Finite Group Databasebeta

← #36369818

Subgroups — #36369818

Order 22144.

280 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C2×#19149739C211yesmaximal, characteristic, complemented
2C2×#19149737C211yesmaximal, characteristic, complemented
3#26012393C212yesmaximal, complemented
4#26012391C212yesmaximal, complemented
5#26012050C211yesmaximal, characteristic
6C2×#14583835—21no—
7C2×C22⋊#10621924—21no—
8#19149739—21no—
9#19149739C2212yes—
10#19149737—21no—
11#19149737C2212yes—
12C22×C22⋊D173C2211yescharacteristic
13#19149662C411yescharacteristic, complemented
14#19149662C2211yescharacteristic
15#19149660—21no—
16#19149651—21no—
17#19149651—21no—
18#19149648C411yescharacteristic, complemented
19#19149648C2211yescharacteristic
20#19149647—21no—
21C2×C22⋊Dic173C411yescharacteristic, complemented
22C22×D4×C173C411yescharacteristic, complemented
23C22×#10621924—41no—
24#14583835—22no—
25#14583835—41no—
26C2×#13063443—41no—
27C22⋊#10621924—22no—
28C22⋊#10621924—41no—
29C23×D173D411yescharacteristic
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C2×C22⋊D173—41no—
33C2×C22⋊D173—41no—
34C2×C22⋊D173C2×C411yescharacteristic
35C2×C22⋊D173D411yescharacteristic
36C2×C22⋊D173D411yescharacteristic
37C2×C22⋊D173C2311yescharacteristic
38C22×Dic173—21no—
39C22×Dic173D411yescharacteristic
40#14583805—21no—
41#14583805—21no—
42C22⋊Dic173—21no—
43C22⋊Dic173—21no—
44C22⋊Dic173—21no—
45C22⋊Dic173C2×C411yescharacteristic, complemented
46C22⋊Dic173C2×C411yescharacteristic, complemented
47#14583803—21no—
48#14583803—21no—
49C24×C173C2×C411yescharacteristic
50C24×C173C2×C411yescharacteristic, complemented
51C2×D4×C173—21no—
52C2×D4×C173—21no—
53C2×D4×C173—41no—
54C2×D4×C173—41no—
55C22×C4×C173C2×C411yescharacteristic, complemented
56C2×#10621924—42no—
57C2×#10621924—82no—
58#13063443—42no—
59C22×D173—21no—
60C22×D173—21no—
61C22×D173—81no—
62C22×D173C22⋊C411yescharacteristic
63C22×D173C2×D411yescharacteristic
64C22⋊D173—41no—
65C22⋊D173—41no—
66C22⋊D173—41no—
67C22⋊D173—41no—
68C22⋊D173—81no—
69C22⋊D173—81no—
70C2×Dic173—21no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—21no—
74C2×Dic173—81no—
75C2×Dic173C22⋊C411yescharacteristic
76C2×Dic173C2×D411yescharacteristic
77C23×C173—21no—
78C23×C173—21no—
79C23×C173—21no—
80C23×C173—41no—
81C23×C173—81no—
82C23×C173—81no—
83C23×C173C22⋊C411yescharacteristic
84C23×C173C22⋊C411yescharacteristic
85C23×C173C22×C411yescharacteristic
86D4×C173—42no—
87D4×C173—41no—
88D4×C173—41no—
89C2×C4×C173—21no—
90C2×C4×C173—41no—
91C22×C173—21no—
92C22×C173—21no—
93C22×C173—41no—
94C22×C173—41no—
95C22×C173—41no—
96C22×C173—41no—
97C22×C173—41no—
98C22×C173—41no—
99C22×C173—81no—
100C22×C173—81no—
101C22×C173—82no—
102C22×C173#9911yescharacteristic
103C22×C173#9911yescharacteristic
104C22×C173C2×C22⋊C411yescharacteristic
105C2×D173—21no—
106C2×D173—21no—
107C2×D173—42no—
108C2×D173—41no—
109C2×D173—41no—
110C2×D173—81no—
111#10621924—82no—
112C4×C173—41no—
113Dic173—21no—
114Dic173—21no—
115Dic173—41no—
116Dic173—41no—
117C2×C173—21no—
118C2×C173—21no—
119C2×C173—21no—
120C2×C173—41no—
121C2×C173—41no—
122C2×C173—81no—
