Finite Group Databasebeta

← #36355563

Subgroups — #36355563

Order 22144.

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

normal subgroups only

This group has 270 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#26012233C211yesmaximal, characteristic, complemented
2#26011678C212yesmaximal, complemented
3#26011286C211yesmaximal, characteristic, complemented
4#26011279C212yesmaximal, complemented
5#455×C173C211yesmaximal, characteristic, complemented
6#19149668—21no—
7#19149667—21no—
8#19149665—22no—
9#19149649—22no—
10#19149647—22no—
11#19149647C2212yes—
12#19149594—22no—
13#19149594C2212yes—
14H(5,𝔽2)×C173C2211yescharacteristic
15#137×C173—21no—
16#124×C173C2211yescharacteristic
17#104×C173—21no—
18#100×C173C2211yescharacteristic
19C23×D173D412yescomplemented
20C2×Dic346—21no—
21C2×D692—21no—
22C2×D692—21no—
23C2×C22⋊D173—22no—
24C2×C22⋊D173—42no—
25C22⋊Dic173—22no—
26C22⋊Dic173D412yescomplemented
27#14583803—22no—
28#14583803—22no—
29#14583803—22no—
30#14583803—42no—
31C4×Dic173—21no—
32#41×C173—21no—
33#41×C173—41no—
34C2×Q8×C173D411yescharacteristic, complemented
35C2×D4×C173—22no—
36C2×D4×C173—41no—
37C2×D4×C173C2311yescharacteristic
38Q16×C173—41no—
39SD16×C173—41no—
40M16×C173—21no—
41C22⋊C4×C173—22no—
42C42×C173D411yescharacteristic, complemented
43C22×D173—22no—
44C22×D173—42no—
45C22×D173—82no—
46Dic346—81no—
47D692—81no—
48D692—81no—
49C22⋊D173—82no—
50C22⋊D173—84no—
51C2×Dic173—22no—
52C2×Dic173—42no—
53C23×C173—22no—
54C23×C173C2×D412yes—
55Q8×C173—21no—
56Q8×C173—41no—
57D4×C173—21no—
58D4×C173—41no—
59D4×C173—42no—
60D4×C173—41no—
61C2×C4×C173—22no—
62C2×C4×C173—21no—
63C2×C4×C173—21no—
64C2×C4×C173—41no—
65C2×C4×C173C2×D411yescharacteristic
66C8×C173—41no—
67C22×C173—22no—
68C22×C173—42no—
69C22×C173—41no—
70C22×C173—82no—
71C22×C173#12011yescharacteristic
72C2×D173—42no—
73C2×D173—42no—
74C2×D173—44no—
75C2×D173—42no—
76C2×D173—82no—
77C4×C173—21no—
78C4×C173—41no—
79C4×C173—41no—
80C4×C173—41no—
81Dic173—42no—
82Dic173—82no—
83C2×C173—21no—
84C2×C173—42no—
85C2×C173—81no—
86C2×C173B(4,𝔽2)11yescharacteristic
87D173—82no—
88C173#219911yescharacteristic, complemented, Sylow
89#2199—1731nomaximal, Sylow
90H(3,𝔽4)—1731no—
91B(4,𝔽2)—1732no—
92#455D17311yescharacteristic, complemented
93#354—1731no—
94AΣL(1,𝔽16)—1732no—
95H(5,𝔽2)C2×D17311yescharacteristic
96#137—21no—
97#124—3462no—
98#124—3461no—
99#124—3461no—
100#124C2×D17311yescharacteristic
101#120—3462no—
102#120—3462no—
103#120—1732no—
104#104—21no—
105#100C2×D17311yescharacteristic
106#99—3462no—
107#99—1732no—
108C24—1732no—
109#41—21no—
110#41—41no—
111C2×Q8—3461no—
112C2×Q8C22⋊D17311yescharacteristic, complemented
113C2×D4—22no—
114C2×D4—41no—
115C2×D4—3461no—
116C2×D4—3461no—
117C2×D4—3462no—
118C2×D4—6922no—
119C2×D4C22×D17311yescharacteristic
120Q16—41no—
121SD16—41no—
122M16—21no—
123C22⋊C4—22no—
124C22⋊C4—3462no—
125C22⋊C4—3462no—
126C22⋊C4—3462no—
127C22⋊C4—3462no—
128C22⋊C4—1732no—
129C22⋊C4—6922no—
130C42—3461no—
131C42C22⋊D17311yescharacteristic, complemented
132C23—22no—
133C23—3462no—
134C23—13842no—
135C23—6922no—
136C23D4×D17312yes—
137Q8—21no—
138Q8—41no—
139Q8—13841no—
140D4—21no—
141D4—41no—
142D4—42no—
143D4—41no—
144D4—13841no—
145D4—13841no—
146D4—13844no—
147D4—13842no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—3462no—
153C2×C4—6922no—
154C2×C4C2×C22⋊D17311yescharacteristic
155C8—41no—
156C22—22no—
157C22—42no—
158C22—41no—
159C22—82no—
160C22—13842no—
161C22—6922no—
162C22—6922no—
163C22—6924no—
164C22—6922no—
165C22#1914964711yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—41no—
170C4—13842no—
171C4—6922no—
172C2—21no—
173C2—42no—
174C2—81no—
175C2—13842no—
176C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#455×C173C2
socleC2×C173B(4,𝔽2)
hypercenter#455D173
hypocenterC173#2199
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #26012233C2semidirect
2. #26011678C2semidirect
3. #26011286C2semidirect
4. #26011279C2semidirect
5. #455×C173C2semidirect
19. C23×D173D4semidirect
26. C22⋊Dic173D4semidirect
34. C2×Q8×C173D4semidirect
42. C42×C173D4semidirect
88. C173#2199semidirect
92. #455D173semidirect
112. C2×Q8C22⋊D173semidirect
131. C42C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260122332
#260116782
#260112862
#260112792
#455×C1732
#2199173