Finite Group Databasebeta

← #36352530

Subgroups — #36352530

Order 22144.

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

normal subgroups only

This group has 282 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#26012068C211yesmaximal, characteristic
2#26011630C211yesmaximal, characteristic, complemented
3#26011410C211yesmaximal, characteristic, complemented
4#26011226C211yesmaximal, characteristic, complemented
5#26011225C211yesmaximal, characteristic, complemented
6#26011216C211yesmaximal, characteristic, complemented
7#448×C173C211yesmaximal, characteristic, complemented
8C2×#14583825—21no—
9C2×#14583813—21no—
10#19149657—21no—
11#19149656—21no—
12#19149653C2211yescharacteristic
13#19149652—21no—
14#19149651—21no—
15#19149651—21no—
16C2×#14583805—21no—
17C2×#14583805C2211yescharacteristic
18#19149603—21no—
19C22⋊#13063436C2211yescharacteristic
20#19149586—21no—
21#19149586C2211yescharacteristic
22C22×Q8×C173C2211yescharacteristic
23C2×SD16×C173—21no—
24#121×C173C2211yescharacteristic
25#103×C173—21no—
26C22⋊C8×C173C2211yescharacteristic
27#14583825—42no—
28#14583825—42no—
29C2×Dic346—21no—
30C2×D692D411yescharacteristic
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C22×Dic173—21no—
35C22×Dic173—21no—
36C22×Dic173D411yescharacteristic
37C22×Dic173D411yescharacteristic
38C2×C4×D173—21no—
39#14583813—42no—
40C2×#13063436—21no—
41C4⋊Dic173D411yescharacteristic
42#14583805—21no—
43#14583805—21no—
44#14583805—21no—
45#14583805—42no—
46C22⋊Dic173—21no—
47#14583803—21no—
48#14583803—21no—
49C2×Q8×C173—21no—
50C2×Q8×C173—21no—
51C2×Q8×C173—42no—
52C2×D4×C173—21no—
53C2×D4×C173D411yescharacteristic
54C22×C4×C173—21no—
55C22×C4×C173C2311yescharacteristic
56SD16×C173—42no—
57C2×C8×C173—21no—
58C4⋊C4×C173D411yescharacteristic
59C22⋊C4×C173—21no—
60C22×D173—21no—
61Dic346—42no—
62D692—41no—
63C22⋊D173—42no—
64C22⋊D173—41no—
65C22⋊D173—41no—
66C22⋊D173—81no—
67C2×Dic173—21no—
68C2×Dic173—21no—
69C2×Dic173—22no—
70C2×Dic173—21no—
71C2×Dic173—21no—
72C2×Dic173—42no—
73C2×Dic173—41no—
74C2×Dic173—41no—
75C2×Dic173—41no—
76C2×Dic173SD1612yescomplemented
77C4×D173—42no—
78#13063436—41no—
79C23×C173—21no—
80C23×C173C2×D411yescharacteristic
81Q8×C173—22no—
82Q8×C173—22no—
83Q8×C173—41no—
84Q8×C173—41no—
85D4×C173—41no—
86D4×C173—41no—
87D4×C173—41no—
88C2×C4×C173—22no—
89C2×C4×C173—21no—
90C2×C4×C173—41no—
91C2×C4×C173—41no—
92C2×C4×C173—41no—
93C2×C4×C173C2×D411yescharacteristic
94C2×C4×C173C2×D411yescharacteristic
95C8×C173—41no—
96C22×C173—21no—
97C22×C173—21no—
98C22×C173—41no—
99C22×C173—41no—
100C22×C173—41no—
101C22×C173#12011yescharacteristic
102C22×C173C2×SD1611yescharacteristic
103C22×C173AGL(1,ℤ/8)11yescharacteristic
104C2×D173—41no—
105C2×D173—41no—
106C2×D173—41no—
107C4×C173—21no—
108C4×C173—21no—
109C4×C173—41no—
110C4×C173—42no—
111Dic173—22no—
112Dic173—42no—
113Dic173—41no—
114Dic173—41no—
115C2×C173—21no—
116C2×C173—21no—
117C2×C173—81no—
