Finite Group Databasebeta

← #36368840

Subgroups — #36368840

Order 22144.

322 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 64 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#26012462C211yesmaximal, characteristic, complemented
2D173⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
3#26012335C214yesmaximal, complemented
4C2×#19149654C211yesmaximal, characteristic
5C2×#14583836—21no—
6C2×D173⋊C8—21no—
7#19149736—21no—
8#19149736—21no—
9C22⋊#13063443C2212yes—
10C22⋊#13063443C2212yes—
11C22×D692C412yes—
12C22×D692C411yescharacteristic
13C22×C4×D173C2211yescharacteristic
14#19149654—21no—
15#19149654—21no—
16#19149654—21no—
17#19149654—21no—
18C2×#14583803C2212yes—
19C2×C4⋊C4×C173C411yescharacteristic
20#14583836—42no—
21D173⋊C8—42no—
22C2×#13063443—22no—
23C2×#13063443—22no—
24C23×D173C2×C411yescharacteristic
25C23×D173C2×C412yes—
26C2×D692—22no—
27C2×D692—22no—
28C2×D692—21no—
29C2×D692—21no—
30C2×D692—42no—
31C2×D692—44no—
32C22×Dic173C2311yescharacteristic
33C2×C4×D173—42no—
34C2×C4×D173D411yescharacteristic
35C2×C4×D173D411yescharacteristic
36C2×C4×D173D411yescharacteristic
37C2×C4×D173D411yescharacteristic
38#14583803—22no—
39#14583803—22no—
40C22×C4×C173C2×C411yescharacteristic
41C22×C4×C173C2×C412yes—
42C4⋊C4×C173—21no—
43C4⋊C4×C173—21no—
44#13063443—42no—
45#13063443—42no—
46C22×D173—22no—
47C22×D173—22no—
48C22×D173—22no—
49C22×D173—24no—
50C22×D173—41no—
51C22×D173—42no—
52C22×D173—42no—
53C22×D173C22⋊C411yescharacteristic
54C22×D173C22⋊C411yescharacteristic
55D692—42no—
56D692—44no—
57D692—44no—
58D692—42no—
59C2×Dic173—41no—
60C2×Dic173C2×D411yescharacteristic
61C2×Dic173C2×D411yescharacteristic
62C4×D173—22no—
63C4×D173—22no—
64C4×D173—41no—
65C4×D173—41no—
66C23×C173C22×C411yescharacteristic
67C2×C4×C173—22no—
68C2×C4×C173—22no—
69C2×C4×C173—42no—
70C2×C4×C173C22⋊C411yescharacteristic
71C2×C4×C173C22⋊C411yescharacteristic
72C22×C173—21no—
73C22×C173—21no—
74C22×C173C2×C22⋊C411yescharacteristic
75C22×C173#13112yes—
76C2×D173—21no—
77C2×D173—21no—
78C2×D173—21no—
79C2×D173—21no—
80C2×D173—21no—
81C2×D173—44no—
82C2×D173—42no—
83C2×D173—42no—
84C2×D173—42no—
85C2×D173—84no—
86C2×D173#10012yes—
87C4×C173—21no—
88C4×C173—42no—
89C4×C173#9912yes—
90Dic173—21no—
91Dic173—21no—
92C2×C173—22no—
93C2×C173#40811yescharacteristic
94C2×C173C2×#9911yescharacteristic
95C2×C173C2×#10011yescharacteristic
96D173—22no—
97D173—41no—
98D173—82no—
99C173#147111yescharacteristic, complemented, Sylow
100#1471—1731nomaximal, Sylow
101C2×#121—1731no—
102#407—1731no—
103C2×C22⋊C8—1731no—
104#323—1734no—
105C22×D4—1732no—
106C22×D4—1731no—
107C23×C4—1731no—
108C2×M16—3461no—
109C22×C8—3461no—
110#121—3461no—
111#121—3461no—
112#121—3461no—
113#121—3461no—
114C2×C4⋊C4#1062192411yescharacteristic
115C2×C22⋊C4—1732no—
116C22⋊C8—3461no—
117C22⋊C8—3461no—
118C22⋊C8—1732no—
119C22⋊C8—1732no—
120C24—1731no—
121C24—1732no—
122C2×D4—3462no—
123C2×D4—3462no—
124C2×D4—3461no—
125C2×D4—3461no—
126C2×D4—6922no—
127C2×D4—6924no—
128C22×C4—1731no—
129C22×C4—1731no—
130C22×C4—1731no—
131C22×C4—1731no—
132C22×C4—1731no—
133C22×C4—6922no—
134C22×C4C2×#1062192412yes—
135C22×C4C2×#1062192411yescharacteristic
136M16—6922no—
137C2×C8—3462no—
138C2×C8—3462no—
139C2×C8—6922no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C22⋊C4—3462no—
143C22⋊C4—3462no—
144C23—3462no—
145C23—3462no—
146C23—3464no—
147C23—3462no—
148C23—1731no—
149C23—1731no—
150C23—6921no—
151C23—6922no—
152C23—6922no—
153C23C22×#1062192411yescharacteristic
154D4—6922no—
155D4—6924no—
156D4—6924no—
157D4—6922no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—3462no—
162C2×C4—3462no—
163C2×C4—1731no—
164C2×C4—1731no—
165C2×C4—6921no—
166C2×C4—6921no—
167C2×C4—6921no—
168C2×C4C22⋊#1062192411yescharacteristic
169C2×C4C22⋊#1062192411yescharacteristic
170C8—6922no—
171C8—6922no—
172C22—21no—
173C22—21no—
174C22—3461no—
175C22—3461no—
176C22—3461no—
177C22—3461no—
178C22—3461no—
179C22—13844no—
180C22—1732no—
181C22—6924no—
182C22—6922no—
183C22—6922no—
184C22—6922no—
185C22C2×C22⋊#1062192411yescharacteristic
186C22#1914976712yes—
187C4—21no—
188C4—42no—
189C4—3461no—
190C4—3461no—
191C4#1914973812yes—
192C2—22no—
193C2—3462no—
194C2—13842no—
195C2—6921no—
196C2#2601246611yescharacteristic
197C2C2×#1914973811yescharacteristic
198C2D173⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×#19149654C2
cosocleC22×Dic173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1471
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #26012462C2semidirect
2. D173⋊(C22⋊C8)C2semidirect
3. #26012335C2semidirect
99. C173#1471semidirect

Maximal subgroups

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

subgroupindex
#260124622
D173⋊(C22⋊C8)2
#260123352
C2×#191496542
#1471173