Finite Group Databasebeta

← #36363492

Subgroups — #36363492

Order 22144.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19149638C211yesmaximal, characteristic, complemented
2#26011829C214yesmaximal, complemented
3#26011812C214yesmaximal, complemented
4#26011534C212yesmaximal, complemented
5#26011492C212yesmaximal, complemented
6#26011455C211yesmaximal, characteristic, complemented
7#525×C173C211yesmaximal, characteristic
8C2×#14583810C2212yes—
9#19149638—22no—
10#19149638C2214yes—
11C2×C4⋊Dic173C2212yes—
12C2×C22⋊Dic173—21no—
13C2×C4×Dic173—21no—
14#19149612C2214yescomplemented
15#19149610C2214yescomplemented
16#19149603C2214yescomplemented
17#19149600C2214yescomplemented
18C22⋊#13063436C2214yes—
19C2×#41×C173C2211yescharacteristic
20C23×C4×C173C2211yescharacteristic
21#123×C173—21no—
22C22⋊Q8×C173C2212yes—
23#121×C173C2212yes—
24#120×C173—21no—
25C4×D4×C173—22no—
26#117×C173C2211yescharacteristic
27C22×Dic173—22no—
28#14583810—24no—
29C2×#13063436C2314yes—
30C4⋊Dic173—22no—
31C4⋊Dic173C2314yes—
32C22⋊Dic173—24no—
33C4×Dic173—24no—
34C24×C173D411yescharacteristic
35#41×C173—24no—
36C2×Q8×C173C2311yescharacteristic
37C2×D4×C173—21no—
38C2×D4×C173—22no—
39C2×D4×C173C2311yescharacteristic
40C22×C4×C173—24no—
41C22×C4×C173—21no—
42C22×C4×C173D411yescharacteristic
43C22×C4×C173D412yes—
44C22×C4×C173C2312yes—
45C22×C4×C173C2311yescharacteristic
46C4⋊C4×C173—22no—
47C4⋊C4×C173C2312yes—
48C22⋊C4×C173—21no—
49C22⋊C4×C173—22no—
50C22⋊C4×C173—22no—
51C42×C173—21no—
52C2×Dic173—24no—
53C2×Dic173—44no—
54#13063436—24no—
55C23×C173—22no—
56C23×C173—21no—
57C23×C173—42no—
58C23×C173C2×D412yes—
59C23×C173C2×D411yescharacteristic
60Q8×C173—22no—
61D4×C173—22no—
62D4×C173—44no—
63D4×C173—42no—
64C2×C4×C173—22no—
65C2×C4×C173—24no—
66C2×C4×C173—24no—
67C2×C4×C173—22no—
68C2×C4×C173—21no—
69C2×C4×C173—42no—
70C2×C4×C173C2×D412yes—
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173#4114yes—
73C2×C4×C173C2411yescharacteristic
74C22×C173—22no—
75C22×C173—22no—
76C22×C173—22no—
77C22×C173—22no—
78C22×C173—44no—
79C22×C173—41no—
80C22×C173—42no—
81C22×C173#12312yes—
82C22×C173C22×D411yescharacteristic
83C4×C173—22no—
84C4×C173—41no—
85C4×C173—42no—
86C4×C173#12312yes—
87C4×C173C2×#4112yes—
88Dic173—44no—
89C2×C173—22no—
90C2×C173—42no—
91C2×C173—81no—
92C2×C173C2×#12311yescharacteristic
93C2×C173#57512yes—
94C173#309311yescharacteristic, complemented, Sylow
95#3093—1731nomaximal, Sylow
96#525D17311yescharacteristic
97C2×#117—1731no—
98#485—1734no—
99#482—1734no—
100#428—1732no—
101#417—1732no—
102#407—1731no—
103C2×#41C2×D17311yescharacteristic
104C23×C4C2×D17311yescharacteristic
105C2×M16—1732no—
106#123—21no—
107C22⋊Q8C2×D17312yes—
108#121C2×D17312yes—
109#120—21no—
110C4×D4—22no—
111#117—3462no—
112#117—1734no—
113#117C2×D17311yescharacteristic
114C2×C4⋊C4—1732no—
115C2×C22⋊C4—3461no—
116C2×C42—3461no—
117#107—1734no—
118#106—1734no—
119#103—1734no—
120#102—1734no—
121C22⋊C8—1734no—
122C24C22⋊D17311yescharacteristic
123#41—24no—
124C2×Q8C22×D17311yescharacteristic
125C2×D4—22no—
126C2×D4—21no—
127C2×D4C22×D17311yescharacteristic
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—3462no—
131C22×C4C22⋊D17311yescharacteristic
132C22×C4C22⋊D17312yes—
133C22×C4C22×D17312yes—
134C22×C4C22×D17311yescharacteristic
135M16—3464no—
136C2×C8—1734no—
137C4⋊C4—22no—
138C4⋊C4—3462no—
139C4⋊C4—1734no—
140C4⋊C4C22×D17312yes—
141C22⋊C4—21no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—3464no—
145C42—21no—
146C42—3464no—
147C23—22no—
148C23—21no—
149C23—42no—
150C23C2×C22⋊D17312yes—
151C23C2×C22⋊D17311yescharacteristic
152Q8—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—3464no—
163C2×C4—6924no—
164C2×C4C2×C22⋊D17312yes—
165C2×C4C2×C22⋊D17311yescharacteristic
166C2×C4#1458382514yes—
167C2×C4C23×D17311yescharacteristic
168C8—3464no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—41no—
175C22—42no—
176C22#1914966012yes—
177C22C22×C22⋊D17311yescharacteristic
178C4—22no—
179C4—41no—
180C4—42no—
181C4—6924no—
182C4#1914966012yes—
183C4C2×#1458382512yes—
184C2—22no—
185C2—42no—
186C2—81no—
187C2C2×#1914966011yescharacteristic
188C2#2601228912yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#525×C173C2
socleC22×C173C22×D4
hypercenter#525D173
hypocenterC173#3093
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#19149638C2semidirect
2. #26011829C2semidirect
3. #26011812C2semidirect
4. #26011534C2semidirect
5. #26011492C2semidirect
6. #26011455C2semidirect
14. #19149612C22semidirect
15. #19149610C22semidirect
16. #19149603C22semidirect
17. #19149600C22semidirect
94. C173#3093semidirect

Maximal subgroups

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

subgroupindex
C2×#191496382
#260118292
#260118122
#260115342
#260114922
#260114552
#525×C1732
#3093173