Finite Group Databasebeta

← #34101078

Subgroups — #34101078

Order 20096.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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#23757826C211yesmaximal, characteristic
2M16×Dic157C211yesmaximal, characteristic
3C4×#14578933C211yesmaximal, characteristic
4#23757034C211yesmaximal, characteristic
5#23757033C211yesmaximal, characteristic
6#23757032C211yesmaximal, characteristic
7#339×C157C211yesmaximal, characteristic
8C2×#14578938C2211yescharacteristic
9#19097770C411yescharacteristic, complemented
10#19097770C411yescharacteristic, complemented
11#19097762C411yescharacteristic, complemented
12#19097762C411yescharacteristic, complemented
13#19097748—21no—
14#19097748—21no—
15C4×Dic314—21no—
16C4×Dic314—21no—
17#19097730C2211yescharacteristic
18C2×#14578933—21no—
19C2×C4×Dic157C2211yescharacteristic
20C2×C4×Dic157C2211yescharacteristic
21#19097681—21no—
22C8×Dic157—21no—
23#19097678—21no—
24C2×M16×C157C2211yescharacteristic
25#117×C157C2211yescharacteristic
26C2×C42×C157C2211yescharacteristic
27C2×Dic314—21no—
28C2×Dic314—21no—
29C22×Dic157—21no—
30C22×Dic157D411yescharacteristic
31C22×Dic157D411yescharacteristic
32#14578938—21no—
33#14578938Q812yescomplemented
34C2×#11929434—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157—21no—
37C4⋊Dic157C2×C411yescharacteristic
38C4⋊Dic157C2×C411yescharacteristic
39#14578933—24no—
40#14578933—21no—
41#14578933—21no—
42#14578933—21no—
43#14578933—21no—
44C22⋊Dic157—21no—
45C4×Dic157—21no—
46C4×Dic157—21no—
47C4×Dic157—21no—
48C4×Dic157C2×C411yescharacteristic
49C4×Dic157C2×C411yescharacteristic
50C4×Dic157D411yescharacteristic
51C4×Dic157D411yescharacteristic
52C22×C4×C157—21no—
53C22×C4×C157C2311yescharacteristic
54M16×C157—21no—
55M16×C157Q812yescomplemented
56C2×C8×C157—21no—
57C4⋊C4×C157C2×C411yescharacteristic
58C4⋊C4×C157C2×C411yescharacteristic
59C22⋊C4×C157—21no—
60C42×C157—21no—
61C42×C157—21no—
62C42×C157—21no—
63Dic314—42no—
64Dic314—42no—
65C2×Dic157—22no—
66C2×Dic157—22no—
67C2×Dic157—21no—
68C2×Dic157—21no—
69C2×Dic157—41no—
70C2×Dic157—41no—
71C2×Dic157—41no—
72C2×Dic157C22⋊C411yescharacteristic
73C2×Dic157C22⋊C411yescharacteristic
74C2×Dic157C22⋊C411yescharacteristic
75C2×Dic157C22⋊C411yescharacteristic
76#11929434—22no—
77C23×C157C2×D411yescharacteristic
78C2×C4×C157—22no—
79C2×C4×C157—21no—
80C2×C4×C157—21no—
81C2×C4×C157—41no—
82C2×C4×C157—41no—
83C2×C4×C157C22×C411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C2×C4×C157C2×Q811yescharacteristic
86C2×C4×C157C2×Q811yescharacteristic
87C2×C4×C157#4111yescharacteristic
88C2×C4×C157#4111yescharacteristic
89C8×C157—22no—
90C22×C157—21no—
91C22×C157—21no—
92C22×C157C2×C22⋊C411yescharacteristic
93C22×C157#12411yescharacteristic
94C22×C157C4⋊Q811yescharacteristic
95C4×C157—42no—
96C4×C157—41no—
97C4×C157—41no—
98C4×C157C4×Q811yescharacteristic
99C4×C157C4×Q811yescharacteristic
100C4×C157C22⋊Q811yescharacteristic
101C4×C157C22⋊Q811yescharacteristic
102Dic157—21no—
103Dic157—21no—
104Dic157—21no—
105Dic157—42no—
106Dic157—41no—
107Dic157—41no—
108Dic157#10412yes—
109C2×C157—21no—
