Finite Group Databasebeta

← #34102087

Subgroups — #34102087

Order 20096.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#23757713C212yesmaximal, complemented
2#23757531C211yesmaximal, characteristic, complemented
3#23757083C212yesmaximal, complemented
4#23757055C211yesmaximal, characteristic, complemented
5#575×C157C211yesmaximal, characteristic
6C2×Dic628C2211yescharacteristic, complemented
7#19097750—21no—
8#19097717—22no—
9#19097715—22no—
10#19097711C2212yescomplemented
11#19097703C2211yescharacteristic, complemented
12#19097696—21no—
13C2×#41×C157—21no—
14#137×C157—22no—
15AGL(1,ℤ/8)×C157—22no—
16#135×C157—21no—
17#135×C157C2212yes—
18C2×M16×C157C2211yescharacteristic
19C2×Dic314—21no—
20Dic628—22no—
21C4⋊Dic157—21no—
22#14578933—41no—
23C22⋊Dic157—41no—
24#14578929—22no—
25#41×C157—22no—
26#41×C157—22no—
27#41×C157—41no—
28#41×C157—41no—
29C2×Q8×C157—21no—
30C2×D4×C157—21no—
31C2×D4×C157—41no—
32C22×C4×C157—41no—
33C22×C4×C157D411yescharacteristic
34Q16×C157—21no—
35Q16×C157D412yes—
36SD16×C157—22no—
37SD16×C157—22no—
38D8×C157—21no—
39D8×C157D412yes—
40M16×C157—22no—
41C2×C8×C157D411yescharacteristic
42C2×C8×C157C2311yescharacteristic
43Dic314—41no—
44Dic314—41no—
45C2×Dic157—41no—
46C2×Dic157—41no—
47C23×C157—41no—
48C23×C157D811yescharacteristic
49Q8×C157—22no—
50Q8×C157—41no—
51D4×C157—22no—
52D4×C157—41no—
53D4×C157—42no—
54D4×C157—42no—
55C2×C4×C157—21no—
56C2×C4×C157—21no—
57C2×C4×C157—42no—
58C2×C4×C157—41no—
59C2×C4×C157—42no—
60C2×C4×C157D811yescharacteristic
61C2×C4×C157C2×D411yescharacteristic
62C8×C157—21no—
63C8×C157C2×D412yes—
64C22×C157—21no—
65C22×C157—41no—
66C22×C157—42no—
67C22×C157—82no—
68C22×C157C2×D811yescharacteristic
69C4×C157—21no—
70C4×C157—42no—
71C4×C157#12011yescharacteristic
72C4×C157AGL(1,ℤ/8)11yescharacteristic
73Dic157—81no—
74Dic157—81no—
75C2×C157—21no—
76C2×C157—41no—
77C2×C157—82no—
78C2×C157#44711yescharacteristic
79C157#219111yescharacteristic, complemented, Sylow
80#2191—1571nomaximal, Sylow
81#575D15711yescharacteristic
82#510—1572no—
83#470—1571no—
84#360—1572no—
85#349—1571no—
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D15712yes—
91C2×Q16—1571no—
92C2×M16C2×D15711yescharacteristic
93C22⋊Q8—3141no—
94Q32—3142no—
95SD32—3142no—
96M32—1572no—
97#106—1571no—
98#103—3141no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—3141no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D15711yescharacteristic
109Q16—21no—
110Q16—3142no—
111Q16C22⋊D15712yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D15712yes—
116M16—22no—
117C2×C8C22⋊D15711yescharacteristic
118C2×C8C22×D15711yescharacteristic
119C4⋊C4—3141no—
120C4⋊C4—6281no—
121C22⋊C4—6281no—
122C16—3142no—
123C23—41no—
124C23#1457893911yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—6281no—
128Q8—6281no—
129D4—22no—
130D4—41no—
131D4—42no—
132D4—42no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—6281no—
139C2×C4—6281no—
140C2×C4#1457893911yescharacteristic
141C2×C4C2×C22⋊D15711yescharacteristic
142C8—21no—
143C8C2×C22⋊D15712yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1457893911yescharacteristic
149C4—21no—
150C4—42no—
151C4—12561no—
152C4—12561no—
153C4#1909774011yescharacteristic
154C4#1909779711yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#2375740711yescharacteristic
subgroupG/H
centerC2#23757407
commutator, cosocleC2×C8×C157C23
FrattiniC2×C8C22×D157
Fitting, prime radical#575×C157C2
socleC2×C157#447
hypercenter#575D157
hypocenterC157#2191
LCSG ▷ C2×C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #23757713C2semidirect
2. #23757531C2semidirect
3. #23757083C2semidirect
4. #23757055C2semidirect
6. C2×Dic628C22semidirect
10. #19097711C22semidirect
11. #19097703C22semidirect
79. C157#2191semidirect

Maximal subgroups

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

subgroupindex
#237577132
#237575312
#237570832
#237570552
#575×C1572
#2191157