Finite Group Databasebeta

← #34112784

Subgroups — #34112784

Order 20096.

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

normal subgroups only

This group has 350 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#23757984C211yesmaximal, characteristic, complemented
3#23757626C212yesmaximal, complemented
4#23757625C211yesmaximal, characteristic, complemented
5#23757583C212yesmaximal, complemented
6#23757487C212yesmaximal
7#23757479C211yesmaximal, characteristic, complemented
8#23757397C211yesmaximal, characteristic, complemented
9#23757360C212yesmaximal, complemented
10C4×#14578939C211yesmaximal, characteristic
11#476×C157C211yesmaximal, characteristic, complemented
12C2×#14578943C2212yes—
13C2×#14578939C2211yescharacteristic
14#19097771C2211yescharacteristic
15#19097764—22no—
16#19097762C2212yes—
17#19097758C2212yes—
18#19097751—21no—
19#19097750—21no—
20#19097747—22no—
21C4×Dic314C2211yescharacteristic
22#19097736C2212yes—
23C4×D628C2211yescharacteristic
24#19097733C2212yes—
25#19097730—21no—
26C4⋊C4×D157—21no—
27#19097704C2211yescharacteristic
28#19097700C2212yes—
29#19097696C2212yes—
30#19097694C2212yes—
31#19097694C2212yes—
32#19097692C2211yescharacteristic
33#19097691C2211yescharacteristic
34C4×#11929434C2211yescharacteristic
35C8×Dic157C2212yes—
36C4⋊Q8×C157C2211yescharacteristic
37C4×D4×C157C2211yescharacteristic
38#106×C157C2212yes—
39C4⋊C8×C157C2211yescharacteristic
40#102×C157C2212yes—
41C2×Dic314D411yescharacteristic
42C2×Dic314C2312yes—
43C2×D628C2311yescharacteristic
44C2×C4×D157—22no—
45C2×C4×D157—21no—
46#14578943—24no—
47#14578939—22no—
48C2×#11929434C2311yescharacteristic
49C2×#11929434C2311yescharacteristic
50C4⋊Dic157—22no—
51C4⋊Dic157—21no—
52C4⋊Dic157—21no—
53C4⋊Dic157D411yescharacteristic
54C4⋊Dic157C2311yescharacteristic
55#14578933—22no—
56#14578933—22no—
57#14578933D412yes—
58C22⋊Dic157—22no—
59C22⋊Dic157—22no—
60#14578931—22no—
61#14578931—21no—
62C4×Dic157—22no—
63C4×Dic157C2312yes—
64C2×Q8×C157—21no—
65C2×D4×C157C2311yescharacteristic
66C22×C4×C157—21no—
67C2×C8×C157C2312yes—
68C4⋊C4×C157—21no—
69C4⋊C4×C157C2312yes—
70C4⋊C4×C157C2311yescharacteristic
71C22⋊C4×C157—21no—
72C42×C157C2311yescharacteristic
73C22×D157—21no—
74Dic314—22no—
75Dic314—22no—
76Dic314—22no—
77D628—21no—
78D628#4112yes—
79C2×Dic157—21no—
80C2×Dic157—22no—
81C2×Dic157C2×D412yes—
82C2×Dic157C2×D412yes—
83C4×D157—42no—
84C4×D157—82no—
85#11929434—21no—
86#11929434—21no—
87C23×C157—21no—
88Q8×C157—81no—
89D4×C157—21no—
90D4×C157—21no—
91C2×C4×C157—22no—
92C2×C4×C157—21no—
93C2×C4×C157—42no—
94C2×C4×C157C2×D411yescharacteristic
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157C2411yescharacteristic
97C8×C157—22no—
98C22×C157—41no—
99C22×C157—41no—
100C22×C157—41no—
101C22×C157C22×D411yescharacteristic
102C2×D157—22no—
103C2×D157—41no—
104C2×D157—41no—
105C4×C157—21no—
106C4×C157—41no—
107C4×C157—42no—
108C4×C157#13512yes—
109C4×C157C2×#4111yescharacteristic
110C4×C1572−1+411yescharacteristic
111Dic157—22no—
112Dic157—22no—
113Dic157—42no—
114Dic157—41no—
115C2×C157—81no—
116C2×C157#54811yescharacteristic
