Finite Group Databasebeta

← #34110526

Subgroups — #34110526

Order 20096.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#23757821C211yesmaximal, characteristic
2C4⋊Q8×D157C211yesmaximal, characteristic, complemented
3#23757631C214yesmaximal, complemented
4#23757625C211yesmaximal, characteristic, complemented
5#23757624C211yesmaximal, characteristic, complemented
6#23757619C211yesmaximal, characteristic, complemented
7#23757271C214yesmaximal, complemented
8C4×#14578937C211yesmaximal, characteristic, complemented
9#486×C157C211yesmaximal, characteristic, complemented
10C2×#14578953C2212yes—
11C2×Q8×D157—21no—
12C2×#14578937C2212yes—
13#19097771C2211yescharacteristic
14#19097771C2211yescharacteristic
15#19097770—21no—
16#19097770C2212yes—
17#19097769—21no—
18#19097758C2212yes—
19#19097742—21no—
20#19097742—21no—
21D4×Dic157—21no—
22D4×Dic157—21no—
23C4⋊C4×D157—21no—
24C4⋊C4×D157C2212yes—
25C42×D157C2211yescharacteristic
26#19097696C2214yes—
27#19097694C2214yes—
28#19097692C2214yes—
29#19097681C2212yes—
30C4×#11929434C2211yescharacteristic
31C4⋊Q8×C157C2211yescharacteristic
32#127×C157C2211yescharacteristic
33#102×C157C2214yes—
34C4×C8×C157C2211yescharacteristic
35#14578953—28no—
36C2×Dic314—22no—
37C2×Dic314—21no—
38C2×Dic314C2312yes—
39Q8×D157—44no—
40C2×C22⋊D157—22no—
41C22×Dic157—22no—
42C2×C4×D157—22no—
43C2×C4×D157D411yescharacteristic
44C2×C4×D157D411yescharacteristic
45C2×C4×D157C2311yescharacteristic
46#14578937—24no—
47C2×#11929434C2312yes—
48C4⋊Dic157—21no—
49C4⋊Dic157—21no—
50C4⋊Dic157—21no—
51C4⋊Dic157C2312yes—
52#14578933—22no—
53#14578933—22no—
54C22⋊Dic157—22no—
55C22⋊Dic157—22no—
56C4×Dic157D411yescharacteristic
57C4×Dic157D411yescharacteristic
58C4×Dic157C2311yescharacteristic
59C2×Q8×C157—21no—
60C2×D4×C157—21no—
61C2×D4×C157—21no—
62C2×D4×C157C2312yes—
63C2×C8×C157C2312yes—
64C4⋊C4×C157—21no—
65C4⋊C4×C157C2312yes—
66C42×C157C2311yescharacteristic
67C22×D157C2×D411yescharacteristic
68Dic314—24no—
69Dic314—44no—
70Dic314—42no—
71C22⋊D157—44no—
72C2×Dic157—22no—
73C2×Dic157—22no—
74C2×Dic157—44no—
75C2×Dic157C2×D411yescharacteristic
76C2×Dic157C2×D411yescharacteristic
77C2×Dic157C2×D411yescharacteristic
78C4×D157—22no—
79C4×D157—22no—
80C4×D157—44no—
81C4×D157#4114yes—
82#11929434—22no—
83C23×C157—22no—
84Q8×C157—42no—
85D4×C157—24no—
86D4×C157—42no—
87D4×C157—42no—
88C2×C4×C157—22no—
89C2×C4×C157C2×D411yescharacteristic
90C2×C4×C157C2×D411yescharacteristic
91C2×C4×C157C2411yescharacteristic
92C8×C157—22no—
93C22×C157—42no—
94C22×C157—42no—
95C22×C157—42no—
96C22×C157C22×D411yescharacteristic
97C2×D157—21no—
98C2×D157—21no—
99C2×D157#12412yes—
100C4×C157—42no—
101C4×C157AGL(1,ℤ/8)12yes—
102C4×C157#13712yes—
103C4×C157C2×#4112yes—
104Dic157—21no—
105Dic157—21no—
106Dic157—42no—
107Dic157—42no—
108Dic157#12412yes—
109C2×C157—82no—
110C2×C157C2×#12411yescharacteristic
