Finite Group Databasebeta

← #36355547

Subgroups — #36355547

Order 22144.

354 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 45 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#26011677C212yesmaximal, complemented
2#26011658C211yesmaximal, characteristic, complemented
3#26011284C211yesmaximal, characteristic, complemented
4#26011278C212yesmaximal, complemented
5D42×C173C211yesmaximal, characteristic, complemented
6#19149724C2211yescharacteristic, complemented
7#19149705—21no—
8#19149648—22no—
9#19149607—21no—
10#19149596C2211yescharacteristic, complemented
11#19149594C2212yescomplemented
12#19149593—22no—
13C22×D4×C173—22no—
14#127×C173C2211yescharacteristic
15#121×C173—22no—
16#120×C173—21no—
17#120×C173C2212yes—
18C4×D4×C173—21no—
19#14583826—21no—
20#14583825—41no—
21C2×C22⋊D173—22no—
22C2×C22⋊D173—42no—
23D4×D173—41no—
24#14583814—41no—
25#14583811—41no—
26#14583810—21no—
27C22⋊Dic173—22no—
28C22⋊Dic173—42no—
29C24×C173—21no—
30C24×C173D412yescomplemented
31C2×D4×C173—21no—
32C2×D4×C173—22no—
33C2×D4×C173—22no—
34C2×D4×C173—22no—
35C2×D4×C173—48no—
36C2×D4×C173D411yescharacteristic, complemented
37C2×D4×C173C2311yescharacteristic
38C22×C4×C173—22no—
39C4⋊C4×C173—21no—
40C22⋊C4×C173—21no—
41C22⋊C4×C173—22no—
42C22⋊C4×C173D412yescomplemented
43C42×C173D411yescharacteristic, complemented
44C22×D173—22no—
45Dic346—21no—
46D692—21no—
47C22⋊D173—42no—
48C22⋊D173—41no—
49C22⋊D173—84no—
50C2×Dic173—22no—
51C2×Dic173—42no—
52C4×D173—41no—
53#13063436—41no—
54C23×C173—22no—
55C23×C173—21no—
56C23×C173—42no—
57C23×C173—44no—
58C23×C173—44no—
59C23×C173—81no—
60C23×C173C2×D412yes—
61D4×C173—24no—
62D4×C173—41no—
63D4×C173—41no—
64D4×C173—44no—
65D4×C173—44no—
66D4×C173—44no—
67C2×C4×C173—21no—
68C2×C4×C173—22no—
69C2×C4×C173—44no—
70C2×C4×C173C2×D411yescharacteristic
71C22×C173—24no—
72C22×C173—22no—
73C22×C173—42no—
74C22×C173—42no—
75C22×C173—42no—
76C22×C173—42no—
77C22×C173—44no—
78C22×C173—41no—
79C22×C173—82no—
80C22×C173—84no—
81C22×C173#12011yescharacteristic
82C2×D173—41no—
83C2×D173—82no—
84C4×C173—22no—
85C4×C173—21no—
86C4×C173—42no—
87Dic173—41no—
88Dic173—82no—
89C2×C173—21no—
90C2×C173—42no—
91C2×C173—44no—
92C2×C173—81no—
93C2×C173B(4,𝔽2)11yescharacteristic
94D173—81no—
95C173#219611yescharacteristic, complemented, Sylow
96#2196—1731nomaximal, Sylow
97D42D17311yescharacteristic, complemented
98B(4,𝔽2)—1732no—
99#453—1731no—
100#353—1731no—
101AΣL(1,𝔽16)—1732no—
102H(5,𝔽2)—1731no—
103C22×D4—22no—
104AGL(1,ℤ/8)—3461no—
105#127C2×D17311yescharacteristic
106#121—22no—
107#120—21no—
108#120—3462no—
109#120C2×D17312yes—
110C4×D4—21no—
111#104—3461no—
112#100—1731no—
113#99—3462no—
114#99—1732no—
115C24—21no—
116C24C22⋊D17312yescomplemented
117#41—3461no—
118#41—6921no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—48no—
124C2×D4—3462no—
125C2×D4—6921no—
126C2×D4—6922no—
127C2×D4C22⋊D17311yescharacteristic, complemented
128C2×D4C22×D17311yescharacteristic
129C22×C4—22no—
130SD16—6921no—
131D8—6921no—
132M16—3461no—
133C4⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—22no—
136C22⋊C4—3462no—
137C22⋊C4—6922no—
138C22⋊C4C22⋊D17312yescomplemented
139C42C22⋊D17311yescharacteristic, complemented
140C23—22no—
141C23—21no—
142C23—42no—
143C23—44no—
144C23—44no—
145C23—81no—
146C23—3462no—
147C23C2×C22⋊D17312yes—
148Q8—3461no—
149D4—24no—
150D4—41no—
151D4—41no—
152D4—44no—
153D4—44no—
154D4—44no—
155D4—3461no—
156D4—13844no—
157D4—6922no—
158D4—6921no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—44no—
162C2×C4—3462no—
163C2×C4—6921no—
164C2×C4—6922no—
165C2×C4C2×C22⋊D17311yescharacteristic
166C8—6921no—
167C22—24no—
168C22—22no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—44no—
173C22—42no—
174C22—41no—
175C22—82no—
176C22—84no—
177C22—13842no—
178C22—6921no—
179C22#1914964811yescharacteristic
180C4—22no—
181C4—21no—
182C4—42no—
183C4—13842no—
184C4—6921no—
185C2—21no—
186C2—42no—
187C2—44no—
188C2—81no—
189C2—13841no—
190C2#2601167811yescharacteristic
subgroupG/H
centerC2#26011678
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
FittingD42×C173C2
socleC2×C173B(4,𝔽2)
hypercenterD42D173
hypocenterC173#2196
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #26011677C2semidirect
2. #26011658C2semidirect
3. #26011284C2semidirect
4. #26011278C2semidirect
5. D42×C173C2semidirect
6. #19149724C22semidirect
10. #19149596C22semidirect
11. #19149594C22semidirect
30. C24×C173D4semidirect
36. C2×D4×C173D4semidirect
42. C22⋊C4×C173D4semidirect
43. C42×C173D4semidirect
95. C173#2196semidirect
97. D42D173semidirect
116. C24C22⋊D173semidirect
127. C2×D4C22⋊D173semidirect
138. C22⋊C4C22⋊D173semidirect
139. C42C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260116772
#260116582
#260112842
#260112782
D42×C1732
#2196173