Finite Group Databasebeta

← #36352484

Subgroups — #36352484

Order 22144.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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#26012116C211yesmaximal, characteristic, complemented
2#26011642C211yesmaximal, characteristic, complemented
3#26011399C211yesmaximal, characteristic
4#26011229C211yesmaximal, characteristic, complemented
5#26011228C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#450×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173—21no—
9C2×#14583814—21no—
10#19149659—21no—
11#19149658C2211yescharacteristic
12#19149657—21no—
13#19149657—21no—
14#19149656—21no—
15#19149655—21no—
16C2×C22⋊Dic173—21no—
17C2×#14583803C2211yescharacteristic
18C22⋊C4×D173—21no—
19#19149600—21no—
20C22⋊#13063436C2211yescharacteristic
21#19149586C2211yescharacteristic
22C22×D4×C173C2211yescharacteristic
23C2×SD16×C173—21no—
24C22⋊Q8×C173C2211yescharacteristic
25#102×C173—21no—
26C22⋊C8×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28C2×Dic346—21no—
29C2×Dic346D411yescharacteristic, complemented
30C2×C22⋊D173—42no—
31C2×C22⋊D173—42no—
32C2×C22⋊D173—42no—
33C22×Dic173—21no—
34C22×Dic173D411yescharacteristic
35C2×C4×D173—21no—
36C2×C4×D173—21no—
37#14583814—42no—
38C2×#13063436—21no—
39C4⋊Dic173—21no—
40C4⋊Dic173D411yescharacteristic, complemented
41#14583805—21no—
42#14583805—21no—
43#14583805—21no—
44C22⋊Dic173—21no—
45C22⋊Dic173—42no—
46#14583803—21no—
47#14583803—21no—
48#14583803—21no—
49#14583803—21no—
50C24×C173—21no—
51C2×Q8×C173D411yescharacteristic, complemented
52C2×D4×C173—21no—
53C2×D4×C173—21no—
54C2×D4×C173—42no—
55C22×C4×C173C2311yescharacteristic
56SD16×C173—42no—
57C2×C8×C173—21no—
58C4⋊C4×C173—21no—
59C4⋊C4×C173D411yescharacteristic, complemented
60C22⋊C4×C173—21no—
61C22×D173—21no—
62C22×D173—21no—
63C22×D173—41no—
64C22×D173—41no—
65C22×D173SD1612yescomplemented
66Dic346—41no—
67Dic346—81no—
68C22⋊D173—48no—
69C2×Dic173—22no—
70C2×Dic173—21no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—41no—
74C2×Dic173—41no—
75C2×Dic173—41no—
76C4×D173—81no—
77C4×D173—81no—
78#13063436—41no—
79C23×C173—22no—
80C23×C173—41no—
81C23×C173—41no—
82C23×C173—41no—
83C23×C173—41no—
84C23×C173—41no—
85C23×C173—41no—
86C23×C173C2×D411yescharacteristic
87Q8×C173—41no—
88D4×C173—22no—
89D4×C173—22no—
90D4×C173—41no—
91D4×C173—41no—
92C2×C4×C173—21no—
93C2×C4×C173—21no—
94C2×C4×C173—41no—
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173C2×D411yescharacteristic
97C8×C173—41no—
98C22×C173—21no—
99C22×C173—21no—
100C22×C173—42no—
101C22×C173—42no—
102C22×C173—42no—
103C22×C173—81no—
104C22×C173—81no—
105C22×C173—81no—
106C22×C173—81no—
107C22×C173#12011yescharacteristic
108C22×C173C2×SD1611yescharacteristic
109C22×C173#13711yescharacteristic
110C2×D173—22no—
111C2×D173—41no—
112C2×D173—41no—
113C2×D173—42no—
114C2×D173—81no—
115C4×C173—21no—
116C4×C173—21no—
117C4×C173—41no—
118C4×C173—41no—
119Dic173—42no—
120Dic173—41no—
121Dic173—41no—
122Dic173—41no—
123C2×C173—21no—
124C2×C173—21no—
125C2×C173—82no—
126C2×C173#44811yescharacteristic
127C2×C173#45311yescharacteristic
