Finite Group Databasebeta

← #36352052

Subgroups — #36352052

Order 22144.

304 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 77 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#26012002C211yesmaximal, characteristic
2C2×C22⋊#13063436C211yesmaximal, characteristic, complemented
3#26011227C212yesmaximal, complemented
4#26011224C212yesmaximal, complemented
5C4×C22⋊C4×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C411yescharacteristic, complemented
7C22×#13063436—21no—
8#19149662—21no—
9#19149658C411yescharacteristic, complemented
10#19149658C2211yescharacteristic
11#19149653C411yescharacteristic, complemented
12#19149653C2211yescharacteristic
13#19149648—21no—
14C4×C22⋊D173—21no—
15C4×C22⋊D173—21no—
16#19149638C411yescharacteristic, complemented
17C22⋊#13063436—21no—
18C22⋊#13063436C2212yes—
19C23×C4×C173C2211yescharacteristic
20C2×C22⋊C4×C173—21no—
21C2×C42×C173—21no—
22#95×C173C2212yes—
23#14583826—42no—
24C2×Dic346C2×C411yescharacteristic
25C2×D692C2×C411yescharacteristic
26C2×C22⋊D173—21no—
27C2×C22⋊D173—21no—
28C2×C22⋊D173—21no—
29C2×C4×D173—21no—
30C2×#13063436—22no—
31C2×#13063436—42no—
32C4⋊Dic173C2×C411yescharacteristic
33C4⋊Dic173C2×C411yescharacteristic
34#14583805—21no—
35#14583805—21no—
36C22⋊Dic173—21no—
37C22⋊Dic173—21no—
38C22⋊Dic173—21no—
39#14583803—21no—
40#14583803—21no—
41C4×Dic173—21no—
42C24×C173D411yescharacteristic
43C22×C4×C173—22no—
44C22×C4×C173—42no—
45C22×C4×C173C2×C411yescharacteristic
46C22×C4×C173C2×C411yescharacteristic
47C22×C4×C173D411yescharacteristic
48C22×C4×C173D411yescharacteristic
49C22×C4×C173D411yescharacteristic
50C22×C4×C173C2311yescharacteristic
51C22⋊C4×C173—24no—
52C42×C173—42no—
53C22×D173—21no—
54Dic346—41no—
55D692—41no—
56C22⋊D173—42no—
57C22⋊D173—42no—
58C22⋊D173—42no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C4×D173—42no—
63#13063436—42no—
64C23×C173—22no—
65C23×C173—22no—
66C23×C173—41no—
67C23×C173C22⋊C411yescharacteristic
68C23×C173C22⋊C411yescharacteristic
69C23×C173C2×D411yescharacteristic
70C2×C4×C173—22no—
71C2×C4×C173—22no—
72C2×C4×C173—22no—
73C2×C4×C173—24no—
74C2×C4×C173—41no—
75C2×C4×C173—41no—
76C2×C4×C173—41no—
77C2×C4×C173—42no—
78C2×C4×C173—42no—
79C2×C4×C173C22⋊C411yescharacteristic
80C2×C4×C173C22⋊C411yescharacteristic
81C2×C4×C173C22×C411yescharacteristic
82C2×C4×C173C2×D411yescharacteristic
83C22×C173—21no—
84C22×C173—21no—
85C22×C173—21no—
86C22×C173—21no—
87C22×C173—21no—
88C22×C173—21no—
89C22×C173—21no—
90C22×C173—44no—
91C22×C173#10412yes—
92C22×C173C2×C22⋊C411yescharacteristic
93C22×C173#13512yes—
94C2×D173—41no—
95C2×D173—41no—
96C2×D173—41no—
97C4×C173—21no—
98C4×C173—21no—
99C4×C173—21no—
100C4×C173—44no—
101C4×C173#9912yes—
102Dic173—41no—
103Dic173—41no—
104Dic173—41no—
105C2×C173—22no—
106C2×C173—22no—
107C2×C173—41no—
108C2×C173C2×#9911yescharacteristic
109C2×C173#41611yescharacteristic
