Finite Group Databasebeta

← #34109666

Subgroups — #34109666

Order 20096.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#19097802C211yesmaximal, characteristic
2C22×#14578944C211yesmaximal, characteristic, complemented
3#23757849C211yesmaximal, characteristic
4#23757495C214yesmaximal, complemented
5#23757483C214yesmaximal, complemented
6#23757300C211yesmaximal, characteristic, complemented
7#23757281C211yesmaximal, characteristic, complemented
8#23757267C211yesmaximal, characteristic
9#519×C157C211yesmaximal, characteristic, complemented
10C2×#14578954C2211yescharacteristic
11C22×Dic314C2211yescharacteristic
12#19097802—24no—
13C2×#14578944—24no—
14C2×#14578944C2212yes—
15C2×#14578944C2214yes—
16C2×#14578941C2212yes—
17C2×#14578938C2211yescharacteristic
18C22×#11929434C2211yescharacteristic
19#19097771C2212yes—
20#19097759C2212yes—
21#19097755—22no—
22#19097751—22no—
23C4⋊#11929434C2214yes—
24#19097694C2212yes—
25#19097694C2214yes—
26#19097691C2212yes—
27C22×Q8×C157C2211yescharacteristic
28C4×Q8×C157—22no—
29C4×Q8×C157C2214yes—
30#117×C157—22no—
31#117×C157C2211yescharacteristic
32#117×C157C2211yescharacteristic
33#14578954—22no—
34#14578954—22no—
35C2×Dic314—21no—
36C2×Dic314—22no—
37C2×Dic314—22no—
38C2×Dic314C2312yes—
39C2×Dic314C2311yescharacteristic
40C2×D628C2311yescharacteristic
41C2×C22⋊D157—21no—
42C22×Dic157—21no—
43C2×C4×D157—21no—
44#14578944—28no—
45#14578944—24no—
46#14578941—24no—
47#14578938—22no—
48C2×#11929434—22no—
49C2×#11929434C2312yes—
50C2×#11929434C2312yes—
51C4⋊Dic157—24no—
52#14578931—24no—
53C2×Q8×C157—21no—
54C2×Q8×C157—22no—
55C2×Q8×C157D414yes—
56C2×Q8×C157C2312yes—
57C22×C4×C157—21no—
58C22×C4×C157C2311yescharacteristic
59C4⋊C4×C157—24no—
60C4⋊C4×C157D412yes—
61C4⋊C4×C157C2312yes—
62C22⋊C4×C157—21no—
63C22⋊C4×C157D412yes—
64C42×C157—24no—
65C42×C157—21no—
66C42×C157C2312yes—
67C22×D157—21no—
68Dic314—22no—
69Dic314—21no—
70Dic314—22no—
71Dic314—21no—
72Dic314—42no—
73D628—21no—
74D628—21no—
75C22⋊D157—42no—
76C2×Dic157—21no—
77C2×Dic157—22no—
78C2×Dic157—42no—
79C4×D157—42no—
80#11929434—22no—
81#11929434—22no—
82C23×C157C2×D411yescharacteristic
83Q8×C157—24no—
84Q8×C157—22no—
85Q8×C157C2×D414yes—
86C2×C4×C157—24no—
87C2×C4×C157—22no—
88C2×C4×C157—22no—
89C2×C4×C157C2×D414yes—
90C2×C4×C157C2×D411yescharacteristic
91C2×C4×C157C2×D412yes—
92C2×C4×C157#4112yes—
93C2×C4×C157C2411yescharacteristic
94C22×C157—22no—
95C22×C157#12112yes—
96C22×C157C22×D411yescharacteristic
97C2×D157—41no—
98C2×D157—42no—
99C4×C157—24no—
100C4×C157—24no—
101C4×C157—42no—
102C4×C157#12112yes—
103C4×C157C22×D411yescharacteristic
104C4×C157C2×#4111yescharacteristic
105Dic157—41no—
106Dic157—42no—
107C2×C157—22no—
108C2×C157C2×#12111yescharacteristic
109C2×C157#57812yes—
110D157—81no—
111C157#304411yescharacteristic, complemented, Sylow
112#3044—1571nomaximal, Sylow
113C2×#137—1571no—
114C22×Q16—1571no—
115#536—1571no—
116#519D15711yescharacteristic, complemented
117#464—1574no—
118#462—1574no—
119#424—1571no—
120#419—1571no—
121#416—1571no—
122C2×#41—1571no—
123C22×Q8—1571no—
124C22×Q8C2×D15711yescharacteristic
125#137—3144no—
126C2×Q16—3144no—
127C2×Q16—1574no—
128C2×Q16—1572no—
129C2×SD16—1572no—
130C2×M16—1571no—
131C22×C8—1571no—
132C4⋊Q8—1572no—
133#124—1572no—
134#123—3142no—
135C22⋊Q8—3142no—
136C4×Q8—22no—
137C4×Q8C2×D15714yes—
138#117—22no—
139#117C2×D15711yescharacteristic
140#117C2×D15711yescharacteristic
141C4⋊C8—1574no—
142#103—1572no—
143#103—1574no—
144#102—1572no—
145#41—3142no—
146#41—3142no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—3141no—
150C2×Q8—3142no—
151C2×Q8—3142no—
152C2×Q8—1572no—
153C2×Q8—1571no—
154C2×Q8D62814yes—
155C2×Q8C22×D15712yes—
156C2×D4—3141no—
157C2×D4—1571no—
158C22×C4—21no—
159C22×C4—3141no—
160C22×C4—3141no—
161C22×C4C22×D15711yescharacteristic
162Q16—3148no—
163Q16—3144no—
164SD16—3144no—
165M16—3142no—
166C2×C8—3142no—
167C2×C8—1572no—
168C2×C8—1572no—
169C4⋊C4—24no—
170C4⋊C4—3144no—
171C4⋊C4C22⋊D15712yes—
172C4⋊C4C22×D15712yes—
173C22⋊C4—21no—
174C22⋊C4—3144no—
175C22⋊C4C22⋊D15712yes—
176C42—24no—
177C42—21no—
178C42C22×D15712yes—
179C23—3141no—
180C23C2×C22⋊D15711yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—3142no—
184Q8—3141no—
185Q8—3142no—
186Q8—3141no—
187Q8—6282no—
188Q8C2×D62814yes—
189D4—3141no—
190D4—3141no—
191D4—6282no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—3142no—
196C2×C4—3141no—
197C2×C4—6282no—
198C2×C4—6282no—
199C2×C4C2×C22⋊D15714yes—
200C2×C4C2×C22⋊D15711yescharacteristic
201C2×C4C2×D62812yes—
202C2×C4#1457895412yes—
203C2×C4C23×D15711yescharacteristic
204C8—3142no—
205C8—3142no—
206C22—22no—
207C22—6282no—
208C22—6281no—
209C22#1909774512yes—
210C22C22×C22⋊D15711yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—6281no—
215C4—6282no—
216C4#1909774512yes—
217C4C22×D62811yescharacteristic
218C4C2×#1457895411yescharacteristic
219C2—22no—
220C2—12561no—
221C2C2×#1909774511yescharacteristic
222C2#2375808812yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#519×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578954C22
hypercenter#519D157
hypocenterC157#3044
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. C22×#14578944C2semidirect
4. #23757495C2semidirect
5. #23757483C2semidirect
6. #23757300C2semidirect
7. #23757281C2semidirect
9. #519×C157C2semidirect
111. C157#3044semidirect
116. #519D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978022
C22×#145789442
#237578492
#237574952
#237574832
#237573002
#237572812
#237572672
#519×C1572
#3044157