117C2×C157C2×#13511yescharacteristic
118C2×C157#57711yescharacteristic
119D157—42no—
120C157#336011yescharacteristic, complemented, Sylow
121#3360—1571nomaximal, Sylow
122#556—1571no—
123#554—1571no—
124#488—1572no—
125#487—1571no—
126#479—1572no—
127#476D15711yescharacteristic, complemented
128#463—1572no—
129#461—1571no—
130C8×Q8—1571no—
131C4×SD16—1572no—
132C4×D8—1571no—
133C2×SD16—1572no—
134C2×D8—1571no—
135C4⋊Q8—1571no—
136C4⋊Q8C2×D15711yescharacteristic
137#126—3142no—
138#125—1572no—
139#124—1572no—
140C22⋊Q8—3141no—
141C22⋊Q8—3141no—
142C22⋊Q8—3142no—
143C4×Q8—1572no—
144C4×Q8—1571no—
145C4×D4—1572no—
146C4×D4—1571no—
147C4×D4C2×D15711yescharacteristic
148#117—3141no—
149C2×C4⋊C4—3141no—
150#107—1571no—
151#106C2×D15712yes—
152C4⋊C8—1572no—
153C4⋊C8C2×D15711yescharacteristic
154#103—1572no—
155#103—1572no—
156#103—1572no—
157#102—1571no—
158#102—1571no—
159#102C2×D15712yes—
160C4×C8—1572no—
161C4×C8—1571no—
162C2×Q8—21no—
163C2×Q8—1572no—
164C2×Q8—1571no—
165C2×D4—1571no—
166C2×D4C22×D15711yescharacteristic
167C22×C4—21no—
168C22×C4—3142no—
169C22×C4—3141no—
170SD16—3144no—
171D8—3142no—
172C2×C8—1571no—
173C2×C8—1571no—
174C2×C8C22×D15712yes—
175C4⋊C4—21no—
176C4⋊C4—3142no—
177C4⋊C4—3142no—
178C4⋊C4—3142no—
179C4⋊C4—3141no—
180C4⋊C4—3141no—
181C4⋊C4—1571no—
182C4⋊C4—1571no—
183C4⋊C4—1572no—
184C4⋊C4C22×D15711yescharacteristic
185C4⋊C4C22×D15712yes—
186C22⋊C4—21no—
187C22⋊C4—3142no—
188C22⋊C4—3142no—
189C22⋊C4—3142no—
190C22⋊C4—3141no—
191C42—3142no—
192C42—1572no—
193C42C22×D15711yescharacteristic
194C23—21no—
195C23—3141no—
196Q8—81no—
197Q8—3142no—
198Q8—3142no—
199Q8—3142no—
200D4—21no—
201D4—21no—
202D4—3141no—
203D4—1572no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—3142no—
208C2×C4—3141no—
209C2×C4—6282no—
210C2×C4—1572no—
211C2×C4—1572no—
212C2×C4—12562no—
213C2×C4D4×D15711yescharacteristic
214C2×C4D4×D15711yescharacteristic
215C2×C4C23×D15711yescharacteristic
216C8—22no—
217C8—3141no—
218C8—3141no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—3142no—
223C22—6281no—
224C22—6281no—
225C22C2×D4×D15711yescharacteristic
226C4—21no—
227C4—42no—
228C4—41no—
229C4—3142no—
230C4—3142no—
231C4—6282no—
232C4—6281no—
233C4#1909779212yes—
234C4#41×D15711yescharacteristic
235C4#1909781911yescharacteristic
236C2—81no—
237C2—6282no—
238C2#2375793311yescharacteristic
239C2C2×#1909779211yescharacteristic
240C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#476×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578939C22
hypercenter#476D157
hypocenterC157#3360
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #23757993C2semidirect
2. #23757984C2semidirect
3. #23757626C2semidirect
4. #23757625C2semidirect
5. #23757583C2semidirect
7. #23757479C2semidirect
8. #23757397C2semidirect
9. #23757360C2semidirect
11. #476×C157C2semidirect
120. C157#3360semidirect
127. #476D157semidirect

Maximal subgroups

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

subgroupindex
#237579932
#237579842
#237576262
#237576252
#237575832
#237574872
#237574792
#237573972
#237573602
C4×#145789392
#476×C1572
#3360157