Finite Group Databasebeta

← #34098574

Subgroups — #34098574

Order 20096.

266 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×#14578955C211yesmaximal, characteristic
2#23757295C211yesmaximal, characteristic, complemented
3C4×#14578937C211yesmaximal, characteristic, complemented
4Q8⋊#11929434C211yesmaximal, characteristic, complemented
5#23757002C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic, complemented
7Q8⋊C8×C157C211yesmaximal, characteristic, complemented
8C2×#14578955C411yescharacteristic
9C2×#14578937—21no—
10C2×#14578937—21no—
11Q8×Dic157C411yescharacteristic
12C4×D628—21no—
13C4×D628C2211yescharacteristic
14#19097733—21no—
15#19097733C411yescharacteristic
16#19097729—21no—
17#19097729C411yescharacteristic
18C42×D157—21no—
19C42×D157C2211yescharacteristic
20C4⋊#11929434C2211yescharacteristic
21#19097700—21no—
22#19097681—21no—
23C4×#11929434C2211yescharacteristic
24C4×Q8×C157C2211yescharacteristic
25C4⋊C8×C157C2211yescharacteristic
26C4×C8×C157C2211yescharacteristic
27#14578955—22no—
28#14578955—22no—
29C2×D628—21no—
30C2×D628C2×C411yescharacteristic
31C2×C4×D157—21no—
32C2×C4×D157—21no—
33C2×C4×D157—21no—
34C2×C4×D157C2×C411yescharacteristic
35C2×C4×D157D411yescharacteristic
36C2×C4×D157D411yescharacteristic
37#14578937—24no—
38#14578937—42no—
39C2×#11929434—21no—
40C2×#11929434—21no—
41C4⋊Dic157—21no—
42C4⋊Dic157C2×C411yescharacteristic
43#14578931—21no—
44#14578931—21no—
45#14578931—21no—
46C4×Dic157—21no—
47C4×Dic157—21no—
48C4×Dic157C2×C411yescharacteristic
49C4×Dic157D411yescharacteristic
50C4×Dic157D411yescharacteristic
51C2×Q8×C157C2×C411yescharacteristic
52C2×C8×C157—21no—
53C2×C8×C157—21no—
54C4⋊C4×C157—21no—
55C4⋊C4×C157C2×C411yescharacteristic
56C42×C157—21no—
57C42×C157C2311yescharacteristic
58C22×D157—21no—
59C22×D157C22⋊C411yescharacteristic
60D628—24no—
61D628—21no—
62D628M1612yescomplemented
63C2×Dic157—21no—
64C2×Dic157C22⋊C411yescharacteristic
65C2×Dic157C22⋊C411yescharacteristic
66C2×Dic157C22⋊C411yescharacteristic
67C4×D157—24no—
68C4×D157—21no—
69C4×D157—21no—
70C4×D157—21no—
71C4×D157—21no—
72C4×D157—22no—
73C4×D157—41no—
74C4×D157—41no—
75C4×D157—42no—
76#11929434—22no—
77#11929434—41no—
78Q8×C157—21no—
79Q8×C157M1612yescomplemented
80C2×C4×C157—21no—
81C2×C4×C157—21no—
82C2×C4×C157C22×C411yescharacteristic
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C8×C157—22no—
86C8×C157—41no—
87C22×C157C2×C22⋊C411yescharacteristic
88C2×D157—21no—
89C2×D157—22no—
90C2×D157—21no—
91C2×D157—21no—
92C2×D157—41no—
93C2×D157—41no—
94C4×C157—22no—
95C4×C157—21no—
96C4×C157—41no—
97C4×C157C2×M1611yescharacteristic
98C4×C157C2×M1611yescharacteristic
99C4×C157AGL(1,ℤ/8)11yescharacteristic
100C4×C157#13711yescharacteristic
101Dic157—21no—
102Dic157—21no—
103Dic157—41no—
104Dic157#10412yes—
105C2×C157#40711yescharacteristic
106C2×C157#41711yescharacteristic
107C2×C157C2×#10411yescharacteristic
108D157—41no—
109D157—42no—
