Finite Group Databasebeta

← #36368890

Subgroups — #36368890

Order 22144.

312 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 74 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#26012508C211yesmaximal, characteristic, complemented
2#26012494C211yesmaximal, characteristic, complemented
3#26012339C214yesmaximal, complemented
4C2×#19149641C211yesmaximal, characteristic
5C2×#14583835—21no—
6#19149746—21no—
7#19149746—21no—
8C4⋊#13063443C2212yescomplemented
9C22⋊#13063443—21no—
10C4×#13063443C2212yescomplemented
11C22×D692C411yescharacteristic
12C22×C4×D173—21no—
13#19149641—21no—
14#19149641—21no—
15#19149641C412yes—
16#19149641C2212yes—
17C2×#14583803—21no—
18C2×C4×Dic173C2211yescharacteristic
19C2×C4⋊C4×C173C411yescharacteristic
20#14583835—42no—
21C2×#13063443—22no—
22C2×#13063443—22no—
23C23×D173—21no—
24C2×D692—24no—
25C2×D692—21no—
26C2×D692C2×C412yes—
27C22×Dic173D411yescharacteristic
28C22×Dic173D411yescharacteristic
29C2×C4×D173—22no—
30C2×C4×D173—22no—
31C2×C4×D173—42no—
32C2×C4×D173—42no—
33#14583803—22no—
34#14583803—22no—
35C4×Dic173C2×C411yescharacteristic
36C4×Dic173D411yescharacteristic
37C4×Dic173D411yescharacteristic
38C4×Dic173C2311yescharacteristic
39C22×C4×C173—21no—
40C22×C4×C173C2×C411yescharacteristic
41C4⋊C4×C173—21no—
42C4⋊C4×C173C2×C412yes—
43#13063443—42no—
44#13063443—42no—
45C22×D173—24no—
46C22×D173—22no—
47C22×D173—42no—
48C22×D173—42no—
49D692—24no—
50D692—22no—
51D692M1614yescomplemented
52C2×Dic173—41no—
53C2×Dic173C22⋊C411yescharacteristic
54C2×Dic173C22⋊C411yescharacteristic
55C2×Dic173D812yes—
56C2×Dic173SD1612yes—
57C2×Dic173C2×D411yescharacteristic
58C2×Dic173C2×D411yescharacteristic
59C4×D173—44no—
60C4×D173—42no—
61C4×D173—42no—
62C23×C173C22⋊C411yescharacteristic
63C2×C4×C173—22no—
64C2×C4×C173—22no—
65C2×C4×C173—42no—
66C2×C4×C173C22⋊C411yescharacteristic
67C2×C4×C173C22×C411yescharacteristic
68C22×C173—21no—
69C22×C173—21no—
70C22×C173#10212yes—
71C22×C173C2×C22⋊C411yescharacteristic
72C2×D173—24no—
73C2×D173—48no—
74C2×D173—42no—
75C2×D173—42no—
76C4×C173—21no—
77C4×C173—42no—
78C4×C173C2×M1612yes—
79Dic173—21no—
80Dic173—21no—
81Dic173#10212yes—
82Dic173C2×D811yescharacteristic
83Dic173C2×SD1611yescharacteristic
84C2×C173—22no—
85C2×C173#40711yescharacteristic
86C2×C173C2×#10211yescharacteristic
87C2×C173#42111yescharacteristic
88D173—44no—
89C173#148611yescharacteristic, complemented, Sylow
90#1486—1731nomaximal, Sylow
91C2×C4×D4—1731no—
92#431—1731no—
93#423—1731no—
94D4⋊C8—1734no—
95C22×D4—1731no—
96C23×C4—3461no—
97C2×M16—3461no—
98C4×D4—3461no—
99C4×D4—3461no—
100C4×D4—1732no—
101C4×D4—1732no—
102C2×C4⋊C4#1062192411yescharacteristic
103C2×C22⋊C4—3461no—
104C2×C42—1731no—
105C4⋊C8—3461no—
106C4⋊C8—3461no—
107C4⋊C8—1732no—
108C22⋊C8—3461no—
109C4×C8—1732no—
110C24—3461no—
111C2×D4—3464no—
112C2×D4—3461no—
113C2×D4—1732no—
114C22×C4—21no—
115C22×C4—3462no—
116C22×C4—3462no—
117C22×C4—1731no—
118C22×C4—1731no—
119C22×C4—6922no—
120C22×C4—6922no—
121C22×C4C2×#1062192411yescharacteristic
122M16—6922no—
123C2×C8—3462no—
124C2×C8—3462no—
125C4⋊C4—21no—
126C4⋊C4C2×#1062192412yes—
127C22⋊C4—3462no—
128C22⋊C4—3462no—
129C42—1731no—
130C42—1731no—
131C42—1731no—
132C42—1731no—
133C23—3464no—
134C23—3462no—
135C23—6922no—
136C23—6922no—
137C23C22⋊#1062192411yescharacteristic
138D4—3464no—
139D4—3462no—
140D4—1734no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—1731no—
145C2×C4—1731no—
146C2×C4—1732no—
147C2×C4—1732no—
148C2×C4—1731no—
149C2×C4—1731no—
150C2×C4—6922no—
151C2×C4—6924no—
152C2×C4—6922no—
153C2×C4—6921no—
154C2×C4C22⋊#1062192411yescharacteristic
155C2×C4C22×#1062192411yescharacteristic
156C8—6922no—
157C8—6922no—
158C22—21no—
159C22—21no—
160C22—3464no—
161C22—6928no—
162C22—6922no—
163C22—6922no—
164C22#1914974112yes—
165C22C2×C22⋊#1062192411yescharacteristic
166C4—21no—
167C4—42no—
168C4—3461no—
169C4—3461no—
170C4—1731no—
171C4—1732no—
172C4—1731no—
173C4D173⋊M1612yes—
174C2—22no—
175C2—6924no—
176C2#2601246311yescharacteristic
177C2C2×#1914974111yescharacteristic
178C2#2601249011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×D692C4
cosocleC4×Dic173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1486
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #26012508C2semidirect
2. #26012494C2semidirect
3. #26012339C2semidirect
8. C4⋊#13063443C22semidirect
10. C4×#13063443C22semidirect
51. D692M16semidirect
89. C173#1486semidirect

Maximal subgroups

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

subgroupindex
#260125082
#260124942
#260123392
C2×#191496412
#1486173