Finite Group Databasebeta

← #34112858

Subgroups — #34112858

Order 20096.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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#23757997C211yesmaximal, characteristic, complemented
2D4×Dic314C211yesmaximal, characteristic, complemented
3#23757634C212yesmaximal, complemented
4#23757623C211yesmaximal, characteristic, complemented
5#23757566C212yesmaximal, complemented
6#23757554C211yesmaximal, characteristic, complemented
7#23757490C212yesmaximal
8#23757400C211yesmaximal, characteristic, complemented
9#23757374C212yesmaximal, complemented
10C4×#14578939C211yesmaximal, characteristic
11#461×C157C211yesmaximal, characteristic, complemented
12C22×Dic314—21no—
13C2×#14578939C2211yescharacteristic
14#19097771C2211yescharacteristic
15#19097770C2212yes—
16#19097766—22no—
17#19097760C2212yes—
18#19097751—21no—
19#19097750—21no—
20#19097748—22no—
21Q8×Dic157C2212yes—
22C4×Dic314C2211yescharacteristic
23C4×D628C2211yescharacteristic
24D4×Dic157C2212yes—
25#19097729—21no—
26#19097705C2212yes—
27#19097704C2211yescharacteristic
28#19097700C2212yes—
29#19097695C2212yes—
30#19097695C2212yes—
31#19097693C2212yes—
32#19097692C2211yescharacteristic
33#19097691C2211yescharacteristic
34#19097681C2212yes—
35C4×#11929434C2211yescharacteristic
36C2×SD16×C157C2212yes—
37C4⋊Q8×C157C2211yescharacteristic
38C4×D4×C157C2211yescharacteristic
39C4⋊C8×C157C2211yescharacteristic
40#103×C157C2212yes—
41C2×Dic314—22no—
42C2×Dic314—21no—
43C2×Dic314—44no—
44C2×Dic314D411yescharacteristic
45C2×D628C2311yescharacteristic
46C22×Dic157—22no—
47C2×C4×D157—21no—
48#14578939—22no—
49C2×#11929434C2311yescharacteristic
50C2×#11929434C2311yescharacteristic
51C4⋊Dic157—22no—
52C4⋊Dic157—21no—
53C4⋊Dic157D411yescharacteristic
54C4⋊Dic157C2311yescharacteristic
55C4⋊Dic157C2312yes—
56#14578933—22no—
57#14578933D412yes—
58C22⋊Dic157—22no—
59#14578931—22no—
60#14578931—22no—
61#14578931—21no—
62C4×Dic157—22no—
63C4×Dic157C2312yes—
64C2×Q8×C157C2312yes—
65C2×D4×C157C2311yescharacteristic
66C22×C4×C157—21no—
67SD16×C157—24no—
68C2×C8×C157C2312yes—
69C4⋊C4×C157—21no—
70C4⋊C4×C157—21no—
71C4⋊C4×C157C2311yescharacteristic
72C22⋊C4×C157—21no—
73C42×C157C2311yescharacteristic
74C22×D157—21no—
75Dic314—22no—
76Dic314—44no—
77Dic314—42no—
78D628—21no—
79D628—21no—
80C2×Dic157—21no—
81C2×Dic157—22no—
82C2×Dic157—44no—
83C2×Dic157C2×D412yes—
84C2×Dic157C2×D412yes—
85C4×D157—42no—
86#11929434—21no—
87#11929434—21no—
88C23×C157—21no—
89Q8×C157—22no—
90Q8×C157—22no—
91D4×C157—21no—
92D4×C157#4112yes—
93C2×C4×C157—21no—
94C2×C4×C157—22no—
95C2×C4×C157—42no—
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157C2411yescharacteristic
99C8×C157—22no—
100C22×C157—22no—
101C22×C157—41no—
102C22×C157—41no—
103C22×C157C22×D411yescharacteristic
104C2×D157—41no—
105C2×D157—41no—
106C2×D157—41no—
107C4×C157—21no—
108C4×C157—42no—
109C4×C157—41no—
110C4×C157AGL(1,ℤ/8)12yes—
111C4×C157C2×#4111yescharacteristic
112C4×C1572−1+411yescharacteristic
113Dic157—22no—
114Dic157—22no—
115Dic157—41no—
116Dic157—42no—
117C2×C157—42no—
118C2×C157#54811yescharacteristic
