Finite Group Databasebeta

← #36355579

Subgroups — #36355579

Order 22144.

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

normal subgroups only

This group has 248 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#26011664C212yesmaximal, complemented
3#26011288C211yesmaximal, characteristic, complemented
4#26011287C212yesmaximal, complemented
5#454×C173C211yesmaximal, characteristic, complemented
6#19149704—22no—
7#19149668C2212yes—
8#19149667—21no—
9#19149665—22no—
10#19149649—22no—
11#19149647—21no—
12#19149606—22no—
13#19149598C2212yescomplemented
142−1+4×C173C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C173—21no—
16#124×C173C2211yescharacteristic
17#104×C173—21no—
18#101×C173C2211yescharacteristic, complemented
19C23×D173—21no—
20C2×Dic346—21no—
21C2×D692—21no—
22C2×D692D412yes—
23C2×C22⋊D173—22no—
24#14583813—42no—
25#14583811—42no—
26#14583810—22no—
27C22⋊Dic173—21no—
28#14583803—22no—
29#14583803—22no—
30#14583803—22no—
31C4×Dic173D412yes—
32#41×C173—21no—
33#41×C173—22no—
34#41×C173—41no—
35C2×Q8×C173—41no—
36C2×Q8×C173C2311yescharacteristic
37C2×D4×C173D411yescharacteristic
38SD16×C173—41no—
39D8×C173—41no—
40M16×C173—21no—
41C22⋊C4×C173—22no—
42C42×C173D411yescharacteristic
43C22×D173—22no—
44C22×D173—82no—
45C22×D173—81no—
46Dic346—81no—
47D692—42no—
48D692—81no—
49C22⋊D173—82no—
50C2×Dic173—22no—
51#13063436—42no—
52C23×C173—21no—
53Q8×C173—22no—
54Q8×C173—21no—
55Q8×C173—41no—
56D4×C173—22no—
57D4×C173—21no—
58D4×C173—41no—
59D4×C173—41no—
60C2×C4×C173—21no—
61C2×C4×C173—41no—
62C2×C4×C173—42no—
63C2×C4×C173C2×D412yes—
64C2×C4×C173C2×D411yescharacteristic
65C8×C173—41no—
66C22×C173—42no—
67C22×C173—41no—
68C22×C173—41no—
69C22×C173#12011yescharacteristic
70C2×D173—42no—
71C2×D173—42no—
72C2×D173—42no—
73C2×D173—84no—
74C4×C173—21no—
75C4×C173—22no—
76C4×C173—41no—
77C4×C173—41no—
78Dic173—42no—
79C2×C173—21no—
80C2×C173—81no—
81C2×C173—81no—
82C2×C173B(4,𝔽2)11yescharacteristic
83D173—82no—
84C173#220211yescharacteristic, complemented, Sylow
85#2202—1731nomaximal, Sylow
86H(3,𝔽4)—1731no—
87#454—1732no—
88#454D17311yescharacteristic, complemented
89#355—1731no—
90#355—1732no—
912−1+4C2×D17311yescharacteristic, complemented
92AGL(1,ℤ/8)—21no—
93AGL(1,ℤ/8)—3462no—
94#124—3462no—
95#124—3461no—
96#124—1732no—
97#124C2×D17311yescharacteristic
98#120—3462no—
99#120—3461no—
100#104—21no—
101#104—3462no—
102#101—1732no—
103#101C2×D17311yescharacteristic, complemented
104C24—3461no—
105#41—22no—
106#41—21no—
107#41—41no—
108C2×Q8—41no—
109C2×Q8—3461no—
110C2×Q8C22×D17311yescharacteristic
111C2×D4—3461no—
112C2×D4—3462no—
113C2×D4—1732no—
114C2×D4C22⋊D17311yescharacteristic
115SD16—41no—
116SD16—6922no—
117D8—41no—
118D8—6922no—
119M16—21no—
120M16—3462no—
121C22⋊C4—22no—
122C22⋊C4—3462no—
123C22⋊C4—3461no—
124C22⋊C4—3462no—
125C22⋊C4—3462no—
126C42—1732no—
127C42C22⋊D17311yescharacteristic
128C23—21no—
129C23—3462no—
130C23—13842no—
131C23—13841no—
132Q8—21no—
133Q8—22no—
134Q8—41no—
135Q8—13841no—
136D4—21no—
137D4—22no—
138D4—41no—
139D4—41no—
140D4—13841no—
141D4—13842no—
142D4—6922no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—3462no—
147C2×C4D4×D17312yes—
148C2×C4C2×C22⋊D17311yescharacteristic
149C8—41no—
150C8—6922no—
151C22—42no—
152C22—41no—
153C22—41no—
154C22—13844no—
155C22—6922no—
156C22—6922no—
157C22—6922no—
158C22#1914964711yescharacteristic
159C4—21no—
160C4—22no—
161C4—41no—
162C4—41no—
163C4—6922no—
164C2—21no—
165C2—81no—
166C2—81no—
167C2—13842no—
168C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×Q8×C173C23
FrattiniC2×Q8C22×D173
Fitting#454×C173C2
socleC2×C173B(4,𝔽2)
hypercenter#454D173
hypocenterC173#2202
LCSG ▷ C2×Q8×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #26012233C2semidirect
2. #26011664C2semidirect
3. #26011288C2semidirect
4. #26011287C2semidirect
5. #454×C173C2semidirect
13. #19149598C22semidirect
14. 2−1+4×C173C22semidirect
18. #101×C173C22semidirect
84. C173#2202semidirect
88. #454D173semidirect
91. 2−1+4C2×D173semidirect
103. #101C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122332
#260116642
#260112882
#260112872
#454×C1732
#2202173