Finite Group Databasebeta

← #36368903

Subgroups — #36368903

Order 22144.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 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#26012513C211yesmaximal, characteristic, complemented
2C4×#14583836C211yesmaximal, characteristic, complemented
3#26012339C212yesmaximal, complemented
4D4⋊#13063443C212yesmaximal, complemented
5C4×D4×D173C211yesmaximal, characteristic
6C2×#14583836—21no—
7C4⋊#13063443C2212yescomplemented
8#19149736—21no—
9#19149734—21no—
10#19149734—21no—
11C4×#13063443C2212yescomplemented
12C2×D4×D173C411yescharacteristic
13C22×C4×D173—21no—
14C4×D692C411yescharacteristic
15D4×Dic173C2211yescharacteristic
16#19149641C2211yescharacteristic
17C4×C22⋊D173—21no—
18#19149639—21no—
19C4⋊C4×D173C411yescharacteristic
20C22⋊C4×D173—21no—
21C42×D173C2211yescharacteristic
22C4×D4×C173C411yescharacteristic
23#14583836—24no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C23×D173—21no—
27C2×D692C2×C411yescharacteristic
28C2×C22⋊D173—21no—
29D4×D173—22no—
30D4×D173—22no—
31C22×Dic173—21no—
32C2×C4×D173—21no—
33C2×C4×D173—21no—
34C2×C4×D173—21no—
35C2×C4×D173—44no—
36C2×C4×D173C2×C411yescharacteristic
37C2×C4×D173D411yescharacteristic
38C2×C4×D173D411yescharacteristic
39C4⋊Dic173C2×C411yescharacteristic
40#14583805—21no—
41C22⋊Dic173—21no—
42#14583803—21no—
43#14583803—21no—
44C4×Dic173D411yescharacteristic
45C4×Dic173D411yescharacteristic
46C4×Dic173C2311yescharacteristic
47C2×D4×C173C2×C411yescharacteristic
48C22×C4×C173—21no—
49C4⋊C4×C173C2×C411yescharacteristic
50C22⋊C4×C173—21no—
51C42×C173C2×C411yescharacteristic
52#13063443—24no—
53#13063443—42no—
54C22×D173—24no—
55C22×D173—21no—
56C22×D173—42no—
57C22×D173—42no—
58C22×D173C22⋊C411yescharacteristic
59D692—21no—
60D692M1612yescomplemented
61C22⋊D173—24no—
62C2×Dic173—21no—
63C2×Dic173—42no—
64C2×Dic173C22⋊C411yescharacteristic
65C2×Dic173C2×D411yescharacteristic
66C2×Dic173C2×D411yescharacteristic
67C4×D173—22no—
68C4×D173—21no—
69C4×D173—21no—
70C4×D173—22no—
71C4×D173—42no—
72C4×D173—42no—
73C4×D173—42no—
74C23×C173—21no—
75D4×C173—21no—
76D4×C173M1612yescomplemented
77C2×C4×C173—21no—
78C2×C4×C173—42no—
79C2×C4×C173C22⋊C411yescharacteristic
80C2×C4×C173C22⋊C411yescharacteristic
81C2×C4×C173C22×C411yescharacteristic
82C22×C173—22no—
83C22×C173—41no—
84C22×C173—41no—
85C22×C173C2×C22⋊C411yescharacteristic
86C2×D173—21no—
87C2×D173—21no—
88C2×D173—22no—
89C2×D173—48no—
90C2×D173—41no—
91C2×D173—41no—
92C2×D173#10412yes—
93C4×C173—21no—
94C4×C173—21no—
95C4×C173—41no—
96C4×C173C2×M1611yescharacteristic
97C4×C173C2×M1611yescharacteristic
98Dic173—21no—
99Dic173—21no—
100Dic173—41no—
101Dic173AGL(1,ℤ/8)12yes—
102C2×C173—42no—
103C2×C173#40711yescharacteristic
104C2×C173#41811yescharacteristic
105C2×C173C2×#10411yescharacteristic
