Finite Group Databasebeta

← #36368978

Subgroups — #36368978

Order 22144.

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

normal subgroups only

This group has 286 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#26012462C211yesmaximal, characteristic, complemented
2#26012433C211yesmaximal, characteristic, complemented
3#26012345C212yesmaximal, complemented
4#26012342C212yesmaximal, complemented
5#26011998C211yesmaximal, characteristic
6C2×#14583836—21no—
7C2×C4⋊#10621924—21no—
8C2×C22⋊#10621924—21no—
9#19149736C2212yes—
10C22⋊#13063443—21no—
11(C2×#10621924)⋊C4C2212yes—
12C22×C4×D173C2211yescharacteristic
13#19149660—21no—
14#19149655C411yescharacteristic, complemented
15#19149655C2211yescharacteristic
16#19149650C411yescharacteristic, complemented
17#19149650C2211yescharacteristic
18#19149647—21no—
19C4×C22⋊D173—21no—
20C4×C22⋊D173—21no—
21#19149638C411yescharacteristic, complemented
22C2×#41×C173C411yescharacteristic, complemented
23C22×#10621924—22no—
24#14583836—42no—
25C2×#13063443—22no—
26C4⋊#10621924—42no—
27C22⋊#10621924—42no—
28C23×D173D411yescharacteristic
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C22×Dic173D411yescharacteristic
32C2×C4×D173—42no—
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—
42C22⋊Dic173—21no—
43C22⋊Dic173—21no—
44#14583803—21no—
45#14583803—21no—
46C4×Dic173—21no—
47#41×C173—42no—
48C2×Q8×C173C2×C411yescharacteristic
49C2×D4×C173—21no—
50C2×D4×C173C2×C411yescharacteristic
51C22×C4×C173—21no—
52C22×C4×C173C2×C411yescharacteristic
53C2×#10621924—42no—
54C2×#10621924—42no—
55C2×#10621924—42no—
56#13063443—42no—
57C22×D173—21no—
58C22×D173—21no—
59C22×D173—41no—
60C22×D173—41no—
61C22×D173C22⋊C411yescharacteristic
62C22×D173C2×D411yescharacteristic
63C22⋊D173—42no—
64C22⋊D173—41no—
65C22⋊D173—41no—
66C2×Dic173—21no—
67C2×Dic173—21no—
68C2×Dic173—41no—
69C2×Dic173C22⋊C411yescharacteristic
70C2×Dic173C2×D411yescharacteristic
71C4×D173—22no—
72C4×D173—22no—
73C4×D173—41no—
74C4×D173—41no—
75C23×C173—21no—
76C23×C173C22⋊C411yescharacteristic
77Q8×C173—41no—
78D4×C173—41no—
79D4×C173—42no—
80C2×C4×C173—22no—
81C2×C4×C173—21no—
82C2×C4×C173—42no—
83C2×C4×C173C22⋊C411yescharacteristic
84C2×C4×C173C22×C411yescharacteristic
85C22×C173—21no—
86C22×C173—21no—
87C22×C173—21no—
88C22×C173—41no—
89C22×C173—41no—
90C22×C173—41no—
91C22×C173C2×C22⋊C411yescharacteristic
92C2×D173—21no—
93C2×D173—21no—
94C2×D173—21no—
95C2×D173—21no—
96C2×D173—21no—
97C2×D173—42no—
98C2×D173—42no—
99C2×D173AGL(1,ℤ/8)11yescharacteristic
100C2×D173#13711yescharacteristic
101#10621924—82no—
102C4×C173—21no—
103C4×C173—41no—
104C4×C173#9912yes—
105Dic173—21no—
106Dic173—21no—
107Dic173—41no—
108Dic173—41no—
109C2×C173—41no—
110C2×C173—81no—
111C2×C173C2×#9911yescharacteristic
112C2×C173#41711yescharacteristic
