Finite Group Databasebeta

← #36354009

Subgroups — #36354009

Order 22144.

358 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 89 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#26012002C211yesmaximal, characteristic
2C2×#19149638C211yesmaximal, characteristic, complemented
3C2×#19149605C212yesmaximal, complemented
4#26011252C212yesmaximal, complemented
5#407×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2211yescharacteristic
7#19149662—21no—
8#19149658C412yescomplemented
9#19149653C412yescomplemented
10#19149648—21no—
11C4×C22⋊D173—22no—
12#19149638—24no—
13#19149638C2211yescharacteristic
14C2×C4⋊Dic173—21no—
15C2×C22⋊Dic173—21no—
16C2×C4×Dic173C2212yes—
17#19149605—24no—
18C23×C4×C173C2211yescharacteristic
19C2×M16×C173C2212yes—
20C22⋊C8×C173—22no—
21#14583826—22no—
22#14583826—22no—
23C2×Dic346C2×C411yescharacteristic
24C2×D692C2×C411yescharacteristic
25C2×C22⋊D173—22no—
26C2×C22⋊D173—21no—
27C22×Dic173—22no—
28C2×C4×D173—21no—
29C4⋊Dic173—24no—
30C4⋊Dic173C2×C412yes—
31#14583805—22no—
32C22⋊Dic173—24no—
33C22⋊Dic173—22no—
34C22⋊Dic173—21no—
35#14583803—22no—
36C4×Dic173—22no—
37C4×Dic173—21no—
38C4×Dic173D414yescomplemented
39C24×C173D411yescharacteristic
40C22×C4×C173—22no—
41C22×C4×C173—22no—
42C22×C4×C173C2×C412yes—
43C22×C4×C173D411yescharacteristic
44C22×C4×C173D412yes—
45C22×C4×C173C2311yescharacteristic
46M16×C173—22no—
47M16×C173—22no—
48C2×C8×C173—22no—
49C22×D173—21no—
50Dic346—22no—
51D692—22no—
52C22⋊D173—44no—
53C22⋊D173—41no—
54C22⋊D173—41no—
55C2×Dic173—21no—
56C2×Dic173—22no—
57C2×Dic173—24no—
58C2×Dic173—44no—
59C4×D173—42no—
60C23×C173—21no—
61C23×C173—21no—
62C23×C173—42no—
63C23×C173C22⋊C412yes—
64C23×C173C2×D411yescharacteristic
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—24no—
68C2×C4×C173—24no—
69C2×C4×C173C22⋊C412yes—
70C2×C4×C173C22×C411yescharacteristic
71C2×C4×C173C2×D412yes—
72C2×C4×C173C2×D411yescharacteristic
73C2×C4×C173#4112yes—
74C8×C173—42no—
75C22×C173—22no—
76C22×C173—22no—
77C22×C173—22no—
78C22×C173—22no—
79C22×C173—42no—
80C22×C173—42no—
81C22×C173C2×C22⋊C411yescharacteristic
82C22×C173#12011yescharacteristic
83C22×C173#12311yescharacteristic
84C2×D173—41no—
85C2×D173—42no—
86C4×C173—22no—
87C4×C173C4×D412yes—
88C4×C173#12112yes—
89Dic173—44no—
90Dic173—41no—
91Dic173—42no—
92C2×C173—21no—
93C2×C173—21no—
94C2×C173—42no—
95C2×C173#38611yescharacteristic
96C2×C173#42112yes—
97D173—81no—
98C173#189711yescharacteristic, complemented, Sylow
99#1897—1731nomaximal, Sylow
100#525—1731no—
101C2×#117—1731no—
102C2×#104—1732no—
103#407D17311yescharacteristic, complemented
104#339—1732no—
105C2×#41—1731no—
106C23×C4C2×D17311yescharacteristic
107C2×M16C2×D17312yes—
108#123—3461no—
109C22⋊Q8—1732no—
110#121—1732no—
111#120—3461no—
112C4×D4—3462no—
113#117—3464no—
114#117—1731no—
115C2×C4⋊C4—3461no—
116C2×C22⋊C4—3461no—
117C2×C42—1732no—
118#104—3464no—
119C22⋊C8—22no—
120C24D69211yescharacteristic
121#41—3462no—
122#41—3462no—
123C2×Q8—1731no—
124C2×D4—3461no—
125C2×D4—3462no—
126C2×D4—1731no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—3462no—
130C22×C4—3461no—
131C22×C4C4×D17312yes—
132C22×C4C22⋊D17312yes—
133C22×C4D69211yescharacteristic
134C22×C4C22×D17311yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—3464no—
139C4⋊C4—3462no—
140C4⋊C4—1732no—
141C22⋊C4—3464no—
142C22⋊C4—3461no—
143C22⋊C4—3462no—
144C22⋊C4—3462no—
145C42—3462no—
146C42—3461no—
147C42—1734no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—3461no—
152C23#1458380312yes—
153C23C2×D69211yescharacteristic
154Q8—3462no—
155D4—3462no—
156D4—6924no—
157D4—6921no—
158D4—6921no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—3462no—
164C2×C4—3461no—
165C2×C4—3464no—
166C2×C4—6924no—
167C2×C4—6922no—
168C2×C4#1458380312yes—
169C2×C4C2×C4×D17311yescharacteristic
170C2×C4D4×D17312yes—
171C2×C4C2×C22⋊D17311yescharacteristic
172C2×C4#1458382512yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—6922no—
181C22—6921no—
182C22C2×#1458380311yescharacteristic
183C22#1914964911yescharacteristic
184C22#1914966311yescharacteristic
185C4—22no—
186C4—6924no—
187C4—6922no—
188C4—6921no—
189C4#1914963912yes—
190C4#1914965112yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—13841no—
195C2#2601136811yescharacteristic
196C2#2601150712yes—
subgroupG/H
centerC2×C4C2×C22⋊D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#407×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012002C2
cosocleC22×C4×C173C23
hypercenter#407D173
hypocenterC173#1897
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#19149638C2semidirect
3. C2×#19149605C2semidirect
4. #26011252C2semidirect
5. #407×C173C2semidirect
8. #19149658C4semidirect
9. #19149653C4semidirect
38. C4×Dic173D4semidirect
98. C173#1897semidirect
103. #407D173semidirect

Maximal subgroups

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

subgroupindex
#260120022
C2×#191496382
C2×#191496052
#260112522
#407×C1732
#1897173