Finite Group Databasebeta

← #36353294

Subgroups — #36353294

Order 22144.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#19149635C212yesmaximal
2C2×#19149635C211yesmaximal, characteristic
3#26011258C211yesmaximal, characteristic
4#26011257C212yesmaximal
5#342×C173C211yesmaximal, characteristic
6C23×Dic173C2211yescharacteristic
7#19149635—22no—
8#19149635—22no—
9#19149635—24no—
10#19149635C418yescomplemented
11C2×#14583805—22no—
12C2×#14583805C412yescomplemented
13C2×C22⋊Dic173C2212yes—
14C2×C22⋊Dic173C2211yescharacteristic
15C2×C4×Dic173—22no—
16C2×C4×Dic173C412yescomplemented
17C2×C22⋊C4×C173C2211yescharacteristic
18C2×C22⋊C4×C173C2212yes—
19C22×Dic173—22no—
20C22×Dic173C2×C412yes—
21C22×Dic173C2×C411yescharacteristic
22C22×Dic173C2×C411yescharacteristic
23C22×Dic173C2×C414yes—
24C22×Dic173D412yes—
25C22×Dic173D412yes—
26C22×Dic173D412yes—
27C22×Dic173Q812yes—
28#14583805—28no—
29#14583805—24no—
30C22⋊Dic173—22no—
31C22⋊Dic173—22no—
32C22⋊Dic173C2×C414yes—
33C4×Dic173—28no—
34C4×Dic173—22no—
35C4×Dic173—22no—
36C24×C173C2311yescharacteristic
37C22×C4×C173—22no—
38C22×C4×C173C2×C412yes—
39C22⋊C4×C173—22no—
40C22⋊C4×C173—22no—
41C22⋊C4×C173C2×C414yes—
42C2×Dic173—24no—
43C2×Dic173—22no—
44C2×Dic173—44no—
45C2×Dic173—42no—
46C2×Dic173C4212yes—
47C2×Dic173C4212yes—
48C2×Dic173C22⋊C414yes—
49C2×Dic173C22⋊C411yescharacteristic
50C2×Dic173C22⋊C411yescharacteristic
51C2×Dic173C22⋊C414yes—
52C2×Dic173C22⋊C412yes—
53C2×Dic173C4⋊C412yes—
54C2×Dic173C4⋊C412yes—
55C2×Dic173C4⋊C414yes—
56C2×Dic173C4⋊C414yes—
57C23×C173—42no—
58C23×C173C22×C411yescharacteristic
59C23×C173C22×C412yes—
60C23×C173C2×D412yes—
61C23×C173C2×D411yescharacteristic
62C23×C173C2×Q811yescharacteristic
63C2×C4×C173—22no—
64C2×C4×C173—24no—
65C2×C4×C173—44no—
66C2×C4×C173—42no—
67C22×C173—22no—
68C22×C173—24no—
69C22×C173—42no—
70C22×C173—41no—
71C22×C173—41no—
72C22×C173C2×C4211yescharacteristic
73C22×C173C2×C22⋊C412yes—
74C22×C173C2×C22⋊C411yescharacteristic
75C22×C173C2×C4⋊C411yescharacteristic
76C22×C173C2×C4⋊C412yes—
77C4×C173—44no—
78C4×C173—42no—
79Dic173—44no—
80Dic173—42no—
81Dic173#9512yes—
82Dic173#9514yes—
83Dic173#9512yes—
84C2×C173—22no—
85C2×C173—21no—
86C2×C173—21no—
87C2×C173—22no—
88C2×C173C2×#9511yescharacteristic
89C2×C173#41012yes—
90C173#174011yescharacteristic, complemented, Sylow
91#1740—1731nomaximal, Sylow
92C2×#117—1732no—
93C2×#117—1731no—
94#342—1732no—
95#342—1731no—
96#342D17311yescharacteristic
97C23×C4—1731no—
98#117—3464no—
99#117—3462no—
100#117—3462no—
101#117—1738no—
102C2×C4⋊C4—3462no—
103C2×C4⋊C4—1732no—
104C2×C22⋊C4—1731no—
105C2×C22⋊C4—1732no—
106C2×C22⋊C4C2×D17311yescharacteristic
107C2×C22⋊C4C2×D17312yes—
108C2×C42—3462no—
109C2×C42—1732no—
110C24C22×D17311yescharacteristic
111C22×C4—22no—
112C22×C4—3462no—
113C22×C4—1732no—
114C22×C4—1731no—
115C22×C4—1731no—
116C22×C4—1734no—
117C22×C4—1732no—
118C22×C4—1732no—
119C22×C4—1732no—
120C22×C4—1732no—
121C22×C4C4×D17312yes—
122C4⋊C4—3468no—
123C4⋊C4—3464no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—3462no—
127C22⋊C4—3462no—
128C22⋊C4—1734no—
129C22⋊C4C4×D17314yes—
130C42—3468no—
131C42—3462no—
132C42—3462no—
133C23—42no—
134C23C2×C4×D17311yescharacteristic
135C23C2×C4×D17312yes—
136C23D4×D17312yes—
137C23D4×D17311yescharacteristic
138C23Q8×D17311yescharacteristic
139C2×C4—24no—
140C2×C4—22no—
141C2×C4—44no—
142C2×C4—42no—
143C2×C4—3464no—
144C2×C4—3462no—
145C2×C4—1732no—
146C2×C4—1732no—
147C2×C4—1734no—
148C2×C4—1734no—
149C2×C4—1731no—
150C2×C4—1731no—
151C2×C4—1732no—
152C2×C4—1732no—
153C2×C4—1732no—
154C2×C4—1734no—
155C2×C4—1734no—
156C2×C4—6924no—
157C2×C4—6922no—
158C22—22no—
159C22—24no—
160C22—42no—
161C22—41no—
162C22—41no—
163C22C42×D17311yescharacteristic
164C22C22⋊C4×D17312yes—
165C22C22⋊C4×D17311yescharacteristic
166C22C4⋊C4×D17311yescharacteristic
167C22C4⋊C4×D17312yes—
168C4—44no—
169C4—42no—
170C4—1732no—
171C4—1734no—
172C4—1732no—
173C4—6924no—
174C4—6922no—
175C2—22no—
176C2—21no—
177C2—21no—
178C2—22no—
179C2#95×D17311yescharacteristic
180C2#2601146312yes—
subgroupG/H
centerC22C22⋊C4×D173
commutatorC22×C173C2×C42
FrattiniC24C22×D173
Fitting#342×C173C2
socleC22×C173C2×C22⋊C4
prime radical, cosocleC24×C173C23
hypercenter#342D173
hypocenterC173#1740
LCSG ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
10. #19149635C4semidirect
12. C2×#14583805C4semidirect
16. C2×C4×Dic173C4semidirect
90. C173#1740semidirect

Maximal subgroups

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

subgroupindex
C2×#191496352
#260112582
#260112572
#342×C1732
#1740173