Finite Group Databasebeta

← #34100554

Subgroups — #34100554

Order 20096.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#23757780C211yesmaximal, characteristic
2#23757257C211yesmaximal, characteristic, complemented
3#23757236C212yesmaximal, complemented
4#23757044C212yesmaximal, complemented
5#568×C157C211yesmaximal, characteristic, complemented
6C22×C4×D157C2211yescharacteristic
7C2×#14578938C2212yes—
8#19097752C412yes—
9#19097747—21no—
10#19097742—21no—
11#19097739C412yes—
12C4×C22⋊D157—22no—
13#19097730C2211yescharacteristic
14#19097690—21no—
15#19097688—21no—
16#19097683—24no—
17C2×#41×C157C2211yescharacteristic
18#131×C157—24no—
19C2×M16×C157—22no—
20C2×M16×C157C2212yes—
21C23×D157C2×C411yescharacteristic
22C2×C22⋊D157—22no—
23C22×Dic157C2×C411yescharacteristic
24C2×C4×D157—22no—
25C2×C4×D157—21no—
26C2×C4×D157—21no—
27C2×C4×D157—22no—
28#14578938—22no—
29#14578938—22no—
30C2×#11929434—22no—
31C4⋊Dic157—21no—
32#14578933—22no—
33C22⋊Dic157—22no—
34C22⋊Dic157C2×C412yes—
35#14578931—22no—
36C4×Dic157—21no—
37#41×C157—22no—
38#41×C157—22no—
39C2×Q8×C157D411yescharacteristic
40C2×D4×C157C2×C412yes—
41C2×D4×C157D411yescharacteristic
42C22×C4×C157D412yes—
43C22×C4×C157C2311yescharacteristic
44M16×C157—24no—
45M16×C157—22no—
46M16×C157—22no—
47C2×C8×C157—24no—
48C2×C8×C157D414yescomplemented
49C22×D157—21no—
50C22×D157—22no—
51C22×D157—42no—
52C22⋊D157—44no—
53C2×Dic157—21no—
54C2×Dic157—22no—
55C2×Dic157—22no—
56C4×D157—41no—
57C4×D157—42no—
58C4×D157—41no—
59#11929434—42no—
60C23×C157C22⋊C412yes—
61C23×C157C22×C411yescharacteristic
62Q8×C157—22no—
63D4×C157—24no—
64D4×C157—22no—
65C2×C4×C157—22no—
66C2×C4×C157—22no—
67C2×C4×C157C22⋊C412yes—
68C2×C4×C157C2×D412yes—
69C2×C4×C157C2×D411yescharacteristic
70C2×C4×C157C2×D411yescharacteristic
71C2×C4×C157#4112yes—
72C8×C157—24no—
73C22×C157—22no—
74C22×C157—41no—
75C22×C157—42no—
76C22×C157C2×C22⋊C411yescharacteristic
77C22×C157C4×D412yes—
78C2×D157—22no—
79C2×D157—24no—
80C2×D157—41no—
81C2×D157—42no—
82C2×D157—41no—
83C4×C157—22no—
84C4×C157#12011yescharacteristic
85C4×C157#12112yes—
86C4×C157#12311yescharacteristic
87Dic157—42no—
88Dic157—41no—
89C2×C157—21no—
90C2×C157—22no—
91C2×C157—42no—
92C2×C157#38611yescharacteristic
93D157—42no—
94C157#189111yescharacteristic, complemented, Sylow
95#1891—1571nomaximal, Sylow
96#568D15711yescharacteristic, complemented
97#525—1571no—
98#413—1571no—
99#407—1572no—
100#344—1572no—
101C2×#41C2×D15711yescharacteristic
102C23×C4—1571no—
103#131—24no—
104C2×M16—22no—
105C2×M16—1572no—
106C2×M16C2×D15712yes—
107#123—1572no—
108C22⋊Q8—3141no—
109#121—3141no—
110#120—1572no—
111C4×D4—3142no—
112#117—1571no—
113#101—3141no—
114#100—3141no—
115C22⋊C8—3144no—
116C24—1571no—
117#41—22no—
118#41—22no—
119C2×Q8D62811yescharacteristic
120C2×D4—3142no—
121C2×D4C4×D15712yes—
122C2×D4D62811yescharacteristic
123C22×C4—3142no—
124C22×C4—3141no—
125C22×C4—3141no—
126C22×C4—3142no—
127C22×C4—1571no—
128C22×C4C22⋊D15712yes—
129C22×C4C22×D15711yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—3142no—
134M16—3142no—
135C2×C8—24no—
136C2×C8—3142no—
137C2×C8C22⋊D15714yescomplemented
138C4⋊C4—3141no—
139C4⋊C4—3142no—
140C22⋊C4—3142no—
141C22⋊C4—3142no—
142C22⋊C4—1572no—
143C42—3141no—
144C23—3141no—
145C23—3142no—
146C23—6282no—
147C23#1457893112yes—
148C23C2×C4×D15711yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—6284no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—3142no—
156C2×C4—3141no—
157C2×C4—3142no—
158C2×C4—6281no—
159C2×C4—6282no—
160C2×C4—6281no—
161C2×C4#1457893112yes—
162C2×C4C2×C22⋊D15712yes—
163C2×C4C2×C22⋊D15711yescharacteristic
164C2×C4C2×D62811yescharacteristic
165C2×C4#1457895412yes—
166C8—24no—
167C8—6282no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—3142no—
172C22—3144no—
173C22—6281no—
174C22—6282no—
175C22—6281no—
176C22C2×#1457893111yescharacteristic
177C22C4×C22⋊D15712yes—
178C4—22no—
179C4—6282no—
180C4—6281no—
181C4#1909774011yescharacteristic
182C4#1909774512yes—
183C4#1909775411yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—6282no—
188C2#2375715211yescharacteristic
subgroupG/H
centerC4#19097740
commutatorC23×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#568×C157C2
socleC2×C157#386
prime radical#23757780C2
cosocleC22×C4×C157C23
hypercenter#568D157
hypocenterC157#1891
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #23757257C2semidirect
3. #23757236C2semidirect
4. #23757044C2semidirect
5. #568×C157C2semidirect
48. C2×C8×C157D4semidirect
94. C157#1891semidirect
96. #568D157semidirect
137. C2×C8C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237577802
#237572572
#237572362
#237570442
#568×C1572
#1891157