Finite Group Databasebeta

← #36352579

Subgroups — #36352579

Order 22144.

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

normal subgroups only

This group has 232 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#26012079C211yesmaximal, characteristic
2#26011822C211yesmaximal, characteristic, complemented
3#26011805C211yesmaximal, characteristic, complemented
4#26011416C211yesmaximal, characteristic, complemented
5#26011229C211yesmaximal, characteristic, complemented
6#26011226C211yesmaximal, characteristic, complemented
7#323×C173C211yesmaximal, characteristic, complemented
8#19149672—21no—
9#19149665—21no—
10#19149661—21no—
11#19149658C2211yescharacteristic
12#19149653C2211yescharacteristic
13#19149652—21no—
14#19149648—21no—
15C4×C22⋊D173—21no—
16C2×C4⋊Dic173C2211yescharacteristic
17C2×C22⋊Dic173—21no—
18#19149613—21no—
19#19149611—21no—
20#19149604—21no—
21#19149601—21no—
22#19149586—21no—
23#19149586C2211yescharacteristic
24C2×C22⋊C4×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26C22⋊C8×C173C2211yescharacteristic
27C2×Dic346D411yescharacteristic
28C2×D692D411yescharacteristic
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C22×Dic173—21no—
33C22×Dic173—21no—
34C2×C4×D173—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173—21no—
37C4⋊Dic173D411yescharacteristic
38C4⋊Dic173D411yescharacteristic
39#14583805—21no—
40#14583805—21no—
41C22⋊Dic173—21no—
42C22⋊Dic173—21no—
43C22⋊Dic173—21no—
44C22⋊Dic173—42no—
45#14583803—21no—
46#14583803—21no—
47C4×Dic173—21no—
48C24×C173D411yescharacteristic
49C22×C4×C173D411yescharacteristic
50C22×C4×C173C2311yescharacteristic
51C2×C8×C173—21no—
52C2×C8×C173—21no—
53C22⋊C4×C173—21no—
54C22⋊C4×C173—21no—
55C22×D173—21no—
56Dic346—41no—
57D692—41no—
58C22⋊D173—41no—
59C22⋊D173—41no—
60C22⋊D173—81no—
61C22⋊D173—81no—
62C2×Dic173—21no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—41no—
66C2×Dic173—41no—
67C2×Dic173—41no—
68C2×Dic173—41no—
69C2×Dic173—41no—
70C2×Dic173—41no—
71C4×D173—81no—
72C23×C173—21no—
73C23×C173—21no—
74C23×C173—21no—
75C23×C173—41no—
76C23×C173—41no—
77C23×C173C2×D411yescharacteristic
78C2×C4×C173—21no—
79C2×C4×C173—41no—
80C2×C4×C173—41no—
81C2×C4×C173C2×D411yescharacteristic
82C2×C4×C173C2×D411yescharacteristic
83C8×C173—41no—
84C8×C173—41no—
85C22×C173—21no—
86C22×C173—21no—
87C22×C173—42no—
88C22×C173—41no—
89C22×C173—41no—
90C22×C173—41no—
91C22×C173—81no—
92C22×C173#12011yescharacteristic
93C22×C173#13511yescharacteristic
94C22×C173AGL(1,ℤ/8)11yescharacteristic
95C2×D173—41no—
96C2×D173—41no—
97C2×D173—41no—
98C4×C173—21no—
99C4×C173—21no—
100C4×C173—41no—
101Dic173—41no—
102Dic173—41no—
103Dic173—41no—
104Dic173—81no—
105Dic173—81no—
106C2×C173—21no—
107C2×C173—21no—
108C2×C173—81no—
109C2×C173#44911yescharacteristic
110C2×C173#45511yescharacteristic
111C2×C173#45811yescharacteristic
112D173—81no—
113C173#161211yescharacteristic, complemented, Sylow
114#1612—1731nomaximal, Sylow
115#538—1731no—
116#483—1731no—
117#480—1731no—
118#397—1731no—
119#328—1731no—
120#327—1731no—
121#323D17311yescharacteristic, complemented
122#126—3461no—
123#124—3461no—
124#123—3461no—
125C22⋊Q8—1731no—
126#121—3461no—
127#121—1731no—
128#120—3461no—
129C4×D4—3461no—
130C2×C4⋊C4—1731no—
131C2×C22⋊C4—3461no—
132C2×C22⋊C4C2×D17311yescharacteristic
133#107—3461no—
134#106—3461no—
135#103—3461no—
136#102—3461no—
137C22⋊C8C2×D17311yescharacteristic
138C22⋊C8C2×D17311yescharacteristic
139#95—3461no—
140#95—1731no—
141C24D69211yescharacteristic
142C2×Q8—1731no—
143C2×D4—3461no—
144C2×D4—3461no—
145C2×D4—3461no—
146C2×D4—1731no—
147C22×C4—3461no—
148C22×C4—3461no—
149C22×C4—3461no—
150C22×C4D69211yescharacteristic
151C22×C4C22×D17311yescharacteristic
152C2×C8—21no—
153C2×C8—21no—
154C4⋊C4—3461no—
155C4⋊C4—3461no—
156C4⋊C4—3461no—
157C4⋊C4—3461no—
158C4⋊C4—1731no—
159C4⋊C4—1731no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—3461no—
163C22⋊C4—3461no—
164C22⋊C4—3461no—
165C22⋊C4—3461no—
166C22⋊C4—3461no—
167C22⋊C4—6922no—
168C42—3461no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23—3461no—
175C23C2×D69211yescharacteristic
176Q8—6921no—
177D4—13841no—
178D4—13841no—
179D4—6921no—
180D4—6921no—
181D4—6921no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—3461no—
186C2×C4—3461no—
187C2×C4—3461no—
188C2×C4—13841no—
189C2×C4—6921no—
190C2×C4—6921no—
191C2×C4—6921no—
192C2×C4—6921no—
193C2×C4—6921no—
194C2×C4—6921no—
195C2×C4D4×D17311yescharacteristic
196C2×C4D4×D17311yescharacteristic
197C8—41no—
198C8—41no—
199C22—21no—
200C22—21no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22—6921no—
207C22—6921no—
208C22—6921no—
209C22#1914964911yescharacteristic
210C22#1914970111yescharacteristic
211C22#1914970711yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—13841no—
216C4—13841no—
217C4—6921no—
218C4—6921no—
219C4—6921no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—13841no—
224C2#2601163811yescharacteristic
225C2#2601167111yescharacteristic
226C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#323×C173C2
socleC22×C173#120
prime radical#26012079C2
hypercenter#323D173
hypocenterC173#1612
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #26011822C2semidirect
3. #26011805C2semidirect
4. #26011416C2semidirect
5. #26011229C2semidirect
6. #26011226C2semidirect
7. #323×C173C2semidirect
113. C173#1612semidirect
121. #323D173semidirect

Maximal subgroups

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

subgroupindex
#260120792
#260118222
#260118052
#260114162
#260112292
#260112262
#323×C1732
#1612173