110C2×C173C2×#10411yescharacteristic
111D173—81no—
112C173#150111yescharacteristic, complemented, Sylow
113#1501—1731nomaximal, Sylow
114#525—1731no—
115C2×C22⋊C8—1731no—
116C4×C22⋊C4D17311yescharacteristic, complemented
117#328—1732no—
118#327—1732no—
119C2×#41—1731no—
120C23×C4C2×D17311yescharacteristic
121C22×C8—3461no—
122#123—3461no—
123C22⋊Q8—1731no—
124C22⋊Q8—1731no—
125#121—1731no—
126#121—1731no—
127#120—3461no—
128C4×D4—3461no—
129C4×D4—3461no—
130#117—1731no—
131C2×C22⋊C4—21no—
132C2×C42—21no—
133C22⋊C8—3461no—
134C22⋊C8—1732no—
135#95C2×D17312yes—
136C24C22⋊D17311yescharacteristic
137#41—6922no—
138C2×Q8—1731no—
139C2×D4—3461no—
140C2×D4—3461no—
141C2×D4—3461no—
142C2×D4—1731no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—3461no—
146C22×C4C4×D17311yescharacteristic
147C22×C4C4×D17311yescharacteristic
148C22×C4C22⋊D17311yescharacteristic
149C22×C4D69211yescharacteristic
150C22×C4D69211yescharacteristic
151C22×C4C22×D17311yescharacteristic
152C2×C8—3462no—
153C2×C8—6922no—
154C4⋊C4—3461no—
155C4⋊C4—3461no—
156C4⋊C4—1731no—
157C4⋊C4—1731no—
158C22⋊C4—24no—
159C22⋊C4—3461no—
160C22⋊C4—3461no—
161C22⋊C4—3461no—
162C22⋊C4—3461no—
163C22⋊C4—3461no—
164C42—42no—
165C42—3461no—
166C23—22no—
167C23—22no—
168C23—41no—
169C23—3461no—
170C23#1458380311yescharacteristic
171C23#1458380311yescharacteristic
172C23C2×C22⋊D17311yescharacteristic
173Q8—6921no—
174D4—6921no—
175D4—6922no—
176D4—6922no—
177D4—6922no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—3461no—
188C2×C4—3461no—
189C2×C4—3461no—
190C2×C4—6922no—
191C2×C4#1458380311yescharacteristic
192C2×C4#1458380311yescharacteristic
193C2×C4C2×C4×D17311yescharacteristic
194C2×C4C2×D69211yescharacteristic
195C8—6922no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—6921no—
205C22—6921no—
206C22—6921no—
207C22#1914960712yes—
208C22C2×#1458380311yescharacteristic
209C22#1914969912yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—44no—
214C4—6921no—
215C4—6921no—
216C4—6921no—
217C4#1914959312yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—13841no—
222C2C2×#1914959311yescharacteristic
223C2#2601148511yescharacteristic
224C2C2×#1914960711yescharacteristic
subgroupG/H
centerC2×C4#14583803
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
FittingC4×C22⋊C4×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012002C2
cosocleC22×C4×C173C23
hypercenterC4×C22⋊C4D173
hypocenterC173#1501
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#13063436C2semidirect
3. #26011227C2semidirect
4. #26011224C2semidirect
5. C4×C22⋊C4×C173C2semidirect
6. C2×#14583826C4semidirect
9. #19149658C4semidirect
11. #19149653C4semidirect
16. #19149638C4semidirect
112. C173#1501semidirect
116. C4×C22⋊C4D173semidirect

Maximal subgroups

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

subgroupindex
#260120022
C2×C22⋊#130634362
#260112272
#260112242
C4×C22⋊C4×C1732
#1501173