119C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C157#57811yescharacteristic
121D157—81no—
122C157#336711yescharacteristic, complemented, Sylow
123#3367—1571nomaximal, Sylow
124#556—1571no—
125D4×Q8—1571no—
126#489—1572no—
127#487—1571no—
128#477—1572no—
129#474—1571no—
130#463—1572no—
131#461D15711yescharacteristic, complemented
132#446—1571no—
133#440—1572no—
134C4×D8—1571no—
135C22×Q8—3141no—
136C2×SD16C2×D15712yes—
137C2×D8—1571no—
138C4⋊Q8—1572no—
139C4⋊Q8—1571no—
140C4⋊Q8C2×D15711yescharacteristic
141#126—3142no—
142#124—1572no—
143C22⋊Q8—3142no—
144C22⋊Q8—3141no—
145C22⋊Q8—3141no—
146C4×Q8—1572no—
147C4×Q8—1571no—
148C4×D4—1572no—
149C4×D4—1571no—
150C4×D4C2×D15711yescharacteristic
151#117—3141no—
152#107—1572no—
153#107—1571no—
154C4⋊C8—1572no—
155C4⋊C8C2×D15711yescharacteristic
156#103—1572no—
157#103—1572no—
158#103C2×D15712yes—
159#102—1571no—
160#102—1572no—
161#102—1571no—
162#97—1572no—
163C4×C8—1571no—
164C2×Q8—3142no—
165C2×Q8—3141no—
166C2×Q8—6284no—
167C2×Q8—1571no—
168C2×Q8C22×D15712yes—
169C2×D4—1571no—
170C2×D4C22×D15711yescharacteristic
171C22×C4—21no—
172C22×C4—3142no—
173C22×C4—3141no—
174SD16—24no—
175D8—3142no—
176C2×C8—1571no—
177C2×C8—1571no—
178C2×C8C22×D15712yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—3142no—
182C4⋊C4—3142no—
183C4⋊C4—3141no—
184C4⋊C4—1572no—
185C4⋊C4—1571no—
186C4⋊C4—1571no—
187C4⋊C4—1572no—
188C4⋊C4C22×D15711yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—3142no—
191C22⋊C4—3142no—
192C22⋊C4—3142no—
193C22⋊C4—3141no—
194C42—3142no—
195C42—1572no—
196C42C22×D15711yescharacteristic
197C23—21no—
198C23—3141no—
199Q8—22no—
200Q8—22no—
201Q8—3142no—
202Q8—6284no—
203Q8—6282no—
204D4—21no—
205D4—3141no—
206D4—3141no—
207D4#1457895312yes—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—3141no—
212C2×C4—3142no—
213C2×C4—6284no—
214C2×C4—6282no—
215C2×C4—1572no—
216C2×C4—1572no—
217C2×C4D4×D15711yescharacteristic
218C2×C4D4×D15711yescharacteristic
219C2×C4C23×D15711yescharacteristic
220C8—22no—
221C8—3141no—
222C8—3141no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—6281no—
227C22—6281no—
228C22—6281no—
229C22C2×D4×D15711yescharacteristic
230C4—21no—
231C4—42no—
232C4—41no—
233C4—3142no—
234C4—3142no—
235C4—6282no—
236C4—6281no—
237C4#1909779812yes—
238C4C2×#1457895311yescharacteristic
239C4#1909781811yescharacteristic
240C2—42no—
241C2—12561no—
242C2#2375792911yescharacteristic
243C2C2×#1909779811yescharacteristic
244C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#461×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578939C22
hypercenter#461D157
hypocenterC157#3367
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #23757997C2semidirect
2. D4×Dic314C2semidirect
3. #23757634C2semidirect
4. #23757623C2semidirect
5. #23757566C2semidirect
6. #23757554C2semidirect
8. #23757400C2semidirect
9. #23757374C2semidirect
11. #461×C157C2semidirect
122. C157#3367semidirect
131. #461D157semidirect

Maximal subgroups

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

subgroupindex
#237579972
D4×Dic3142
#237576342
#237576232
#237575662
#237575542
#237574902
#237574002
#237573742
C4×#145789392
#461×C1572
#3367157