Finite Group Databasebeta

← #34109933

Subgroups — #34109933

Order 20096.

440 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 153 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#23757830C211yesmaximal, characteristic, complemented
2#23757573C214yesmaximal, complemented
3#23757563C212yesmaximal, complemented
4#23757553C212yesmaximal, complemented
5#23757313C212yesmaximal, complemented
6#23757297C211yesmaximal, characteristic, complemented
7#23757278C212yesmaximal
8C2×C4×D4×C157C211yesmaximal, characteristic
9C2×#14578938C2212yes—
10#19097771C2211yescharacteristic, complemented
11#19097771C2211yescharacteristic, complemented
12#19097763C2212yescomplemented
13#19097750—21no—
14#19097750—21no—
15C4×Dic314—21no—
16C4×Dic314—21no—
17#19097730C2212yes—
18#19097704C2214yescomplemented
19#19097702C2214yescomplemented
20C4⋊#11929434C2212yescomplemented
21C4⋊#11929434C2212yescomplemented
22#19097692C2214yes—
23#19097692C2214yes—
24C22×D4×C157C2211yescharacteristic
25C23×C4×C157—21no—
26C4×D4×C157—21no—
27C4×D4×C157—21no—
28C4×D4×C157C2212yes—
29C4×D4×C157C2212yes—
30C2×C4⋊C4×C157C2211yescharacteristic
31C2×C22⋊C4×C157—21no—
32C2×C42×C157C2211yescharacteristic
33C2×Dic314—21no—
34C2×Dic314—21no—
35#14578938—24no—
36C2×#11929434C2314yes—
37C4⋊Dic157—21no—
38C4⋊Dic157—21no—
39C4⋊Dic157C2312yes—
40C4⋊Dic157C2312yes—
41#14578933—22no—
42#14578933—22no—
43C22⋊Dic157—22no—
44C4×Dic157—22no—
45C24×C157—21no—
46C2×D4×C157—21no—
47C2×D4×C157—22no—
48C2×D4×C157Q814yescomplemented
49C2×D4×C157C2312yes—
50C22×C4×C157—21no—
51C22×C4×C157—22no—
52C22×C4×C157—22no—
53C22×C4×C157—44no—
54C22×C4×C157D411yescharacteristic
55C22×C4×C157D411yescharacteristic
56C22×C4×C157C2311yescharacteristic
57C4⋊C4×C157—21no—
58C4⋊C4×C157C2312yes—
59C22⋊C4×C157—22no—
60C22⋊C4×C157—22no—
61C42×C157D411yescharacteristic
62C42×C157D411yescharacteristic
63C42×C157C2311yescharacteristic
64C42×C157C2311yescharacteristic
65Dic314—42no—
66Dic314—42no—
67C2×Dic157—22no—
68C2×Dic157—22no—
69#11929434—24no—
70C23×C157—24no—
71C23×C157—22no—
72C23×C157—42no—
73C23×C157—42no—
74C23×C157C2×D411yescharacteristic
75D4×C157—24no—
76D4×C157—22no—
77D4×C157C2×Q814yes—
78C2×C4×C157—22no—
79C2×C4×C157—22no—
80C2×C4×C157—41no—
81C2×C4×C157—44no—
82C2×C4×C157—42no—
83C2×C4×C157—44no—
84C2×C4×C157C2×D411yescharacteristic
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157C2×D411yescharacteristic
87C2×C4×C157C2×D411yescharacteristic
88C2×C4×C157C2×D411yescharacteristic
89C2×C4×C157C2×Q812yes—
90C2×C4×C157#4112yes—
91C2×C4×C157C2411yescharacteristic
92C22×C157—21no—
93C22×C157—21no—
94C22×C157—24no—
95C22×C157—48no—
96C22×C157—42no—
97C22×C157—42no—
98C22×C157C22⋊Q812yes—
99C22×C157C22×D411yescharacteristic
100C4×C157—21no—
101C4×C157—21no—
102C4×C157—21no—
103C4×C157—42no—
104C4×C157C22⋊Q812yes—
105C4×C157AGL(1,ℤ/8)12yes—
106C4×C157C22×Q811yescharacteristic
107C4×C157C2×#4111yescharacteristic
108Dic157—42no—
109Dic157—42no—
110C2×C157—22no—
111C2×C157—44no—
112C2×C157C2×C22⋊Q811yescharacteristic
113C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
114C2×C157#57511yescharacteristic
115C157#307711yescharacteristic, complemented, Sylow
