Finite Group Databasebeta

← #36369004

Subgroups — #36369004

Order 22144.

218 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 56 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#26012510C211yesmaximal, characteristic, complemented
2#26012436C211yesmaximal, characteristic, complemented
3#26012345C212yesmaximal, complemented
4#26012342C212yesmaximal
5#26012020C211yesmaximal, characteristic
6C4⋊#13063443—21no—
7#19149736C2212yes—
8C4×#13063443—21no—
9(C2×#10621924)⋊C4—21no—
10(C2×#10621924)⋊C4—21no—
11(C2×#10621924)⋊C4C2212yes—
12#19149672—21no—
13#19149671C411yescharacteristic, complemented
14#19149661—21no—
15#19149655C2211yescharacteristic
16#19149650C2211yescharacteristic
17Q8×Dic173C411yescharacteristic, complemented
18D4×Dic173C411yescharacteristic, complemented
19#19149639—21no—
20#19149635—21no—
21C42×D173C2211yescharacteristic
22#124×C173C411yescharacteristic, complemented
23C22×#10621924—22no—
24C2×#13063443—22no—
25C2×C22⋊D173—21no—
26C22×Dic173—21no—
27C2×C4×D173D411yescharacteristic
28C2×C4×D173D411yescharacteristic
29C2×C4×D173C2311yescharacteristic
30C4⋊Dic173—21no—
31C4⋊Dic173C2×C411yescharacteristic
32C4⋊Dic173C2×C411yescharacteristic
33#14583805—21no—
34#14583805—21no—
35#14583805—21no—
36C22⋊Dic173—21no—
37C22⋊Dic173—21no—
38#14583803—21no—
39C4×Dic173—21no—
40C4×Dic173C2×C411yescharacteristic
41C4×Dic173D411yescharacteristic
42C4×Dic173D411yescharacteristic
43C2×Q8×C173C2×C411yescharacteristic
44C2×D4×C173C2×C411yescharacteristic
45C22⋊C4×C173—21no—
46C22⋊C4×C173—21no—
47C42×C173C2×C411yescharacteristic
48C2×#10621924—42no—
49C2×#10621924—42no—
50C2×#10621924—42no—
51#13063443—42no—
52C22×D173C2×D411yescharacteristic
53C22⋊D173—42no—
54C2×Dic173—21no—
55C2×Dic173—21no—
56C2×Dic173—42no—
57C2×Dic173C22⋊C411yescharacteristic
58C2×Dic173C22⋊C411yescharacteristic
59C2×Dic173C2×D411yescharacteristic
60C4×D173—22no—
61C4×D173—41no—
62C4×D173—41no—
63C23×C173—21no—
64Q8×C173—41no—
65D4×C173—41no—
66C2×C4×C173—21no—
67C2×C4×C173C22⋊C411yescharacteristic
68C2×C4×C173C22⋊C411yescharacteristic
69C2×C4×C173C22×C411yescharacteristic
70C22×C173—41no—
71C22×C173—41no—
72C22×C173—41no—
73C22×C173C2×C22⋊C411yescharacteristic
74C2×D173—21no—
75C2×D173—21no—
76C2×D173#13512yes—
77#10621924—82no—
78C4×C173—21no—
79C4×C173—21no—
80C4×C173—21no—
81C4×C173—41no—
82Dic173—21no—
83Dic173—21no—
84Dic173—21no—
85Dic173—41no—
86Dic173—41no—
87C2×C173—81no—
88C2×C173#41011yescharacteristic
89C2×C173#41611yescharacteristic
90C2×C173#42111yescharacteristic
91D173—22no—
92C173#152111yescharacteristic, complemented, Sylow
93#1521—1731nomaximal, Sylow
94#529—1731no—
95#431—1731no—
96#383—1731no—
97#328—1732no—
98#327—1732no—
99#126—3461no—
100#125—1731no—
101#124#1062192411yescharacteristic, complemented
102#123—3461no—
103C22⋊Q8—1731no—
104#121—1731no—
105C4×Q8—1731no—
106C4×D4—3461no—
107C4×D4—1731no—
108#117—3461no—
109C2×C42—1731no—
110C4⋊C8—3461no—
111C22⋊C8—1732no—
112C4×C8—3461no—
113#95—3461no—
114#95—3461no—
115#95—1732no—
116C2×Q8C2×#1062192411yescharacteristic
117C2×D4—3461no—
118C2×D4C2×#1062192411yescharacteristic
119C22×C4—3461no—
120C22×C4—3462no—
121C22×C4—1731no—
122C22×C4—1731no—
123C22×C4—1731no—
124C2×C8—3462no—
125C4⋊C4—3461no—
126C4⋊C4—3461no—
127C4⋊C4—3461no—
128C4⋊C4—3461no—
129C4⋊C4—1731no—
130C4⋊C4—1731no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—3461no—
134C22⋊C4—3461no—
135C22⋊C4—3461no—
136C42—3461no—
137C42—1731no—
138C42—1731no—
139C42—1731no—
140C42C2×#1062192411yescharacteristic
141C23—21no—
142C23—1731no—
143Q8—41no—
144D4—41no—
145D4—6922no—
146C2×C4—21no—
147C2×C4—3462no—
148C2×C4—3461no—
149C2×C4—3461no—
150C2×C4—1731no—
151C2×C4—1731no—
152C2×C4—1731no—
153C2×C4—6921no—
154C2×C4—6921no—
155C2×C4—6922no—
156C2×C4—6922no—
157C2×C4—6922no—
158C2×C4—6922no—
159C2×C4C22⋊#1062192411yescharacteristic
160C2×C4C22⋊#1062192411yescharacteristic
161C2×C4C22×#1062192411yescharacteristic
162C8—6922no—
163C22—41no—
164C22—41no—
165C22—41no—
166C22—3461no—
167C22—3461no—
168C22—1732no—
169C22C2×C22⋊#1062192411yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—41no—
174C4—3461no—
175C4—3461no—
176C4—3461no—
177C4—13842no—
178C4—6921no—
179C4—6921no—
180C2—81no—
181C2—3462no—
182C2#2601247111yescharacteristic
183C2#2601248111yescharacteristic
184C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC22×C173C2×C22⋊C4
prime radical#19149650C22
cosocleC2×C4×D173C23
hypercenter#124#10621924
hypocenterC173#1521
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #26012510C2semidirect
2. #26012436C2semidirect
3. #26012345C2semidirect
13. #19149671C4semidirect
17. Q8×Dic173C4semidirect
18. D4×Dic173C4semidirect
22. #124×C173C4semidirect
92. C173#1521semidirect
101. #124#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125102
#260124362
#260123452
#260123422
#260120202
#1521173