Finite Group Databasebeta

← #36369520

Subgroups — #36369520

Order 22144.

252 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 66 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#26012569C211yesmaximal, characteristic
2#26012506C212yesmaximal, complemented
3#26012465C211yesmaximal, characteristic
4#26012366C212yesmaximal, complemented
5#26011472C211yesmaximal, characteristic
6#19149766—22no—
7#19149766—22no—
8C2×#14583836C2211yescharacteristic
9D173⋊M16C2212yes—
10C2×#14583835—22no—
11C2×D173⋊C8C2211yescharacteristic
12#19149750—22no—
13#19149749—22no—
14C22⋊#13063443—22no—
15C2×#14583825C2211yescharacteristic
16M16×D173C2212yes—
17#19149597C412yes—
18#19149596C411yescharacteristic
19#100×C173C411yescharacteristic
20#14583836—22no—
21#14583836D412yes—
22#14583836D412yes—
23#14583835—24no—
24#14583835—22no—
25D173⋊C8—22no—
26D173⋊C8—22no—
27C2×#13063443—22no—
28C2×#13063443—22no—
29C2×#13063443—21no—
30C2×#13063443—21no—
31#14583825—22no—
32#14583825—22no—
33C2×Dic346C2×C411yescharacteristic
34C2×C22⋊D173Q812yes—
35C22×Dic173D412yes—
36C2×C4×D173C2311yescharacteristic
37#14583810C2×C412yes—
38#14583809—22no—
39C8×D173—22no—
40C2×D4×C173C2×C411yescharacteristic
41M16×C173C2×C412yes—
42#13063443—22no—
43#13063443—22no—
44#13063443—41no—
45#13063443—41no—
46C22×D173C2×Q811yescharacteristic
47Dic346—22no—
48C22⋊D173—24no—
49C2×Dic173—22no—
50C2×Dic173—22no—
51C2×Dic173C4⋊C412yes—
52C2×Dic173C2×D411yescharacteristic
53C4×D173C2×D412yes—
54C4×D173#4112yes—
55#13063436—22no—
56C23×C173C4⋊C412yes—
57D4×C173—22no—
58C2×C4×C173C22×C411yescharacteristic
59C8×C173—22no—
60C22×C173—22no—
61C22×C173—42no—
62C22×C173C2×C4⋊C411yescharacteristic
63C2×D173—41no—
64C2×D173C22⋊Q812yes—
65C4×C173C4×D412yes—
66Dic173—22no—
67Dic173#12011yescharacteristic
68Dic173#12311yescharacteristic
69C2×C173—21no—
70C2×C173—42no—
71C2×C173#38511yescharacteristic
72D173—22no—
73C173#185811yescharacteristic, complemented, Sylow
74#1858—1731nomaximal, Sylow
75#568—1731no—
76#430—1732no—
77#413—1731no—
78#408—1731no—
79#341—1732no—
80C2×#41—1731no—
81#131—3462no—
82#131—3462no—
83C2×M16—3462no—
84C2×M16—1731no—
85C2×M16—1732no—
86C2×M16—1732no—
87C22×C8—1731no—
88#108—3462no—
89#108—3462no—
90#101—1732no—
91#100—1731no—
92#100#1062192411yescharacteristic
93C22⋊C8—3462no—
94#41—3462no—
95#41—3462no—
96C2×Q8—1731no—
97C2×D4—1732no—
98C2×D4C2×#1062192411yescharacteristic
99C22×C4—1732no—
100C22×C4—1731no—
101M16—3462no—
102M16—3464no—
103M16—3462no—
104M16—3462no—
105M16—1732no—
106M16—1732no—
107M16—1732no—
108M16C2×#1062192412yes—
109C2×C8—3462no—
110C2×C8—3462no—
111C2×C8—3462no—
112C2×C8—3462no—
113C2×C8—3461no—
114C2×C8—3461no—
115C2×C8—3462no—
116C23—1731no—
117C23C4⋊#1062192412yes—
118Q8—3462no—
119D4—22no—
120D4—3464no—
121C2×C4—3462no—
122C2×C4—3462no—
123C2×C4—1732no—
124C2×C4—1731no—
125C2×C4—1732no—
126C2×C4—1732no—
127C2×C4C22×#1062192411yescharacteristic
128C8—22no—
129C8—3462no—
130C8—3462no—
131C8—3462no—
132C8—6921no—
133C8—6921no—
134C22—22no—
135C22—42no—
136C22—1732no—
137C22—6921no—
138C22C2×C4⋊#1062192411yescharacteristic
139C4—3462no—
140C4—1731no—
141C4—1731no—
142C4D4×#1062192412yes—
143C2—21no—
144C2—42no—
145C2—3462no—
146C2#2601243811yescharacteristic
subgroupG/H
centerC2#26012438
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#100×C173C4
socleC2×C173#385
prime radicalC2×#14583825C22
cosocleC2×C4×D173C23
hypercenter#100#10621924
hypocenterC173#1858
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. #26012506C2semidirect
4. #26012366C2semidirect
73. C173#1858semidirect

Maximal subgroups

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

subgroupindex
#260125692
#260125062
#260124652
#260123662
#260114722
#1858173