Finite Group Databasebeta

← #36352515

Subgroups — #36352515

Order 22144.

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

normal subgroups only

This group has 308 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#26011629C212yesmaximal, complemented
2#26011403C211yesmaximal, characteristic, complemented
3#26011225C212yesmaximal, complemented
4#26011215C211yesmaximal, characteristic, complemented
5#534×C173C211yesmaximal, characteristic
6C22×Dic346C2211yescharacteristic, complemented
7C2×#14583814—22no—
8C2×C22⋊Dic173—21no—
9#19149602—22no—
10C22⋊#13063436C2212yescomplemented
11#19149586—21no—
12#19149586C2211yescharacteristic, complemented
13C22×D4×C173—21no—
14#123×C173—22no—
15#121×C173—21no—
16#121×C173C2212yes—
17#120×C173—22no—
18C2×C22⋊C4×C173C2211yescharacteristic
19C2×Dic346—22no—
20C2×Dic346—42no—
21C22×Dic173—21no—
22C22×Dic173—21no—
23#14583814—44no—
24C2×#13063436—22no—
25C22⋊Dic173—42no—
26C24×C173—21no—
27C24×C173D411yescharacteristic
28C2×D4×C173—22no—
29C2×D4×C173—22no—
30C2×D4×C173—21no—
31C2×D4×C173—42no—
32C2×D4×C173—42no—
33C2×D4×C173D412yes—
34C22×C4×C173—21no—
35C22×C4×C173D411yescharacteristic
36C22×C4×C173C2311yescharacteristic
37C4⋊C4×C173—21no—
38C4⋊C4×C173D412yes—
39C22⋊C4×C173—22no—
40C22⋊C4×C173—22no—
41C22⋊C4×C173—22no—
42Dic346—24no—
43Dic346—42no—
44C2×Dic173—22no—
45C2×Dic173—41no—
46C2×Dic173—41no—
47C2×Dic173—41no—
48C2×Dic173—41no—
49C2×Dic173—41no—
50#13063436—42no—
51C23×C173—21no—
52C23×C173—21no—
53C23×C173—22no—
54C23×C173—41no—
55C23×C173—41no—
56C23×C173—42no—
57C23×C173—81no—
58C23×C173—81no—
59C23×C173SD1612yescomplemented
60C23×C173C2×D411yescharacteristic
61D4×C173—42no—
62D4×C173—44no—
63D4×C173—42no—
64D4×C173—42no—
65D4×C173—81no—
66C2×C4×C173—21no—
67C2×C4×C173—22no—
68C2×C4×C173—41no—
69C2×C4×C173—42no—
70C2×C4×C173—41no—
71C2×C4×C173C2×D412yes—
72C22×C173—21no—
73C22×C173—21no—
74C22×C173—22no—
75C22×C173—42no—
76C22×C173—41no—
77C22×C173—41no—
78C22×C173—42no—
79C22×C173—42no—
80C22×C173—42no—
81C22×C173—81no—
82C22×C173—84no—
83C22×C173#12011yescharacteristic
84C22×C173C2×SD1611yescharacteristic
85C22×C173AGL(1,ℤ/8)11yescharacteristic
86C4×C173—22no—
87C4×C173—42no—
88C4×C173—41no—
89Dic173—42no—
90Dic173—81no—
91C2×C173—21no—
92C2×C173—21no—
93C2×C173—42no—
94C2×C173—82no—
95C2×C173#45011yescharacteristic
96C2×C173#45511yescharacteristic
97C2×C173B(4,𝔽2)11yescharacteristic
98C173#160111yescharacteristic, complemented, Sylow
99#1601—1731nomaximal, Sylow
100#534D17311yescharacteristic
101#448—1732no—
102#393—1731no—
103#327—1732no—
104#323—1731no—
105C22×Q8—1731no—
106C22×D4—21no—
107C2×SD16—3462no—
108#123—22no—
109#121—21no—
110#121C2×D17312yes—
111#120—22no—
112C2×C22⋊C4—3461no—
113C2×C22⋊C4C2×D17311yescharacteristic
114#103—3462no—
115C22⋊C8—1732no—
116#95—3461no—
117#95—1731no—
118C24—21no—
119C24C22⋊D17311yescharacteristic
120C2×Q8—3462no—
121C2×Q8—6922no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4C22⋊D17312yes—
128C22×C4—21no—
129C22×C4—3461no—
130C22×C4—3461no—
131C22×C4C22⋊D17311yescharacteristic
132C22×C4C22×D17311yescharacteristic
133SD16—6924no—
134C2×C8—3462no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D17312yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—6922no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—41no—
145C23—41no—
146C23—42no—
147C23—81no—
148C23—81no—
149C23#1458381412yescomplemented
150C23C2×C22⋊D17311yescharacteristic
151Q8—3464no—
152Q8—6922no—
153D4—42no—
154D4—44no—
155D4—42no—
156D4—42no—
157D4—81no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—3462no—
164C2×C4—6921no—
165C2×C4—6921no—
166C2×C4—6921no—
167C2×C4—6921no—
168C2×C4—6921no—
169C2×C4C2×C22⋊D17312yes—
170C8—6922no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—81no—
181C22—84no—
182C22#1914964811yescharacteristic
183C22C2×#1458381411yescharacteristic
184C22#1914970511yescharacteristic
185C4—22no—
186C4—41no—
187C4—42no—
188C4—13841no—
189C4—6922no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—82no—
194C2#2601164211yescharacteristic
195C2#2601166911yescharacteristic
196C2#2601167811yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting, prime radical#534×C173C2
socleC22×C173#120
hypercenter#534D173
hypocenterC173#1601
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #26011629C2semidirect
2. #26011403C2semidirect
3. #26011225C2semidirect
4. #26011215C2semidirect
6. C22×Dic346C22semidirect
10. C22⋊#13063436C22semidirect
12. #19149586C22semidirect
59. C23×C173SD16semidirect
98. C173#1601semidirect
149. C23#14583814semidirect

Maximal subgroups

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

subgroupindex
#260116292
#260114032
#260112252
#260112152
#534×C1732
#1601173