110C2×C157—21no—
111C2×C157#39111yescharacteristic
112C2×C157C2×#10411yescharacteristic
113C2×C157#42111yescharacteristic
114C157#198611yescharacteristic, complemented, Sylow
115#1986—1571nomaximal, Sylow
116#533—1571no—
117C4×M16—1571no—
118C4×C4⋊C4—1571no—
119#339—1571no—
120#339—1571no—
121#339—1571no—
122#339D15711yescharacteristic
123C2×M16—1571no—
124C2×M16C2×D15711yescharacteristic
125C4⋊Q8—1571no—
126C4⋊Q8—1571no—
127#125—1571no—
128#125—1571no—
129C22⋊Q8—3141no—
130C22⋊Q8—3141no—
131C4×Q8—3141no—
132C4×Q8—3141no—
133#117—1571no—
134#117C2×D15711yescharacteristic
135C2×C4⋊C4—3141no—
136C2×C42—1571no—
137C2×C42—1571no—
138C2×C42C2×D15711yescharacteristic
139#97—3141no—
140C4×C8—3141no—
141#95—3141no—
142C2×Q8—3141no—
143C2×Q8—3141no—
144C22×C4—21no—
145C22×C4—3141no—
146C22×C4—1571no—
147C22×C4—1571no—
148C22×C4C22×D15711yescharacteristic
149M16—21no—
150M16—3141no—
151M16—1572no—
152M16Dic31412yescomplemented
153C2×C8—21no—
154C2×C8—3141no—
155C4⋊C4—3144no—
156C4⋊C4—3141no—
157C4⋊C4—3141no—
158C4⋊C4—3141no—
159C4⋊C4—3141no—
160C4⋊C4—3141no—
161C4⋊C4—3141no—
162C4⋊C4—1571no—
163C4⋊C4—1571no—
164C4⋊C4C4×D15711yescharacteristic
165C4⋊C4C4×D15711yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—3141no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—3141no—
172C42—3141no—
173C42—3141no—
174C42—1571no—
175C42—1571no—
176C42—1571no—
177C42—1571no—
178C23D4×D15711yescharacteristic
179Q8—6282no—
180Q8—6282no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3142no—
187C2×C4—3142no—
188C2×C4—3141no—
189C2×C4—3141no—
190C2×C4—6281no—
191C2×C4—6281no—
192C2×C4—6281no—
193C2×C4—1571no—
194C2×C4—1571no—
195C2×C4—1571no—
196C2×C4—1571no—
197C2×C4C2×C4×D15711yescharacteristic
198C2×C4D4×D15711yescharacteristic
199C2×C4Q8×D15711yescharacteristic
200C2×C4C2×Dic31411yescharacteristic
201C2×C4#1457895311yescharacteristic
202C2×C4#1457895411yescharacteristic
203C8—22no—
204C8—3142no—
205C22—21no—
206C22—21no—
207C22C22⋊C4×D15711yescharacteristic
208C22#1909775711yescharacteristic
209C22#1909777011yescharacteristic
210C4—42no—
211C4—41no—
212C4—41no—
213C4—3141no—
214C4—3141no—
215C4—3141no—
216C4—6282no—
217C4—6281no—
218C4—6281no—
219C4—1572no—
220C4#1909773611yescharacteristic
221C4C4×Dic31411yescharacteristic
222C4#1909774811yescharacteristic
223C4#1909774911yescharacteristic
224C2—21no—
225C2—21no—
226C2#2375717711yescharacteristic
227C2#104×D15711yescharacteristic
228C2#2375728811yescharacteristic
subgroupG/H
centerC2×C4D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#339×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC23×C157C2×D4
cosocleC22×C4×C157C23
hypercenter#339D157
hypocenterC157#1986
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
9. #19097770C4semidirect
10. #19097770C4semidirect
11. #19097762C4semidirect
12. #19097762C4semidirect
33. #14578938Q8semidirect
55. M16×C157Q8semidirect
114. C157#1986semidirect
152. M16Dic314semidirect

Maximal subgroups

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

subgroupindex
#237578262
M16×Dic1572
C4×#145789332
#237570342
#237570332
#237570322
#339×C1572
#1986157