Finite Group Databasebeta

← #34100658

Subgroups — #34100658

Order 20096.

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

normal subgroups only

This group has 312 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#23757844C211yesmaximal, characteristic
2#23757283C212yesmaximal, complemented
3#23757247C211yesmaximal, characteristic
4#23757030C212yesmaximal, complemented
5#568×C157C211yesmaximal, characteristic
6C22×Dic314C2211yescharacteristic
7#19097769C412yescomplemented
8#19097758C412yescomplemented
9#19097752—22no—
10#19097750—22no—
11#19097730C2212yes—
12#19097730C2211yescharacteristic
13#19097696—24no—
14#19097686—22no—
15C2×#41×C157C2211yescharacteristic
16#131×C157—24no—
17C2×M16×C157—22no—
18C2×M16×C157C2212yes—
19C2×Dic314—21no—
20C2×Dic314—22no—
21C2×Dic314—22no—
22C2×Dic314C2×C412yes—
23C22×Dic157—21no—
24C4⋊Dic157—21no—
25C4⋊Dic157—22no—
26#14578933—24no—
27C22⋊Dic157—24no—
28C22⋊Dic157—21no—
29C22⋊Dic157—22no—
30C4×Dic157—22no—
31C4×Dic157C2×C412yes—
32#41×C157—22no—
33#41×C157—22no—
34C2×Q8×C157C2×C411yescharacteristic
35C2×D4×C157C2×C411yescharacteristic
36C2×D4×C157D412yes—
37C22×C4×C157D412yes—
38C22×C4×C157C2311yescharacteristic
39M16×C157—24no—
40M16×C157—22no—
41M16×C157—22no—
42C2×C8×C157—24no—
43C2×C8×C157D414yes—
44Dic314—24no—
45Dic314—41no—
46Dic314—41no—
47C2×Dic157—22no—
48C2×Dic157—22no—
49C2×Dic157—42no—
50C2×Dic157—42no—
51C23×C157C22⋊C412yes—
52C23×C157C2×D411yescharacteristic
53Q8×C157—22no—
54D4×C157—24no—
55D4×C157—22no—
56C2×C4×C157—22no—
57C2×C4×C157—22no—
58C2×C4×C157C22⋊C412yes—
59C2×C4×C157C22×C411yescharacteristic
60C2×C4×C157C2×D412yes—
61C2×C4×C157C2×D411yescharacteristic
62C2×C4×C157#4112yes—
63C8×C157—24no—
64C22×C157—22no—
65C22×C157—41no—
66C22×C157—42no—
67C22×C157C2×C22⋊C411yescharacteristic
68C22×C157#12112yes—
69C4×C157—22no—
70C4×C157C4×D412yes—
71C4×C157#12011yescharacteristic
72C4×C157#12311yescharacteristic
73Dic157—42no—
74Dic157—42no—
75Dic157—82no—
76C2×C157—21no—
77C2×C157—22no—
78C2×C157—42no—
79C2×C157#38611yescharacteristic
80C157#190911yescharacteristic, complemented, Sylow
81#1909—1571nomaximal, Sylow
82#568D15711yescharacteristic
83#536—1571no—
84#419—1572no—
85#410—1571no—
86H(3,ℤ/4)—1572no—
87C2×#41C2×D15711yescharacteristic
88C22×Q8—1571no—
89#131—24no—
90C2×M16—22no—
91C2×M16C2×D15712yes—
92C4⋊Q8—1572no—
93#124—1572no—
94#123—3142no—
95C22⋊Q8—3142no—
96#117—1572no—
97#117—1571no—
98#103—3144no—
99#99—3142no—
100#41—22no—
101#41—22no—
102C2×Q8—3141no—
103C2×Q8—3142no—
104C2×Q8—3142no—
105C2×Q8—1572no—
106C2×Q8C4×D15711yescharacteristic
107C2×D4C4×D15711yescharacteristic
108C2×D4D62812yes—
109C22×C4—3141no—
110C22×C4C22⋊D15712yes—
111C22×C4C22×D15711yescharacteristic
112M16—24no—
113M16—22no—
114M16—22no—
115C2×C8—24no—
116C2×C8C22⋊D15714yes—
117C4⋊C4—3141no—
118C4⋊C4—3144no—
119C4⋊C4—3142no—
120C22⋊C4—3141no—
121C22⋊C4—3144no—
122C22⋊C4—3142no—
123C42—3142no—
124C42—1572no—
125C23#1457893112yes—
126C23C2×D62811yescharacteristic
127Q8—22no—
128Q8—3144no—
129Q8—6281no—
130Q8—6281no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—3142no—
136C2×C4—3142no—
137C2×C4—6282no—
138C2×C4—6282no—
139C2×C4#1457893112yes—
140C2×C4C2×C4×D15711yescharacteristic
141C2×C4C2×C22⋊D15712yes—
142C2×C4C2×C22⋊D15711yescharacteristic
143C2×C4#1457895412yes—
144C8—24no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×#1457893111yescharacteristic
149C22#1909774512yes—
150C4—22no—
151C4—6282no—
152C4—6282no—
153C4—12562no—
154C4C4×C22⋊D15712yes—
155C4#1909774011yescharacteristic
156C4#1909775411yescharacteristic
157C2—21no—
158C2—22no—
159C2—42no—
160C2#2375715211yescharacteristic
subgroupG/H
centerC2#23757152
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#568×C157C2
socleC2×C157#386
prime radicalC2×#41×C157C22
cosocleC22×C4×C157C23
hypercenter#568D157
hypocenterC157#1909
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #23757283C2semidirect
4. #23757030C2semidirect
7. #19097769C4semidirect
8. #19097758C4semidirect
80. C157#1909semidirect

Maximal subgroups

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

subgroupindex
#237578442
#237572832
#237572472
#237570302
#568×C1572
#1909157