes—
195C22×C4—3141no—
196C22×C4—3141no—
197C22×C4—3142no—
198C22×C4—1571no—
199C22×C4—1571no—
200C22×C4—1571no—
201SD16—24no—
202SD16—3144no—
203C2×C8—1572no—
204C2×C8C22×D15712yes—
205C4⋊C4—21no—
206C4⋊C4—3141no—
207C4⋊C4—3141no—
208C4⋊C4—3141no—
209C4⋊C4—3141no—
210C4⋊C4—3141no—
211C4⋊C4—1571no—
212C4⋊C4—1572no—
213C4⋊C4C22×D15711yescharacteristic
214C22⋊C4—3142no—
215C22⋊C4—3142no—
216C42—21no—
217C42—3141no—
218C42—3141no—
219C42—3141no—
220C42—1571no—
221C42—1571no—
222C42—1571no—
223C42C22×D15711yescharacteristic
224C23—22no—
225C23—1571no—
226Q8—21no—
227Q8—3144no—
228Q8—3141no—
229Q8—6282no—
230Q8—1572no—
231Q8#1457895312yescomplemented
232D4—22no—
233D4—22no—
234D4—42no—
235D4—42no—
236D4—6284no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—3144no—
240C2×C4—3142no—
241C2×C4—3141no—
242C2×C4—3142no—
243C2×C4—3141no—
244C2×C4—3141no—
245C2×C4—3141no—
246C2×C4—3142no—
247C2×C4—6282no—
248C2×C4—6284no—
249C2×C4—1571no—
250C2×C4—1571no—
251C2×C4—1571no—
252C2×C4D4×D15711yescharacteristic
253C2×C4D4×D15711yescharacteristic
254C2×C4C23×D15711yescharacteristic
255C8—22no—
256C8—3142no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—3141no—
261C22—3141no—
262C22—1572no—
263C22C2×D4×D15711yescharacteristic
264C4—22no—
265C4—21no—
266C4—41no—
267C4—3142no—
268C4—3141no—
269C4—3141no—
270C4—3141no—
271C4—6282no—
272C4—6281no—
273C4#1909780012yes—
274C4#41×D15711yescharacteristic
275C4C2×#1457895311yescharacteristic
276C2—82no—
277C2—3142no—
278C2#2375778311yescharacteristic
279C2AGL(1,ℤ/8)×D15711yescharacteristic
280C2C2×#1909780011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#460×C157C2
socleC22×C157C22×D4
prime radical#23757821C2
hypercenter#460D157
hypocenterC157#3112
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D157C2semidirect
3. #23757585C2semidirect
4. #23757569C2semidirect
5. #23757568C2semidirect
6. #23757470C2semidirect
7. #23757410C2semidirect
8. #23757374C2semidirect
9. #23757370C2semidirect
10. #23757329C2semidirect
11. #460×C157C2semidirect
87. Dic314#41semidirect
105. Q8×C157#41semidirect
140. C157#3112semidirect
148. #460D157semidirect
231. Q8#14578953semidirect

Maximal subgroups

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

subgroupindex
#237578212
C4×Q8×D1572
#237575852
#237575692
#237575682
#237574702
#237574102
#237573742
#237573702
#237573292
#460×C1572
#3112157