Finite Group Databasebeta

← #36352580

Subgroups — #36352580

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#26012081C211yesmaximal, characteristic
2#26011819C211yesmaximal, characteristic, complemented
3#26011805C211yesmaximal, characteristic, complemented
4#26011418C211yesmaximal, characteristic, complemented
5#26011224C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#328×C173C211yesmaximal, characteristic, complemented
8#19149674—21no—
9#19149668—21no—
10#19149664—21no—
11#19149654—21no—
12#19149653C2211yescharacteristic
13#19149649—21no—
14#19149639—21no—
15C2×C4⋊Dic173C2211yescharacteristic
16C2×#14583803—21no—
17C2×#14583803C2211yescharacteristic
18#19149613—21no—
19#19149610—21no—
20#19149603—21no—
21#19149601—21no—
22C22⋊#13063436C2211yescharacteristic
23#19149586—21no—
24C22⋊Q8×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28C2×D692—21no—
29C2×D692—21no—
30C2×D692D411yescharacteristic
31C2×C22⋊D173—21no—
32C22×Dic173—21no—
33C22×Dic173D411yescharacteristic
34C2×C4×D173—21no—
35C2×#13063436—21no—
36C4⋊Dic173—21no—
37C4⋊Dic173—21no—
38C4⋊Dic173D411yescharacteristic
39#14583805—21no—
40#14583803—21no—
41#14583803—21no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—42no—
47C4×Dic173—21no—
48C2×Q8×C173D411yescharacteristic
49C22×C4×C173—21no—
50C22×C4×C173C2311yescharacteristic
51C2×C8×C173—21no—
52C4⋊C4×C173—21no—
53C4⋊C4×C173D411yescharacteristic
54C22⋊C4×C173—21no—
55C22×D173—21no—
56C22×D173—21no—
57C22×D173—21no—
58C22×D173—21no—
59C22×D173—41no—
60C22×D173—41no—
61D692—41no—
62D692—81no—
63D692—81no—
64C22⋊D173—41no—
65C22⋊D173—41no—
66C2×Dic173—21no—
67C2×Dic173—21no—
68C2×Dic173—41no—
69C2×Dic173—41no—
70C2×Dic173—41no—
71C2×Dic173—41no—
72C4×D173—81no—
73#13063436—41no—
74C23×C173C2×D411yescharacteristic
75Q8×C173—41no—
76C2×C4×C173—21no—
77C2×C4×C173—21no—
78C2×C4×C173—41no—
79C2×C4×C173—41no—
80C2×C4×C173—41no—
81C2×C4×C173—41no—
82C2×C4×C173C2×D411yescharacteristic
83C2×C4×C173C2×D411yescharacteristic
84C8×C173—41no—
85C22×C173—21no—
86C22×C173—21no—
87C22×C173#12011yescharacteristic
88C22×C173#13511yescharacteristic
89C22×C173AGL(1,ℤ/8)11yescharacteristic
90C2×D173—42no—
91C2×D173—41no—
92C2×D173—41no—
93C2×D173—41no—
94C2×D173—41no—
95C2×D173—41no—
96C2×D173—41no—
97C2×D173—81no—
98C4×C173—21no—
99C4×C173—21no—
100C4×C173—41no—
101C4×C173—41no—
102C4×C173—81no—
103Dic173—41no—
104Dic173—41no—
105Dic173—81no—
106C2×C173—21no—
107C2×C173—21no—
108C2×C173#44911yescharacteristic
109C2×C173#45511yescharacteristic
110C2×C173#45811yescharacteristic
111D173—81no—
112D173—81no—
113C173#161211yescharacteristic, complemented, Sylow
114#1612—1731nomaximal, Sylow
115#538—1731no—
116#483—1731no—
117#480—1731no—
118#397—1731no—
119#328D17311yescharacteristic, complemented
120#327—1731no—
121#323—1731no—
122#126—3461no—
123#124—3461no—
124#123—3461no—
125C22⋊Q8C2×D17311yescharacteristic
126#121—3461no—
127#121—1731no—
128#120—3461no—
129C4×D4—3461no—
130C2×C4⋊C4—1731no—
131C2×C22⋊C4—3461no—
132C2×C22⋊C4—1731no—
133#107—3461no—
134#106—3461no—
135#103—3461no—
136#102—3461no—
137C22⋊C8—1731no—
138C22⋊C8C2×D17311yescharacteristic
139#95—3461no—
140#95C2×D17311yescharacteristic
141C24—1731no—
142C2×Q8D69211yescharacteristic
143C2×D4—3461no—
144C2×D4—3461no—
145C2×D4—3461no—
146C2×D4—1731no—
147C22×C4—21no—
148C22×C4—3461no—
149C22×C4—3461no—
150C22×C4—1731no—
151C22×C4C22×D17311yescharacteristic
152C2×C8—21no—
153C2×C8—3461no—
154C4⋊C4—21no—
155C4⋊C4—3461no—
156C4⋊C4—3461no—
157C4⋊C4—3461no—
158C4⋊C4—1731no—
159C4⋊C4D69211yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—3461no—
162C22⋊C4—3461no—
163C22⋊C4—3461no—
164C22⋊C4—3461no—
165C22⋊C4—3461no—
166C22⋊C4—3461no—
167C22⋊C4—6922no—
168C42—3461no—
169C23—3461no—
170C23—3461no—
171C23—3461no—
172C23—3461no—
173C23—6921no—
174C23—6921no—
175C23D4×D17311yescharacteristic
176Q8—41no—
177D4—13841no—
178D4—13841no—
179D4—6921no—
180D4—6921no—
181D4—6921no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—3461no—
189C2×C4—3461no—
190C2×C4—13841no—
191C2×C4—6921no—
192C2×C4—6921no—
193C2×C4—6921no—
194C2×C4—6921no—
195C2×C4D4×D17311yescharacteristic
196C2×C4C2×D69211yescharacteristic
197C8—41no—
198C8—6921no—
199C22—21no—
200C22—21no—
201C22—13841no—
202C22—6922no—
203C22—6921no—
204C22—6921no—
205C22—6921no—
206C22—6921no—
207C22—6921no—
208C22—6921no—
209C22#1914964911yescharacteristic
210C22#1914970211yescharacteristic
211C22#1914970611yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—81no—
217C4—13841no—
218C4—6921no—
219C4—6921no—
220C2—21no—
221C2—21no—
222C2—13841no—
223C2—13841no—
224C2#2601163911yescharacteristic
225C2#2601167011yescharacteristic
226C2#2601168211yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173#120
prime radical#26012081C2
hypercenter#328D173
hypocenterC173#1612
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #26011819C2semidirect
3. #26011805C2semidirect
4. #26011418C2semidirect
5. #26011224C2semidirect
6. #26011214C2semidirect
7. #328×C173C2semidirect
113. C173#1612semidirect
119. #328D173semidirect

Maximal subgroups

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

subgroupindex
#260120812
#260118192
#260118052
#260114182
#260112242
#260112142
#328×C1732
#1612173