Finite Group Databasebeta

← #36368953

Subgroups — #36368953

Order 22144.

272 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 58 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#26012427C211yesmaximal, characteristic, complemented
3#26012343C212yesmaximal, complemented
4#26012342C212yesmaximal, complemented
5#26012038C211yesmaximal, characteristic
6C2×C4×#10621924—21no—
7#19149746—21no—
8#19149736C2212yes—
9#19149734—21no—
10(C2×#10621924)⋊C4—21no—
11(C2×#10621924)⋊C4C2212yes—
12C2×#14583826—21no—
13#19149678C411yescharacteristic, complemented
14#19149675C411yescharacteristic, complemented
15#19149665—21no—
16#19149654C2211yescharacteristic
17#19149652—21no—
18#19149650C2211yescharacteristic
19C4×D692C411yescharacteristic, complemented
20C4×C22⋊D173—21no—
21C42×D173C2211yescharacteristic
22C4×D4×C173C411yescharacteristic, complemented
23C22×#10621924—22no—
24C2×#13063443—22no—
25C4×#10621924—24no—
26#14583826—44no—
27C2×Dic346—21no—
28C2×D692—21no—
29C2×D692C2×C411yescharacteristic
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C4×D173—21no—
34C2×C4×D173D411yescharacteristic
35C2×C4×D173D411yescharacteristic
36C2×C4×D173C2311yescharacteristic
37C4⋊Dic173C2×C411yescharacteristic
38#14583805—21no—
39C22⋊Dic173—21no—
40#14583803—21no—
41#14583803—21no—
42C4×Dic173C2×C411yescharacteristic
43C4×Dic173D411yescharacteristic
44C4×Dic173D411yescharacteristic
45C2×D4×C173C2×C411yescharacteristic
46C22×C4×C173—21no—
47C4⋊C4×C173C2×C411yescharacteristic
48C22⋊C4×C173—21no—
49C42×C173C2×C411yescharacteristic
50C2×#10621924—24no—
51C2×#10621924—42no—
52C2×#10621924—42no—
53#13063443—42no—
54C22×D173—21no—
55C22×D173C2×D411yescharacteristic
56Dic346—42no—
57D692—41no—
58D692—42no—
59C22⋊D173—42no—
60C22⋊D173—42no—
61C22⋊D173—42no—
62C2×Dic173—21no—
63C2×Dic173C22⋊C411yescharacteristic
64C2×Dic173C22⋊C411yescharacteristic
65C2×Dic173C2×D411yescharacteristic
66C4×D173—22no—
67C4×D173—22no—
68C4×D173—41no—
69C4×D173—42no—
70C23×C173—21no—
71D4×C173—41no—
72C2×C4×C173—21no—
73C2×C4×C173—42no—
74C2×C4×C173C22⋊C411yescharacteristic
75C2×C4×C173C22⋊C411yescharacteristic
76C2×C4×C173C22×C411yescharacteristic
77C22×C173—41no—
78C22×C173—41no—
79C22×C173—41no—
80C22×C173C2×C22⋊C411yescharacteristic
81C2×D173—21no—
82C2×D173—21no—
83C2×D173—41no—
84C2×D173—41no—
85C2×D173—41no—
86C2×D173AGL(1,ℤ/8)12yes—
87#10621924—44no—
88C4×C173—21no—
89C4×C173—21no—
90C4×C173—21no—
91C4×C173—41no—
92Dic173—21no—
93Dic173—21no—
94Dic173—41no—
95Dic173#10412yes—
96C2×C173—81no—
97C2×C173#41011yescharacteristic
98C2×C173#41811yescharacteristic
99C2×C173C2×#10411yescharacteristic
100D173—22no—
101D173—81no—
102C173#150511yescharacteristic, complemented, Sylow
103#1505—1731nomaximal, Sylow
104#532—1731no—
105#432—1731no—
106#376—1731no—
107#327—1732no—
108#327—1732no—
109C2×#41—3461no—
110C4⋊Q8—1731no—
111#127—1731no—
112#124—3461no—
113#121—3461no—
114#121—1731no—
115#121—1731no—
116C4×D4—3461no—
117C4×D4—1731no—
118C4×D4#1062192411yescharacteristic, complemented
119C2×C42—3461no—
120C2×C42—1731no—
121C4⋊C8—3461no—
122C22⋊C8—1732no—
123#97—3461no—
124#95—3461no—
125#95—1732no—
126#41—6924no—
127C2×Q8—3461no—
128C2×D4—3461no—
129C2×D4—3461no—
130C2×D4—3461no—
131C2×D4—3461no—
132C2×D4—1731no—
133C2×D4C2×#1062192411yescharacteristic
134C22×C4—21no—
135C22×C4—3462no—
136C22×C4—3461no—
137C22×C4—1731no—
138C22×C4—1731no—
139C22×C4—1731no—
140C2×C8—3462no—
141C4⋊C4—3461no—
142C4⋊C4—1731no—
143C4⋊C4C2×#1062192411yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—3461no—
146C22⋊C4—3461no—
147C22⋊C4—3461no—
148C42—3464no—
149C42—1731no—
150C42—1731no—
151C42—1731no—
152C42C2×#1062192411yescharacteristic
153C23—21no—
154C23—3461no—
155C23—1731no—
156Q8—6922no—
157D4—41no—
158D4—6921no—
159D4—6922no—
160D4—6922no—
161D4—6922no—
162D4—6922no—
163C2×C4—21no—
164C2×C4—42no—
165C2×C4—3462no—
166C2×C4—3462no—
167C2×C4—3464no—
168C2×C4—3461no—
169C2×C4—1731no—
170C2×C4—1731no—
171C2×C4—1731no—
172C2×C4—6921no—
173C2×C4—6922no—
174C2×C4—6922no—
175C2×C4—6922no—
176C2×C4C22⋊#1062192411yescharacteristic
177C2×C4C22⋊#1062192411yescharacteristic
178C2×C4C22×#1062192411yescharacteristic
179C8—6922no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—3461no—
184C22—3461no—
185C22—1732no—
186C22—6921no—
187C22—6921no—
188C22—6921no—
189C22C2×C22⋊#1062192411yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—41no—
194C4—3461no—
195C4—3461no—
196C4—1732no—
197C4—6924no—
198C4—6921no—
199C2—81no—
200C2—3462no—
201C2—13841no—
202C2#2601247011yescharacteristic
203C2#2601248311yescharacteristic
204C2D173⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4×D4×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012038C2
cosocleC2×C4×D173C23
hypercenterC4×D4#10621924
hypocenterC173#1505
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #26012514C2semidirect
2. #26012427C2semidirect
3. #26012343C2semidirect
4. #26012342C2semidirect
13. #19149678C4semidirect
14. #19149675C4semidirect
19. C4×D692C4semidirect
22. C4×D4×C173C4semidirect
102. C173#1505semidirect
118. C4×D4#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125142
#260124272
#260123432
#260123422
#260120382
#1505173