Finite Group Databasebeta

← #36354630

Subgroups — #36354630

Order 22144.

394 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 69 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#26012291C211yesmaximal, characteristic, complemented
2#26012002C212yesmaximal, complemented
3#26011507C212yesmaximal, complemented
4#26011248C211yesmaximal, characteristic, complemented
5#407×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2212yes—
7#19149712—22no—
8#19149707—22no—
9#19149701—22no—
10#19149662—22no—
11#19149658—22no—
12#19149653—22no—
13#19149648—22no—
14C4×C22⋊D173—24no—
15#19149638C2212yes—
16#19149605—24no—
17C23×C4×C173C2211yescharacteristic
18C2×M16×C173C2211yescharacteristic
19C2×M16×C173C2211yescharacteristic
20C22⋊C8×C173—22no—
21#14583826—22no—
22#14583826—22no—
23#14583826—22no—
24#14583826—22no—
25C2×Dic346D412yescomplemented
26C2×D692D412yescomplemented
27C2×C22⋊D173—24no—
28C2×C22⋊D173—22no—
29C2×C4×D173—22no—
30Dic692—22no—
31#14583815—24no—
32D1384—22no—
33C4⋊Dic173—22no—
34#14583805—24no—
35C22⋊Dic173—24no—
36C22⋊Dic173—22no—
37#14583803—24no—
38C4×Dic173D414yescomplemented
39C24×C173D411yescharacteristic
40C22×C4×C173—22no—
41C22×C4×C173—22no—
42C22×C4×C173—21no—
43C22×C4×C173—21no—
44C22×C4×C173D411yescharacteristic
45C22×C4×C173C2311yescharacteristic
46M16×C173—22no—
47M16×C173—22no—
48C2×C8×C173—21no—
49C2×C8×C173D412yescomplemented
50C22×D173—22no—
51Dic346—22no—
52Dic346—22no—
53D692—22no—
54D692—22no—
55C22⋊D173—48no—
56C22⋊D173—42no—
57C22⋊D173—42no—
58C2×Dic173—22no—
59C2×Dic173—24no—
60C4×D173—42no—
61C4×D173—42no—
62C23×C173—22no—
63C23×C173—21no—
64C23×C173—42no—
65C23×C173C2×D411yescharacteristic
66C2×C4×C173—22no—
67C2×C4×C173—21no—
68C2×C4×C173—41no—
69C2×C4×C173—42no—
70C2×C4×C173—41no—
71C2×C4×C173C2×D412yes—
72C2×C4×C173C2×D412yes—
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173#4111yescharacteristic
75C8×C173—22no—
76C8×C173—41no—
77C22×C173—22no—
78C22×C173—24no—
79C22×C173—41no—
80C22×C173—42no—
81C22×C173—41no—
82C22×C173—41no—
83C22×C173#12012yes—
84C22×C173#12111yescharacteristic
85C2×D173—44no—
86C2×D173—42no—
87C4×C173—41no—
88C4×C173#12011yescharacteristic
89C4×C173#12112yes—
90C4×C173#12711yescharacteristic
91Dic173—44no—
92Dic173—42no—
93C2×C173—22no—
94C2×C173—21no—
95C2×C173—42no—
96C2×C173#39211yescharacteristic
97D173—82no—
98C173#201011yescharacteristic, complemented, Sylow
99#2010—1731nomaximal, Sylow
100#575—1731no—
101#525—1732no—
102#421—1732no—
103#407D17311yescharacteristic, complemented
104H(3,ℤ/4)—1731no—
105C2×#41—1732no—
106C23×C4C2×D17311yescharacteristic
107#137—3462no—
108AGL(1,ℤ/8)—3462no—
109#135—3462no—
110C2×M16C2×D17311yescharacteristic
111C2×M16C2×D17311yescharacteristic
112#123—3462no—
113C22⋊Q8—3462no—
114#121—3462no—
115#120—3462no—
116C4×D4—3464no—
117#117—1732no—
118#104—3464no—
119C22⋊C8—22no—
120C24D69211yescharacteristic
121#41—3462no—
122#41—3462no—
123#41—3462no—
124#41—3462no—
125C2×Q8—1732no—
126C2×D4—3464no—
127C2×D4—3462no—
128C2×D4—1732no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—3462no—
134C22×C4D69211yescharacteristic
135C22×C4C22×D17311yescharacteristic
136Q16—3462no—
137SD16—3464no—
138D8—3462no—
139M16—22no—
140M16—22no—
141C2×C8—21no—
142C2×C8C22⋊D17312yescomplemented
143C4⋊C4—3464no—
144C4⋊C4—3462no—
145C22⋊C4—3464no—
146C22⋊C4—3464no—
147C22⋊C4—3462no—
148C42—1734no—
149C23—22no—
150C23—21no—
151C23—42no—
152C23—3462no—
153C23C2×D69211yescharacteristic
154Q8—3462no—
155Q8—3462no—
156D4—3462no—
157D4—3462no—
158D4—6928no—
159D4—6922no—
160D4—6922no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—3464no—
167C2×C4—3462no—
168C2×C4—6922no—
169C2×C4—6922no—
170C2×C4D4×D17312yes—
171C2×C4D4×D17312yes—
172C2×C4C2×C22⋊D17311yescharacteristic
173C2×C4#1458382611yescharacteristic
174C8—22no—
175C8—41no—
176C22—24no—
177C22—22no—
178C22—41no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—6924no—
183C22—6922no—
184C22#1914964912yes—
185C22#1914965311yescharacteristic
186C4—41no—
187C4—6924no—
188C4—6922no—
189C4#1914964711yescharacteristic
190C4#1914965212yes—
191C4#1914967511yescharacteristic
192C2—22no—
193C2—21no—
194C2—42no—
195C2—13842no—
196C2#2601140011yescharacteristic
subgroupG/H
centerC4#19149675
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#407×C173C2
socleC2×C173#392
hypercenter#407D173
hypocenterC173#2010
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #26012291C2semidirect
2. #26012002C2semidirect
3. #26011507C2semidirect
4. #26011248C2semidirect
5. #407×C173C2semidirect
25. C2×Dic346D4semidirect
26. C2×D692D4semidirect
38. C4×Dic173D4semidirect
49. C2×C8×C173D4semidirect
98. C173#2010semidirect
103. #407D173semidirect
142. C2×C8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260122912
#260120022
#260115072
#260112482
#407×C1732
#2010173