Finite Group Databasebeta

← #34110531

Subgroups — #34110531

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#23757824C211yesmaximal, characteristic
2C4⋊Q8×D157C211yesmaximal, characteristic, complemented
3#23757634C214yesmaximal, complemented
4#23757623C211yesmaximal, characteristic, complemented
5#23757621C211yesmaximal, characteristic, complemented
6#23757620C211yesmaximal, characteristic, complemented
7#23757273C214yesmaximal, complemented
8C4×#14578937C211yesmaximal, characteristic, complemented
9#487×C157C211yesmaximal, characteristic, complemented
10C2×#14578955C2212yes—
11C2×Q8×D157—21no—
12C2×#14578937C2212yes—
13#19097771C2211yescharacteristic
14#19097770—21no—
15#19097770C2212yes—
16#19097769—21no—
17#19097768C2211yescharacteristic
18#19097760C2212yes—
19#19097746—21no—
20#19097746—21no—
21#19097733—21no—
22#19097733—21no—
23C4⋊C4×D157—21no—
24C4⋊C4×D157C2212yes—
25C42×D157C2211yescharacteristic
26#19097695C2214yes—
27#19097693C2214yes—
28#19097691C2214yes—
29#19097681C2212yes—
30C4×#11929434C2211yescharacteristic
31C4⋊Q8×C157C2211yescharacteristic
32C4⋊Q8×C157C2211yescharacteristic
33#103×C157C2214yes—
34C4×C8×C157C2211yescharacteristic
35#14578955—28no—
36C2×Dic314—22no—
37C2×Dic314—21no—
38Q8×D157—44no—
39C2×D628—22no—
40C2×D628—21no—
41C2×D628—21no—
42C2×D628C2312yes—
43C2×C4×D157—22no—
44C2×C4×D157—22no—
45C2×C4×D157D411yescharacteristic
46C2×C4×D157D411yescharacteristic
47C2×C4×D157C2311yescharacteristic
48#14578937—24no—
49C2×#11929434C2312yes—
50C4⋊Dic157—21no—
51C4⋊Dic157C2312yes—
52#14578933—22no—
53#14578933—22no—
54#14578931—22no—
55#14578931—22no—
56C4×Dic157D411yescharacteristic
57C4×Dic157D411yescharacteristic
58C4×Dic157C2311yescharacteristic
59C2×Q8×C157—21no—
60C2×Q8×C157C2312yes—
61C2×C8×C157C2312yes—
62C4⋊C4×C157—21no—
63C4⋊C4×C157—21no—
64C4⋊C4×C157—21no—
65C4⋊C4×C157C2312yes—
66C42×C157C2311yescharacteristic
67C22×D157—22no—
68C22×D157C2×D411yescharacteristic
69Dic314—44no—
70Dic314—42no—
71D628—24no—
72D628—44no—
73D628—42no—
74D628—42no—
75C2×Dic157—22no—
76C2×Dic157C2×D411yescharacteristic
77C2×Dic157C2×D411yescharacteristic
78C2×Dic157C2×D411yescharacteristic
79C4×D157—22no—
80C4×D157—22no—
81C4×D157—44no—
82C4×D157—44no—
83C4×D157#4114yes—
84#11929434—22no—
85Q8×C157—24no—
86Q8×C157—42no—
87C2×C4×C157—22no—
88C2×C4×C157—22no—
89C2×C4×C157C2×D411yescharacteristic
90C2×C4×C157C2×D411yescharacteristic
91C2×C4×C157C2411yescharacteristic
92C8×C157—22no—
93C22×C157C22×D411yescharacteristic
94C2×D157—21no—
95C2×D157—21no—
96C2×D157—42no—
97C2×D157—42no—
98C2×D157—42no—
99C2×D157#12412yes—
100C4×C157—42no—
101C4×C157—42no—
102C4×C157AGL(1,ℤ/8)12yes—
103C4×C157#13712yes—
104C4×C157C2×#4112yes—
105Dic157—21no—
106Dic157—21no—
107Dic157—42no—
108Dic157#12412yes—
109C2×C157C2×#12411yescharacteristic
110C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C157C2×#13711yescharacteristic
112D157—22no—
113D157—82no—
114C157#314111yescharacteristic, complemented, Sylow
115#3141—1571nomaximal, Sylow
116#532—1571no—
117C2×C4⋊Q8—1571no—
118#489—1574no—
119#487—1571no—
120#487D15711yescharacteristic, complemented
121#486—1571no—
122#486—1571no—
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⋊Q8C2×D15711yescharacteristic
133C4⋊Q8C2×D15711yescharacteristic
134#127—1571no—
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#103C2×D15714yes—
145#102—1574no—
146#102—1574no—
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×Q8C22×D15712yes—
156C2×D4—3142no—
157C2×D4—3141no—
158C2×D4—3141no—
159C2×D4—1572no—
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—21no—
170C4⋊C4—21no—
171C4⋊C4—3142no—
172C4⋊C4—3142no—
173C4⋊C4—3141no—
174C4⋊C4—1572no—
175C4⋊C4C22×D15712yes—
176C22⋊C4—3142no—
177C22⋊C4—3142no—
178C42—1571no—
179C42—1571no—
180C42—1571no—
181C42C22×D15711yescharacteristic
182C23—3142no—
183C23—1571no—
184Q8—24no—
185Q8—42no—
186Q8—6284no—
187Q8—6282no—
188D4—3144no—
189D4—6284no—
190D4—6282no—
191D4—6282no—
192C2×C4—22no—
193C2×C4—22no—
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—3141no—
209C22—3141no—
210C22—6282no—
211C22—6282no—
212C22—6282no—
213C22—1572no—
214C22C2×D4×D15711yescharacteristic
215C4—42no—
216C4—42no—
217C4—3141no—
218C4—3141no—
219C4—6282no—
220C4—1572no—
221C4#1909779812yes—
222C4#1909780312yes—
223C4#41×D15712yes—
224C2—3142no—
225C2—12562no—
226C2#124×D15711yescharacteristic
227C2C2×#1909779811yescharacteristic
228C2C2×#1909780311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#487×C157C2
socleC22×C157C22×D4
prime radical#23757824C2
hypercenter#487D157
hypocenterC157#3141
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D157C2semidirect
3. #23757634C2semidirect
4. #23757623C2semidirect
5. #23757621C2semidirect
6. #23757620C2semidirect
7. #23757273C2semidirect
8. C4×#14578937C2semidirect
9. #487×C157C2semidirect
114. C157#3141semidirect
120. #487D157semidirect

Maximal subgroups

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

subgroupindex
#237578242
C4⋊Q8×D1572
#237576342
#237576232
#237576212
#237576202
#237572732
C4×#145789372
#487×C1572
#3141157