118C2×C173#45011yescharacteristic
119C2×C173#45611yescharacteristic
120C2×C173B(4,𝔽2)11yescharacteristic
121D173—81no—
122C173#160411yescharacteristic, complemented, Sylow
123#1604—1731nomaximal, Sylow
124#537—1731no—
125#448—1731no—
126#448D17311yescharacteristic, complemented
127#395—1731no—
128#327—1731no—
129#327—1731no—
130#324—1731no—
131C2×#41—3461no—
132C22×Q8C2×D17311yescharacteristic
133C2×SD16—21no—
134C2×SD16—3461no—
135C22⋊Q8—3461no—
136C22⋊Q8—3461no—
137#121—3461no—
138#121—3461no—
139#121—3461no—
140#121—1731no—
141#121C2×D17311yescharacteristic
142C2×C4⋊C4—3461no—
143C2×C4⋊C4—1731no—
144#103—21no—
145#103—3461no—
146C22⋊C8—1731no—
147C22⋊C8C2×D17311yescharacteristic
148#95—3461no—
149#95—1731no—
150#41—6922no—
151#41—6922no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—42no—
155C2×Q8—3461no—
156C2×D4—21no—
157C2×D4—3461no—
158C2×D4—3461no—
159C2×D4—3461no—
160C2×D4—1731no—
161C2×D4C22⋊D17311yescharacteristic
162C22×C4—21no—
163C22×C4—3461no—
164C22×C4—3461no—
165C22×C4—3461no—
166C22×C4—1731no—
167C22×C4—1731no—
168C22×C4C22×D17311yescharacteristic
169SD16—42no—
170SD16—6922no—
171C2×C8—21no—
172C2×C8—3461no—
173C4⋊C4—3461no—
174C4⋊C4—3461no—
175C4⋊C4—3461no—
176C4⋊C4—1731no—
177C4⋊C4—6922no—
178C4⋊C4C22⋊D17311yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—3461no—
181C22⋊C4—3461no—
182C22⋊C4—3461no—
183C23—21no—
184C23—3461no—
185C23D4×D17311yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—6922no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—13841no—
195D4—6921no—
196D4—6922no—
197D4—6921no—
198D4—6921no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—3461no—
205C2×C4—3461no—
206C2×C4—3462no—
207C2×C4—3461no—
208C2×C4—3461no—
209C2×C4—1732no—
210C2×C4—6922no—
211C2×C4—6922no—
212C2×C4—6921no—
213C2×C4—6921no—
214C2×C4—6921no—
215C2×C4D4×D17311yescharacteristic
216C2×C4C2×C22⋊D17311yescharacteristic
217C8—41no—
218C8—6921no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—6921no—
225C22—6921no—
226C22—6921no—
227C22#1914964711yescharacteristic
228C22SD16×D17311yescharacteristic
229C22#1914970611yescharacteristic
230C4—21no—
231C4—21no—
232C4—42no—
233C4—41no—
234C4—3462no—
235C4—6922no—
236C4—6921no—
237C4—6921no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—13841no—
242C2#2601164011yescharacteristic
243C2#2601167211yescharacteristic
244C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#448×C173C2
socleC22×C173#120
prime radical#26012068C2
hypercenter#448D173
hypocenterC173#1604
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #26011630C2semidirect
3. #26011410C2semidirect
4. #26011226C2semidirect
5. #26011225C2semidirect
6. #26011216C2semidirect
7. #448×C173C2semidirect
76. C2×Dic173SD16semidirect
122. C173#1604semidirect
126. #448D173semidirect

Maximal subgroups

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

subgroupindex
#260120682
#260116302
#260114102
#260112262
#260112252
#260112162
#448×C1732
#1604173