Finite Group Databasebeta

← #34110687

Subgroups — #34110687

Order 20096.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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#23757828C211yesmaximal, characteristic
2#125×D157C211yesmaximal, characteristic
3#23757683C211yesmaximal, characteristic, complemented
4#23757682C211yesmaximal, characteristic, complemented
5#23757679C211yesmaximal, characteristic, complemented
6#23757674C212yesmaximal, complemented
7#23757673C212yesmaximal, complemented
8#23757314C212yesmaximal
9#23757312C212yesmaximal
10#23757225C211yesmaximal, characteristic
11#501×C157C211yesmaximal, characteristic, complemented
12C2×#14578937C2212yes—
13#19097771C2211yescharacteristic
14#19097763C2211yescharacteristic
15#19097762C2212yes—
16#19097762C2211yescharacteristic
17#19097762C2211yescharacteristic
18#19097761—21no—
19#19097761—21no—
20#19097751—21no—
21#19097747—21no—
22Q8×Dic157—21no—
23#19097736—21no—
24#19097729C2212yes—
25C4⋊C4×D157—21no—
26C4⋊C4×D157—21no—
27C4⋊C4×D157C2211yescharacteristic
28C4⋊C4×D157C2211yescharacteristic
29C42×D157C2211yescharacteristic
30#19097705C2212yes—
31#19097704C2212yes—
32#19097704C2212yes—
33#19097703C2212yes—
34#19097702C2212yes—
35#19097702C2212yes—
36#19097682C2211yescharacteristic
37C8×Dic157C2212yes—
38C4⋊Q8×C157C2211yescharacteristic
39#125×C157C2211yescharacteristic
40#107×C157C2212yes—
41#106×C157C2212yes—
42#97×C157C2211yescharacteristic
43C2×Dic314—21no—
44C2×C4×D157—21no—
45C2×C4×D157—21no—
46C2×C4×D157D411yescharacteristic
47C2×C4×D157D411yescharacteristic
48C2×C4×D157C2311yescharacteristic
49#14578937Q818yescomplemented
50C2×#11929434C2312yes—
51C4⋊Dic157—21no—
52C4⋊Dic157—21no—
53C4⋊Dic157—22no—
54C4⋊Dic157—21no—
55C4⋊Dic157C2311yescharacteristic
56C4⋊Dic157C2312yes—
57C4⋊Dic157C2311yescharacteristic
58#14578933—21no—
59#14578933—21no—
60#14578933—21no—
61#14578933—21no—
62#14578933—22no—
63#14578933—21no—
64#14578933—21no—
65#14578931—22no—
66C4×Dic157—22no—
67C4×Dic157D411yescharacteristic
68C4×Dic157D411yescharacteristic
69C4×Dic157C2311yescharacteristic
70C2×Q8×C157—21no—
71C2×C8×C157C2312yes—
72C4⋊C4×C157—21no—
73C4⋊C4×C157—21no—
74C4⋊C4×C157—21no—
75C4⋊C4×C157C2311yescharacteristic
76C4⋊C4×C157C2312yes—
77C4⋊C4×C157C2311yescharacteristic
78C42×C157C2311yescharacteristic
79C22×D157C2×D411yescharacteristic
80Dic314—42no—
81C2×Dic157—21no—
82C2×Dic157—21no—
83C2×Dic157—22no—
84C2×Dic157C2×D411yescharacteristic
85C2×Dic157C2×D411yescharacteristic
86C2×Dic157C2×D411yescharacteristic
87C4×D157—41no—
88C4×D157—41no—
89C4×D157—42no—
90C4×D157—42no—
91C4×D157C2×Q814yes—
92#11929434C2×Q814yes—
93Q8×C157—42no—
94C2×C4×C157—21no—
95C2×C4×C157—21no—
96C2×C4×C157—22no—
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157C2×D411yescharacteristic
99C2×C4×C157C2411yescharacteristic
100C8×C157C2×Q814yes—
101C22×C157C22×D411yescharacteristic
102C2×D157—21no—
103C2×D157—21no—
104C2×D157C4⋊Q812yes—
105C4×C157—21no—
106C4×C157—21no—
107C4×C157—42no—
108C4×C157—41no—
109C4×C157—41no—
110C4×C157C22×Q812yes—
111Dic157—21no—
112Dic157—21no—
113Dic157—41no—
114Dic157—41no—
