Finite Group Databasebeta

← #36368971

Subgroups — #36368971

Order 22144.

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

normal subgroups only

This group has 274 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#26012463C211yesmaximal, characteristic
2#26012434C211yesmaximal, characteristic
3#26012344C214yesmaximal, complemented
4C2×#19149659C211yesmaximal, characteristic
5C2×#14583835—21no—
6C2×C22⋊#10621924—21no—
7#19149736—21no—
8#19149736C2212yes—
9(C2×#10621924)⋊C4—21no—
10(C2×#10621924)⋊C4C2212yes—
11C22×Dic346C411yescharacteristic, complemented
12C22×C4×D173C2211yescharacteristic
13#19149659—21no—
14#19149659—21no—
15#19149659C412yescomplemented
16#19149659C2212yes—
17C2×C4⋊Dic173—21no—
18C2×#14583803—21no—
19C2×C4⋊C4×C173C411yescharacteristic, complemented
20C22×#10621924—22no—
21#14583835—42no—
22C2×#13063443—22no—
23C22⋊#10621924—42no—
24C23×D173D411yescharacteristic
25C2×Dic346—21no—
26C2×Dic346—42no—
27C2×Dic346C2×C412yes—
28C22×Dic173—21no—
29C22×Dic173D411yescharacteristic
30C2×C4×D173—42no—
31C2×C4×D173C2×C411yescharacteristic
32C2×C4×D173D411yescharacteristic
33C2×C4×D173D411yescharacteristic
34C2×C4×D173C2311yescharacteristic
35C4⋊Dic173—22no—
36C4⋊Dic173—22no—
37#14583803—22no—
38#14583803—22no—
39C22×C4×C173—21no—
40C22×C4×C173C2×C411yescharacteristic
41C4⋊C4×C173—21no—
42C4⋊C4×C173C2×C412yes—
43C2×#10621924—42no—
44C2×#10621924—42no—
45C2×#10621924—42no—
46#13063443—42no—
47C22×D173—22no—
48C22×D173—22no—
49C22×D173—41no—
50C22×D173C22⋊C411yescharacteristic
51C22×D173C2×D411yescharacteristic
52Dic346—42no—
53Dic346—42no—
54C2×Dic173—22no—
55C2×Dic173—22no—
56C2×Dic173—42no—
57C2×Dic173C22⋊C411yescharacteristic
58C2×Dic173C2×D411yescharacteristic
59C4×D173—41no—
60C4×D173—41no—
61C4×D173—41no—
62C4×D173—41no—
63C23×C173C22⋊C411yescharacteristic
64C2×C4×C173—22no—
65C2×C4×C173—22no—
66C2×C4×C173—42no—
67C2×C4×C173C22⋊C411yescharacteristic
68C2×C4×C173C22×C411yescharacteristic
69C22×C173—21no—
70C22×C173—21no—
71C22×C173#9912yes—
72C22×C173C2×C22⋊C411yescharacteristic
73C2×D173—21no—
74C2×D173—21no—
75C2×D173—21no—
76C2×D173—21no—
77C2×D173—21no—
78C2×D173—44no—
79C2×D173#13712yes—
80#10621924—82no—
81C4×C173—21no—
82C4×C173—21no—
83C4×C173—42no—
84Dic173—21no—
85Dic173—21no—
86Dic173—42no—
87C2×C173—22no—
88C2×C173C2×#9911yescharacteristic
89C2×C173#41911yescharacteristic
90C2×C173#42111yescharacteristic
91D173—22no—
92D173—41no—
93C173#151111yescharacteristic, complemented, Sylow
94#1511—1731nomaximal, Sylow
95C2×C22⋊Q8—1731no—
96#407—1731no—
97#381—1731no—
98#328—1734no—
99C22×Q8—1731no—
100C23×C4—1731no—
101C2×M16—3461no—
102C22⋊Q8—3461no—
103C22⋊Q8—3461no—
104C22⋊Q8—1732no—
105C22⋊Q8—1732no—
106C2×C4⋊C4—3461no—
107C2×C4⋊C4#1062192411yescharacteristic, complemented
108C2×C22⋊C4—3461no—
109C2×C22⋊C4—3461no—
110C22⋊C8—3461no—
111C22⋊C8—1732no—
112#95—3461no—
113#95—1732no—
114C24—1731no—
115C2×Q8—3461no—
116C2×Q8—1732no—
117C2×Q8—6922no—
118C22×C4—21no—
119C22×C4—3461no—
120C22×C4—3462no—
121C22×C4—1731no—
122C22×C4—1731no—
123C22×C4—1731no—
124C22×C4—1731no—
125C22×C4—1731no—
126C22×C4—6922no—
127C22×C4C2×#1062192411yescharacteristic
128M16—6922no—
129C2×C8—3462no—
130C4⋊C4—21no—
131C4⋊C4—3462no—
132C4⋊C4—3462no—
133C4⋊C4C2×#1062192412yes—
134C22⋊C4—3462no—
135C22⋊C4—3462no—
136C22⋊C4—6922no—
137C23—3462no—
138C23—3462no—
139C23—1731no—
140C23—1731no—
141C23—6921no—
142C23C22⋊#1062192411yescharacteristic
143Q8—6922no—
144Q8—6922no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—42no—
148C2×C4—3462no—
149C2×C4—3462no—
150C2×C4—1731no—
151C2×C4—1731no—
152C2×C4—6921no—
153C2×C4—6921no—
154C2×C4—6921no—
155C2×C4—6921no—
156C2×C4—6922no—
157C2×C4—6922no—
158C2×C4—6922no—
159C2×C4—6922no—
160C2×C4C22⋊#1062192411yescharacteristic
161C2×C4C22×#1062192411yescharacteristic
162C8—6922no—
163C22—21no—
164C22—21no—
165C22—3461no—
166C22—3461no—
167C22—3461no—
168C22—3461no—
169C22—3461no—
170C22—1732no—
171C22—6924no—
172C22#1914973812yes—
173C22C2×C22⋊#1062192411yescharacteristic
174C4—21no—
175C4—21no—
176C4—42no—
177C4—3461no—
178C4—3461no—
179C4—13842no—
180C4—6922no—
181C2—22no—
182C2—3462no—
183C2—6921no—
184C2C2×#1914973811yescharacteristic
185C2#2601248411yescharacteristic
186C2#2601248911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC23×D173D4
cosocleC2×C4×D173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1511
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #26012344C2semidirect
11. C22×Dic346C4semidirect
15. #19149659C4semidirect
19. C2×C4⋊C4×C173C4semidirect
93. C173#1511semidirect
107. C2×C4⋊C4#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124632
#260124342
#260123442
C2×#191496592
#1511173