Finite Group Databasebeta

← #36352494

Subgroups — #36352494

Order 22144.

272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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#26012088C211yesmaximal, characteristic, complemented
2#26011644C211yesmaximal, characteristic, complemented
3#26011627C211yesmaximal, characteristic
4#26011408C211yesmaximal, characteristic, complemented
5#26011229C211yesmaximal, characteristic, complemented
6#26011226C211yesmaximal, characteristic, complemented
7#324×C173C211yesmaximal, characteristic, complemented
8C22×D692C2211yescharacteristic
9C2×#14583815—21no—
10C2×D1384—21no—
11#19149674—21no—
12#19149669—21no—
13#19149664—21no—
14#19149662—21no—
15#19149658C2211yescharacteristic
16#19149657—21no—
17#19149653C2211yescharacteristic
18C4×C22⋊D173—21no—
19C2×#14583803—21no—
20C2×#14583803—21no—
21#19149601—21no—
22#19149601—21no—
23#19149586C2211yescharacteristic
24C2×C4⋊C4×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26C22⋊C8×C173C2211yescharacteristic
27C23×D173—21no—
28C2×Dic346D411yescharacteristic
29C2×D692—21no—
30C2×D692—21no—
31C2×D692—42no—
32C2×D692D411yescharacteristic
33C2×C22⋊D173—21no—
34C22×Dic173—21no—
35C2×C4×D173—21no—
36#14583815—42no—
37D1384—42no—
38C4⋊Dic173D411yescharacteristic
39C4⋊Dic173D411yescharacteristic
40#14583805—21no—
41#14583805—21no—
42#14583805—21no—
43#14583805—21no—
44C22⋊Dic173—21no—
45#14583803—21no—
46#14583803—21no—
47#14583803—21no—
48#14583803—21no—
49#14583803—42no—
50#14583803—42no—
51C4×Dic173—21no—
52C22×C4×C173D411yescharacteristic
53C22×C4×C173D411yescharacteristic
54C22×C4×C173C2311yescharacteristic
55C2×C8×C173—21no—
56C2×C8×C173—21no—
57C4⋊C4×C173—21no—
58C4⋊C4×C173—21no—
59C22×D173—22no—
60C22×D173—21no—
61C22×D173—41no—
62C22×D173—41no—
63C22×D173—41no—
64C22×D173—41no—
65C22×D173—41no—
66C22×D173—41no—
67Dic346—41no—
68D692—22no—
69D692—22no—
70D692—41no—
71D692—41no—
72D692—41no—
73C22⋊D173—41no—
74C22⋊D173—41no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173—41no—
79C2×Dic173—41no—
80C2×Dic173—41no—
81C4×D173—81no—
82C23×C173C2×D411yescharacteristic
83C2×C4×C173—21no—
84C2×C4×C173—21no—
85C2×C4×C173—21no—
86C2×C4×C173—21no—
87C2×C4×C173—41no—
88C2×C4×C173—41no—
89C2×C4×C173C2×D411yescharacteristic
90C2×C4×C173C2×D411yescharacteristic
91C8×C173—41no—
92C8×C173—41no—
93C22×C173—21no—
94C22×C173—21no—
95C22×C173#12011yescharacteristic
96C22×C173#13511yescharacteristic
97C22×C173AGL(1,ℤ/8)11yescharacteristic
98C2×D173—42no—
99C2×D173—42no—
100C2×D173—41no—
101C2×D173—41no—
102C2×D173—41no—
103C2×D173—42no—
104C2×D173—81no—
105C2×D173—81no—
106C2×D173—81no—
107C2×D173—81no—
108C4×C173—21no—
109C4×C173—21no—
110C4×C173—41no—
111C4×C173—41no—
112Dic173—41no—
113Dic173—41no—
114Dic173—41no—
115Dic173—81no—
116C2×C173—21no—
117C2×C173—21no—
118C2×C173#44911yescharacteristic
119C2×C173#45411yescharacteristic
