Finite Group Databasebeta

← #36368961

Subgroups — #36368961

Order 22144.

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

normal subgroups only

This group has 294 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
2C4×C22⋊#10621924C211yesmaximal, characteristic
3#26012345C212yesmaximal, complemented
4#26012344C212yesmaximal, complemented
5C22⋊Q8×D173C211yesmaximal, characteristic
6C2×#14583835—21no—
7C2×C22⋊#10621924—21no—
8C2×C4×#10621924—21no—
9#19149736—21no—
10#19149736C2212yes—
11(C2×#10621924)⋊C4C2212yes—
12C2×Q8×D173C411yescharacteristic, complemented
13C22×C4×D173C2211yescharacteristic
14#19149659C2211yescharacteristic
15#19149658C411yescharacteristic, complemented
16#19149657—21no—
17#19149656—21no—
18#19149655C2211yescharacteristic
19C4⋊C4×D173—21no—
20C4⋊C4×D173C411yescharacteristic, complemented
21C22⋊C4×D173—21no—
22C22⋊Q8×C173C411yescharacteristic, complemented
23C22×#10621924—22no—
24#14583835—42no—
25C2×#13063443—22no—
26C22⋊#10621924—24no—
27C4×#10621924—42no—
28C23×D173D411yescharacteristic
29C2×Dic346—21no—
30C2×Dic346C2×C411yescharacteristic
31Q8×D173—42no—
32C22×Dic173D411yescharacteristic
33C2×C4×D173—21no—
34C2×C4×D173—21no—
35C2×C4×D173—42no—
36C2×C4×D173C2×C411yescharacteristic
37C2×C4×D173D411yescharacteristic
38C2×C4×D173D411yescharacteristic
39C2×C4×D173C2311yescharacteristic
40C4⋊Dic173—21no—
41C4⋊Dic173C2×C411yescharacteristic
42#14583805—21no—
43#14583805—21no—
44#14583805—21no—
45C22⋊Dic173—21no—
46#14583803—21no—
47#14583803—21no—
48C2×Q8×C173C2×C411yescharacteristic
49C22×C4×C173C2×C411yescharacteristic
50C4⋊C4×C173—21no—
51C4⋊C4×C173C2×C411yescharacteristic
52C22⋊C4×C173—21no—
53C2×#10621924—24no—
54C2×#10621924—42no—
55C2×#10621924—42no—
56#13063443—42no—
57C22×D173—22no—
58C22×D173—22no—
59C22×D173—41no—
60C22×D173C22⋊C411yescharacteristic
61C22×D173C2×D411yescharacteristic
62Dic346—42no—
63Dic346—41no—
64C2×Dic173—22no—
65C2×Dic173—21no—
66C2×Dic173—21no—
67C2×Dic173C22⋊C411yescharacteristic
68C2×Dic173C2×D411yescharacteristic
69C4×D173—22no—
70C4×D173—22no—
71C4×D173—41no—
72C4×D173—41no—
73C4×D173—42no—
74C4×D173—42no—
75C23×C173C22⋊C411yescharacteristic
76Q8×C173—41no—
77C2×C4×C173—21no—
78C2×C4×C173—21no—
79C2×C4×C173—41no—
80C2×C4×C173C22⋊C411yescharacteristic
81C2×C4×C173C22×C411yescharacteristic
82C22×C173—21no—
83C22×C173—21no—
84C22×C173—21no—
85C22×C173C2×C22⋊C411yescharacteristic
86C2×D173—21no—
87C2×D173—21no—
88C2×D173—21no—
89C2×D173—21no—
90C2×D173—44no—
91C2×D173#10412yes—
92C2×D173#13712yes—
93#10621924—44no—
94C4×C173—21no—
95C4×C173—21no—
96C4×C173—41no—
97C4×C173—41no—
98Dic173—21no—
99Dic173—41no—
100Dic173—41no—
101Dic173#9912yes—
102C2×C173—41no—
103C2×C173C2×#9911yescharacteristic
