Finite Group Databasebeta

← #36368662

Subgroups — #36368662

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#26012478C211yesmaximal, characteristic
2#26012465C211yesmaximal, characteristic
3#26011463C211yesmaximal, characteristic
4C2×#14583836C411yescharacteristic, complemented
5C2×#14583835—21no—
6C2×D173⋊C8C411yescharacteristic, complemented
7#19149740C412yescomplemented
8#19149739—21no—
9C22⋊#13063443—21no—
10C22⋊#13063443—21no—
11C2×#14583825C2211yescharacteristic
12#19149635—21no—
13#19149594C411yescharacteristic
14#19149593—21no—
15#99×C173C411yescharacteristic
16#14583836—41no—
17#14583835—22no—
18#14583835—41no—
19D173⋊C8—41no—
20C2×#13063443—21no—
21C2×#13063443—21no—
22C2×#13063443—41no—
23#14583825—41no—
24#14583825—41no—
25C2×Dic346C2×C411yescharacteristic
26C2×C22⋊D173D411yescharacteristic
27C2×C22⋊D173Q811yescharacteristic
28C22×Dic173D411yescharacteristic
29C22×Dic173D411yescharacteristic
30C2×C4×D173C2×C411yescharacteristic
31#14583805—41no—
32C22⋊Dic173—21no—
33C4×Dic173—41no—
34C2×D4×C173C2×C411yescharacteristic
35C22⋊C4×C173—21no—
36#13063443—42no—
37#13063443—41no—
38#13063443—41no—
39C22×D173C4⋊C411yescharacteristic
40Dic346—41no—
41C22⋊D173—41no—
42C22⋊D173—41no—
43C2×Dic173—21no—
44C2×Dic173—21no—
45C2×Dic173—21no—
46C2×Dic173—21no—
47C2×Dic173—41no—
48C2×Dic173C22⋊C411yescharacteristic
49C2×Dic173C22⋊C411yescharacteristic
50C2×Dic173C4⋊C411yescharacteristic
51C4×D173—21no—
52C4×D173—21no—
53C23×C173C22⋊C411yescharacteristic
54C23×C173C4⋊C411yescharacteristic
55D4×C173—41no—
56C2×C4×C173—41no—
57C2×C4×C173C4211yescharacteristic
58C22×C173—21no—
59C22×C173—21no—
60C22×C173—41no—
61C22×C173—41no—
62C22×C173#9511yescharacteristic
63C2×D173—21no—
64C2×D173—41no—
65C4×C173—21no—
66C4×C173—81no—
67Dic173—21no—
68Dic173—21no—
69Dic173—81no—
70Dic173#9911yescharacteristic
71Dic173#9911yescharacteristic
72C2×C173—21no—
73C2×C173—41no—
74C2×C173—41no—
75C2×C173#34211yescharacteristic
76D173—41no—
77C173#139511yescharacteristic, complemented, Sylow
78#1395—1731nomaximal, Sylow
79#413—1731no—
80#410—1731no—
81#408—1731no—
82C2×#41—1731no—
83C2×M16—3461no—
84C2×M16—1731no—
85C22×C8—1731no—
86#117—3461no—
87#101—1732no—
88#100—3461no—
89#99—3461no—
90#99—1731no—
91#99#1062192411yescharacteristic
92C22⋊C8—3461no—
93C22⋊C8—3461no—
94#41—6921no—
95#41—6921no—
96C2×Q8—1731no—
97C2×D4—1731no—
98C2×D4—1731no—
99C2×D4C2×#1062192411yescharacteristic
100C22×C4—1731no—
101C22×C4—1731no—
102C22×C4—1731no—
103M16—3462no—
104M16—6921no—
105M16—6921no—
106C2×C8—3461no—
107C2×C8—3461no—
108C2×C8—6921no—
109C2×C8—6921no—
110C4⋊C4—6921no—
111C22⋊C4—21no—
112C22⋊C4—3461no—
113C42—6921no—
114C23—1731no—
115C23C22⋊#1062192411yescharacteristic
116C23C4⋊#1062192411yescharacteristic
117Q8—6921no—
118D4—41no—
119D4—6921no—
120D4—6921no—
121C2×C4—41no—
122C2×C4—3461no—
123C2×C4—3461no—
124C2×C4—3461no—
125C2×C4—3461no—
126C2×C4—3461no—
127C2×C4—3461no—
128C2×C4—1731no—
129C2×C4—1731no—
130C2×C4—1731no—
131C2×C4—6921no—
132C2×C4C4×#1062192411yescharacteristic
133C8—6922no—
134C8—6921no—
135C8—6921no—
136C22—21no—
137C22—21no—
138C22—41no—
139C22—41no—
140C22—3461no—
141C22—6921no—
142C22(C2×#10621924)⋊C411yescharacteristic
143C4—21no—
144C4—81no—
145C4—3461no—
146C4—3461no—
147C4—13841no—
148C4—1731no—
149C4—1731no—
150C2—21no—
151C2—41no—
152C2—41no—
153C2—6921no—
154C2#2601236811yescharacteristic
subgroupG/H
centerC2#26012368
commutatorC2×C4×C173C42
FrattiniC2×D4C2×#10621924
Fitting#99×C173C4
socleC2×C173#342
prime radical, cosocleC2×#14583825C22
hypercenter#99#10621924
hypocenterC173#1395
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #99

Complements

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

NH ≅ G/Ntype
4. C2×#14583836C4semidirect
6. C2×D173⋊C8C4semidirect
7. #19149740C4semidirect
77. C173#1395semidirect

Maximal subgroups

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

subgroupindex
#260124782
#260124652
#260114632
#1395173