113C2×C173#42111yescharacteristic
114D173—21no—
115D173—21no—
116D173—41no—
117C173#151311yescharacteristic, complemented, Sylow
118#1513—1731nomaximal, Sylow
119#525—1731no—
120#407—1731no—
121#380—1731no—
122#328—1732no—
123#327—1732no—
124C2×#41#1062192411yescharacteristic, complemented
125C23×C4—1731no—
126C2×M16—3461no—
127#123—3461no—
128C22⋊Q8—1731no—
129C22⋊Q8—1731no—
130#121—1731no—
131#121—1731no—
132#120—3461no—
133C4×D4—3461no—
134C4×D4—3461no—
135#117—1731no—
136C2×C4⋊C4—3461no—
137C2×C22⋊C4—3461no—
138C22⋊C8—3461no—
139C22⋊C8—1732no—
140#95—1732no—
141C24—1731no—
142#41—42no—
143C2×Q8C2×#1062192411yescharacteristic
144C2×D4—21no—
145C2×D4—3461no—
146C2×D4—3461no—
147C2×D4C2×#1062192411yescharacteristic
148C22×C4—21no—
149C22×C4—3462no—
150C22×C4—1731no—
151C22×C4—1731no—
152C22×C4—1731no—
153C22×C4—1731no—
154C22×C4—1731no—
155C22×C4—6922no—
156C22×C4C2×#1062192411yescharacteristic
157M16—6922no—
158C2×C8—3462no—
159C4⋊C4—3461no—
160C4⋊C4—3461no—
161C4⋊C4—1731no—
162C4⋊C4—1731no—
163C4⋊C4—6922no—
164C22⋊C4—3461no—
165C22⋊C4—3461no—
166C22⋊C4—3461no—
167C22⋊C4—3461no—
168C22⋊C4—3461no—
169C22⋊C4—6922no—
170C42—3461no—
171C23—21no—
172C23—3461no—
173C23—3461no—
174C23—1731no—
175C23—1731no—
176C23—6921no—
177C23—6921no—
178C23C22⋊#1062192411yescharacteristic
179Q8—41no—
180D4—41no—
181D4—42no—
182D4—6922no—
183D4—6921no—
184D4—6921no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—42no—
188C2×C4—3462no—
189C2×C4—3462no—
190C2×C4—3461no—
191C2×C4—3461no—
192C2×C4—1731no—
193C2×C4—1731no—
194C2×C4—6921no—
195C2×C4—6921no—
196C2×C4—6921no—
197C2×C4—6922no—
198C2×C4—6922no—
199C2×C4—6922no—
200C2×C4C22⋊#1062192411yescharacteristic
201C2×C4C22×#1062192411yescharacteristic
202C8—6922no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—3461no—
210C22—3461no—
211C22—3461no—
212C22—3461no—
213C22—3461no—
214C22—1731no—
215C22—1731no—
216C22—6922no—
217C22—6922no—
218C22C2×C22⋊#1062192411yescharacteristic
219C4—21no—
220C4—41no—
221C4—3461no—
222C4—3461no—
223C4—13842no—
224C4—6921no—
225C4—6921no—
226C4#1914973712yes—
227C2—41no—
228C2—81no—
229C2—3461no—
230C2—3461no—
231C2—6921no—
232C2C2×#1914973711yescharacteristic
233C2#2601248211yescharacteristic
234C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×#41×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26011998C2
cosocleC2×C4×D173C23
hypercenterC2×#41#10621924
hypocenterC173#1513
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #26012462C2semidirect
2. #26012433C2semidirect
3. #26012345C2semidirect
4. #26012342C2semidirect
14. #19149655C4semidirect
16. #19149650C4semidirect
21. #19149638C4semidirect
22. C2×#41×C173C4semidirect
117. C173#1513semidirect
124. C2×#41#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124622
#260124332
#260123452
#260123422
#260119982
#1513173