123C2×C173C2×#9911yescharacteristic
124D173—41no—
125D173—41no—
126C173#212111yescharacteristic, complemented, Sylow
127#2121—1731nomaximal, Sylow
128#534—1731no—
129C2×#100—1731no—
130C2×#99—1731no—
131#352—1732no—
132AΣL(1,𝔽16)—1732no—
133C22×D4—1731no—
134C22×D4#1062192411yescharacteristic, complemented
135C2×M16—3461no—
136#123—3461no—
137#123—1731no—
138#123—1731no—
139#121—3461no—
140#121—3461no—
141#120—3461no—
142#120—1731no—
143#120—1731no—
144C2×C22⋊C4—3461no—
145C2×C22⋊C4—1731no—
146#100—3461no—
147#100—1732no—
148#99—3461no—
149#99—1732no—
150C24—1731no—
151C24C2×#1062192411yescharacteristic
152C24C2×#1062192411yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—3461no—
158C2×D4—3461no—
159C2×D4—1731no—
160C2×D4—1731no—
161C2×D4—1731no—
162C2×D4—1731no—
163C2×D4—6921no—
164C2×D4—6921no—
165C22×C4—3461no—
166C22×C4—1731no—
167C22×C4—6921no—
168C22×C4C2×#1062192411yescharacteristic, complemented
169M16—3462no—
170M16—6921no—
171C2×C8—6921no—
172C4⋊C4—3461no—
173C4⋊C4—3461no—
174C22⋊C4—3461no—
175C22⋊C4—3461no—
176C22⋊C4—3461no—
177C22⋊C4—3461no—
178C22⋊C4—3461no—
179C22⋊C4—3462no—
180C22⋊C4—1731no—
181C22⋊C4—1731no—
182C22⋊C4—6921no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—41no—
187C23—81no—
188C23—81no—
189C23—3461no—
190C23—3461no—
191C23—13841no—
192C23—1731no—
193C23—1731no—
194C23C22⋊#1062192411yescharacteristic
195C23C22⋊#1062192411yescharacteristic
196C23C22×#1062192411yescharacteristic
197D4—42no—
198D4—41no—
199D4—41no—
200D4—13841no—
201D4—13841no—
202D4—6921no—
203D4—6921no—
204D4—6921no—
205D4—6921no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—3461no—
209C2×C4—3461no—
210C2×C4—3461no—
211C2×C4—3461no—
212C2×C4—13842no—
213C2×C4—13841no—
214C2×C4—1731no—
215C2×C4—1731no—
216C2×C4—6922no—
217C8—6922no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—81no—
227C22—81no—
228C22—82no—
229C22—3461no—
230C22—3461no—
231C22—13841no—
232C22—6921no—
233C22—6921no—
234C22—6922no—
235C22#1914973711yescharacteristic
236C22#1914973711yescharacteristic
237C22C2×C22⋊#1062192411yescharacteristic
238C4—41no—
239C4—3461no—
240C4—3461no—
241C4—13842no—
242C4—6921no—
243C4—6921no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—81no—
250C2—6921no—
251C2—6921no—
252C2C2×#1914973711yescharacteristic
subgroupG/H
centerC2C2×#19149737
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC22×D4×C173C4
socleC2×C173C2×#99
prime radical#26012050C2
cosocleC2×C22⋊D173C23
hypercenterC22×D4#10621924
hypocenterC173#2121
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19149739C2semidirect
2. C2×#19149737C2semidirect
3. #26012393C2semidirect
4. #26012391C2semidirect
13. #19149662C4semidirect
18. #19149648C4semidirect
21. C2×C22⋊Dic173C4semidirect
22. C22×D4×C173C4semidirect
45. C22⋊Dic173C2×C4semidirect
46. C22⋊Dic173C2×C4semidirect
50. C24×C173C2×C4semidirect
55. C22×C4×C173C2×C4semidirect
126. C173#2121semidirect
134. C22×D4#10621924semidirect
152. C24C2×#10621924semidirect
168. C22×C4C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
C2×#191497392
C2×#191497372
#260123932
#260123912
#260120502
#2121173