Finite Group Databasebeta

← #36368665

Subgroups — #36368665

Order 22144.

162 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 38 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#26012472C211yesmaximal, characteristic
2#26012466C211yesmaximal, characteristic
3#26011474C211yesmaximal, characteristic
4C2×#14583836C411yescharacteristic, complemented
5C2×D173⋊C8C411yescharacteristic, complemented
6#19149758—21no—
7#19149738—21no—
8#19149738C412yes—
9#19149736—21no—
10#19149736—21no—
11C2×#14583827C2211yescharacteristic
12M16×D173—21no—
13#19149598C411yescharacteristic, complemented
14#19149595—21no—
15#101×C173C411yescharacteristic, complemented
16#14583836—41no—
17D173⋊C8—41no—
18C2×#13063443—21no—
19C2×#13063443—21no—
20C4⋊#10621924—41no—
21C22⋊#10621924—22no—
22C4×#10621924—41no—
23#14583827—41no—
24#14583827—41no—
25C2×D692C2×C411yescharacteristic
26C2×D692D411yescharacteristic
27C2×D692Q811yescharacteristic
28C2×C4×D173C2×C411yescharacteristic
29C2×C4×D173D411yescharacteristic
30C2×C4×D173D411yescharacteristic
31#14583810—21no—
32#14583809—41no—
33C8×D173—41no—
34C2×Q8×C173C2×C411yescharacteristic
35M16×C173—21no—
36C2×#10621924—42no—
37#13063443—41no—
38#13063443—41no—
39C22×D173C22⋊C411yescharacteristic
40C22×D173C4⋊C411yescharacteristic
41C22×D173C4⋊C411yescharacteristic
42D692—41no—
43D692—41no—
44D692—41no—
45C2×Dic173C22⋊C411yescharacteristic
46C4×D173—21no—
47C4×D173—21no—
48C4×D173—21no—
49C4×D173—21no—
50C4×D173—21no—
51C4×D173—21no—
52#13063436—41no—
53Q8×C173—41no—
54C2×C4×C173C4211yescharacteristic
55C2×C4×C173C22⋊C411yescharacteristic
56C2×C4×C173C4⋊C411yescharacteristic
57C8×C173—41no—
58C22×C173#9511yescharacteristic
59C2×D173—21no—
60C2×D173—21no—
61C2×D173—21no—
62C2×D173—41no—
63C2×D173—41no—
64C2×D173—41no—
65#10621924—82no—
66C4×C173—21no—
67C4×C173—21no—
68C4×C173—21no—
69Dic173#9911yescharacteristic
70Dic173#9911yescharacteristic
71C2×C173—21no—
72C2×C173#34211yescharacteristic
73D173—41no—
74D173—41no—
75D173—41no—
76C173#139511yescharacteristic, complemented, Sylow
77#1395—1731nomaximal, Sylow
78#413—1731no—
79#410—1731no—
80#408—1731no—
81C2×#41—1731no—
82C2×M16—3461no—
83C2×M16—1731no—
84C22×C8—1731no—
85#117—3461no—
86#101—1731no—
87#101#1062192411yescharacteristic, complemented
88#100—3461no—
89#99—3461no—
90#99—1732no—
91C22⋊C8—3461no—
92C22⋊C8—3461no—
93#41—6921no—
94#41—6921no—
95C2×Q8C2×#1062192411yescharacteristic
96C2×D4—1731no—
97C2×D4—1731no—
98C2×D4—1731no—
99C22×C4—1731no—
100C22×C4—1731no—
101C22×C4—1731no—
102M16—21no—
103M16—3461no—
104M16—6921no—
105M16—6921no—
106C2×C8—3461no—
107C2×C8—3461no—
108C2×C8—6921no—
109C2×C8—6921no—
110C4⋊C4—6921no—
111C22⋊C4—3462no—
112C42—6921no—
113C23—1731no—
114C23—1731no—
115C23—1731no—
116Q8—41no—
117D4—6921no—
118D4—6921no—
119D4—6921no—
120C2×C4—3461no—
121C2×C4—3461no—
122C2×C4—3461no—
123C2×C4—3461no—
124C2×C4—3461no—
125C2×C4—3461no—
126C2×C4—1731no—
127C2×C4—6922no—
128C2×C4C4×#1062192411yescharacteristic
129C2×C4C22⋊#1062192411yescharacteristic
130C2×C4C4⋊#1062192411yescharacteristic
131C8—41no—
132C8—6921no—
133C8—6921no—
134C8—6921no—
135C22—3461no—
136C22—3461no—
137C22—3461no—
138C22—6921no—
139C22—6921no—
140C22—6921no—
141C22(C2×#10621924)⋊C411yescharacteristic
142C4—21no—
143C4—21no—
144C4—21no—
145C4—13842no—
146C4—1731no—
147C4—1731no—
148C2—21no—
149C2—6921no—
150C2—6921no—
151C2—6921no—
152C2#2601236811yescharacteristic
subgroupG/H
centerC2#26012368
commutatorC2×C4×C173C42
FrattiniC2×Q8C2×#10621924
Fitting#101×C173C4
socleC2×C173#342
prime radical, cosocleC2×#14583827C22
hypercenter#101#10621924
hypocenterC173#1395
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #101

Complements

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

NH ≅ G/Ntype
4. C2×#14583836C4semidirect
5. C2×D173⋊C8C4semidirect
13. #19149598C4semidirect
15. #101×C173C4semidirect
76. C173#1395semidirect
87. #101#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124722
#260124662
#260114742
#1395173