Finite Group Databasebeta

← #36354719

Subgroups — #36354719

Order 22144.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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#26012122C211yesmaximal, characteristic, complemented
2#26011496C212yesmaximal
3#26011260C211yesmaximal, characteristic, complemented
4#26011251C212yesmaximal, complemented
5#390×C173C211yesmaximal, characteristic
6C2×#14583810C2212yes—
7#19149678—22no—
8#19149670—22no—
9#19149658—22no—
10#19149638C2212yes—
11#19149600—22no—
12#19149600—22no—
13C22×D4×C173C2211yescharacteristic
14C2×C4⋊C4×C173C2211yescharacteristic
15C2×C22⋊C4×C173—22no—
16C2×C42×C173C2211yescharacteristic
17C2×Dic346—22no—
18#14583810—22no—
19#14583810—22no—
20C2×#13063436—22no—
21C4⋊Dic173—22no—
22C4⋊Dic173D412yes—
23C4⋊Dic173D412yes—
24#14583805—24no—
25C22⋊Dic173—22no—
26C4×Dic173—22no—
27C24×C173—21no—
28C2×D4×C173—22no—
29C2×D4×C173—22no—
30C2×D4×C173—21no—
31C2×D4×C173Q812yescomplemented
32C22×C4×C173—21no—
33C22×C4×C173D412yes—
34C22×C4×C173C2311yescharacteristic
35C4⋊C4×C173—22no—
36C22⋊C4×C173—44no—
37C42×C173—22no—
38Dic346—82no—
39C2×Dic173—22no—
40C2×Dic173—22no—
41#13063436—42no—
42C23×C173—22no—
43C23×C173—42no—
44C23×C173—42no—
45C23×C173—41no—
46C23×C173—41no—
47C23×C173C2×D411yescharacteristic
48D4×C173—24no—
49D4×C173—42no—
50C2×C4×C173—22no—
51C2×C4×C173—22no—
52C2×C4×C173—41no—
53C2×C4×C173—41no—
54C2×C4×C173—42no—
55C2×C4×C173C2×D412yes—
56C2×C4×C173C2×Q811yescharacteristic
57C2×C4×C173#4112yes—
58C2×C4×C173#4111yescharacteristic
59C22×C173—21no—
60C22×C173—21no—
61C22×C173—44no—
62C22×C173—42no—
63C22×C173—82no—
64C22×C173—81no—
65C22×C173—81no—
66C22×C173#12011yescharacteristic
67C22×C173C22⋊Q811yescharacteristic
68C22×C173#12411yescharacteristic
69C4×C173—42no—
70C4×C173—42no—
71C4×C173C22⋊Q812yes—
72C4×C173#12412yes—
73Dic173—42no—
74Dic173—42no—
75C2×C173—21no—
76C2×C173—21no—
77C2×C173—82no—
78C2×C173#39311yescharacteristic
79C2×C173#45311yescharacteristic
80C2×C173#45411yescharacteristic
81C173#202711yescharacteristic, complemented, Sylow
82#2027—1731nomaximal, Sylow
83#544—1731no—
84#418—1732no—
85#390D17311yescharacteristic
86#343—1731no—
87#339—1732no—
88C22×D4C2×D17311yescharacteristic
89C2×M16—1732no—
90C4⋊Q8—3462no—
91#125—3462no—
92C22⋊Q8—3462no—
93#117—1732no—
94C2×C4⋊C4C2×D17311yescharacteristic
95C2×C22⋊C4—22no—
96C2×C42C2×D17311yescharacteristic
97#102—3462no—
98#102—3462no—
99C24—21no—
100C2×Q8—3462no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4Dic34612yescomplemented
105C22×C4—21no—
106C22×C4C22⋊D17312yes—
107C22×C4C22×D17311yescharacteristic
108M16—3462no—
109M16—3462no—
110C2×C8—3462no—
111C4⋊C4—22no—
112C4⋊C4—3462no—
113C4⋊C4—3464no—
114C4⋊C4—1732no—
115C4⋊C4—1732no—
116C22⋊C4—44no—
117C22⋊C4—3462no—
118C42—22no—
119C42—3462no—
120C23—22no—
121C23—42no—
122C23—42no—
123C23—41no—
124C23—41no—
125C23C2×C22⋊D17311yescharacteristic
126Q8—13842no—
127D4—24no—
128D4—42no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—3462no—
135C2×C4—3462no—
136C2×C4D4×D17312yes—
137C2×C4C2×Dic34611yescharacteristic
138C2×C4#1458382512yes—
139C2×C4#1458382611yescharacteristic
140C8—6922no—
141C22—21no—
142C22—21no—
143C22—44no—
144C22—42no—
145C22—82no—
146C22—81no—
147C22—81no—
148C22#1914964711yescharacteristic
149C22#1914965811yescharacteristic
150C22#1914966611yescharacteristic
151C4—42no—
152C4—42no—
153C4—6922no—
154C4—6922no—
155C4#1914965612yes—
156C4#1914966512yes—
157C2—21no—
158C2—21no—
159C2—82no—
160C2#2601140311yescharacteristic
161C2#2601165811yescharacteristic
162C2#2601166311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#390×C173C2
socleC22×C173#120
prime radicalC22×D4×C173C22
hypercenter#390D173
hypocenterC173#2027
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #26012122C2semidirect
3. #26011260C2semidirect
4. #26011251C2semidirect
31. C2×D4×C173Q8semidirect
81. C173#2027semidirect
104. C2×D4Dic346semidirect

Maximal subgroups

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

subgroupindex
#260121222
#260114962
#260112602
#260112512
#390×C1732
#2027173