Finite Group Databasebeta

← #36368845

Subgroups — #36368845

Order 22144.

214 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 58 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#26012337C212yesmaximal, complemented
4#26012334C212yesmaximal
5C2×#19149672C211yesmaximal, characteristic
6#19149746—21no—
7C4⋊#13063443—21no—
8C22⋊#13063443C2212yes—
9C22⋊#13063443C2212yes—
10#19149734—21no—
11C4×#13063443—21no—
12#19149672—21no—
13#19149672—21no—
14#19149672—21no—
15#19149672—21no—
16C2×C4⋊Dic173C411yescharacteristic
17C2×#14583805C411yescharacteristic
18C2×#14583805C2211yescharacteristic
19C2×C22⋊Dic173C411yescharacteristic
20C2×C22⋊Dic173C2211yescharacteristic
21C2×C4×Dic173C2211yescharacteristic
22C2×C22⋊C4×C173C411yescharacteristic
23C2×#13063443—22no—
24C2×#13063443—22no—
25C22×Dic173C2×C411yescharacteristic
26C22×Dic173C2×C411yescharacteristic
27C22×Dic173C2×C411yescharacteristic
28C22×Dic173C2311yescharacteristic
29C4⋊Dic173—21no—
30C4⋊Dic173—21no—
31#14583805—21no—
32#14583805—21no—
33#14583805—21no—
34#14583805—21no—
35C22⋊Dic173—21no—
36C22⋊Dic173—21no—
37C22⋊Dic173—21no—
38C22⋊Dic173—21no—
39C4×Dic173D411yescharacteristic
40C4×Dic173D411yescharacteristic
41C4×Dic173D411yescharacteristic
42C4×Dic173D411yescharacteristic
43C24×C173C2×C411yescharacteristic
44C22×C4×C173C2×C411yescharacteristic
45C22×C4×C173C2×C411yescharacteristic
46C22⋊C4×C173—21no—
47C22⋊C4×C173—21no—
48#13063443—42no—
49#13063443—42no—
50C2×Dic173—21no—
51C2×Dic173—21no—
52C2×Dic173—22no—
53C2×Dic173—41no—
54C2×Dic173—41no—
55C2×Dic173—41no—
56C2×Dic173C22⋊C411yescharacteristic
57C2×Dic173C22⋊C411yescharacteristic
58C2×Dic173C2×D411yescharacteristic
59C2×Dic173C2×D411yescharacteristic
60C23×C173—21no—
61C23×C173—22no—
62C23×C173—41no—
63C23×C173—41no—
64C23×C173C22×C411yescharacteristic
65C2×C4×C173—22no—
66C2×C4×C173—21no—
67C2×C4×C173—41no—
68C2×C4×C173C22⋊C411yescharacteristic
69C2×C4×C173C22⋊C411yescharacteristic
70C22×C173—21no—
71C22×C173—21no—
72C22×C173—41no—
73C22×C173—41no—
74C22×C173—41no—
75C22×C173—82no—
76C22×C173C2×C22⋊C411yescharacteristic
77C22×C173#13112yes—
78C4×C173—21no—
79C4×C173—21no—
80C4×C173—41no—
81Dic173—21no—
82Dic173—21no—
83Dic173—21no—
84Dic173—21no—
85Dic173—41no—
86Dic173—41no—
87C2×C173—22no—
88C2×C173—81no—
89C2×C173#40811yescharacteristic
90C2×C173#41011yescharacteristic
91C2×C173#41311yescharacteristic
92C173#147311yescharacteristic, complemented, Sylow
93#1473—1731nomaximal, Sylow
94C2×#126—1731no—
95#432—1731no—
96#431—1731no—
97#324—1732no—
98#323—1732no—
99#126—3461no—
100#126—3461no—
101#126—3461no—
102#126—3461no—
103C2×C4⋊C4—1731no—
104C2×C4⋊C4—1731no—
105C2×C4⋊C4—1731no—
106C2×C22⋊C4—1731no—
107C2×C22⋊C4—1731no—
108C2×C22⋊C4#1062192411yescharacteristic
109C2×C42—1731no—
110C4⋊C8—3461no—
111C4⋊C8—3461no—
112C22⋊C8—1732no—
113C22⋊C8—1732no—
114#97—3461no—
115C4×C8—3461no—
116C24C2×#1062192411yescharacteristic
117C22×C4—1731no—
118C22×C4—1731no—
119C22×C4—1731no—
120C22×C4—1731no—
121C22×C4C2×#1062192411yescharacteristic
122C22×C4C2×#1062192411yescharacteristic
123C2×C8—3462no—
124C2×C8—3462no—
125C4⋊C4—3461no—
126C4⋊C4—3461no—
127C4⋊C4—3461no—
128C4⋊C4—3461no—
129C4⋊C4—3461no—
130C4⋊C4—3461no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—3461no—
134C22⋊C4—3461no—
135C22⋊C4—3461no—
136C22⋊C4—3461no—
137C42—1731no—
138C42—1731no—
139C42—1731no—
140C42—1731no—
141C23—21no—
142C23—22no—
143C23—41no—
144C23—41no—
145C23C22×#1062192411yescharacteristic
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—3461no—
150C2×C4—3461no—
151C2×C4—3462no—
152C2×C4—1731no—
153C2×C4—1731no—
154C2×C4—1731no—
155C2×C4—1731no—
156C2×C4—6921no—
157C2×C4—6921no—
158C2×C4—6921no—
159C2×C4C22⋊#1062192411yescharacteristic
160C2×C4C22⋊#1062192411yescharacteristic
161C8—6922no—
162C8—6922no—
163C22—21no—
164C22—21no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—82no—
169C22C2×C22⋊#1062192411yescharacteristic
170C22#1914976612yes—
171C4—21no—
172C4—21no—
173C4—41no—
174C4—3461no—
175C4—3461no—
176C4—3461no—
177C4—3461no—
178C4—6921no—
179C4—6921no—
180C2—22no—
181C2—81no—
182C2#2601246411yescharacteristic
183C2#2601247111yescharacteristic
184C2#2601247711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C22⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC24×C173C2×C4
cosocleC22×Dic173C23
hypercenterC2×C22⋊C4#10621924
hypocenterC173#1473
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. #26012337C2semidirect
92. C173#1473semidirect

Maximal subgroups

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

subgroupindex
#260125112
#260125072
#260123372
#260123342
C2×#191496722
#1473173