Finite Group Databasebeta

← #36369468

Subgroups — #36369468

Order 22144.

300 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 74 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
1D173⋊(C22⋊C8)C211yesmaximal, characteristic
2#26012433C211yesmaximal, characteristic
3#26012364C212yesmaximal, complemented
4#26012364C212yesmaximal, complemented
5#117×D173C211yesmaximal, characteristic
6C2×D173⋊C8C2212yes—
7C2×C4⋊#10621924C2212yes—
8C2×C22⋊#10621924—21no—
9#19149736—21no—
10C22⋊#13063443—21no—
11(C2×#10621924)⋊C4—21no—
12C22×C4×D173C2211yescharacteristic
13#19149638C411yescharacteristic, complemented
14#19149637C411yescharacteristic, complemented
15#19149637C411yescharacteristic, complemented
16#19149636—21no—
17#19149635—21no—
18C4⋊C4×D173C2211yescharacteristic
19C4⋊C4×D173C2211yescharacteristic
20C22⋊C4×D173—21no—
21C42×D173—21no—
22#117×C173C411yescharacteristic, complemented
23C22×#10621924—22no—
24D173⋊C8—24no—
25C2×#13063443—22no—
26C4⋊#10621924—24no—
27C22⋊#10621924—42no—
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—42no—
53C2×#10621924—42no—
54C2×#10621924—42no—
55#13063443—42no—
56C22×D173—22no—
57C22×D173—22no—
58C22×D173—21no—
59C22×D173—21no—
60C22×D173C22⋊C411yescharacteristic
61C22×D173C2×D411yescharacteristic
62C2×Dic173—22no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173C22⋊C411yescharacteristic
66C2×Dic173C2×D411yescharacteristic
67C4×D173—22no—
68C4×D173—22no—
69C4×D173—22no—
70C4×D173—42no—
71C4×D173—41no—
72C4×D173—41no—
73C4×D173#4114yes—
74C23×C173C22⋊C411yescharacteristic
75C2×C4×C173—22no—
76C2×C4×C173—21no—
77C2×C4×C173—21no—
78C2×C4×C173C22⋊C411yescharacteristic
79C2×C4×C173C22×C411yescharacteristic
80C22×C173—21no—
81C22×C173—21no—
82C22×C173—21no—
83C22×C173C2×C22⋊C411yescharacteristic
84C2×D173—21no—
85C2×D173—21no—
86C2×D173—21no—
87C2×D173—42no—
88C2×D173—42no—
89C2×D173#12312yes—
90C2×D173#13512yes—
91C2×D173AGL(1,ℤ/8)11yescharacteristic
92C2×D173#13711yescharacteristic
93#10621924—82no—
94C4×C173—21no—
95C4×C173—41no—
96C4×C173—41no—
97C4×C173#11712yes—
98Dic173—21no—
99Dic173—41no—
100Dic173—41no—
101Dic173#12312yes—
102C2×C173—41no—
103C2×C173#38111yescharacteristic
104C2×C173#41611yescharacteristic
105C2×C173#41711yescharacteristic
106D173—41no—
107D173#48212yes—
108D173#48512yes—
109C173#182511yescharacteristic, complemented, Sylow
110#1825—1731nomaximal, Sylow
111C2×#117—1731no—
112C2×C22⋊C8—1731no—
113#380—1731no—
114#340—1732no—
115#340—1732no—
116C23×C4—1731no—
117C22×C8—1732no—
118#117—3461no—
119#117—3461no—
120#117—1731no—
121#117—1731no—
122#117—1731no—
123#117#1062192411yescharacteristic, complemented
124C2×C4⋊C4—1732no—
125C2×C4⋊C4—1731no—
126C2×C4⋊C4—1731no—
127C2×C22⋊C4—3461no—
128C2×C22⋊C4—3461no—
129C2×C42—3461no—
130C22⋊C8—3461no—
131C22⋊C8—3461no—
132#95—3461no—
133C24—1731no—
134C22×C4—3464no—
135C22×C4—3461no—
136C22×C4—3461no—
137C22×C4—3462no—
138C22×C4—1731no—
139C22×C4—1731no—
140C22×C4—1731no—
141C22×C4—1731no—
142C22×C4—1731no—
143C22×C4C2×#1062192411yescharacteristic
144C2×C8—3462no—
145C2×C8—3464no—
146C4⋊C4—3461no—
147C4⋊C4—3461no—
148C4⋊C4—3464no—
149C4⋊C4—1731no—
150C4⋊C4—1731no—
151C4⋊C4C2×#1062192411yescharacteristic
152C4⋊C4C2×#1062192411yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—3461no—
155C22⋊C4—3461no—
156C22⋊C4—3461no—
157C22⋊C4—6922no—
158C42—21no—
159C42—3461no—
160C42—3461no—
161C42—3461no—
162C23—3461no—
163C23—3462no—
164C23—3462no—
165C23—3461no—
166C23—1731no—
167C23—1731no—
168C23C22⋊#1062192411yescharacteristic
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—3462no—
173C2×C4—3462no—
174C2×C4—3462no—
175C2×C4—3462no—
176C2×C4—3461no—
177C2×C4—3461no—
178C2×C4—1731no—
179C2×C4—1731no—
180C2×C4—1734no—
181C2×C4—6922no—
182C2×C4—6921no—
183C2×C4—6921no—
184C2×C4—6922no—
185C2×C4—6922no—
186C2×C4—6922no—
187C2×C4C22⋊#1062192411yescharacteristic
188C2×C4C22×#1062192411yescharacteristic
189C8—6922no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—3461no—
194C22—3461no—
195C22—3461no—
196C22—1732no—
197C22—1732no—
198C22—1731no—
199C22—1731no—
200C22—6922no—
201C22—6922no—
202C22C2×C22⋊#1062192411yescharacteristic
203C4—21no—
204C4—41no—
205C4—41no—
206C4—3461no—
207C4—13842no—
208C4—1732no—
209C4—6921no—
210C4—6921no—
211C4#1914975812yes—
212C2—41no—
213C2—1732no—
214C2—1732no—
215C2—6921no—
216C2#2601243411yescharacteristic
217C2#2601248111yescharacteristic
218C2#2601248211yescharacteristic
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#1825
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. #26012364C2semidirect
4. #26012364C2semidirect
13. #19149638C4semidirect
14. #19149637C4semidirect
15. #19149637C4semidirect
22. #117×C173C4semidirect
109. C173#1825semidirect
123. #117#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊(C22⋊C8)2
#260124332
#260123642
#117×D1732
#1825173