Finite Group Databasebeta

← #36368972

Subgroups — #36368972

Order 22144.

274 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 58 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
2#26012434C211yesmaximal, characteristic
3#26012345C212yesmaximal, complemented
4#26012344C212yesmaximal, complemented
5C22⋊Q8×D173C211yesmaximal, characteristic
6C2×#14583836—21no—
7C2×C22⋊#10621924—21no—
8#19149736C2212yes—
9C22⋊#13063443—21no—
10(C2×#10621924)⋊C4—21no—
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#14583836—42no—
25C2×#13063443—22no—
26C22⋊#10621924—42no—
27C23×D173D411yescharacteristic
28C2×Dic346—21no—
29C2×Dic346C2×C411yescharacteristic
30Q8×D173—42no—
31C22×Dic173D411yescharacteristic
32C2×C4×D173—21no—
33C2×C4×D173—21no—
34C2×C4×D173—42no—
35C2×C4×D173C2×C411yescharacteristic
36C2×C4×D173D411yescharacteristic
37C2×C4×D173D411yescharacteristic
38C2×C4×D173C2311yescharacteristic
39C4⋊Dic173—21no—
40C4⋊Dic173C2×C411yescharacteristic
41#14583805—21no—
42#14583805—21no—
43#14583805—21no—
44C22⋊Dic173—21no—
45#14583803—21no—
46#14583803—21no—
47C2×Q8×C173C2×C411yescharacteristic
48C22×C4×C173C2×C411yescharacteristic
49C4⋊C4×C173—21no—
50C4⋊C4×C173C2×C411yescharacteristic
51C22⋊C4×C173—21no—
52C2×#10621924—42no—
53C2×#10621924—42no—
54C2×#10621924—42no—
55#13063443—42no—
56C22×D173—22no—
57C22×D173—22no—
58C22×D173—41no—
59C22×D173C22⋊C411yescharacteristic
60C22×D173C2×D411yescharacteristic
61Dic346—42no—
62Dic346—41no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—41no—
66C2×Dic173C22⋊C411yescharacteristic
67C2×Dic173C2×D411yescharacteristic
68C4×D173—22no—
69C4×D173—22no—
70C4×D173—41no—
71C4×D173—41no—
72C4×D173—42no—
73C4×D173—42no—
74C23×C173C22⋊C411yescharacteristic
75Q8×C173—41no—
76C2×C4×C173—21no—
77C2×C4×C173—21no—
78C2×C4×C173—41no—
79C2×C4×C173C22⋊C411yescharacteristic
80C2×C4×C173C22×C411yescharacteristic
81C22×C173—21no—
82C22×C173—21no—
83C22×C173—21no—
84C22×C173C2×C22⋊C411yescharacteristic
85C2×D173—21no—
86C2×D173—21no—
87C2×D173—21no—
88C2×D173—21no—
89C2×D173—44no—
90C2×D173#9912yes—
91C2×D173#13712yes—
92#10621924—82no—
93C4×C173—21no—
94C4×C173—21no—
95C4×C173—41no—
96C4×C173—41no—
97Dic173—21no—
98Dic173—21no—
99Dic173—41no—
100Dic173—41no—
101C2×C173—41no—
102C2×C173C2×#9911yescharacteristic
103C2×C173#41911yescharacteristic
104C2×C173#42111yescharacteristic
105D173—22no—
106D173—22no—
107C173#151111yescharacteristic, complemented, Sylow
108#1511—1731nomaximal, Sylow
109C2×C22⋊Q8—1731no—
110#407—1731no—
111#381—1731no—
112#328—1732no—
113#328—1732no—
114C22×Q8—1731no—
115C23×C4—1731no—
116C2×M16—3461no—
117C22⋊Q8—3461no—
118C22⋊Q8—3461no—
119C22⋊Q8—1731no—
120C22⋊Q8—1731no—
121C22⋊Q8—1731no—
122C22⋊Q8#1062192411yescharacteristic, complemented
123C2×C4⋊C4—3461no—
124C2×C4⋊C4—1731no—
125C2×C22⋊C4—3461no—
126C2×C22⋊C4—3461no—
127C22⋊C8—3461no—
128C22⋊C8—1732no—
129#95—3461no—
130#95—1732no—
131C24—1731no—
132C2×Q8—3461no—
133C2×Q8—1731no—
134C2×Q8—6922no—
135C2×Q8C2×#1062192411yescharacteristic
136C22×C4—3461no—
137C22×C4—3461no—
138C22×C4—3462no—
139C22×C4—1731no—
140C22×C4—1731no—
141C22×C4—1731no—
142C22×C4—1731no—
143C22×C4—1731no—
144C22×C4—6922no—
145C22×C4C2×#1062192411yescharacteristic
146M16—6922no—
147C2×C8—3462no—
148C4⋊C4—21no—
149C4⋊C4—3461no—
150C4⋊C4—3461no—
151C4⋊C4—3461no—
152C4⋊C4—3461no—
153C4⋊C4—1731no—
154C4⋊C4C2×#1062192411yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3461no—
157C22⋊C4—3461no—
158C22⋊C4—3461no—
159C22⋊C4—6922no—
160C23—3462no—
161C23—3462no—
162C23—1731no—
163C23—1731no—
164C23—6921no—
165C23C22⋊#1062192411yescharacteristic
166Q8—41no—
167Q8—6921no—
168Q8—6922no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—3462no—
173C2×C4—3462no—
174C2×C4—3461no—
175C2×C4—3461no—
176C2×C4—1731no—
177C2×C4—1731no—
178C2×C4—6921no—
179C2×C4—6921no—
180C2×C4—6922no—
181C2×C4—6922no—
182C2×C4—6921no—
183C2×C4—6922no—
184C2×C4—6922no—
185C2×C4—6922no—
186C2×C4C22⋊#1062192411yescharacteristic
187C2×C4C22×#1062192411yescharacteristic
188C8—6922no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—3461no—
193C22—3461no—
194C22—3461no—
195C22—3461no—
196C22—1732no—
197C22—1732no—
198C22—6924no—
199C22C2×C22⋊#1062192411yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—3461no—
205C4—3461no—
206C4—13842no—
207C4—6921no—
208C4—6921no—
209C2—41no—
210C2—3462no—
211C2—3462no—
212C2D173⋊#9911yescharacteristic
213C2#2601248411yescharacteristic
214C2#2601248811yescharacteristic
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#1511
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
107. C173#1511semidirect
122. C22⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124622
#260124342
#260123452
#260123442
C22⋊Q8×D1732
#1511173