Finite Group Databasebeta

← #36369017

Subgroups — #36369017

Order 22144.

270 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C4×#14583835C212yesmaximal, complemented
2#26012347C212yesmaximal, complemented
3#26012346C212yesmaximal, complemented
4#26012037C211yesmaximal, characteristic
5C2×#14583835—22no—
6#19149734C2214yes—
7C4×#13063443—22no—
8C2×#14583826C411yescharacteristic
9#19149677C411yescharacteristic
10#19149675C411yescharacteristic
11#19149668C2211yescharacteristic
12#19149666C2211yescharacteristic
13#19149651—22no—
14C4×C22⋊D173—22no—
15C2×C4×Dic173C2211yescharacteristic
16C2×#41×C173C411yescharacteristic
17#14583835—28no—
18C2×#13063443—24no—
19#14583826—42no—
20C2×Dic346C2×C411yescharacteristic
21C2×D692C2×C411yescharacteristic
22C2×C22⋊D173—22no—
23C2×C22⋊D173—21no—
24C22×Dic173D412yes—
25C2×C4×D173—21no—
26#14583805—22no—
27C22⋊Dic173—22no—
28#14583803—22no—
29C4×Dic173C2×C411yescharacteristic
30C4×Dic173D412yes—
31C4×Dic173C2311yescharacteristic
32#41×C173—42no—
33C2×Q8×C173C2×C411yescharacteristic
34C2×D4×C173—21no—
35C2×D4×C173C2×C411yescharacteristic
36C22×C4×C173—21no—
37C22×C4×C173C2×C411yescharacteristic
38#13063443—28no—
39C22×D173—21no—
40Dic346—41no—
41D692—41no—
42C22⋊D173—42no—
43C22⋊D173—44no—
44C2×Dic173—24no—
45C2×Dic173—21no—
46C2×Dic173C22⋊C412yes—
47C2×Dic173C2×D412yes—
48C4×D173—42no—
49C23×C173—21no—
50C23×C173C22⋊C411yescharacteristic
51Q8×C173—41no—
52D4×C173—41no—
53D4×C173—42no—
54C2×C4×C173—22no—
55C2×C4×C173—21no—
56C2×C4×C173—42no—
57C2×C4×C173C22⋊C411yescharacteristic
58C2×C4×C173C22×C411yescharacteristic
59C22×C173—21no—
60C22×C173—22no—
61C22×C173—41no—
62C22×C173—42no—
63C22×C173C2×C22⋊C411yescharacteristic
64C2×D173—41no—
65C2×D173—42no—
66C4×C173—21no—
67C4×C173—41no—
68C4×C173#10012yes—
69Dic173—22no—
70Dic173—41no—
71Dic173#10414yes—
72C2×C173—41no—
73C2×C173—81no—
74C2×C173C2×#10011yescharacteristic
75C2×C173C2×#10412yes—
76D173—81no—
77C173#152511yescharacteristic, complemented, Sylow
78#1525—1731nomaximal, Sylow
79#532—1731no—
80C4×M16—1732no—
81#329—1732no—
82#329—1732no—
83C2×#41—1731no—
84C2×#41#1062192411yescharacteristic
85C2×M16—3462no—
86C4⋊Q8—1731no—
87#127—1731no—
88#124—1731no—
89#124—1731no—
90#121—3462no—
91C4×D4—3462no—
92C2×C42—1731no—
93#97—1734no—
94C4×C8—3462no—
95#41—42no—
96#41—6922no—
97C2×Q8—1731no—
98C2×Q8C2×#1062192411yescharacteristic
99C2×D4—21no—
100C2×D4—3461no—
101C2×D4—3462no—
102C2×D4—1731no—
103C2×D4C2×#1062192411yescharacteristic
104C22×C4—21no—
105C22×C4—3461no—
106C22×C4—1732no—
107C22×C4C2×#1062192411yescharacteristic
108M16—3468no—
109C2×C8—3464no—
110C4⋊C4—3462no—
111C22⋊C4—3462no—
112C22⋊C4—3462no—
113C42—1732no—
114C42—1731no—
115C42—1731no—
116C23—21no—
117C23—3461no—
118C23C22⋊#1062192411yescharacteristic
119Q8—41no—
120Q8—6921no—
121D4—41no—
122D4—42no—
123D4—6921no—
124D4—6924no—
125D4—6922no—
126C2×C4—22no—
127C2×C4—21no—
128C2×C4—42no—
129C2×C4—3464no—
130C2×C4—3461no—
131C2×C4—1732no—
132C2×C4—1732no—
133C2×C4—6922no—
134C2×C4C22⋊#1062192411yescharacteristic
135C2×C4C22×#1062192411yescharacteristic
136C8—3468no—
137C22—21no—
138C22—22no—
139C22—41no—
140C22—42no—
141C22—6921no—
142C22—6922no—
143C22C2×C22⋊#1062192411yescharacteristic
144C4—21no—
145C4—41no—
146C4—3462no—
147C4—1734no—
148C4—6921no—
149C4#1914973912yes—
150C2—41no—
151C2—81no—
152C2—13841no—
153C2C2×#1914973911yescharacteristic
154C2D173⋊#10412yes—
subgroupG/H
centerC2×C4C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×#41×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012037C2
cosocleC4×Dic173C23
hypercenterC2×#41#10621924
hypocenterC173#1525
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C4×#14583835C2semidirect
2. #26012347C2semidirect
3. #26012346C2semidirect
77. C173#1525semidirect

Maximal subgroups

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

subgroupindex
C4×#145838352
#260123472
#260123462
#260120372
#1525173