Finite Group Databasebeta

← #36352604

Subgroups — #36352604

Order 22144.

202 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 53 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
2#26011826C212yesmaximal
3#26011227C212yesmaximal
4#26011217C211yesmaximal, characteristic
5#400×C173C211yesmaximal, characteristic
6#19149678—22no—
7#19149670—22no—
8#19149658—21no—
9#19149658C2212yes—
10#19149638—21no—
11#19149612—22no—
12#19149602—22no—
13C22⋊#13063436C2212yes—
14C2×C4⋊C4×C173C2211yescharacteristic
15C2×C4⋊C4×C173C2211yescharacteristic
16#95×C173—21no—
17#95×C173—21no—
18#95×C173C2211yescharacteristic
19C2×Dic346—21no—
20C2×Dic346D412yes—
21C2×#13063436—22no—
22C4⋊Dic173—21no—
23C4⋊Dic173—21no—
24C4⋊Dic173—21no—
25C4⋊Dic173D412yes—
26#14583805—22no—
27#14583805—22no—
28C22⋊Dic173—22no—
29C4×Dic173—22no—
30C22×C4×C173—21no—
31C22×C4×C173—21no—
32C22×C4×C173D411yescharacteristic
33C22×C4×C173D411yescharacteristic
34C22×C4×C173C2311yescharacteristic
35C4⋊C4×C173—22no—
36C4⋊C4×C173—22no—
37Dic346—42no—
38Dic346—81no—
39C2×Dic173—22no—
40C2×Dic173—22no—
41#13063436—42no—
42C23×C173C2×D411yescharacteristic
43C2×C4×C173—21no—
44C2×C4×C173—21no—
45C2×C4×C173—21no—
46C2×C4×C173—41no—
47C2×C4×C173—41no—
48C2×C4×C173—41no—
49C2×C4×C173—41no—
50C2×C4×C173—42no—
51C2×C4×C173—41no—
52C2×C4×C173—41no—
53C2×C4×C173Q1612yes—
54C2×C4×C173C2×D412yes—
55C22×C173—21no—
56C22×C173—21no—
57C22×C173#12011yescharacteristic
58C22×C173C2×Q1611yescharacteristic
59C22×C173#13711yescharacteristic
60C4×C173—22no—
61C4×C173—22no—
62C4×C173—41no—
63C4×C173—81no—
64C4×C173—81no—
65Dic173—42no—
66Dic173—42no—
67C2×C173—21no—
68C2×C173—21no—
69C2×C173#45111yescharacteristic
70C2×C173#45411yescharacteristic
71C2×C173#45811yescharacteristic
72C173#161711yescharacteristic, complemented, Sylow
73#1617—1731nomaximal, Sylow
74#544—1731no—
75#484—1732no—
76#400D17311yescharacteristic
77#328—1732no—
78#324—1731no—
79C4⋊Q8—3462no—
80#125—3462no—
81C22⋊Q8—3461no—
82C22⋊Q8—1732no—
83#117—3461no—
84C2×C4⋊C4C2×D17311yescharacteristic
85C2×C4⋊C4C2×D17311yescharacteristic
86#107—3462no—
87#103—3462no—
88C22⋊C8—1732no—
89#95—21no—
90#95—21no—
91#95C2×D17311yescharacteristic
92C2×Q8—3461no—
93C2×Q8—1732no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4C22⋊D17311yescharacteristic
97C22×C4C22⋊D17311yescharacteristic
98C22×C4C22×D17311yescharacteristic
99C2×C8—3462no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—3461no—
103C4⋊C4—3461no—
104C4⋊C4—3461no—
105C4⋊C4—3462no—
106C4⋊C4—3462no—
107C4⋊C4—1732no—
108C22⋊C4—3462no—
109C42—3462no—
110C23C2×C22⋊D17311yescharacteristic
111Q8—13841no—
112Q8—6922no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—3462no—
124C2×C4—3462no—
125C2×C4#1458381612yes—
126C2×C4D4×D17312yes—
127C8—6922no—
128C22—21no—
129C22—21no—
130C22#1914964711yescharacteristic
131C22C2×#1458381611yescharacteristic
132C22#1914970911yescharacteristic
133C4—22no—
134C4—22no—
135C4—41no—
136C4—81no—
137C4—81no—
138C4—6922no—
139C4—6922no—
140C2—21no—
141C2—21no—
142C2#2601164611yescharacteristic
143C2#2601166311yescharacteristic
144C2#2601168011yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#400×C173C2
socleC22×C173#120
prime radicalC23×C173C2×D4
hypercenter#400D173
hypocenterC173#1617
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
72. C173#1617semidirect

Maximal subgroups

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

subgroupindex
#260121222
#260118262
#260112272
#260112172
#400×C1732
#1617173