Finite Group Databasebeta

← #36355559

Subgroups — #36355559

Order 22144.

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

normal subgroups only

This group has 246 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#26011680C212yesmaximal, complemented
2#26011658C211yesmaximal, characteristic
3#26011284C211yesmaximal, characteristic, complemented
4#26011281C212yesmaximal, complemented
5#559×C173C211yesmaximal, characteristic, complemented
6#19149724C2211yescharacteristic
7#19149705—21no—
8#19149662—22no—
9#19149607—21no—
10#19149596C2211yescharacteristic
11#19149594C2212yes—
12#19149593—22no—
13#127×C173C2211yescharacteristic
14#124×C173—22no—
15#123×C173—21no—
16#123×C173C2212yes—
17C4×Q8×C173—21no—
18C4×D4×C173—22no—
19#14583826—21no—
20#14583825—41no—
21C2×C22⋊D173—22no—
22D4×D173—41no—
23#14583814—41no—
24#14583811—41no—
25#14583810—21no—
26#14583805—42no—
27C22⋊Dic173—22no—
28#14583803—42no—
29C2×Q8×C173—21no—
30C2×D4×C173—22no—
31C2×D4×C173D411yescharacteristic
32C2×D4×C173C2311yescharacteristic
33C22×C4×C173—21no—
34C22×C4×C173D412yes—
35C4⋊C4×C173—21no—
36C4⋊C4×C173—22no—
37C22⋊C4×C173—22no—
38C22⋊C4×C173—21no—
39C22⋊C4×C173—22no—
40C22⋊C4×C173D412yes—
41C42×C173—22no—
42C42×C173D411yescharacteristic
43C22×D173—22no—
44Dic346—21no—
45D692—21no—
46C22⋊D173—42no—
47C22⋊D173—41no—
48C2×Dic173—22no—
49C2×Dic173—42no—
50C4×D173—41no—
51#13063436—41no—
52C23×C173—21no—
53C23×C173C2×D412yes—
54Q8×C173—42no—
55D4×C173—41no—
56D4×C173—41no—
57D4×C173—44no—
58C2×C4×C173—21no—
59C2×C4×C173—22no—
60C2×C4×C173—22no—
61C2×C4×C173—42no—
62C2×C4×C173—81no—
63C2×C4×C173C2×D411yescharacteristic
64C22×C173—22no—
65C22×C173—42no—
66C22×C173—42no—
67C22×C173—41no—
68C22×C173#12011yescharacteristic
69C2×D173—41no—
70C2×D173—82no—
71C4×C173—22no—
72C4×C173—21no—
73C4×C173—42no—
74C4×C173—42no—
75Dic173—41no—
76Dic173—82no—
77C2×C173—21no—
78C2×C173—42no—
79C2×C173—81no—
80C2×C173B(4,𝔽2)11yescharacteristic
81D173—81no—
82C173#219811yescharacteristic, complemented, Sylow
83#2198—1731nomaximal, Sylow
84#559D17311yescharacteristic, complemented
85#458—1732no—
86#453—1731no—
87#353—1731no—
88#352—1732no—
89H(5,𝔽2)—1731no—
90AGL(1,ℤ/8)—3461no—
91#127C2×D17311yescharacteristic
92#124—22no—
93#123—21no—
94#123—3462no—
95#123C2×D17312yes—
96C4×Q8—21no—
97C4×D4—22no—
98#104—3461no—
99#100—1731no—
100#99—3462no—
101#99—1732no—
102#41—3461no—
103#41—6921no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—3462no—
107C2×D4—6921no—
108C2×D4C22⋊D17311yescharacteristic
109C2×D4C22×D17311yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D17312yes—
112SD16—6921no—
113D8—6921no—
114M16—3461no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—6922no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—3462no—
122C22⋊C4—6922no—
123C22⋊C4C22⋊D17312yes—
124C42—22no—
125C42C22⋊D17311yescharacteristic
126C23—21no—
127C23—3462no—
128C23C2×C22⋊D17312yes—
129Q8—42no—
130Q8—3461no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—3461no—
135D4—6922no—
136D4—6921no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—3462no—
143C2×C4—6921no—
144C2×C4—6922no—
145C2×C4C2×C22⋊D17311yescharacteristic
146C8—6921no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—13842no—
152C22—6921no—
153C22#1914964811yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—13842no—
159C4—6921no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—13841no—
164C2#2601167811yescharacteristic
subgroupG/H
centerC2#26011678
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#559×C173C2
socleC2×C173B(4,𝔽2)
prime radical#26011658C2
hypercenter#559D173
hypocenterC173#2198
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #26011680C2semidirect
3. #26011284C2semidirect
4. #26011281C2semidirect
5. #559×C173C2semidirect
82. C173#2198semidirect
84. #559D173semidirect

Maximal subgroups

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

subgroupindex
#260116802
#260116582
#260112842
#260112812
#559×C1732
#2198173