Finite Group Databasebeta

← #36355546

Subgroups — #36355546

Order 22144.

354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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#26012126C211yesmaximal, characteristic, complemented
2#26011678C212yesmaximal, complemented
3#26011284C211yesmaximal, characteristic, complemented
4#26011279C212yesmaximal, complemented
5#453×C173C211yesmaximal, characteristic, complemented
6C2×D4×D173—22no—
7#19149650—22no—
8#19149647—21no—
9#19149647—22no—
10#19149647C2212yes—
11C4×D692—21no—
12#19149594—22no—
13#19149594C2212yescomplemented
14H(5,𝔽2)×C173C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C173—21no—
16#127×C173C2211yescharacteristic
17#104×C173—21no—
18#100×C173C2211yescharacteristic, complemented
19C23×D173—21no—
20C23×D173D412yescomplemented
21C2×D692—21no—
22C2×C22⋊D173—22no—
23C2×C22⋊D173—22no—
24C2×C22⋊D173—42no—
25D4×D173—48no—
26C2×C4×D173—22no—
27C4⋊Dic173—21no—
28C22⋊Dic173—21no—
29C22⋊Dic173—22no—
30C22⋊Dic173D412yescomplemented
31#14583803—22no—
32#14583803—42no—
33#41×C173—21no—
34#41×C173—41no—
35C2×D4×C173—22no—
36C2×D4×C173—22no—
37C2×D4×C173—41no—
38C2×D4×C173D411yescharacteristic, complemented
39C2×D4×C173C2311yescharacteristic
40SD16×C173—41no—
41D8×C173—41no—
42M16×C173—21no—
43C42×C173D411yescharacteristic, complemented
44C22×D173—22no—
45C22×D173—44no—
46C22×D173—44no—
47C22×D173—42no—
48C22×D173—81no—
49D692—24no—
50C22⋊D173—44no—
51C22⋊D173—44no—
52C22⋊D173—84no—
53C2×Dic173—22no—
54C2×Dic173—42no—
55C4×D173—44no—
56C23×C173—22no—
57C23×C173—21no—
58C23×C173C2×D412yes—
59Q8×C173—21no—
60D4×C173—21no—
61D4×C173—41no—
62D4×C173—41no—
63D4×C173—44no—
64D4×C173—42no—
65D4×C173—41no—
66C2×C4×C173—21no—
67C2×C4×C173—22no—
68C2×C4×C173—41no—
69C2×C4×C173C2×D411yescharacteristic
70C8×C173—41no—
71C22×C173—22no—
72C22×C173—42no—
73C22×C173—42no—
74C22×C173—41no—
75C22×C173—41no—
76C22×C173—82no—
77C22×C173#12011yescharacteristic
78C2×D173—24no—
79C2×D173—42no—
80C2×D173—42no—
81C2×D173—44no—
82C2×D173—82no—
83C2×D173—84no—
84C4×C173—22no—
85C4×C173—21no—
86C4×C173—41no—
87Dic173—42no—
88Dic173—82no—
89C2×C173—21no—
90C2×C173—42no—
91C2×C173—81no—
92C2×C173—81no—
93C2×C173B(4,𝔽2)11yescharacteristic
94D173—44no—
95C173#219611yescharacteristic, complemented, Sylow
96#2196—1731nomaximal, Sylow
97D42—1731no—
98B(4,𝔽2)—1732no—
99#453D17311yescharacteristic, complemented
100#353—1731no—
101AΣL(1,𝔽16)—1732no—
102H(5,𝔽2)C2×D17311yescharacteristic, complemented
103C22×D4—3462no—
104AGL(1,ℤ/8)—21no—
105#127C2×D17311yescharacteristic
106#121—3462no—
107#120—3461no—
108#120—3462no—
109#120—1732no—
110C4×D4—3461no—
111#104—21no—
112#100C2×D17311yescharacteristic, complemented
113#99—3462no—
114#99—1732no—
115C24—3461no—
116C24—1732no—
117#41—21no—
118#41—41no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—41no—
122C2×D4—3461no—
123C2×D4—3462no—
124C2×D4—3462no—
125C2×D4—6928no—
126C2×D4—6922no—
127C2×D4C22⋊D17311yescharacteristic, complemented
128C2×D4C22×D17311yescharacteristic
129C22×C4—3462no—
130SD16—41no—
131D8—41no—
132M16—21no—
133C4⋊C4—3461no—
134C22⋊C4—3461no—
135C22⋊C4—3462no—
136C22⋊C4—3462no—
137C22⋊C4—1732no—
138C22⋊C4—6922no—
139C42C22⋊D17311yescharacteristic, complemented
140C23—22no—
141C23—21no—
142C23—3462no—
143C23—13841no—
144C23—6922no—
145C23—6924no—
146C23—6924no—
147C23D4×D17312yes—
148Q8—21no—
149D4—21no—
150D4—41no—
151D4—41no—
152D4—44no—
153D4—42no—
154D4—41no—
155D4—3464no—
156D4—13844no—
157D4—6924no—
158D4—6924no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—3462no—
163C2×C4—6924no—
164C2×C4—6922no—
165C2×C4C2×C22⋊D17311yescharacteristic
166C8—41no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—41no—
171C22—41no—
172C22—82no—
173C22—3464no—
174C22—13842no—
175C22—13844no—
176C22—6922no—
177C22—6924no—
178C22—6922no—
179C22#1914964711yescharacteristic
180C4—22no—
181C4—21no—
182C4—41no—
183C4—13842no—
184C4—6922no—
185C2—21no—
186C2—42no—
187C2—81no—
188C2—81no—
189C2—6924no—
190C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#453×C173C2
socleC2×C173B(4,𝔽2)
hypercenter#453D173
hypocenterC173#2196
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #26012126C2semidirect
2. #26011678C2semidirect
3. #26011284C2semidirect
4. #26011279C2semidirect
5. #453×C173C2semidirect
13. #19149594C22semidirect
14. H(5,𝔽2)×C173C22semidirect
18. #100×C173C22semidirect
20. C23×D173D4semidirect
30. C22⋊Dic173D4semidirect
38. C2×D4×C173D4semidirect
43. C42×C173D4semidirect
95. C173#2196semidirect
99. #453D173semidirect
102. H(5,𝔽2)C2×D173semidirect
112. #100C2×D173semidirect
127. C2×D4C22⋊D173semidirect
139. C42C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260121262
#260116782
#260112842
#260112792
#453×C1732
#2196173