Finite Group Databasebeta

← #36352520

Subgroups — #36352520

Order 22144.

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

normal subgroups only

This group has 256 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, complemented
2#26011403C211yesmaximal, characteristic
3#26011228C212yesmaximal, complemented
4#26011215C211yesmaximal, characteristic
5#543×C173C211yesmaximal, characteristic
6C22×Dic346C2211yescharacteristic
7C2×#14583816—22no—
8C2×C22⋊Dic173—21no—
9#19149602—22no—
10C22⋊#13063436C2212yes—
11#19149586—21no—
12#19149586C2211yescharacteristic
13C22⋊Q8×C173—22no—
14C22⋊Q8×C173—21no—
15C22⋊Q8×C173—22no—
16C22⋊Q8×C173C2212yes—
17C2×C22⋊C4×C173—21no—
18C2×C22⋊C4×C173C2211yescharacteristic
19C2×Dic346—22no—
20C2×Dic346—42no—
21C22×Dic173—21no—
22C22×Dic173—21no—
23#14583816—44no—
24C2×#13063436—22no—
25C22⋊Dic173—42no—
26C24×C173D411yescharacteristic
27C2×Q8×C173—21no—
28C2×Q8×C173D412yes—
29C22×C4×C173—21no—
30C22×C4×C173—21no—
31C22×C4×C173D411yescharacteristic
32C22×C4×C173C2311yescharacteristic
33C4⋊C4×C173—21no—
34C4⋊C4×C173—22no—
35C4⋊C4×C173—22no—
36C4⋊C4×C173D412yes—
37C22⋊C4×C173—22no—
38C22⋊C4×C173—22no—
39C22⋊C4×C173—22no—
40Dic346—24no—
41Dic346—42no—
42C2×Dic173—22no—
43C2×Dic173—41no—
44C2×Dic173—41no—
45C2×Dic173—41no—
46C2×Dic173—41no—
47C2×Dic173—41no—
48#13063436—42no—
49C23×C173—21no—
50C23×C173—21no—
51C23×C173—41no—
52C23×C173—41no—
53C23×C173Q1612yescomplemented
54C23×C173C2×D411yescharacteristic
55Q8×C173—42no—
56Q8×C173—81no—
57C2×C4×C173—22no—
58C2×C4×C173—21no—
59C2×C4×C173—22no—
60C2×C4×C173—41no—
61C2×C4×C173—41no—
62C2×C4×C173—81no—
63C2×C4×C173—81no—
64C2×C4×C173C2×D412yes—
65C22×C173—21no—
66C22×C173—21no—
67C22×C173—22no—
68C22×C173—42no—
69C22×C173—41no—
70C22×C173—41no—
71C22×C173—81no—
72C22×C173#12011yescharacteristic
73C22×C173C2×Q1611yescharacteristic
74C22×C173#13711yescharacteristic
75C4×C173—22no—
76C4×C173—41no—
77C4×C173—42no—
78C4×C173—42no—
79Dic173—42no—
80Dic173—81no—
81C2×C173—21no—
82C2×C173—21no—
83C2×C173—42no—
84C2×C173#45111yescharacteristic
85C2×C173#45511yescharacteristic
86C2×C173B(4,𝔽2)11yescharacteristic
87C173#160211yescharacteristic, complemented, Sylow
88#1602—1731nomaximal, Sylow
89#543D17311yescharacteristic
90#451—1732no—
91#393—1731no—
92#328—1732no—
93#323—1731no—
94C22×Q8—1731no—
95C2×Q16—3462no—
96C22⋊Q8—22no—
97C22⋊Q8—21no—
98C22⋊Q8—22no—
99C22⋊Q8C2×D17312yes—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—3461no—
102C2×C22⋊C4C2×D17311yescharacteristic
103#103—3462no—
104C22⋊C8—1732no—
105#95—3461no—
106#95—1731no—
107C24C22⋊D17311yescharacteristic
108C2×Q8—21no—
109C2×Q8—3462no—
110C2×Q8—6922no—
111C2×Q8C22⋊D17312yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—3461no—
115C22×C4—3461no—
116C22×C4C22⋊D17311yescharacteristic
117C22×C4C22×D17311yescharacteristic
118Q16—6924no—
119C2×C8—3462no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4C22⋊D17312yes—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—6922no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1458381612yescomplemented
133C23C2×C22⋊D17311yescharacteristic
134Q8—42no—
135Q8—81no—
136Q8—3464no—
137Q8—6922no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—3462no—
146C2×C4—6921no—
147C2×C4—6921no—
148C2×C4—6921no—
149C2×C4—6921no—
150C2×C4—6921no—
151C2×C4C2×C22⋊D17312yes—
152C8—6922no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1914964811yescharacteristic
161C22C2×#1458381611yescharacteristic
162C22#1914971011yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—42no—
167C4—13841no—
168C4—6922no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2601164711yescharacteristic
173C2#2601166911yescharacteristic
174C2#2601167811yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#543×C173C2
socleC22×C173#120
prime radicalC24×C173D4
hypercenter#543D173
hypocenterC173#1602
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #26011646C2semidirect
3. #26011228C2semidirect
53. C23×C173Q16semidirect
87. C173#1602semidirect
132. C23#14583816semidirect

Maximal subgroups

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

subgroupindex
#260116462
#260114032
#260112282
#260112152
#543×C1732
#1602173