Finite Group Databasebeta

← #36361868

Subgroups — #36361868

Order 22144.

556 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 169 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×C42×D173C211yesmaximal, characteristic
2#26011387C212yesmaximal, complemented
3#26011372C214yesmaximal, complemented
4#26011362C214yesmaximal
5#26011357C211yesmaximal, characteristic, complemented
6#26011348C212yesmaximal
7C2×#125×C173C211yesmaximal, characteristic, complemented
8C22×C4×D173C2212yes—
9C22×C4×D173C2211yescharacteristic
10C2×C4⋊Dic173C2212yes—
11C2×#14583805C2214yes—
12C2×#14583803C2214yes—
13C2×C4×Dic173C2212yes—
14C2×C4×Dic173C2211yescharacteristic
15C42×D173—24no—
16#19149586C2214yes—
17#19149586C2214yes—
18#19149586C2214yes—
19#125×C173—24no—
20C2×C4⋊C4×C173C2212yes—
21C2×C4⋊C4×C173C2214yes—
22C2×C42×C173C2211yescharacteristic
23C23×D173C2311yescharacteristic
24C22×Dic173C2314yes—
25C22×Dic173C2311yescharacteristic
26C22×Dic173C2312yes—
27C2×C4×D173—24no—
28C2×C4×D173—24no—
29C2×C4×D173—24no—
30C2×C4×D173—22no—
31C2×C4×D173—22no—
32C2×C4×D173Q814yes—
33C4⋊Dic173—24no—
34#14583805—28no—
35#14583803—28no—
36C4×Dic173—24no—
37C4×Dic173—22no—
38C22×C4×C173C2314yes—
39C22×C4×C173C2312yes—
40C22×C4×C173C2311yescharacteristic
41C4⋊C4×C173—28no—
42C4⋊C4×C173—24no—
43C42×C173D414yes—
44C22×D173—21no—
45C22×D173—22no—
46C22×D173—21no—
47C22×D173C2×Q812yes—
48C22×D173#4114yes—
49C2×Dic173—21no—
50C2×Dic173—22no—
51C2×Dic173—22no—
52C2×Dic173—21no—
53C2×Dic173—24no—
54C2×Dic173—24no—
55C2×Dic173—24no—
56C2×Dic173C2×Q812yes—
57C2×Dic173#4114yes—
58C4×D173—216no—
59C4×D173—28no—
60C23×C173C2411yescharacteristic
61C2×C4×C173—22no—
62C2×C4×C173—22no—
63C2×C4×C173—24no—
64C2×C4×C173—24no—
65C2×C4×C173—24no—
66C2×C4×C173C2×D414yes—
67C2×C4×C173C2×D412yes—
68C2×C4×C173C2×Q812yes—
69C2×C4×C173#4112yes—
70C22×C173C22×D411yescharacteristic
71C22×C173C22×Q811yescharacteristic
72C22×C173C2×#4111yescharacteristic
73C22×C173C2×#4112yes—
74C22×C173C2×#4112yes—
75C2×D173—22no—
76C2×D173—24no—
77C2×D173—24no—
78C2×D173—22no—
79C2×D173#12514yescomplemented
80C4×C173—24no—
81C4×C173—44no—
82C4×C173C22⋊Q814yes—
83Dic173—22no—
84Dic173—24no—
85Dic173—44no—
86C2×C173C2×C22⋊Q811yescharacteristic
87C2×C173#52512yes—
88C2×C173C2×#12511yescharacteristic
89C2×C173#52912yes—
90C2×C173#53311yescharacteristic
91D173—24no—
92C173#285111yescharacteristic, complemented, Sylow
93#2851—1731nomaximal, Sylow
94C2×#125D17311yescharacteristic, complemented
95C22×C42—1731no—
96#389—1732no—
97#387—1734no—
98#385—1734no—
99#382—1731no—
100#380—1732no—
101C23×C4—1732no—
102C23×C4—1731no—
103#125—24no—
104C2×C4⋊C4—1734no—
105C2×C4⋊C4—1732no—
106C2×C4⋊C4C2×D17312yes—
107C2×C4⋊C4C2×D17314yes—
108C2×C22⋊C4—1734no—
109C2×C42—3464no—
110C2×C42—1732no—
111C2×C42—1731no—
112C2×C42C2×D17311yescharacteristic
113#95—1734no—
114#95—1734no—
115#95—1734no—
116C24—1731no—
117C22×C4—3464no—
118C22×C4—3464no—
119C22×C4—3464no—
120C22×C4—3462no—
121C22×C4—3462no—
122C22×C4—1734no—
123C22×C4—1731no—
124C22×C4—1732no—
125C22×C4—1734no—
126C22×C4C22×D17314yes—
127C22×C4C22×D17312yes—
128C22×C4C22×D17311yescharacteristic
129C4⋊C4—28no—
130C4⋊C4—24no—
131C4⋊C4—3468no—
132C4⋊C4—3464no—
133C22⋊C4—3468no—
134C42—3464no—
135C42—3462no—
136C42C22⋊D17314yes—
137C23—3461no—
138C23—3462no—
139C23—3461no—
140C23—1732no—
141C23—1734no—
142C23C23×D17311yescharacteristic
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—34616no—
149C2×C4—3461no—
150C2×C4—3462no—
151C2×C4—3462no—
152C2×C4—3461no—
153C2×C4—3468no—
154C2×C4—3464no—
155C2×C4—3464no—
156C2×C4—3464no—
157C2×C4—1732no—
158C2×C4—1734no—
159C2×C4C2×C22⋊D17314yes—
160C2×C4C2×C22⋊D17312yes—
161C2×C4Q8×D17312yes—
162C2×C4#1458382712yes—
163C22—3462no—
164C22—3464no—
165C22—3464no—
166C22—3462no—
167C22—1734no—
168C22C22×C22⋊D17311yescharacteristic
169C22C2×Q8×D17311yescharacteristic
170C22#41×D17312yes—
171C22#41×D17312yes—
172C22C2×#1458382711yescharacteristic
173C4—24no—
174C4—44no—
175C4—3462no—
176C4—3464no—
177C4—6924no—
178C4#1914965514yes—
179C2—3464no—
180C2C2×#1914965511yescharacteristic
181C2#2601199812yes—
182C2#125×D17311yescharacteristic
183C2#2601202612yes—
184C2#2601204711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
FittingC2×#125×C173C2
prime radicalC23×D173C23
hypercenterC2×#125D173
hypocenterC173#2851
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #26011387C2semidirect
3. #26011372C2semidirect
5. #26011357C2semidirect
7. C2×#125×C173C2semidirect
79. C2×D173#125semidirect
92. C173#2851semidirect
94. C2×#125D173semidirect

Maximal subgroups

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

subgroupindex
C2×C42×D1732
#260113872
#260113722
#260113622
#260113572
#260113482
C2×#125×C1732
#2851173