120C2×C173B(4,𝔽2)11yescharacteristic
121D173—81no—
122D173—82no—
123C173#159711yescharacteristic, complemented, Sylow
124#1597—1731nomaximal, Sylow
125#539—1731no—
126#450—1731no—
127#447—1731no—
128#394—1731no—
129#328—1731no—
130#327—1731no—
131#324D17311yescharacteristic, complemented
132C22×D4—1731no—
133C2×SD16—3461no—
134C2×D8—3461no—
135#126—3461no—
136#125—3461no—
137#123—3461no—
138#123—3461no—
139C22⋊Q8—3461no—
140C22⋊Q8—1731no—
141#121—1731no—
142C4×D4—3461no—
143C2×C4⋊C4C2×D17311yescharacteristic
144C2×C22⋊C4—3461no—
145C2×C22⋊C4—3461no—
146#102—3461no—
147#102—3461no—
148C22⋊C8C2×D17311yescharacteristic
149C22⋊C8C2×D17311yescharacteristic
150#95—1731no—
151C24—3461no—
152C2×Q8—1731no—
153C2×D4—3461no—
154C2×D4—3461no—
155C2×D4—3461no—
156C2×D4—1731no—
157C2×D4—6922no—
158C22×C4—3461no—
159C22×C4—3461no—
160C22×C4D69211yescharacteristic
161C22×C4D69211yescharacteristic
162C22×C4C22×D17311yescharacteristic
163SD16—6922no—
164D8—6922no—
165C2×C8—21no—
166C2×C8—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—3461no—
170C4⋊C4—3461no—
171C4⋊C4—3461no—
172C4⋊C4—3461no—
173C4⋊C4—1731no—
174C4⋊C4—1731no—
175C22⋊C4—3461no—
176C22⋊C4—3461no—
177C22⋊C4—3461no—
178C22⋊C4—3461no—
179C22⋊C4—3461no—
180C22⋊C4—6922no—
181C22⋊C4—6922no—
182C42—3461no—
183C23—3462no—
184C23—3461no—
185C23—6921no—
186C23—6921no—
187C23—6921no—
188C23—6921no—
189C23—6921no—
190C23—6921no—
191C23C2×D69211yescharacteristic
192Q8—6921no—
193D4—3462no—
194D4—3462no—
195D4—6921no—
196D4—6921no—
197D4—6921no—
198D4—6921no—
199D4—6921no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—3461no—
207C2×C4—3461no—
208C2×C4—3461no—
209C2×C4—13841no—
210C2×C4—6921no—
211C2×C4—6921no—
212C2×C4—6921no—
213C2×C4D4×D17311yescharacteristic
214C2×C4D4×D17311yescharacteristic
215C8—41no—
216C8—41no—
217C22—21no—
218C22—21no—
219C22—13841no—
220C22—13841no—
221C22—13841no—
222C22—13841no—
223C22—6922no—
224C22—6922no—
225C22—6921no—
226C22—6921no—
227C22—6922no—
228C22—6921no—
229C22#1914964911yescharacteristic
230C22#1914970111yescharacteristic
231C22#1914970711yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—13841no—
237C4—6921no—
238C4—6921no—
239C4—6921no—
240C2—21no—
241C2—21no—
242C2—13842no—
243C2—13841no—
244C2#2601163811yescharacteristic
245C2#2601166611yescharacteristic
246C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#324×C173C2
socleC22×C173#120
prime radical#26011627C2
hypercenter#324D173
hypocenterC173#1597
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #26012088C2semidirect
2. #26011644C2semidirect
4. #26011408C2semidirect
5. #26011229C2semidirect
6. #26011226C2semidirect
7. #324×C173C2semidirect
123. C173#1597semidirect
131. #324D173semidirect

Maximal subgroups

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

subgroupindex
#260120882
#260116442
#260116272
#260114082
#260112292
#260112262
#324×C1732
#1597173