Finite Group Databasebeta

← #36354098

Subgroups — #36354098

Order 22144.

314 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 85 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#26012065C211yesmaximal, characteristic
2#26011969C211yesmaximal, characteristic, complemented
3#26011507C212yesmaximal
4#26011248C212yesmaximal, complemented
5C2×#101×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2211yescharacteristic
7#19149677C412yescomplemented
8#19149668C412yescomplemented
9#19149662—22no—
10#19149655—22no—
11Q8×Dic173—24no—
12#19149638—22no—
13#19149638C2211yescharacteristic
14#19149638C2212yes—
15#19149605—24no—
16C22×Q8×C173C2211yescharacteristic
17C2×M16×C173C2212yes—
18#101×C173—22no—
19#14583826—22no—
20#14583826—22no—
21C2×Dic346C2×C411yescharacteristic
22C2×D692C2×C411yescharacteristic
23C2×C22⋊D173—21no—
24C2×C4×D173—21no—
25C4⋊Dic173—24no—
26C4⋊Dic173—22no—
27C4⋊Dic173—21no—
28#14583805—24no—
29C22⋊Dic173—22no—
30C22⋊Dic173—21no—
31#14583803—24no—
32C4×Dic173—24no—
33C4×Dic173C2×C412yes—
34C4×Dic173D414yes—
35C2×Q8×C173—22no—
36C2×Q8×C173—22no—
37C2×Q8×C173C2×C412yes—
38C2×Q8×C173D412yes—
39C22×C4×C173D412yes—
40C22×C4×C173C2311yescharacteristic
41M16×C173—22no—
42M16×C173—22no—
43C2×C8×C173—22no—
44C22×D173—21no—
45Dic346—22no—
46D692—22no—
47C22⋊D173—41no—
48C22⋊D173—41no—
49C2×Dic173—21no—
50C2×Dic173—22no—
51C2×Dic173—24no—
52C4×D173—42no—
53C23×C173C2×D411yescharacteristic
54Q8×C173—24no—
55Q8×C173—24no—
56C2×C4×C173—22no—
57C2×C4×C173—22no—
58C2×C4×C173C22⋊C414yes—
59C2×C4×C173C22×C411yescharacteristic
60C2×C4×C173C2×D412yes—
61C2×C4×C173C2×D411yescharacteristic
62C2×C4×C173#4112yes—
63C8×C173—42no—
64C22×C173—41no—
65C22×C173C2×C22⋊C411yescharacteristic
66C22×C173#12011yescharacteristic
67C22×C173#12311yescharacteristic
68C2×D173—42no—
69C2×D173—41no—
70C4×C173—24no—
71C4×C173C4×D412yes—
72C4×C173#12112yes—
73Dic173—44no—
74Dic173—42no—
75Dic173—41no—
76C2×C173—21no—
77C2×C173—21no—
78C2×C173—21no—
79C2×C173#38611yescharacteristic
80D173—81no—
81C173#191211yescharacteristic, complemented, Sylow
82#1912—1731nomaximal, Sylow
83#536—1731no—
84#519—1731no—
85#421—1732no—
86C2×#101D17311yescharacteristic, complemented
87H(3,ℤ/4)—1732no—
88C2×#41—1731no—
89C22×Q8C2×D17311yescharacteristic
90C2×M16C2×D17312yes—
91C4⋊Q8—1732no—
92#124—1732no—
93#123—3462no—
94C22⋊Q8—3462no—
95C4×Q8—3464no—
96#117—3462no—
97#117—1732no—
98#117—1731no—
99#104—3464no—
100#101—22no—
101#41—3462no—
102#41—3462no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—1731no—
106C2×Q8C4×D17312yes—
107C2×Q8C22⋊D17312yes—
108C2×D4—3461no—
109C2×D4—1731no—
110C22×C4—3461no—
111C22×C4D69212yes—
112C22×C4C22×D17311yescharacteristic
113M16—22no—
114M16—22no—
115C2×C8—22no—
116C4⋊C4—3464no—
117C4⋊C4—3462no—
118C4⋊C4—3461no—
119C4⋊C4—3464no—
120C22⋊C4—3462no—
121C22⋊C4—3461no—
122C22⋊C4—3464no—
123C42—3464no—
124C42—1732no—
125C42—1734no—
126C23—3461no—
127C23C2×D69211yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—3462no—
131D4—3462no—
132D4—6921no—
133D4—6921no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—3462no—
137C2×C4—3461no—
138C2×C4—3464no—
139C2×C4—6922no—
140C2×C4#1458380314yes—
141C2×C4C2×C4×D17311yescharacteristic
142C2×C4D4×D17312yes—
143C2×C4C2×C22⋊D17311yescharacteristic
144C2×C4#1458382512yes—
145C8—42no—
146C22—41no—
147C22—6922no—
148C22—6921no—
149C22C2×#1458380311yescharacteristic
150C22#1914964911yescharacteristic
151C22#1914966311yescharacteristic
152C4—24no—
153C4—6924no—
154C4—6922no—
155C4—6921no—
156C4#1914963912yes—
157C4#1914965112yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—13841no—
162C2#2601136811yescharacteristic
subgroupG/H
centerC2#26011368
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
FittingC2×#101×C173C2
socleC2×C173#386
prime radicalC2×#14583826C22
cosocleC22×C4×C173C23
hypercenterC2×#101D173
hypocenterC173#1912
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #26011969C2semidirect
4. #26011248C2semidirect
5. C2×#101×C173C2semidirect
7. #19149677C4semidirect
8. #19149668C4semidirect
81. C173#1912semidirect
86. C2×#101D173semidirect

Maximal subgroups

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

subgroupindex
#260120652
#260119692
#260115072
#260112482
C2×#101×C1732
#1912173