Finite Group Databasebeta

← #36352537

Subgroups — #36352537

Order 22144.

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

normal subgroups only

This group has 238 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#26011646C212yesmaximal
2#26011411C211yesmaximal, characteristic
3#26011228C212yesmaximal
4#26011217C211yesmaximal, characteristic
5#544×C173C211yesmaximal, characteristic
6C22×Dic346C2211yescharacteristic
7C2×#14583816—22no—
8C2×#14583805—21no—
9#19149602—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
19C2×Dic346—22no—
20C2×Dic346—42no—
21C22×Dic173—21no—
22C22×Dic173—21no—
23#14583816—44no—
24C2×#13063436—22no—
25#14583805—42no—
26C2×Q8×C173—21no—
27C2×Q8×C173D412yes—
28C22×C4×C173D411yescharacteristic
29C22×C4×C173D411yescharacteristic
30C22×C4×C173C2311yescharacteristic
31C4⋊C4×C173—21no—
32C4⋊C4×C173—22no—
33C4⋊C4×C173—21no—
34C4⋊C4×C173—21no—
35C4⋊C4×C173—22no—
36C4⋊C4×C173—22no—
37C4⋊C4×C173D412yes—
38C22⋊C4×C173—22no—
39C42×C173—22no—
40Dic346—24no—
41Dic346—42no—
42C2×Dic173—22no—
43C2×Dic173—22no—
44C2×Dic173—41no—
45C2×Dic173—41no—
46C2×Dic173—41no—
47C2×Dic173—41no—
48#13063436—42no—
49C23×C173C2×D411yescharacteristic
50Q8×C173—42no—
51Q8×C173—81no—
52C2×C4×C173—21no—
53C2×C4×C173—21no—
54C2×C4×C173—22no—
55C2×C4×C173—22no—
56C2×C4×C173—21no—
57C2×C4×C173—41no—
58C2×C4×C173—41no—
59C2×C4×C173Q1612yescomplemented
60C2×C4×C173C2×D412yes—
61C22×C173—21no—
62C22×C173—21no—
63C22×C173#12011yescharacteristic
64C22×C173C2×Q1611yescharacteristic
65C22×C173#13711yescharacteristic
66C4×C173—22no—
67C4×C173—22no—
68C4×C173—42no—
69C4×C173—41no—
70C4×C173—42no—
71Dic173—42no—
72Dic173—42no—
73C2×C173—21no—
74C2×C173—21no—
75C2×C173#45111yescharacteristic
76C2×C173#45611yescharacteristic
77C2×C173B(4,𝔽2)11yescharacteristic
78C173#160511yescharacteristic, complemented, Sylow
79#1605—1731nomaximal, Sylow
80#544D17311yescharacteristic
81#451—1732no—
82#395—1731no—
83#328—1732no—
84#324—1731no—
85C22×Q8—1731no—
86C2×Q16—3462no—
87C4⋊Q8—22no—
88#125—22no—
89C22⋊Q8—21no—
90C22⋊Q8C2×D17312yes—
91#117—21no—
92C2×C4⋊C4—3461no—
93C2×C4⋊C4C2×D17311yescharacteristic
94#103—3462no—
95C22⋊C8—1732no—
96#95—3461no—
97#95—1731no—
98C2×Q8—21no—
99C2×Q8—3462no—
100C2×Q8—6922no—
101C2×Q8C22⋊D17312yes—
102C22×C4—3461no—
103C22×C4—3461no—
104C22×C4C22⋊D17311yescharacteristic
105C22×C4C22⋊D17311yescharacteristic
106C22×C4C22×D17311yescharacteristic
107Q16—6924no—
108C2×C8—3462no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—6922no—
116C4⋊C4C22⋊D17312yes—
117C22⋊C4—22no—
118C42—22no—
119C23C2×C22⋊D17311yescharacteristic
120Q8—42no—
121Q8—81no—
122Q8—3464no—
123Q8—6922no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—3462no—
132C2×C4—3462no—
133C2×C4—6921no—
134C2×C4—6921no—
135C2×C4—6921no—
136C2×C4—6921no—
137C2×C4#1458381612yescomplemented
138C2×C4C2×C22⋊D17312yes—
139C8—6922no—
140C22—21no—
141C22—21no—
142C22#1914964811yescharacteristic
143C22C2×#1458381611yescharacteristic
144C22#1914971011yescharacteristic
145C4—22no—
146C4—22no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—6922no—
151C4—6922no—
152C2—21no—
153C2—21no—
154C2#2601164711yescharacteristic
155C2#2601167411yescharacteristic
156C2#2601167811yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#544×C173C2
socleC22×C173#120
prime radicalC23×C173C2×D4
hypercenter#544D173
hypocenterC173#1605
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
59. C2×C4×C173Q16semidirect
78. C173#1605semidirect
137. C2×C4#14583816semidirect

Maximal subgroups

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

subgroupindex
#260116462
#260114112
#260112282
#260112172
#544×C1732
#1605173