106D173—22no—
107D173—42no—
108C173#149011yescharacteristic, complemented, Sylow
109#1490—1731nomaximal, Sylow
110C2×C4×D4—1731no—
111#432—1731no—
112C4×M16—1731no—
113D4⋊C8—1732no—
114D4⋊C8—1732no—
115C22×D4—1731no—
116C23×C4—3461no—
117C2×M16—3461no—
118C4×D4—3461no—
119C4×D4—3461no—
120C4×D4—1731no—
121C4×D4—1731no—
122C4×D4—1731no—
123C4×D4#1062192411yescharacteristic
124C2×C4⋊C4—1731no—
125C2×C22⋊C4—3461no—
126C2×C42—1731no—
127C4⋊C8—1732no—
128C22⋊C8—3461no—
129#97—3461no—
130#97—3461no—
131C4×C8—1732no—
132C24—3461no—
133C2×D4—3462no—
134C2×D4—3462no—
135C2×D4—3461no—
136C2×D4—1731no—
137C2×D4C2×#1062192411yescharacteristic
138C22×C4—21no—
139C22×C4—3461no—
140C22×C4—3461no—
141C22×C4—3461no—
142C22×C4—3461no—
143C22×C4—1731no—
144C22×C4—1731no—
145C22×C4—1731no—
146C22×C4—6924no—
147M16—3464no—
148C2×C8—3462no—
149C2×C8—3462no—
150C4⋊C4—3461no—
151C4⋊C4—1731no—
152C4⋊C4C2×#1062192411yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—3461no—
155C22⋊C4—3461no—
156C22⋊C4—3461no—
157C42—1731no—
158C42—1731no—
159C42—1731no—
160C42C2×#1062192411yescharacteristic
161C23—21no—
162C23—3464no—
163C23—3461no—
164C23—1731no—
165C23—6922no—
166C23—6922no—
167D4—21no—
168D4—3464no—
169D4—3461no—
170D4—1732no—
171D4#1458383612yescomplemented
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—3462no—
175C2×C4—3462no—
176C2×C4—3461no—
177C2×C4—3461no—
178C2×C4—3461no—
179C2×C4—1731no—
180C2×C4—1731no—
181C2×C4—1731no—
182C2×C4—6922no—
183C2×C4—6922no—
184C2×C4—6922no—
185C2×C4—6922no—
186C2×C4C22⋊#1062192411yescharacteristic
187C2×C4C22⋊#1062192411yescharacteristic
188C2×C4C22×#1062192411yescharacteristic
189C8—3464no—
190C8—6922no—
191C22—22no—
192C22—41no—
193C22—41no—
194C22—3461no—
195C22—3461no—
196C22—3462no—
197C22—1732no—
198C22—6928no—
199C22—6921no—
200C22—6921no—
201C22C2×C22⋊#1062192411yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—3461no—
206C4—3461no—
207C4—1732no—
208C4—6921no—
209C4D173⋊M1611yescharacteristic
210C4C2×#1458383611yescharacteristic
211C2—42no—
212C2—3462no—
213C2—6922no—
214C2#2601246211yescharacteristic
215C2#2601248311yescharacteristic
216C2D173⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4×D4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×D4×D173C4
cosocleC4×Dic173C23
hypercenterC4×D4#10621924
hypocenterC173#1490
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #26012513C2semidirect
2. C4×#14583836C2semidirect
3. #26012339C2semidirect
4. D4⋊#13063443C2semidirect
7. C4⋊#13063443C22semidirect
11. C4×#13063443C22semidirect
60. D692M16semidirect
76. D4×C173M16semidirect
108. C173#1490semidirect
171. D4#14583836semidirect

Maximal subgroups

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

subgroupindex
#260125132
C4×#145838362
#260123392
D4⋊#130634432
C4×D4×D1732
#1490173