111C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C157C2×#13711yescharacteristic
113D157—22no—
114C157#314111yescharacteristic, complemented, Sylow
115#3141—1571nomaximal, Sylow
116#532—1571no—
117C2×C4⋊Q8—1571no—
118#489—1574no—
119#487—1571no—
120#487—1571no—
121#486—1571no—
122#486D15711yescharacteristic, complemented
123#417—1574no—
124C4×M16—1571no—
125C2×#41—1572no—
126C22×Q8—3141no—
127C2×M16—1572no—
128C4⋊Q8—3141no—
129C4⋊Q8—3141no—
130C4⋊Q8—1572no—
131C4⋊Q8—1571no—
132C4⋊Q8—1571no—
133C4⋊Q8C2×D15711yescharacteristic
134#127C2×D15711yescharacteristic
135#124—1572no—
136#121—3141no—
137#121—3141no—
138C4×D4—3141no—
139C4×D4—3141no—
140C2×C4⋊C4—3141no—
141C2×C4⋊C4—1572no—
142C2×C42—1571no—
143#103—1574no—
144#103—1574no—
145#102—1574no—
146#102C2×D15714yes—
147#97—1572no—
148C4×C8—1571no—
149C4×C8C2×D15711yescharacteristic
150#41—3148no—
151C2×Q8—21no—
152C2×Q8—3142no—
153C2×Q8—3141no—
154C2×Q8—6284no—
155C2×Q8—1572no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—3142no—
159C2×D4C22×D15712yes—
160C22×C4—3142no—
161C22×C4—3142no—
162C22×C4—1571no—
163C22×C4—1571no—
164C22×C4—1571no—
165M16—3144no—
166C2×C8—1572no—
167C2×C8C22×D15712yes—
168C4⋊C4—21no—
169C4⋊C4—3142no—
170C4⋊C4—3142no—
171C4⋊C4—3141no—
172C4⋊C4—3141no—
173C4⋊C4—3141no—
174C4⋊C4—1572no—
175C4⋊C4C22×D15712yes—
176C22⋊C4—3142no—
177C22⋊C4—3142no—
178C42—1571no—
179C42—1571no—
180C42—1571no—
181C42C22×D15711yescharacteristic
182C23—22no—
183C23—1571no—
184Q8—42no—
185Q8—3144no—
186Q8—6284no—
187Q8—6282no—
188D4—24no—
189D4—42no—
190D4—42no—
191D4—6284no—
192C2×C4—22no—
193C2×C4—3142no—
194C2×C4—3142no—
195C2×C4—3142no—
196C2×C4—3142no—
197C2×C4—6284no—
198C2×C4—6284no—
199C2×C4—1571no—
200C2×C4—1571no—
201C2×C4—1571no—
202C2×C4—1574no—
203C2×C4D4×D15711yescharacteristic
204C2×C4D4×D15711yescharacteristic
205C2×C4C23×D15711yescharacteristic
206C8—22no—
207C8—3142no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—3141no—
212C22—3141no—
213C22—1572no—
214C22C2×D4×D15711yescharacteristic
215C4—42no—
216C4—3141no—
217C4—3141no—
218C4—6282no—
219C4—6282no—
220C4—1572no—
221C4#1909779512yes—
222C4#1909780012yes—
223C4#41×D15712yes—
224C2—82no—
225C2—3142no—
226C2#124×D15711yescharacteristic
227C2C2×#1909779511yescharacteristic
228C2C2×#1909780011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#486×C157C2
socleC22×C157C22×D4
prime radical#23757821C2
hypercenter#486D157
hypocenterC157#3141
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D157C2semidirect
3. #23757631C2semidirect
4. #23757625C2semidirect
5. #23757624C2semidirect
6. #23757619C2semidirect
7. #23757271C2semidirect
8. C4×#14578937C2semidirect
9. #486×C157C2semidirect
114. C157#3141semidirect
122. #486D157semidirect

Maximal subgroups

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

subgroupindex
#237578212
C4⋊Q8×D1572
#237576312
#237576252
#237576242
#237576192
#237572712
C4×#145789372
#486×C1572
#3141157