Finite Group Databasebeta

← #36368832

Subgroups — #36368832

Order 22144.

256 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 60 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#26012511C211yesmaximal, characteristic, complemented
2#26012507C211yesmaximal, characteristic, complemented
3#26012335C212yesmaximal, complemented
4#26012334C212yesmaximal, complemented
5C2×#19149665C211yesmaximal, characteristic
6#19149746—21no—
7C4⋊#13063443—21no—
8C22⋊#13063443C2212yes—
9C22⋊#13063443C2212yes—
10#19149734—21no—
11C4×#13063443—21no—
12C22×Dic346C411yescharacteristic
13C22×C22⋊D173C411yescharacteristic
14#19149665—21no—
15#19149665—21no—
16#19149665—21no—
17#19149665—21no—
18C2×C22⋊Dic173C2211yescharacteristic
19C2×#14583803C411yescharacteristic
20C2×#14583803C2211yescharacteristic
21C2×C4×Dic173C2211yescharacteristic
22C2×C22⋊C4×C173C411yescharacteristic
23C2×#13063443—22no—
24C2×#13063443—22no—
25C23×D173C2×C411yescharacteristic
26C2×Dic346—21no—
27C2×Dic346—21no—
28C2×Dic346—42no—
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—42no—
32C22×Dic173C2×C411yescharacteristic
33C22×Dic173C2×C411yescharacteristic
34C22×Dic173C2311yescharacteristic
35C22⋊Dic173—21no—
36C22⋊Dic173—21no—
37#14583803—21no—
38#14583803—21no—
39#14583803—21no—
40#14583803—21no—
41C4×Dic173D411yescharacteristic
42C4×Dic173D411yescharacteristic
43C4×Dic173D411yescharacteristic
44C4×Dic173D411yescharacteristic
45C24×C173C2×C411yescharacteristic
46C22×C4×C173C2×C411yescharacteristic
47C22×C4×C173C2×C411yescharacteristic
48C22⋊C4×C173—21no—
49C22⋊C4×C173—21no—
50#13063443—42no—
51#13063443—42no—
52C22×D173—21no—
53C22×D173—22no—
54C22×D173—41no—
55C22×D173—41no—
56Dic346—42no—
57Dic346—42no—
58C22⋊D173—42no—
59C22⋊D173—42no—
60C2×Dic173—22no—
61C2×Dic173—21no—
62C2×Dic173—41no—
63C2×Dic173—41no—
64C2×Dic173C22⋊C411yescharacteristic
65C2×Dic173C22⋊C411yescharacteristic
66C2×Dic173C2×D411yescharacteristic
67C2×Dic173C2×D411yescharacteristic
68C23×C173—21no—
69C23×C173—22no—
70C23×C173—41no—
71C23×C173—41no—
72C23×C173C22×C411yescharacteristic
73C2×C4×C173—22no—
74C2×C4×C173—21no—
75C2×C4×C173—41no—
76C2×C4×C173C22⋊C411yescharacteristic
77C2×C4×C173C22⋊C411yescharacteristic
78C22×C173—21no—
79C22×C173—21no—
80C22×C173—41no—
81C22×C173—41no—
82C22×C173—41no—
83C22×C173—82no—
84C22×C173C2×C22⋊C411yescharacteristic
85C22×C173#13112yes—
86C2×D173—41no—
87C2×D173—41no—
88C2×D173—41no—
89C2×D173—82no—
90C4×C173—21no—
91C4×C173—21no—
92C4×C173—41no—
93Dic173—21no—
94Dic173—21no—
95Dic173—21no—
96Dic173—41no—
97Dic173#10012yes—
98C2×C173—22no—
99C2×C173—81no—
100C2×C173#40811yescharacteristic
101C2×C173#41011yescharacteristic
102C2×C173C2×#10011yescharacteristic
103D173—81no—
104C173#146911yescharacteristic, complemented, Sylow
105#1469—1731nomaximal, Sylow
106C2×#124—1731no—
107#432—1731no—
108#431—1731no—
109#323—1732no—
110#323—1732no—
111C22×Q8—1731no—
112C22×D4—1731no—
113#124—3461no—
114#124—3461no—
115#124—3461no—
116#124—3461no—
117C2×C22⋊C4—1731no—
118C2×C22⋊C4—1731no—
119C2×C22⋊C4—1731no—
120C2×C22⋊C4#1062192411yescharacteristic
121C2×C42—1731no—
122C4⋊C8—3461no—
123C4⋊C8—3461no—
124C22⋊C8—1732no—
125C22⋊C8—1732no—
126#97—3461no—
127C4×C8—3461no—
128C24—1731no—
129C24C2×#1062192411yescharacteristic
130C2×Q8—3461no—
131C2×Q8—3461no—
132C2×Q8—6922no—
133C2×D4—3461no—
134C2×D4—3461no—
135C2×D4—6922no—
136C22×C4—1731no—
137C22×C4—1731no—
138C22×C4—1731no—
139C22×C4C2×#1062192411yescharacteristic
140C22×C4C2×#1062192411yescharacteristic
141C2×C8—3462no—
142C2×C8—3462no—
143C22⋊C4—21no—
144C22⋊C4—21no—
145C22⋊C4—3461no—
146C22⋊C4—3461no—
147C22⋊C4—3461no—
148C22⋊C4—3461no—
149C22⋊C4—3461no—
150C22⋊C4—3461no—
151C42—1731no—
152C42—1731no—
153C42—1731no—
154C42—1731no—
155C23—21no—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—3462no—
160C23—3461no—
161C23—6921no—
162C23—6921no—
163C23C22×#1062192411yescharacteristic
164Q8—6922no—
165Q8—6922no—
166D4—6922no—
167D4—6922no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—3462no—
172C2×C4—3461no—
173C2×C4—1731no—
174C2×C4—1731no—
175C2×C4—1731no—
176C2×C4—1731no—
177C2×C4—6921no—
178C2×C4—6921no—
179C2×C4C22⋊#1062192411yescharacteristic
180C2×C4C22⋊#1062192411yescharacteristic
181C8—6922no—
182C8—6922no—
183C22—21no—
184C22—21no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—82no—
189C22—13842no—
190C22—6921no—
191C22—6921no—
192C22—6921no—
193C22C2×C22⋊#1062192411yescharacteristic
194C22#1914976612yes—
195C4—21no—
196C4—21no—
197C4—41no—
198C4—3461no—
199C4—3461no—
200C4—3461no—
201C4—1732no—
202C4—6921no—
203C2—22no—
204C2—81no—
205C2—13841no—
206C2#2601246411yescharacteristic
207C2#2601247011yescharacteristic
208C2D173⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C22⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×C22⋊D173C4
cosocleC22×Dic173C23
hypercenterC2×C22⋊C4#10621924
hypocenterC173#1469
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #26012511C2semidirect
2. #26012507C2semidirect
3. #26012335C2semidirect
4. #26012334C2semidirect
104. C173#1469semidirect

Maximal subgroups

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

subgroupindex
#260125112
#260125072
#260123352
#260123342
C2×#191496652
#1469173