Finite Group Databasebeta

← #36368976

Subgroups — #36368976

Order 22144.

222 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 56 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#26012514C211yesmaximal, characteristic, complemented
2#26012435C211yesmaximal, characteristic, complemented
3#26012345C212yesmaximal, complemented
4#26012343C212yesmaximal
5#26012028C211yesmaximal, characteristic
6C2×C4⋊#10621924—21no—
7#19149746—21no—
8#19149736C2212yes—
9#19149734—21no—
10(C2×#10621924)⋊C4—21no—
11(C2×#10621924)⋊C4C2212yes—
12#19149674—21no—
13#19149670C411yescharacteristic, complemented
14#19149668C411yescharacteristic, complemented
15#19149664—21no—
16#19149655C2211yescharacteristic
17#19149654C2211yescharacteristic
18C4×D692—21no—
19C4×D692C411yescharacteristic, complemented
20#19149636—21no—
21C42×D173C2211yescharacteristic
22C4×Q8×C173C411yescharacteristic, complemented
23C22×#10621924—22no—
24C2×#13063443—22no—
25C4⋊#10621924—42no—
26C2×D692—21no—
27C2×D692C2×C411yescharacteristic
28C2×C4×D173—21no—
29C2×C4×D173D411yescharacteristic
30C2×C4×D173D411yescharacteristic
31C2×C4×D173C2311yescharacteristic
32C4⋊Dic173—21no—
33C4⋊Dic173C2×C411yescharacteristic
34#14583805—21no—
35#14583805—21no—
36#14583803—21no—
37#14583803—21no—
38#14583803—21no—
39#14583803—21no—
40#14583803—21no—
41C4×Dic173C2×C411yescharacteristic
42C4×Dic173D411yescharacteristic
43C4×Dic173D411yescharacteristic
44C2×Q8×C173C2×C411yescharacteristic
45C4⋊C4×C173—21no—
46C4⋊C4×C173C2×C411yescharacteristic
47C42×C173—21no—
48C42×C173C2×C411yescharacteristic
49C2×#10621924—42no—
50C2×#10621924—42no—
51C2×#10621924—42no—
52#13063443—42no—
53C22×D173—21no—
54C22×D173C2×D411yescharacteristic
55D692—41no—
56D692—41no—
57D692—41no—
58C2×Dic173—21no—
59C2×Dic173C22⋊C411yescharacteristic
60C2×Dic173C22⋊C411yescharacteristic
61C2×Dic173C2×D411yescharacteristic
62C4×D173—21no—
63C4×D173—21no—
64C4×D173—41no—
65C4×D173—41no—
66C4×D173—42no—
67Q8×C173—41no—
68C2×C4×C173—21no—
69C2×C4×C173—21no—
70C2×C4×C173C22⋊C411yescharacteristic
71C2×C4×C173C22⋊C411yescharacteristic
72C2×C4×C173C22×C411yescharacteristic
73C22×C173C2×C22⋊C411yescharacteristic
74C2×D173—21no—
75C2×D173—21no—
76C2×D173—41no—
77C2×D173—41no—
78C2×D173—41no—
79C2×D173AGL(1,ℤ/8)11yescharacteristic
80C2×D173#13711yescharacteristic
81#10621924—82no—
82C4×C173—21no—
83C4×C173—21no—
84C4×C173—21no—
85C4×C173—41no—
86C4×C173—41no—
87Dic173—21no—
88Dic173—21no—
89Dic173—21no—
90Dic173—41no—
91C2×C173#41011yescharacteristic
92C2×C173#41711yescharacteristic
93C2×C173#42111yescharacteristic
94D173—21no—
95D173—21no—
96D173—81no—
97C173#151211yescharacteristic, complemented, Sylow
98#1512—1731nomaximal, Sylow
99#529—1731no—
100#432—1731no—
101#382—1731no—
102#328—1732no—
103#327—1732no—
104#126—3461no—
105#125—1731no—
106#124—1731no—
107#123—3461no—
108C22⋊Q8—1731no—
109#121—1731no—
110C4×Q8#1062192411yescharacteristic, complemented
111C4×D4—3461no—
112C4×D4—1731no—
113#117—3461no—
114C2×C4⋊C4—3461no—
115C2×C42—1731no—
116C4⋊C8—3461no—
117C22⋊C8—1732no—
118#97—3461no—
119#95—3461no—
120#95—1732no—
121C2×Q8C2×#1062192411yescharacteristic
122C2×D4—3461no—
123C2×D4—1731no—
124C22×C4—3461no—
125C22×C4—3462no—
126C22×C4—1731no—
127C22×C4—1731no—
128C22×C4—1731no—
129C2×C8—3462no—
130C4⋊C4—21no—
131C4⋊C4—3461no—
132C4⋊C4—3461no—
133C4⋊C4—3461no—
134C4⋊C4—1731no—
135C4⋊C4—6922no—
136C4⋊C4C2×#1062192411yescharacteristic
137C22⋊C4—3461no—
138C22⋊C4—3461no—
139C22⋊C4—3461no—
140C22⋊C4—3461no—
141C22⋊C4—3461no—
142C42—21no—
143C42—1731no—
144C42—1731no—
145C42—1731no—
146C42C2×#1062192411yescharacteristic
147C23—3461no—
148C23—1731no—
149Q8—41no—
150D4—6921no—
151D4—6921no—
152D4—6921no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—3461no—
156C2×C4—3461no—
157C2×C4—3461no—
158C2×C4—1731no—
159C2×C4—1731no—
160C2×C4—1731no—
161C2×C4—6921no—
162C2×C4—6921no—
163C2×C4—6922no—
164C2×C4—6922no—
165C2×C4—6922no—
166C2×C4—6922no—
167C2×C4C22⋊#1062192411yescharacteristic
168C2×C4C22⋊#1062192411yescharacteristic
169C2×C4C22×#1062192411yescharacteristic
170C8—6922no—
171C22—3461no—
172C22—3461no—
173C22—1731no—
174C22—1731no—
175C22—6921no—
176C22—6921no—
177C22—6921no—
178C22C2×C22⋊#1062192411yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—41no—
183C4—41no—
184C4—3461no—
185C4—3461no—
186C4—3461no—
187C4—13842no—
188C4—6921no—
189C2—3461no—
190C2—3461no—
191C2—13841no—
192C2#2601247011yescharacteristic
193C2#2601248211yescharacteristic
194C2#2601249011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4×Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radical#19149654C22
cosocleC2×C4×D173C23
hypercenterC4×Q8#10621924
hypocenterC173#1512
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #26012514C2semidirect
2. #26012435C2semidirect
3. #26012345C2semidirect
13. #19149670C4semidirect
14. #19149668C4semidirect
19. C4×D692C4semidirect
22. C4×Q8×C173C4semidirect
97. C173#1512semidirect
110. C4×Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125142
#260124352
#260123452
#260123432
#260120282
#1512173