104C2×C173#41911yescharacteristic
105C2×C173C2×#10411yescharacteristic
106D173—22no—
107D173—22no—
108C173#150811yescharacteristic, complemented, Sylow
109#1508—1731nomaximal, Sylow
110C2×C22⋊Q8—1731no—
111#407—1731no—
112C4×C22⋊C4—1731no—
113#328—1732no—
114#328—1732no—
115C22×Q8—1731no—
116C23×C4—1731no—
117C2×M16—3461no—
118C22⋊Q8—3461no—
119C22⋊Q8—3461no—
120C22⋊Q8—1731no—
121C22⋊Q8—1731no—
122C22⋊Q8—1731no—
123C22⋊Q8#1062192411yescharacteristic, complemented
124C2×C4⋊C4—3461no—
125C2×C4⋊C4—1731no—
126C2×C22⋊C4—3461no—
127C2×C22⋊C4—3461no—
128C2×C42—3461no—
129C22⋊C8—3461no—
130C22⋊C8—1732no—
131#95—1732no—
132C24—1731no—
133C2×Q8—3461no—
134C2×Q8—1731no—
135C2×Q8—6922no—
136C2×Q8C2×#1062192411yescharacteristic
137C22×C4—3461no—
138C22×C4—3461no—
139C22×C4—3462no—
140C22×C4—1731no—
141C22×C4—1731no—
142C22×C4—1731no—
143C22×C4—1731no—
144C22×C4—1731no—
145C22×C4—6922no—
146C22×C4C2×#1062192411yescharacteristic
147M16—6922no—
148C2×C8—3462no—
149C4⋊C4—21no—
150C4⋊C4—3461no—
151C4⋊C4—3461no—
152C4⋊C4—3461no—
153C4⋊C4—3461no—
154C4⋊C4—1731no—
155C4⋊C4C2×#1062192411yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—3461no—
158C22⋊C4—3461no—
159C22⋊C4—3464no—
160C22⋊C4—3461no—
161C42—6922no—
162C23—3462no—
163C23—3462no—
164C23—1731no—
165C23—1731no—
166C23—6921no—
167C23C22⋊#1062192411yescharacteristic
168Q8—41no—
169Q8—6921no—
170Q8—6922no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—3462no—
175C2×C4—3462no—
176C2×C4—3462no—
177C2×C4—3464no—
178C2×C4—3461no—
179C2×C4—3461no—
180C2×C4—1731no—
181C2×C4—1731no—
182C2×C4—6921no—
183C2×C4—6921no—
184C2×C4—6922no—
185C2×C4—6922no—
186C2×C4—6922no—
187C2×C4—6922no—
188C2×C4C22⋊#1062192411yescharacteristic
189C2×C4C22×#1062192411yescharacteristic
190C8—6922no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—3461no—
195C22—3461no—
196C22—3461no—
197C22—3461no—
198C22—1732no—
199C22—1732no—
200C22—6924no—
201C22C2×C22⋊#1062192411yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—3461no—
207C4—1732no—
208C4—6924no—
209C4—6921no—
210C4—6921no—
211C2—41no—
212C2—3462no—
213C2—3462no—
214C2D173⋊#9911yescharacteristic
215C2#2601248411yescharacteristic
216C2D173⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22⋊Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC23×D173D4
cosocleC2×C4×D173C23
hypercenterC22⋊Q8#10621924
hypocenterC173#1508
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #26012345C2semidirect
4. #26012344C2semidirect
12. C2×Q8×D173C4semidirect
15. #19149658C4semidirect
20. C4⋊C4×D173C4semidirect
22. C22⋊Q8×C173C4semidirect
108. C173#1508semidirect
123. C22⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124632
C4×C22⋊#106219242
#260123452
#260123442
C22⋊Q8×D1732
#1508173