Finite Group Databasebeta

← #36369471

Subgroups — #36369471

Order 22144.

306 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 68 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#26012462C211yesmaximal, characteristic
2C4×C22⋊#10621924C211yesmaximal, characteristic
3#26012361C212yesmaximal, complemented
4#26012361C212yesmaximal, complemented
5#117×D173C211yesmaximal, characteristic
6C2×#14583836C2212yes—
7C2×C22⋊#10621924—21no—
8C2×C4×#10621924C2212yes—
9#19149736—21no—
10C22⋊#13063443—21no—
11(C2×#10621924)⋊C4—21no—
12C22×C4×D173C2211yescharacteristic
13#19149638C411yescharacteristic, complemented
14#19149637C2211yescharacteristic
15#19149637C2211yescharacteristic
16#19149636—21no—
17#19149635—21no—
18C4⋊C4×D173C411yescharacteristic, complemented
19C4⋊C4×D173C411yescharacteristic, complemented
20C22⋊C4×D173—21no—
21C42×D173—21no—
22#117×C173C411yescharacteristic, complemented
23C22×#10621924—22no—
24#14583836—24no—
25C2×#13063443—22no—
26C22⋊#10621924—24no—
27C4×#10621924—24no—
28C23×D173D411yescharacteristic
29C22×Dic173D411yescharacteristic
30C2×C4×D173—24no—
31C2×C4×D173—21no—
32C2×C4×D173—21no—
33C2×C4×D173C2×C411yescharacteristic
34C2×C4×D173D411yescharacteristic
35C2×C4×D173D411yescharacteristic
36C2×C4×D173C2311yescharacteristic
37C4⋊Dic173C2×C411yescharacteristic
38C4⋊Dic173C2×C411yescharacteristic
39#14583805—21no—
40#14583805—21no—
41C22⋊Dic173—21no—
42#14583803—21no—
43#14583803—21no—
44C4×Dic173—21no—
45C4×Dic173—21no—
46C4×Dic173—21no—
47C22×C4×C173C2×C411yescharacteristic
48C4⋊C4×C173C2×C411yescharacteristic
49C4⋊C4×C173C2×C411yescharacteristic
50C22⋊C4×C173—21no—
51C42×C173—21no—
52C2×#10621924—24no—
53C2×#10621924—42no—
54C2×#10621924—42no—
55#13063443—42no—
56C22×D173—22no—
57C22×D173—22no—
58C22×D173—41no—
59C22×D173C22⋊C411yescharacteristic
60C22×D173C2×D411yescharacteristic
61C2×Dic173—22no—
62C2×Dic173—21no—
63C2×Dic173—21no—
64C2×Dic173C22⋊C411yescharacteristic
65C2×Dic173C2×D411yescharacteristic
66C4×D173—22no—
67C4×D173—22no—
68C4×D173—22no—
69C4×D173—42no—
70C4×D173—42no—
71C4×D173#4114yes—
72C23×C173C22⋊C411yescharacteristic
73C2×C4×C173—22no—
74C2×C4×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173C22⋊C411yescharacteristic
77C2×C4×C173C22×C411yescharacteristic
78C22×C173—21no—
79C22×C173—21no—
80C22×C173—21no—
81C22×C173C2×C22⋊C411yescharacteristic
82C2×D173—21no—
83C2×D173—21no—
84C2×D173—21no—
85C2×D173—21no—
86C2×D173—44no—
87C2×D173#10412yes—
88C2×D173#12312yes—
89#10621924—44no—
90C4×C173—21no—
91C4×C173—41no—
92C4×C173—41no—
93C4×C173#11712yes—
94Dic173—21no—
95Dic173—41no—
96Dic173—41no—
97Dic173#12312yes—
98C2×C173—41no—
99C2×C173#38111yescharacteristic
100C2×C173C2×#10411yescharacteristic
101C2×C173#42111yescharacteristic
102D173—22no—
103D173—22no—
104C173#182611yescharacteristic, complemented, Sylow
105#1826—1731nomaximal, Sylow
106C2×#117—1731no—
107#407—1731no—
108C4×C22⋊C4—1731no—
109#339—1732no—
110#339—1732no—
111C23×C4—1731no—
112C2×M16—1732no—
113#117—3461no—
114#117—3461no—
115#117—1731no—
116#117—1731no—
117#117—1731no—
118#117#1062192411yescharacteristic, complemented
119C2×C4⋊C4—1731no—
120C2×C4⋊C4—1731no—
121C2×C22⋊C4—3461no—
122C2×C22⋊C4—3461no—
123C2×C42—3461no—
124C2×C42—1732no—
125C22⋊C8—3461no—
126C22⋊C8—3461no—
127#95—3461no—
128C24—1731no—
129C22×C4—3464no—
130C22×C4—3461no—
131C22×C4—3461no—
132C22×C4—3462no—
133C22×C4—1731no—
134C22×C4—1731no—
135C22×C4—1731no—
136C22×C4—1731no—
137C22×C4—1731no—
138C22×C4C2×#1062192411yescharacteristic
139M16—3464no—
140C2×C8—3462no—
141C4⋊C4—3461no—
142C4⋊C4—3461no—
143C4⋊C4—1731no—
144C4⋊C4—1731no—
145C4⋊C4C2×#1062192411yescharacteristic
146C4⋊C4C2×#1062192411yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—3461no—
149C22⋊C4—3461no—
150C22⋊C4—3464no—
151C22⋊C4—3461no—
152C42—21no—
153C42—3461no—
154C42—3461no—
155C42—3461no—
156C42—3464no—
157C23—3462no—
158C23—3462no—
159C23—1731no—
160C23—1731no—
161C23—6921no—
162C23C22⋊#1062192411yescharacteristic
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—3462no—
167C2×C4—3462no—
168C2×C4—3462no—
169C2×C4—3462no—
170C2×C4—3464no—
171C2×C4—3461no—
172C2×C4—3461no—
173C2×C4—1731no—
174C2×C4—1731no—
175C2×C4—1734no—
176C2×C4—6922no—
177C2×C4—6922no—
178C2×C4—6922no—
179C2×C4—6922no—
180C2×C4C22⋊#1062192411yescharacteristic
181C2×C4C22×#1062192411yescharacteristic
182C8—6922no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—3461no—
187C22—3461no—
188C22—3461no—
189C22—3461no—
190C22—1732no—
191C22—1732no—
192C22—6924no—
193C22C2×C22⋊#1062192411yescharacteristic
194C4—21no—
195C4—41no—
196C4—41no—
197C4—3461no—
198C4—1732no—
199C4—6924no—
200C4—6921no—
201C4—6921no—
202C4#1914975812yes—
203C2—41no—
204C2—3462no—
205C2—3462no—
206C2#2601243411yescharacteristic
207C2D173⋊#10411yescharacteristic
208C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#117×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC23×D173D4
cosocleC2×C4×D173C23
hypercenter#117#10621924
hypocenterC173#1826
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #26012361C2semidirect
4. #26012361C2semidirect
13. #19149638C4semidirect
18. C4⋊C4×D173C4semidirect
19. C4⋊C4×D173C4semidirect
22. #117×C173C4semidirect
104. C173#1826semidirect
118. #117#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124622
C4×C22⋊#106219242
#260123612
#117×D1732
#1826173