Finite Group Databasebeta

← #36352571

Subgroups — #36352571

Order 22144.

210 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 57 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#26011819C212yesmaximal
2#26011421C211yesmaximal, characteristic
3#26011228C212yesmaximal
4#26011217C211yesmaximal, characteristic
5#544×C173C211yesmaximal, characteristic
6C2×C4⋊Dic173C2211yescharacteristic
7C2×#14583805—21no—
8#19149610—22no—
9#19149603—22no—
10C22⋊#13063436C2212yes—
11#19149586—21no—
12#19149586C2211yescharacteristic
13C4⋊Q8×C173—22no—
14#125×C173—22no—
15C22⋊Q8×C173—21no—
16C22⋊Q8×C173C2212yes—
17#117×C173—21no—
18C2×C4⋊C4×C173C2211yescharacteristic
19C22×Dic173—21no—
20C22×Dic173—21no—
21C2×#13063436—22no—
22C4⋊Dic173—22no—
23#14583805—42no—
24C2×Q8×C173—21no—
25C2×Q8×C173D412yes—
26C22×C4×C173D411yescharacteristic
27C22×C4×C173D411yescharacteristic
28C22×C4×C173C2311yescharacteristic
29C4⋊C4×C173—21no—
30C4⋊C4×C173—22no—
31C4⋊C4×C173—21no—
32C4⋊C4×C173—21no—
33C4⋊C4×C173—22no—
34C4⋊C4×C173—22no—
35C4⋊C4×C173D412yes—
36C22⋊C4×C173—22no—
37C42×C173—22no—
38C2×Dic173—22no—
39C2×Dic173—41no—
40C2×Dic173—41no—
41C2×Dic173—41no—
42C2×Dic173—41no—
43C2×Dic173—41no—
44#13063436—42no—
45C23×C173C2×D411yescharacteristic
46Q8×C173—42no—
47Q8×C173—81no—
48C2×C4×C173—21no—
49C2×C4×C173—21no—
50C2×C4×C173—22no—
51C2×C4×C173—22no—
52C2×C4×C173—21no—
53C2×C4×C173—41no—
54C2×C4×C173—41no—
55C2×C4×C173SD1612yes—
56C2×C4×C173C2×D412yes—
57C22×C173—21no—
58C22×C173—21no—
59C22×C173#12011yescharacteristic
60C22×C173C2×SD1611yescharacteristic
61C22×C173#13711yescharacteristic
62C4×C173—22no—
63C4×C173—22no—
64C4×C173—42no—
65C4×C173—41no—
66C4×C173—42no—
67Dic173—42no—
68Dic173—81no—
69C2×C173—21no—
70C2×C173—21no—
71C2×C173#44811yescharacteristic
72C2×C173#45611yescharacteristic
73C2×C173#45811yescharacteristic
74C173#161111yescharacteristic, complemented, Sylow
75#1611—1731nomaximal, Sylow
76#544D17311yescharacteristic
77#483—1732no—
78#398—1731no—
79#328—1732no—
80#324—1731no—
81C4⋊Q8—22no—
82#125—22no—
83C22⋊Q8—21no—
84C22⋊Q8C2×D17312yes—
85#117—21no—
86C2×C4⋊C4—3461no—
87C2×C4⋊C4—1731no—
88C2×C4⋊C4C2×D17311yescharacteristic
89#106—3462no—
90#103—3462no—
91C22⋊C8—1732no—
92#95—3461no—
93#95—1731no—
94C2×Q8—21no—
95C2×Q8C22⋊D17312yes—
96C22×C4—3461no—
97C22×C4—3461no—
98C22×C4C22⋊D17311yescharacteristic
99C22×C4C22⋊D17311yescharacteristic
100C22×C4C22×D17311yescharacteristic
101C2×C8—3462no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—3462no—
109C4⋊C4—6922no—
110C4⋊C4C22⋊D17312yes—
111C22⋊C4—22no—
112C42—22no—
113C23C2×C22⋊D17311yescharacteristic
114Q8—42no—
115Q8—81no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—21no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—3462no—
124C2×C4—6921no—
125C2×C4—6921no—
126C2×C4—6921no—
127C2×C4—6921no—
128C2×C4—6921no—
129C2×C4#1458381312yes—
130C2×C4C2×C22⋊D17312yes—
131C8—6922no—
132C22—21no—
133C22—21no—
134C22#1914964811yescharacteristic
135C22C2×#1458381311yescharacteristic
136C22#1914971011yescharacteristic
137C4—22no—
138C4—22no—
139C4—42no—
140C4—41no—
141C4—42no—
142C4—13841no—
143C4—6922no—
144C2—21no—
145C2—21no—
146C2#2601163011yescharacteristic
147C2#2601167411yescharacteristic
148C2#2601168111yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#544×C173C2
socleC22×C173#120
prime radicalC23×C173C2×D4
hypercenter#544D173
hypocenterC173#1611
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
74. C173#1611semidirect

Maximal subgroups

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

subgroupindex
#260118192
#260114212
#260112282
#260112172
#544×C1732
#1611173