Finite Group Databasebeta

← #34109668

Subgroups — #34109668

Order 20096.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14578944C211yesmaximal, characteristic, complemented
2#23757844C211yesmaximal, characteristic
3#23757751C211yesmaximal, characteristic, complemented
4#23757493C214yesmaximal, complemented
5#23757481C214yesmaximal, complemented
6#23757301C211yesmaximal, characteristic, complemented
7#23757283C211yesmaximal, characteristic, complemented
8#23757269C211yesmaximal, characteristic
9C2×#137×C157C211yesmaximal, characteristic
10C22×Dic314C2211yescharacteristic
11C2×#14578944—24no—
12C2×#14578944C2214yes—
13C22×#11929434C2211yescharacteristic
14#19097769C2212yes—
15#19097758C2212yes—
16#19097752—22no—
17#19097750—22no—
18Q8×Dic157—22no—
19Q8×Dic157C2214yes—
20#19097730—22no—
21#19097730C2211yescharacteristic
22#19097730C2211yescharacteristic
23#19097700C2214yes—
24#19097696C2214yes—
25#19097695C2212yes—
26#19097692C2212yes—
27C2×#41×C157C2211yescharacteristic
28C22×Q8×C157C2211yescharacteristic
29#137×C157—24no—
30C2×Q16×C157C2212yes—
31C2×SD16×C157C2212yes—
32C2×M16×C157C2211yescharacteristic
33C2×Dic314—21no—
34C2×Dic314—22no—
35C2×Dic314—22no—
36C2×Dic314C2312yes—
37C22×Dic157—21no—
38#14578944—28no—
39C2×#11929434—22no—
40C2×#11929434C2312yes—
41C4⋊Dic157—24no—
42C4⋊Dic157D412yes—
43C4⋊Dic157C2312yes—
44#14578933—24no—
45C22⋊Dic157—24no—
46C22⋊Dic157—21no—
47C22⋊Dic157D412yes—
48C4×Dic157—24no—
49C4×Dic157—21no—
50C4×Dic157C2312yes—
51#41×C157—22no—
52#41×C157—22no—
53C2×Q8×C157—21no—
54C2×Q8×C157—22no—
55C2×Q8×C157D414yes—
56C2×Q8×C157C2311yescharacteristic
57C2×Q8×C157C2312yes—
58C2×D4×C157—21no—
59C2×D4×C157C2311yescharacteristic
60C22×C4×C157—21no—
61C22×C4×C157—21no—
62C22×C4×C157C2311yescharacteristic
63Q16×C157—24no—
64SD16×C157—24no—
65M16×C157—22no—
66C2×C8×C157C2312yes—
67Dic314—22no—
68Dic314—22no—
69Dic314—42no—
70C2×Dic157—22no—
71C2×Dic157—22no—
72C2×Dic157—42no—
73C2×Dic157C2×D414yes—
74#11929434—22no—
75C23×C157—21no—
76C23×C157C2×D411yescharacteristic
77Q8×C157—24no—
78Q8×C157—22no—
79Q8×C157—21no—
80Q8×C157—21no—
81Q8×C157C2×D414yes—
82D4×C157—21no—
83D4×C157—21no—
84D4×C157—42no—
85C2×C4×C157—24no—
86C2×C4×C157—22no—
87C2×C4×C157—21no—
88C2×C4×C157—42no—
89C2×C4×C157C2×D411yescharacteristic
90C2×C4×C157C2×D412yes—
91C2×C4×C157#4112yes—
92C2×C4×C157C2411yescharacteristic
93C8×C157—22no—
94C22×C157—22no—
95C22×C157—41no—
96C22×C157—42no—
97C22×C157#12112yes—
98C22×C157C22×D411yescharacteristic
99C4×C157—24no—
100C4×C157—41no—
101C4×C157#12112yes—
102C4×C157C22×D411yescharacteristic
103C4×C157C2×#4111yescharacteristic
104Dic157—24no—
105Dic157—42no—
106Dic157—42no—
107C2×C157—22no—
108C2×C157—81no—
109C2×C157C2×#12111yescharacteristic
110C2×C157#57812yes—
111C157#304411yescharacteristic, complemented, Sylow
112#3044—1571nomaximal, Sylow
113C2×#137D15711yescharacteristic
114C22×Q16—1571no—
115#536—1571no—
116#519—1571no—
117#464—1574no—
118#462—1574no—
119#424—1571no—
120#419—1571no—
121#416—1571no—
122C2×#41C2×D15711yescharacteristic
123C22×Q8—1571no—
124C22×Q8C2×D15711yescharacteristic
125#137—24no—
126C2×Q16—3144no—
127C2×Q16—1574no—
128C2×Q16C2×D15712yes—
129C2×SD16C2×D15712yes—
130C2×M16C2×D15711yescharacteristic
131C22×C8—1571no—
132C4⋊Q8—1572no—
133#124—1572no—
134#123—3142no—
135C22⋊Q8—3142no—
136C4×Q8—3142no—
137C4×Q8—1574no—
138#117—3142no—
139#117—1571no—
140#117—1571no—
141C4⋊C8—1574no—
142#103—1572no—
143#103—1574no—
144#102—1572no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—3141no—
150C2×Q8—3142no—
151C2×Q8—3142no—
152C2×Q8—1572no—
153C2×Q8C22⋊D15714yes—
154C2×Q8C22×D15711yescharacteristic
155C2×Q8C22×D15712yes—
156C2×D4—21no—
157C2×D4C22×D15711yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—3141no—
161C22×C4C22×D15711yescharacteristic
162Q16—24no—
163Q16—3148no—
164SD16—24no—
165M16—22no—
166C2×C8—3142no—
167C2×C8—1572no—
168C2×C8C22×D15712yes—
169C4⋊C4—3144no—
170C4⋊C4—3144no—
171C4⋊C4—1572no—
172C4⋊C4—1572no—
173C22⋊C4—3141no—
174C22⋊C4—3144no—
175C22⋊C4—1572no—
176C42—3144no—
177C42—3141no—
178C42—1572no—
179C23—21no—
180C23D4×D15711yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—21no—
184Q8—21no—
185Q8—3142no—
186Q8—3142no—
187Q8—6282no—
188Q8C2×C22⋊D15714yes—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—3142no—
197C2×C4—3142no—
198C2×C4—6282no—
199C2×C4—1574no—
200C2×C4D4×D15711yescharacteristic
201C2×C4C2×C22⋊D15712yes—
202C2×C4#1457895312yes—
203C2×C4C23×D15711yescharacteristic
204C8—22no—
205C8—3142no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#1909774312yes—
210C22C2×D4×D15711yescharacteristic
211C4—24no—
212C4—41no—
213C4—3144no—
214C4—6282no—
215C4—6282no—
216C4#1909774312yes—
217C4C22×C22⋊D15711yescharacteristic
218C4C2×#1457895311yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1909774311yescharacteristic
222C2#2375808912yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#137×C157C2
socleC22×C157C22×D4
prime radicalC2×#41×C157C22
hypercenterC2×#137D157
hypocenterC157#3044
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#14578944C2semidirect
3. #23757751C2semidirect
4. #23757493C2semidirect
5. #23757481C2semidirect
6. #23757301C2semidirect
7. #23757283C2semidirect
111. C157#3044semidirect

Maximal subgroups

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

subgroupindex
C22×#145789442
#237578442
#237577512
#237574932
#237574812
#237573012
#237572832
#237572692
C2×#137×C1572
#3044157