Finite Group Databasebeta

← #36354731

Subgroups — #36354731

Order 22144.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#26012117C211yesmaximal, characteristic
2#26011352C211yesmaximal, characteristic
3#26011352C212yesmaximal
4#26011257C212yesmaximal
5#342×C173C211yesmaximal, characteristic
6C23×Dic173C2211yescharacteristic
7#19149658—22no—
8#19149656—24no—
9C2×C22⋊Dic173C2211yescharacteristic
10C2×C22⋊Dic173C2212yes—
11#19149586—22no—
12#19149586—24no—
13C2×C22⋊C4×C173C2212yes—
14C2×C22⋊C4×C173C2211yescharacteristic
15C2×Dic346—22no—
16C22×Dic173—22no—
17C22×Dic173—21no—
18C22×Dic173—21no—
19C22×Dic173—21no—
20C22×Dic173—22no—
21C22×Dic173D414yes—
22C22×Dic173Q812yescomplemented
23C4⋊Dic173—22no—
24#14583805—24no—
25C22⋊Dic173—22no—
26C22⋊Dic173—22no—
27C22⋊Dic173—22no—
28C24×C173C2311yescharacteristic
29C22×C4×C173—22no—
30C22×C4×C173D412yes—
31C22⋊C4×C173—22no—
32C22⋊C4×C173—22no—
33C22⋊C4×C173—22no—
34Dic346—82no—
35C2×Dic173—24no—
36C2×Dic173—24no—
37C2×Dic173—22no—
38C2×Dic173—41no—
39C2×Dic173—42no—
40C2×Dic173—41no—
41C2×Dic173—44no—
42C2×Dic173—42no—
43C2×Dic173—41no—
44C23×C173—42no—
45C23×C173C2×D412yes—
46C23×C173C2×D411yescharacteristic
47C23×C173C2×Q811yescharacteristic
48C23×C173#4111yescharacteristic
49C23×C173#4112yes—
50C2×C4×C173—22no—
51C2×C4×C173—42no—
52C2×C4×C173—44no—
53C2×C4×C173—42no—
54C22×C173—22no—
55C22×C173—24no—
56C22×C173—41no—
57C22×C173—42no—
58C22×C173—41no—
59C22×C173#12011yescharacteristic
60C22×C173C22⋊Q812yes—
61C22×C173C22⋊Q811yescharacteristic
62C22×C173#12412yes—
63C22×C173#12411yescharacteristic
64C4×C173—42no—
65C4×C173—82no—
66Dic173—42no—
67Dic173—44no—
68Dic173—81no—
69C2×C173—21no—
70C2×C173—22no—
71C2×C173—22no—
72C2×C173—21no—
73C2×C173#39311yescharacteristic
74C2×C173#45812yes—
75C173#203011yescharacteristic, complemented, Sylow
76#2030—1731nomaximal, Sylow
77#543—1731no—
78#381—1732no—
79#381—1731no—
80#342—1732no—
81#342D17311yescharacteristic
82C23×C4—1731no—
83C22⋊Q8—3464no—
84C22⋊Q8—3462no—
85C2×C22⋊C4—1731no—
86C2×C22⋊C4—1732no—
87C2×C22⋊C4C2×D17312yes—
88C2×C22⋊C4C2×D17311yescharacteristic
89#95—3464no—
90#95—3462no—
91C24C22×D17311yescharacteristic
92C2×Q8—3462no—
93C22×C4—22no—
94C22×C4—3462no—
95C22×C4—3461no—
96C22×C4—3461no—
97C22×C4—3462no—
98C22×C4—3461no—
99C22×C4—1732no—
100C22×C4—1734no—
101C22×C4D69212yes—
102C4⋊C4—3464no—
103C4⋊C4—3462no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—3462no—
108C22⋊C4—3462no—
109C22⋊C4—3462no—
110C23—42no—
111C23D4×D17312yes—
112C23C2×D69211yescharacteristic
113C23Q8×D17311yescharacteristic
114C23#1458382511yescharacteristic
115C23#1458382612yes—
116Q8—13842no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—3464no—
122C2×C4—3464no—
123C2×C4—3462no—
124C2×C4—6922no—
125C2×C4—6921no—
126C2×C4—6921no—
127C2×C4—6922no—
128C2×C4—6921no—
129C2×C4—6924no—
130C22—24no—
131C22—22no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#1914964911yescharacteristic
136C22#1914965712yes—
137C22#1914965911yescharacteristic
138C22#1914966512yes—
139C22#1914966711yescharacteristic
140C4—42no—
141C4—82no—
142C4—13841no—
143C4—6922no—
144C4—6924no—
145C2—22no—
146C2—21no—
147C2—22no—
148C2—21no—
149C2#2601140411yescharacteristic
150C2#2601168212yes—
subgroupG/H
centerC22#19149649
commutator, prime radical, cosocleC24×C173C23
FrattiniC24C22×D173
Fitting#342×C173C2
socleC22×C173#120
hypercenter#342D173
hypocenterC173#2030
LCSG ▷ C24×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
22. C22×Dic173Q8semidirect
75. C173#2030semidirect

Maximal subgroups

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

subgroupindex
#260121172
#260113522
#260112572
#342×C1732
#2030173