Finite Group Databasebeta

← #36368986

Subgroups — #36368986

Order 22144.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 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#26012514C211yesmaximal, characteristic
2#26012436C211yesmaximal, characteristic
3#26012344C214yesmaximal
4#26012046C211yesmaximal, characteristic
5#19149746—21no—
6#19149736C2212yes—
7#19149734—21no—
8(C2×#10621924)⋊C4—21no—
9(C2×#10621924)⋊C4—21no—
10(C2×#10621924)⋊C4C2212yes—
11#19149679C411yescharacteristic, complemented
12#19149670—21no—
13#19149659C2212yes—
14#19149655—21no—
15Q8×Dic173—21no—
16#19149644C412yescomplemented
17#19149637—21no—
18C42×D173C2211yescharacteristic
19C4⋊Q8×C173C411yescharacteristic, complemented
20C22×#10621924—22no—
21C2×#13063443—22no—
22C2×Dic346C2×C412yes—
23C2×C4×D173D411yescharacteristic
24C2×C4×D173D411yescharacteristic
25C2×C4×D173C2311yescharacteristic
26C4⋊Dic173—22no—
27C4⋊Dic173—21no—
28C4⋊Dic173—21no—
29#14583805—22no—
30#14583803—22no—
31C4×Dic173—22no—
32C4×Dic173C2×C411yescharacteristic
33C4×Dic173D411yescharacteristic
34C4×Dic173D411yescharacteristic
35C2×Q8×C173—21no—
36C4⋊C4×C173—21no—
37C4⋊C4×C173C2×C412yes—
38C42×C173C2×C411yescharacteristic
39C2×#10621924—42no—
40C2×#10621924—42no—
41C2×#10621924—42no—
42#13063443—42no—
43C22×D173C2×D411yescharacteristic
44Dic346—42no—
45C2×Dic173—22no—
46C2×Dic173C22⋊C411yescharacteristic
47C2×Dic173C22⋊C411yescharacteristic
48C2×Dic173C2×D411yescharacteristic
49C4×D173—22no—
50C4×D173—41no—
51C4×D173—41no—
52Q8×C173—42no—
53C2×C4×C173—22no—
54C2×C4×C173C22⋊C411yescharacteristic
55C2×C4×C173C22⋊C411yescharacteristic
56C2×C4×C173C22×C411yescharacteristic
57C22×C173C2×C22⋊C411yescharacteristic
58C2×D173—21no—
59C2×D173—21no—
60C2×D173#13712yes—
61#10621924—82no—
62C4×C173—21no—
63C4×C173—21no—
64C4×C173—21no—
65C4×C173—42no—
66Dic173—21no—
67Dic173—21no—
68Dic173—21no—
69Dic173—42no—
70C2×C173#41011yescharacteristic
71C2×C173#41911yescharacteristic
72C2×C173#42111yescharacteristic
73D173—22no—
74C173#151511yescharacteristic, complemented, Sylow
75#1515—1731nomaximal, Sylow
76#533—1731no—
77#432—1731no—
78#383—1731no—
79#328—1734no—
80C4⋊Q8—1731no—
81C4⋊Q8#1062192411yescharacteristic, complemented
82#125—3461no—
83C22⋊Q8—3461no—
84C22⋊Q8—1732no—
85C4×Q8—3461no—
86C4×Q8—1732no—
87#117—3461no—
88C2×C42—1731no—
89C4⋊C8—3461no—
90C22⋊C8—1732no—
91#97—3461no—
92#95—3461no—
93#95—3461no—
94#95—1732no—
95C2×Q8—21no—
96C2×Q8—1732no—
97C22×C4—3462no—
98C22×C4—1731no—
99C22×C4—1731no—
100C22×C4—1731no—
101C2×C8—3462no—
102C4⋊C4—21no—
103C4⋊C4—3462no—
104C4⋊C4—3462no—
105C4⋊C4—3461no—
106C4⋊C4—3461no—
107C4⋊C4C2×#1062192412yes—
108C22⋊C4—3462no—
109C42—3462no—
110C42—1731no—
111C42—1731no—
112C42—1731no—
113C42C2×#1062192411yescharacteristic
114C23—1731no—
115Q8—42no—
116Q8—6922no—
117C2×C4—22no—
118C2×C4—3462no—
119C2×C4—3462no—
120C2×C4—1731no—
121C2×C4—1731no—
122C2×C4—1731no—
123C2×C4—6921no—
124C2×C4—6921no—
125C2×C4—6922no—
126C2×C4—6922no—
127C2×C4—6922no—
128C2×C4C22⋊#1062192411yescharacteristic
129C2×C4C22⋊#1062192411yescharacteristic
130C2×C4C22×#1062192411yescharacteristic
131C8—6922no—
132C22—3461no—
133C22—3461no—
134C22—1732no—
135C22C2×C22⋊#1062192411yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—42no—
140C4—3461no—
141C4—3461no—
142C4—3461no—
143C4—13842no—
144C4—6922no—
145C2—3462no—
146C2#2601247211yescharacteristic
147C2#2601248411yescharacteristic
148C2#2601248911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4⋊Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×D173C2×D4
cosocleC2×C4×D173C23
hypercenterC4⋊Q8#10621924
hypocenterC173#1515
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. #19149679C4semidirect
16. #19149644C4semidirect
19. C4⋊Q8×C173C4semidirect
74. C173#1515semidirect
81. C4⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125142
#260124362
#260123442
#260120462
#1515173