Finite Group Databasebeta

← #36368839

Subgroups — #36368839

Order 22144.

322 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 64 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#26012461C211yesmaximal, characteristic, complemented
2C2×C22⋊#13063443C211yesmaximal, characteristic, complemented
3#26012335C212yesmaximal, complemented
4#26012334C212yesmaximal, complemented
5C2×#19149651C211yesmaximal, characteristic
6C2×#14583835—21no—
7C22×#13063443—21no—
8C22⋊#13063443—21no—
9C22⋊#13063443—21no—
10C22⋊#13063443C2212yes—
11C22⋊#13063443C2212yes—
12C22×C22⋊D173C411yescharacteristic
13C22×C22⋊D173C411yescharacteristic
14C23×Dic173C2211yescharacteristic
15#19149651—21no—
16#19149651—21no—
17#19149651—21no—
18#19149651—21no—
19C2×#14583805C411yescharacteristic
20C2×C22⋊Dic173C2211yescharacteristic
21C2×#14583803C2211yescharacteristic
22C22×D4×C173C411yescharacteristic
23#14583835—42no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C2×#13063443—42no—
27C23×D173C2×C411yescharacteristic
28C2×C22⋊D173—21no—
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C2×C22⋊D173—42no—
33C2×C22⋊D173—42no—
34C22×Dic173—42no—
35C22×Dic173C2×C411yescharacteristic
36C22×Dic173C2×C411yescharacteristic
37C22×Dic173D411yescharacteristic
38C22×Dic173D411yescharacteristic
39C22×Dic173D411yescharacteristic
40C22×Dic173D411yescharacteristic
41C22×Dic173C2311yescharacteristic
42#14583805—21no—
43#14583805—21no—
44C22⋊Dic173—21no—
45C22⋊Dic173—21no—
46#14583803—21no—
47#14583803—21no—
48C24×C173C2×C411yescharacteristic
49C24×C173C2×C411yescharacteristic
50C2×D4×C173—21no—
51C2×D4×C173—21no—
52C2×D4×C173—42no—
53C22×C4×C173C2×C411yescharacteristic
54#13063443—42no—
55#13063443—42no—
56C22×D173—21no—
57C22×D173—22no—
58C22×D173—41no—
59C22×D173—41no—
60C22⋊D173—42no—
61C22⋊D173—42no—
62C22⋊D173—42no—
63C22⋊D173—42no—
64C2×Dic173—22no—
65C2×Dic173—22no—
66C2×Dic173—22no—
67C2×Dic173—21no—
68C2×Dic173—41no—
69C2×Dic173—41no—
70C2×Dic173—41no—
71C2×Dic173—41no—
72C2×Dic173C22⋊C411yescharacteristic
73C2×Dic173C22⋊C411yescharacteristic
74C2×Dic173C2×D411yescharacteristic
75C2×Dic173C2×D411yescharacteristic
76C23×C173—22no—
77C23×C173—22no—
78C23×C173—21no—
79C23×C173—22no—
80C23×C173—41no—
81C23×C173—41no—
82C23×C173—41no—
83C23×C173C22⋊C411yescharacteristic
84C23×C173C22⋊C411yescharacteristic
85C23×C173C22×C411yescharacteristic
86D4×C173—42no—
87D4×C173—42no—
88C2×C4×C173—21no—
89C2×C4×C173—41no—
90C22×C173—21no—
91C22×C173—21no—
92C22×C173—21no—
93C22×C173—21no—
94C22×C173—21no—
95C22×C173—21no—
96C22×C173—21no—
97C22×C173—44no—
98C22×C173—41no—
99C22×C173—41no—
100C22×C173—41no—
101C22×C173—82no—
102C22×C173#10012yes—
103C22×C173C2×C22⋊C411yescharacteristic
104C22×C173#13112yes—
105C2×D173—41no—
106C2×D173—41no—
107C2×D173—41no—
108C2×D173—82no—
109C4×C173—41no—
110Dic173—21no—
111Dic173—21no—
112Dic173—21no—
113Dic173—41no—
114Dic173#9912yes—
115C2×C173—22no—
116C2×C173—22no—
