Finite Group Databasebeta

← #36368541

Subgroups — #36368541

Order 22144.

222 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 30 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#26012369C211yesmaximal, characteristic, complemented
2#26012335C211yesmaximal, characteristic, complemented
3#26011399C211yesmaximal, characteristic
4C2×C22⋊#10621924—21no—
5C22⋊#13063443—21no—
6C22×C22⋊D173—21no—
7C2×C22⋊Dic173—21no—
8C2×#14583803C2211yescharacteristic
9#19149586C411yescharacteristic, complemented
10C22×D4×C173C411yescharacteristic, complemented
11C22×#10621924—41no—
12C2×#13063443—41no—
13C22⋊#10621924—22no—
14C22⋊#10621924—41no—
15C23×D173D411yescharacteristic
16C2×C22⋊D173—42no—
17C2×C22⋊D173—42no—
18C2×C22⋊D173—41no—
19C2×C22⋊D173—41no—
20C22×Dic173—21no—
21C22×Dic173D411yescharacteristic
22C22⋊Dic173—42no—
23#14583803—41no—
24C24×C173—21no—
25C2×D4×C173—41no—
26C2×D4×C173—42no—
27C22×C4×C173C2×C411yescharacteristic
28C2×#10621924—42no—
29C2×#10621924—81no—
30C2×#10621924—81no—
31#13063443—81no—
32C22×D173—21no—
33C22×D173—21no—
34C22×D173—41no—
35C22×D173—41no—
36C22×D173D811yescharacteristic
37C22×D173SD1611yescharacteristic
38C22⋊D173—44no—
39C22⋊D173—44no—
40C2×Dic173—22no—
41C2×Dic173—21no—
42C2×Dic173—41no—
43C2×Dic173—41no—
44C2×Dic173—41no—
45C23×C173—22no—
46C23×C173—41no—
47C23×C173—41no—
48C23×C173—41no—
49C23×C173—41no—
50C23×C173—41no—
51C23×C173—41no—
52C23×C173C22⋊C411yescharacteristic
53D4×C173—42no—
54D4×C173—42no—
55C2×C4×C173—21no—
56C2×C4×C173—41no—
57C22×C173—21no—
58C22×C173—21no—
59C22×C173—42no—
60C22×C173—42no—
61C22×C173—42no—
62C22×C173—81no—
63C22×C173—81no—
64C22×C173—81no—
65C22×C173—81no—
66C22×C173#9911yescharacteristic
67C22×C173#10211yescharacteristic
68C22×C173#10411yescharacteristic
69C2×D173—21no—
70C2×D173—21no—
71C2×D173—41no—
72C2×D173—41no—
73C2×D173—41no—
74C2×D173—41no—
75C2×D173—81no—
76#10621924—82no—
77C4×C173—41no—
78Dic173—42no—
79Dic173—41no—
80C2×C173—21no—
81C2×C173—21no—
82C2×C173—82no—
83C2×C173#32711yescharacteristic
84C2×C173AΣL(1,𝔽16)11yescharacteristic
85C2×C173#35311yescharacteristic
86D173—41no—
87D173—41no—
88C173#134311yescharacteristic, complemented, Sylow
89#1343—1731nomaximal, Sylow
90#392—1731no—
91#342—1731no—
92#323—1731no—
93C22×D4—3461no—
94C22×D4#1062192411yescharacteristic, complemented
95C2×C22⋊C4—3461no—
96C2×C22⋊C4—3461no—
97C2×C22⋊C4—1731no—
98C22⋊C8—3461no—
99#95—1731no—
100C24—21no—
101C24—1731no—
102C2×D4—41no—
103C2×D4—42no—
104C2×D4—6922no—
105C2×D4—6922no—
106C2×D4—6921no—
107C2×D4—6921no—
108C22×C4—3461no—
109C22×C4—1731no—
110C22×C4—6921no—
111C22×C4C2×#1062192411yescharacteristic
112C2×C8—6921no—
113C22⋊C4—3462no—
114C22⋊C4—6921no—
115C22⋊C4—6922no—
116C22⋊C4—6921no—
117C23—22no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—3461no—
125C23—3461no—
126C23—1731no—
127C23—1731no—
128C23—6921no—
129C23—6921no—
130C23C22⋊#1062192411yescharacteristic
131D4—42no—
132D4—42no—
133D4—6924no—
134D4—6924no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—3462no—
138C2×C4—3461no—
139C2×C4—13841no—
140C2×C4—13841no—
141C2×C4—6922no—
142C2×C4—6921no—
143C2×C4—6921no—
144C2×C4—6921no—
145C8—13841no—
146C22—21no—
147C22—21no—
148C22—42no—
149C22—42no—
150C22—42no—
151C22—81no—
152C22—81no—
153C22—81no—
154C22—81no—
155C22—3461no—
156C22—3461no—
157C22—13841no—
158C22—6921no—
159C22—6921no—
160C22—6921no—
161C22—6921no—
162C22#1914973711yescharacteristic
163C22#1914974111yescharacteristic
164C22#1914974311yescharacteristic
165C4—41no—
166C4—13842no—
167C4—6922no—
168C4—6921no—
169C2—21no—
170C2—21no—
171C2—82no—
172C2—6921no—
173C2—6921no—
174C2#2601234211yescharacteristic
175C2#2601239111yescharacteristic
176C2#2601239511yescharacteristic
subgroupG/H
centerC22#19149737
commutatorC22×C4×C173C2×C4
FrattiniC22×C4C2×#10621924
FittingC22×D4×C173C4
socleC22×C173#99
prime radical#26011399C2
cosocleC2×#14583803C22
hypercenterC22×D4#10621924
hypocenterC173#1343
LCSG ▷ C22×C4×C173 ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #26012369C2semidirect
2. #26012335C2semidirect
9. #19149586C4semidirect
10. C22×D4×C173C4semidirect
88. C173#1343semidirect
94. C22×D4#10621924semidirect

Maximal subgroups

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

subgroupindex
#260123692
#260123352
#260113992
#1343173