Finite Group Databasebeta

← #34112853

Subgroups — #34112853

Order 20096.

372 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 115 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#23757993C211yesmaximal, characteristic, complemented
2#23757938C211yesmaximal, characteristic, complemented
3#23757631C212yesmaximal, complemented
4#23757625C211yesmaximal, characteristic, complemented
5#23757567C212yesmaximal, complemented
6#23757487C212yesmaximal
7#23757479C211yesmaximal, characteristic, complemented
8#23757400C211yesmaximal, characteristic, complemented
9#23757372C212yesmaximal, complemented
10C4×#14578939C211yesmaximal, characteristic
11#474×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157—21no—
13C2×#14578943C2212yes—
14C2×#14578939C2211yescharacteristic
15#19097771C2211yescharacteristic
16#19097770C2212yes—
17#19097764—22no—
18#19097758C2212yes—
19#19097750—21no—
20#19097749—22no—
21#19097747—21no—
22C4×Dic314C2211yescharacteristic
23#19097736C2212yes—
24C4×D628C2211yescharacteristic
25#19097733C2212yes—
26#19097730—21no—
27#19097704C2211yescharacteristic
28#19097700C2212yes—
29#19097696C2212yes—
30#19097694C2212yes—
31#19097694C2212yes—
32#19097692C2211yescharacteristic
33#19097691C2211yescharacteristic
34#19097681C2212yes—
35C4×#11929434C2211yescharacteristic
36C4⋊Q8×C157C2211yescharacteristic
37C4×D4×C157C2211yescharacteristic
38#107×C157C2212yes—
39C4⋊C8×C157C2211yescharacteristic
40#102×C157C2212yes—
41C2×Dic314—22no—
42C2×Dic314D411yescharacteristic
43C2×Dic314C2312yes—
44Q8×D157—44no—
45C2×D628C2311yescharacteristic
46C2×C4×D157—22no—
47C2×C4×D157—21no—
48#14578943—24no—
49#14578939—22no—
50C2×#11929434C2311yescharacteristic
51C2×#11929434C2311yescharacteristic
52C4⋊Dic157—21no—
53C4⋊Dic157—21no—
54C4⋊Dic157D411yescharacteristic
55C4⋊Dic157C2311yescharacteristic
56#14578933—22no—
57#14578933—22no—
58#14578933D412yes—
59C22⋊Dic157—22no—
60C22⋊Dic157—22no—
61#14578931—22no—
62#14578931—21no—
63C4×Dic157—22no—
64C4×Dic157C2312yes—
65C2×Q8×C157—21no—
66C2×D4×C157C2311yescharacteristic
67C22×C4×C157—21no—
68C2×C8×C157C2312yes—
69C4⋊C4×C157—21no—
70C4⋊C4×C157C2312yes—
71C4⋊C4×C157C2311yescharacteristic
72C22⋊C4×C157—21no—
73C42×C157C2311yescharacteristic
74C22×D157—21no—
75Dic314—22no—
76Dic314—22no—
77Dic314—22no—
78Dic314—44no—
79D628—21no—
80D628#4112yes—
81C2×Dic157—21no—
82C2×Dic157—22no—
83C2×Dic157C2×D412yes—
84C2×Dic157C2×D412yes—
85C4×D157—44no—
86C4×D157—42no—
87#11929434—21no—
88#11929434—21no—
89C23×C157—21no—
90Q8×C157—42no—
91D4×C157—21no—
92D4×C157—21no—
93C2×C4×C157—22no—
94C2×C4×C157—21no—
95C2×C4×C157—42no—
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157C2411yescharacteristic
99C8×C157—22no—
100C22×C157—41no—
101C22×C157—41no—
102C22×C157—41no—
103C22×C157C22×D411yescharacteristic
104C2×D157—22no—
105C2×D157—41no—
106C2×D157—41no—
107C4×C157—21no—
108C4×C157—41no—
109C4×C157—42no—
110C4×C157AGL(1,ℤ/8)12yes—
111C4×C157C2×#4111yescharacteristic
112C4×C1572−1+411yescharacteristic
113Dic157—22no—
114Dic157—22no—
115Dic157—42no—
116Dic157—41no—
117C2×C157—81no—
118C2×C157#54811yescharacteristic
119C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C157#57811yescharacteristic
121D157—42no—
122C157#336711yescharacteristic, complemented, Sylow
123#3367—1571nomaximal, Sylow
124#556—1571no—
125D4×Q8—1571no—
126#489—1572no—
127#487—1571no—
128#477—1572no—
129#474D15711yescharacteristic, complemented
130#463—1572no—
131#461—1571no—
132#446—1571no—
133#440—1572no—
134C4×D8—1571no—
135C22×Q8—3141no—
136C2×SD16—1572no—
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—1571no—
153#107C2×D15712yes—
154C4⋊C8—1572no—
155C4⋊C8C2×D15711yescharacteristic
156#103—1572no—
157#103—1572no—
158#103—1572no—
159#102—1571no—
160#102—1571no—
161#102C2×D15712yes—
162#97—1572no—
163C4×C8—1571no—
164C2×Q8—21no—
165C2×Q8—3142no—
166C2×Q8—6284no—
167C2×Q8—1572no—
168C2×Q8—1571no—
169C2×D4—1571no—
170C2×D4C22×D15711yescharacteristic
171C22×C4—21no—
172C22×C4—3142no—
173C22×C4—3141no—
174SD16—3144no—
175D8—3142no—
176C2×C8—1571no—
177C2×C8—1571no—
178C2×C8C22×D15712yes—
179C4⋊C4—21no—
180C4⋊C4—3142no—
181C4⋊C4—3142no—
182C4⋊C4—3141no—
183C4⋊C4—3141no—
184C4⋊C4—1571no—
185C4⋊C4—1571no—
186C4⋊C4—1572no—
187C4⋊C4C22×D15711yescharacteristic
188C4⋊C4C22×D15712yes—
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—42no—
200Q8—3142no—
201Q8—3142no—
202Q8—3142no—
203Q8—6284no—
204D4—21no—
205D4—21no—
206D4—3141no—
207D4—1572no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—3142no—
212C2×C4—3141no—
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—41no—
224C22—41no—
225C22—41no—
226C22—3142no—
227C22—6281no—
228C22—6281no—
229C22C2×D4×D15711yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—3142no—
234C4—3142no—
235C4—6282no—
236C4—6281no—
237C4#1909779512yes—
238C4#41×D15711yescharacteristic
239C4#1909781911yescharacteristic
240C2—81no—
241C2—6282no—
242C2#2375793311yescharacteristic
243C2C2×#1909779511yescharacteristic
244C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#474×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578939C22
hypercenter#474D157
hypocenterC157#3367
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #23757993C2semidirect
2. #23757938C2semidirect
3. #23757631C2semidirect
4. #23757625C2semidirect
5. #23757567C2semidirect
7. #23757479C2semidirect
8. #23757400C2semidirect
9. #23757372C2semidirect
11. #474×C157C2semidirect
122. C157#3367semidirect
129. #474D157semidirect

Maximal subgroups

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

subgroupindex
#237579932
#237579382
#237576312
#237576252
#237575672
#237574872
#237574792
#237574002
#237573722
C4×#145789392
#474×C1572
#3367157