117C2×C173—41no—
118C2×C173—81no—
119C2×C173#40811yescharacteristic
120C2×C173C2×#9911yescharacteristic
121C2×C173C2×#10011yescharacteristic
122D173—81no—
123C173#147111yescharacteristic, complemented, Sylow
124#1471—1731nomaximal, Sylow
125C2×#121—1731no—
126#407—1731no—
127C2×C22⋊C8—1731no—
128#323—1732no—
129#323—1732no—
130C22×D4—1731no—
131C22×D4—1731no—
132C22×D4#1062192411yescharacteristic
133C23×C4—1731no—
134C2×M16—3461no—
135C22×C8—3461no—
136#121—3461no—
137#121—3461no—
138#121—3461no—
139#121—3461no—
140C2×C4⋊C4—1731no—
141C2×C22⋊C4—1731no—
142C2×C22⋊C4—1731no—
143C22⋊C8—3461no—
144C22⋊C8—3461no—
145C22⋊C8—1732no—
146C22⋊C8—1732no—
147C24—1731no—
148C24C2×#1062192411yescharacteristic
149C24C2×#1062192411yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C2×D4—3461no—
154C2×D4—3461no—
155C2×D4—3461no—
156C2×D4—3461no—
157C2×D4—6922no—
158C2×D4—6922no—
159C22×C4—1731no—
160C22×C4—1731no—
161C22×C4—1731no—
162C22×C4—1731no—
163C22×C4—1731no—
164C22×C4—1731no—
165C22×C4—1731no—
166C22×C4—6922no—
167C22×C4C2×#1062192411yescharacteristic
168M16—6922no—
169C2×C8—3462no—
170C2×C8—3462no—
171C2×C8—6922no—
172C4⋊C4—3461no—
173C4⋊C4—3461no—
174C22⋊C4—3461no—
175C22⋊C4—3461no—
176C22⋊C4—3461no—
177C22⋊C4—3461no—
178C23—22no—
179C23—22no—
180C23—21no—
181C23—22no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—3462no—
186C23—3461no—
187C23—6921no—
188C23—6921no—
189C23C22⋊#1062192411yescharacteristic
190C23C22⋊#1062192411yescharacteristic
191C23C22×#1062192411yescharacteristic
192D4—42no—
193D4—42no—
194D4—6922no—
195D4—6922no—
196D4—6922no—
197D4—6922no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—3462no—
201C2×C4—3462no—
202C2×C4—3462no—
203C2×C4—3461no—
204C2×C4—1731no—
205C2×C4—1731no—
206C2×C4—1731no—
207C2×C4—1731no—
208C2×C4—6921no—
209C2×C4—6921no—
210C2×C4—6921no—
211C2×C4—6921no—
212C8—6922no—
213C8—6922no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—44no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—82no—
226C22—13842no—
227C22—6921no—
228C22—6921no—
229C22—6921no—
230C22#1914973912yes—
231C22C2×C22⋊#1062192411yescharacteristic
232C22#1914976612yes—
233C4—41no—
234C4—3461no—
235C4—3461no—
236C4—3461no—
237C4—1732no—
238C4—6921no—
239C2—22no—
240C2—22no—
241C2—41no—
242C2—81no—
243C2—13841no—
244C2#2601246411yescharacteristic
245C2D173⋊#9911yescharacteristic
246C2C2×#1914973911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC22×D4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×#19149651C2
cosocleC22×Dic173C23
hypercenterC22×D4#10621924
hypocenterC173#1471
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #26012461C2semidirect
2. C2×C22⋊#13063443C2semidirect
3. #26012335C2semidirect
4. #26012334C2semidirect
123. C173#1471semidirect

Maximal subgroups

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

subgroupindex
#260124612
C2×C22⋊#130634432
#260123352
#260123342
C2×#191496512
#1471173