115Dic157—42no—
116Dic157C4⋊Q812yes—
117C2×C157C2×C4⋊Q811yescharacteristic
118C2×C157#57511yescharacteristic
119C2×C157#57511yescharacteristic
120D157—22no—
121C157#316611yescharacteristic, complemented, Sylow
122#3166—1571nomaximal, Sylow
123#533—1571no—
124C2×#125—1571no—
125#501—1571no—
126#501—1571no—
127#501—1571no—
128#501D15711yescharacteristic, complemented
129#499—1572no—
130#499—1572no—
131#428—1572no—
132#428—1572no—
133C4×M16—1571no—
134C2×M16—1572no—
135C4⋊Q8—1571no—
136C4⋊Q8C2×D15711yescharacteristic
137#125—3141no—
138#125—3141no—
139#125—1571no—
140#125—1571no—
141#125—1572no—
142#125—1571no—
143#125C2×D15711yescharacteristic
144C22⋊Q8—3141no—
145C22⋊Q8—3141no—
146C4×Q8—3141no—
147C4×Q8—3141no—
148#117—1572no—
149C2×C4⋊C4—3141no—
150C2×C4⋊C4—3141no—
151C2×C4⋊C4—1571no—
152C2×C4⋊C4—1571no—
153C2×C42—1571no—
154#107—1572no—
155#107—1572no—
156#107—1572no—
157#107C2×D15712yes—
158#106—1572no—
159#106—1572no—
160#106—1572no—
161#106C2×D15712yes—
162#97—1571no—
163#97C2×D15711yescharacteristic
164C4×C8—1572no—
165C2×Q8—21no—
166C2×Q8—3141no—
167C22×C4—3141no—
168C22×C4—3141no—
169C22×C4—1571no—
170C22×C4—1571no—
171C22×C4—1571no—
172M16—1578no—
173C2×C8—1572no—
174C2×C8C22×D15712yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—3141no—
179C4⋊C4—3141no—
180C4⋊C4—3141no—
181C4⋊C4—3141no—
182C4⋊C4—3141no—
183C4⋊C4—3141no—
184C4⋊C4—3142no—
185C4⋊C4—3142no—
186C4⋊C4—3141no—
187C4⋊C4—3141no—
188C4⋊C4—3141no—
189C4⋊C4—1571no—
190C4⋊C4—1571no—
191C4⋊C4—1572no—
192C4⋊C4C22×D15711yescharacteristic
193C4⋊C4C22×D15712yes—
194C4⋊C4C22×D15711yescharacteristic
195C22⋊C4—3142no—
196C42—3142no—
197C42—1571no—
198C42—1571no—
199C42—1571no—
200C42C22×D15711yescharacteristic
201C23—1571no—
202Q8—42no—
203Q8—6282no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—3142no—
208C2×C4—3141no—
209C2×C4—3141no—
210C2×C4—6281no—
211C2×C4—6281no—
212C2×C4—6282no—
213C2×C4—6282no—
214C2×C4—1571no—
215C2×C4—1571no—
216C2×C4—1571no—
217C2×C4—1574no—
218C2×C4D4×D15711yescharacteristic
219C2×C4D4×D15711yescharacteristic
220C2×C4C23×D15711yescharacteristic
221C8—1574no—
222C8Q8×D15714yes—
223C22—3141no—
224C22—3141no—
225C22—1572no—
226C22C2×D4×D15711yescharacteristic
227C4—21no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—41no—
232C4—3141no—
233C4—3141no—
234C4—6281no—
235C4—6281no—
236C4—6282no—
237C4—1572no—
238C4C2×Q8×D15712yes—
239C2—3142no—
240C2C4⋊Q8×D15711yescharacteristic
241C2#2375806911yescharacteristic
242C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#501×C157C2
socleC22×C157C22×D4
prime radicalC22×D157C2×D4
hypercenter#501D157
hypocenterC157#3166
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #23757683C2semidirect
4. #23757682C2semidirect
5. #23757679C2semidirect
6. #23757674C2semidirect
7. #23757673C2semidirect
11. #501×C157C2semidirect
49. #14578937Q8semidirect
121. C157#3166semidirect
128. #501D157semidirect

Maximal subgroups

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

subgroupindex
#237578282
#125×D1572
#237576832
#237576822
#237576792
#237576742
#237576732
#237573142
#237573122
#237572252
#501×C1572
#3166157