110C157#148911yescharacteristic, complemented, Sylow
111#1489—1571nomaximal, Sylow
112C4×#41—1571no—
113#423—1571no—
114C4×M16—1571no—
115Q8⋊C8—1571no—
116Q8⋊C8D15711yescharacteristic, complemented
117D4⋊C8—1571no—
118D4⋊C8—1571no—
119C2×#41—1571no—
120C2×M16—3141no—
121C2×M16—3141no—
122C4×Q8—1571no—
123C4×Q8C2×D15711yescharacteristic
124C4×D4—3141no—
125C4×D4—3141no—
126C4×D4—1571no—
127C4×D4—1571no—
128#117—3141no—
129#117—1571no—
130C2×C42—3141no—
131C2×C42—1571no—
132C4⋊C8—3141no—
133C4⋊C8—1571no—
134C4⋊C8C2×D15711yescharacteristic
135#97—3141no—
136C4×C8—1571no—
137C4×C8C2×D15711yescharacteristic
138#41—3142no—
139#41—3142no—
140C2×Q8C4×D15711yescharacteristic
141C2×D4—3141no—
142C2×D4—1571no—
143C22×C4—3141no—
144C22×C4—3141no—
145C22×C4—3141no—
146C22×C4—1571no—
147C22×C4—1571no—
148C22×C4—1571no—
149M16—3144no—
150M16—6282no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—3141no—
154C2×C8—3141no—
155C4⋊C4—21no—
156C4⋊C4—3141no—
157C4⋊C4—1571no—
158C4⋊C4C4×D15711yescharacteristic
159C22⋊C4—3141no—
160C22⋊C4—3141no—
161C22⋊C4—3141no—
162C42—21no—
163C42—3141no—
164C42—3141no—
165C42—1571no—
166C42—1571no—
167C42—1571no—
168C42C22×D15711yescharacteristic
169C23—3141no—
170C23—1571no—
171Q8—21no—
172Q8#1457893712yescomplemented
173D4—3144no—
174D4—3141no—
175D4—1572no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—3144no—
179C2×C4—3142no—
180C2×C4—3141no—
181C2×C4—3141no—
182C2×C4—3141no—
183C2×C4—3141no—
184C2×C4—3141no—
185C2×C4—6282no—
186C2×C4—6281no—
187C2×C4—6281no—
188C2×C4—1571no—
189C2×C4—1571no—
190C2×C4—1571no—
191C2×C4C2×C4×D15711yescharacteristic
192C2×C4D4×D15711yescharacteristic
193C2×C4D4×D15711yescharacteristic
194C8—22no—
195C8—41no—
196C8—3142no—
197C8—6281no—
198C22—3141no—
199C22—3142no—
200C22—3141no—
201C22—3141no—
202C22—6281no—
203C22—6281no—
204C22C22⋊C4×D15711yescharacteristic
205C4—22no—
206C4—21no—
207C4—41no—
208C4—3141no—
209C4—3141no—
210C4—6281no—
211C4—1572no—
212C4M16×D15711yescharacteristic
213C4C2×#1457893711yescharacteristic
214C4#1909779811yescharacteristic
215C4#1909780311yescharacteristic
216C2—6281no—
217C2—6282no—
218C2#2375723511yescharacteristic
219C2#2375727311yescharacteristic
220C2#104×D15711yescharacteristic
subgroupG/H
centerC2×C4D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC42C22×D157
FittingQ8⋊C8×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC2×#14578955C4
cosocleC42×C157C23
hypercenterQ8⋊C8D157
hypocenterC157#1489
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #23757295C2semidirect
3. C4×#14578937C2semidirect
4. Q8⋊#11929434C2semidirect
5. #23757002C2semidirect
6. #23757000C2semidirect
7. Q8⋊C8×C157C2semidirect
62. D628M16semidirect
79. Q8×C157M16semidirect
110. C157#1489semidirect
116. Q8⋊C8D157semidirect
172. Q8#14578937semidirect

Maximal subgroups

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

subgroupindex
C4×#145789552
#237572952
C4×#145789372
Q8⋊#119294342
#237570022
#237570002
Q8⋊C8×C1572
#1489157