Finite Group Databasebeta

← #36352598

Subgroups — #36352598

Order 22144.

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

normal subgroups only

This group has 246 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#26011809C211yesmaximal, characteristic, complemented
3#26011429C211yesmaximal, characteristic, complemented
4#26011226C211yesmaximal, characteristic, complemented
5#26011225C211yesmaximal, characteristic, complemented
6#26011216C211yesmaximal, characteristic, complemented
7#481×C173C211yesmaximal, characteristic, complemented
8C2×#14583825—21no—
9#19149657—21no—
10#19149656—21no—
11#19149653C2211yescharacteristic
12#19149652—21no—
13#19149651—21no—
14#19149651—21no—
15C2×#14583805C2211yescharacteristic
16#19149610—21no—
17#19149599—21no—
18C22⋊#13063436C2211yescharacteristic
19#19149586—21no—
20#19149586—21no—
21#19149586C2211yescharacteristic
22#121×C173C2211yescharacteristic
23C2×C4⋊C4×C173C2211yescharacteristic
24#106×C173—21no—
25#102×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—
39C2×#13063436—21no—
40C4⋊Dic173D411yescharacteristic
41#14583805—21no—
42#14583805—21no—
43#14583805—21no—
44C22⋊Dic173—21no—
45#14583803—21no—
46#14583803—21no—
47C2×D4×C173—21no—
48C2×D4×C173D411yescharacteristic
49C22×C4×C173—21no—
50C22×C4×C173C2311yescharacteristic
51C2×C8×C173—21no—
52C4⋊C4×C173—21no—
53C4⋊C4×C173—21no—
54C4⋊C4×C173D411yescharacteristic
55C22⋊C4×C173—21no—
56C22×D173—21no—
57Dic346—42no—
58D692—41no—
59C22⋊D173—42no—
60C22⋊D173—41no—
61C22⋊D173—41no—
62C22⋊D173—81no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C2×Dic173—21no—
67C2×Dic173—42no—
68C2×Dic173—41no—
69C2×Dic173—41no—
70C2×Dic173—41no—
71C2×Dic173—41no—
72C2×Dic173SD1612yes—
73C4×D173—42no—
74#13063436—41no—
75C23×C173—21no—
76C23×C173C2×D411yescharacteristic
77D4×C173—41no—
78D4×C173—41no—
79D4×C173—41no—
80C2×C4×C173—21no—
81C2×C4×C173—41no—
82C2×C4×C173—42no—
83C2×C4×C173—41no—
84C2×C4×C173C2×D411yescharacteristic
85C2×C4×C173C2×D411yescharacteristic
86C8×C173—41no—
87C22×C173—21no—
88C22×C173—21no—
89C22×C173—41no—
90C22×C173—41no—
91C22×C173—41no—
92C22×C173#12011yescharacteristic
93C22×C173C2×SD1611yescharacteristic
94C22×C173AGL(1,ℤ/8)11yescharacteristic
95C2×D173—41no—
96C2×D173—41no—
97C2×D173—41no—
98C4×C173—21no—
99C4×C173—21no—
100C4×C173—41no—
101C4×C173—81no—
102Dic173—22no—
103Dic173—41no—
104Dic173—41no—
105Dic173—81no—
106C2×C173—21no—
107C2×C173—21no—
108C2×C173—81no—
109C2×C173#45011yescharacteristic
110C2×C173#45411yescharacteristic
111C2×C173#45811yescharacteristic
112D173—81no—
113C173#161611yescharacteristic, complemented, Sylow
114#1616—1731nomaximal, Sylow
115#537—1731no—
116#481—1731no—
117#481D17311yescharacteristic, complemented
118#400—1731no—
119#327—1731no—
120#327—1731no—
121#324—1731no—
122C2×#41—3461no—
123C22⋊Q8—3461no—
124C22⋊Q8—3461no—
125#121—3461no—
126#121—3461no—
127#121—3461no—
128#121—1731no—
129#121C2×D17311yescharacteristic
130C2×C4⋊C4—1731no—
131C2×C4⋊C4C2×D17311yescharacteristic
132#106—21no—
133#106—3461no—
134#102—21no—
135#102—3461no—
136C22⋊C8—1731no—
137C22⋊C8C2×D17311yescharacteristic
138#95—3461no—
139#95—3461no—
140#95—1731no—
141#41—6922no—
142#41—6922no—
143C2×Q8—3461no—
144C2×D4—21no—
145C2×D4—3461no—
146C2×D4—3461no—
147C2×D4—3461no—
148C2×D4—1731no—
149C2×D4C22⋊D17311yescharacteristic
150C22×C4—21no—
151C22×C4—3461no—
152C22×C4—3461no—
153C22×C4—3461no—
154C22×C4—1731no—
155C22×C4—1731no—
156C22×C4C22×D17311yescharacteristic
157C2×C8—21no—
158C2×C8—3461no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—3461no—
162C4⋊C4—3461no—
163C4⋊C4—3461no—
164C4⋊C4—1731no—
165C4⋊C4C22⋊D17311yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—3461no—
168C22⋊C4—3461no—
169C22⋊C4—3461no—
170C23—21no—
171C23—3461no—
172C23D4×D17311yescharacteristic
173Q8—6922no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—13841no—
178D4—6921no—
179D4—6922no—
180D4—6921no—
181D4—6921no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—3461no—
187C2×C4—3461no—
188C2×C4—3461no—
189C2×C4—3461no—
190C2×C4—1732no—
191C2×C4—6922no—
192C2×C4—6922no—
193C2×C4—6921no—
194C2×C4—6921no—
195C2×C4—6921no—
196C2×C4—6921no—
197C2×C4D4×D17311yescharacteristic
198C2×C4C2×C22⋊D17311yescharacteristic
199C8—41no—
200C8—6921no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—6921no—
207C22—6921no—
208C22—6921no—
209C22#1914964711yescharacteristic
210C22SD16×D17311yescharacteristic
211C22#1914970611yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—3462no—
217C4—13841no—
218C4—6921no—
219C4—6921no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—13841no—
224C2#2601164011yescharacteristic
225C2#2601166211yescharacteristic
226C2#2601168011yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#481×C173C2
socleC22×C173#120
prime radical#26012068C2
hypercenter#481D173
hypocenterC173#1616
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. #26011809C2semidirect
3. #26011429C2semidirect
4. #26011226C2semidirect
5. #26011225C2semidirect
6. #26011216C2semidirect
7. #481×C173C2semidirect
113. C173#1616semidirect
117. #481D173semidirect

Maximal subgroups

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

subgroupindex
#260120682
#260118092
#260114292
#260112262
#260112252
#260112162
#481×C1732
#1616173