128C2×C173B(4,𝔽2)11yescharacteristic
129D173—42no—
130C173#159611yescharacteristic, complemented, Sylow
131#1596—1731nomaximal, Sylow
132#543—1731no—
133#450—1731no—
134#450D17311yescharacteristic, complemented
135#392—1731no—
136#328—1731no—
137#328—1731no—
138#323—1731no—
139C22×D4—3461no—
140C22×D4C2×D17311yescharacteristic
141C2×SD16—21no—
142C2×SD16—3461no—
143C22⋊Q8—3461no—
144C22⋊Q8—3461no—
145C22⋊Q8—3461no—
146C22⋊Q8—3461no—
147C22⋊Q8—3461no—
148C22⋊Q8—1731no—
149C22⋊Q8C2×D17311yescharacteristic
150C2×C22⋊C4—3461no—
151C2×C22⋊C4—3461no—
152C2×C22⋊C4—1731no—
153#102—21no—
154#102—3461no—
155C22⋊C8—1731no—
156C22⋊C8C2×D17311yescharacteristic
157#95—1731no—
158C24—21no—
159C24—1731no—
160C2×Q8—3461no—
161C2×Q8—1731no—
162C2×Q8C22⋊D17311yescharacteristic, complemented
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—6922no—
167C2×D4—6922no—
168C2×D4—6922no—
169C22×C4—3461no—
170C22×C4—3461no—
171C22×C4—3461no—
172C22×C4—1731no—
173C22×C4C22×D17311yescharacteristic
174SD16—42no—
175SD16—6922no—
176C2×C8—21no—
177C2×C8—3461no—
178C4⋊C4—21no—
179C4⋊C4—3461no—
180C4⋊C4—3461no—
181C4⋊C4—3461no—
182C4⋊C4—3461no—
183C4⋊C4—1731no—
184C4⋊C4C22⋊D17311yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—3461no—
187C22⋊C4—3461no—
188C22⋊C4—3461no—
189C22⋊C4—3461no—
190C22⋊C4—3461no—
191C22⋊C4—6922no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—3461no—
200C23—3461no—
201C23—1732no—
202C23—6921no—
203C23—6921no—
204C23D4×D17311yescharacteristic
205Q8—41no—
206Q8—13841no—
207Q8—6921no—
208D4—22no—
209D4—22no—
210D4—41no—
211D4—41no—
212D4—6928no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—3462no—
217C2×C4—3461no—
218C2×C4—3461no—
219C2×C4—3461no—
220C2×C4—13841no—
221C2×C4—13841no—
222C2×C4—6921no—
223C2×C4—6921no—
224C2×C4—6921no—
225C2×C4D4×D17311yescharacteristic
226C2×C4C2×C22⋊D17311yescharacteristic
227C8—41no—
228C8—6921no—
229C22—21no—
230C22—21no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—3462no—
239C22—13841no—
240C22—6922no—
241C22—6921no—
242C22—6921no—
243C22#1914964711yescharacteristic
244C22SD16×D17311yescharacteristic
245C22#1914970811yescharacteristic
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—6922no—
251C4—6921no—
252C4—6921no—
253C4—6921no—
254C2—21no—
255C2—21no—
256C2—82no—
257C2—6922no—
258C2#2601162811yescharacteristic
259C2#2601165711yescharacteristic
260C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#450×C173C2
socleC22×C173#120
prime radical#26011399C2
hypercenter#450D173
hypocenterC173#1596
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26012116C2semidirect
2. #26011642C2semidirect
4. #26011229C2semidirect
5. #26011228C2semidirect
6. #26011214C2semidirect
7. #450×C173C2semidirect
29. C2×Dic346D4semidirect
40. C4⋊Dic173D4semidirect
51. C2×Q8×C173D4semidirect
59. C4⋊C4×C173D4semidirect
65. C22×D173SD16semidirect
130. C173#1596semidirect
134. #450D173semidirect
162. C2×Q8C22⋊D173semidirect
184. C4⋊C4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260121162
#260116422
#260113992
#260112292
#260112282
#260112142
#450×C1732
#1596173