116#3077—1571nomaximal, Sylow
117#533—1571no—
118C2×C4×D4D15711yescharacteristic
119#478—1574no—
120#476—1572no—
121#474—1572no—
122#428—1572no—
123#423—1571no—
124#418—1572no—
125C22×D4C2×D15711yescharacteristic
126C23×C4—21no—
127C2×M16—1572no—
128C4⋊Q8—1571no—
129C4⋊Q8—1571no—
130#125—1572no—
131C22⋊Q8—3141no—
132C22⋊Q8—3141no—
133C4×Q8—3141no—
134C4×Q8—3141no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4C2×D15712yes—
138C4×D4C2×D15712yes—
139#117—1572no—
140C2×C4⋊C4C2×D15711yescharacteristic
141C2×C22⋊C4—21no—
142C2×C42C2×D15711yescharacteristic
143#107—1574no—
144#106—1574no—
145C4⋊C8—1572no—
146C4⋊C8—1572no—
147#102—1574no—
148#102—1574no—
149C24—21no—
150C2×Q8—3141no—
151C2×Q8—3141no—
152C2×D4—21no—
153C2×D4—22no—
154C2×D4Dic31414yescomplemented
155C2×D4C22×D15712yes—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—44no—
160C22×C4C22⋊D15711yescharacteristic
161C22×C4C22⋊D15711yescharacteristic
162C22×C4C22×D15711yescharacteristic
163M16—3144no—
164C2×C8—1574no—
165C4⋊C4—21no—
166C4⋊C4—3142no—
167C4⋊C4—3142no—
168C4⋊C4—3141no—
169C4⋊C4—3141no—
170C4⋊C4—1572no—
171C4⋊C4—1572no—
172C4⋊C4C22×D15712yes—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—3142no—
176C42—3142no—
177C42C22⋊D15711yescharacteristic
178C42C22⋊D15711yescharacteristic
179C42C22×D15711yescharacteristic
180C42C22×D15711yescharacteristic
181C23—24no—
182C23—22no—
183C23—42no—
184C23—42no—
185C23C2×C22⋊D15711yescharacteristic
186Q8—6282no—
187Q8—6282no—
188D4—24no—
189D4—22no—
190D4C2×Dic31414yes—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—3142no—
198C2×C4—3142no—
199C2×C4C2×C22⋊D15711yescharacteristic
200C2×C4C2×C22⋊D15711yescharacteristic
201C2×C4C2×C22⋊D15711yescharacteristic
202C2×C4C2×C22⋊D15711yescharacteristic
203C2×C4C2×C22⋊D15711yescharacteristic
204C2×C4C2×Dic31412yes—
205C2×C4#1457895412yes—
206C2×C4C23×D15711yescharacteristic
207C8—3144no—
208C22—21no—
209C22—21no—
210C22—24no—
211C22—48no—
212C22—42no—
213C22—42no—
214C22#1909775012yes—
215C22C22×C22⋊D15711yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—6282no—
221C4—6282no—
222C4#1909775012yes—
223C4#1909779712yes—
224C4C22×Dic31411yescharacteristic
225C4C2×#1457895411yescharacteristic
226C2—22no—
227C2—44no—
228C2C2×#1909775011yescharacteristic
229C2C2×#1909779711yescharacteristic
230C2#2375807111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×C4×D4×C157C2
socleC22×C157C22×D4
prime radicalC22×D4×C157C22
hypercenterC2×C4×D4D157
hypocenterC157#3077
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #23757830C2semidirect
2. #23757573C2semidirect
3. #23757563C2semidirect
4. #23757553C2semidirect
5. #23757313C2semidirect
6. #23757297C2semidirect
10. #19097771C22semidirect
11. #19097771C22semidirect
12. #19097763C22semidirect
18. #19097704C22semidirect
19. #19097702C22semidirect
20. C4⋊#11929434C22semidirect
21. C4⋊#11929434C22semidirect
48. C2×D4×C157Q8semidirect
115. C157#3077semidirect
154. C2×D4Dic314semidirect

Maximal subgroups

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

subgroupindex
#237578302
#237575732
#237575632
#237575532
#237573132
#237572972
#237572782
C2×C4×D4×C1572
#3077157