Finite Group Databasebeta

← #36368882

Subgroups — #36368882

Order 22144.

262 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 62 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#26012507C211yesmaximal, characteristic, complemented
2C4×#14583835C211yesmaximal, characteristic, complemented
3#26012339C212yesmaximal, complemented
4D4⋊#13063443C212yesmaximal, complemented
5#26012036C211yesmaximal, characteristic
6C2×#14583835—21no—
7C4⋊#13063443C2212yes—
8C22⋊#13063443—21no—
9#19149734—21no—
10C4×#13063443—21no—
11C4×#13063443C2212yes—
12C2×#14583825—21no—
13#19149678C411yescharacteristic
14#19149675C411yescharacteristic
15#19149665—21no—
16#19149653C411yescharacteristic
17#19149651—21no—
18D4×Dic173C2211yescharacteristic
19#19149641C2211yescharacteristic
20#19149639—21no—
21C2×C4×Dic173C2211yescharacteristic
22#121×C173C411yescharacteristic
23#14583835—24no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26#14583825—44no—
27C2×Dic346—21no—
28C2×D692C2×C411yescharacteristic
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C22×Dic173—21no—
33C22×Dic173D411yescharacteristic
34C22×Dic173D411yescharacteristic
35C2×C4×D173—21no—
36C4⋊Dic173C2×C411yescharacteristic
37#14583805—21no—
38C22⋊Dic173—21no—
39#14583803—21no—
40#14583803—21no—
41C4×Dic173C2×C411yescharacteristic
42C4×Dic173D411yescharacteristic
43C4×Dic173D411yescharacteristic
44C4×Dic173C2311yescharacteristic
45C2×D4×C173—21no—
46C2×D4×C173C2×C411yescharacteristic
47C22×C4×C173C2×C411yescharacteristic
48C4⋊C4×C173C2×C411yescharacteristic
49C22⋊C4×C173—21no—
50#13063443—24no—
51#13063443—42no—
52C22×D173—21no—
53Dic346—42no—
54D692—22no—
55C22⋊D173—42no—
56C22⋊D173—42no—
57C22⋊D173—42no—
58C2×Dic173—22no—
59C2×Dic173—22no—
60C2×Dic173—21no—
61C2×Dic173—42no—
62C2×Dic173C22⋊C411yescharacteristic
63C2×Dic173C22⋊C411yescharacteristic
64C2×Dic173C2×D411yescharacteristic
65C2×Dic173C2×D411yescharacteristic
66C4×D173—42no—
67C23×C173—21no—
68C23×C173C22⋊C411yescharacteristic
69D4×C173—22no—
70D4×C173—42no—
71C2×C4×C173—22no—
72C2×C4×C173—21no—
73C2×C4×C173C22⋊C411yescharacteristic
74C2×C4×C173C22×C411yescharacteristic
75C22×C173—21no—
76C22×C173—21no—
77C22×C173—21no—
78C22×C173—41no—
79C22×C173—41no—
80C22×C173—41no—
81C22×C173C2×C22⋊C411yescharacteristic
82C2×D173—41no—
83C2×D173—41no—
84C2×D173—41no—
85C4×C173—21no—
86C4×C173—41no—
87C4×C173#13112yes—
88Dic173—21no—
89Dic173—21no—
90Dic173—41no—
91Dic173#10412yes—
92Dic173AGL(1,ℤ/8)12yes—
93C2×C173—41no—
94C2×C173—81no—
95C2×C173#40811yescharacteristic
96C2×C173#41811yescharacteristic
97C2×C173C2×#10411yescharacteristic
98D173—81no—
99C173#148411yescharacteristic, complemented, Sylow
100#1484—1731nomaximal, Sylow
101#532—1731no—
102#431—1731no—
103C4×M16—1731no—
104D4⋊C8—1732no—
105D4⋊C8—1732no—
106C2×#41—3461no—
107C2×M16—3461no—
108C4⋊Q8—1731no—
109#127—1731no—
110#124—3461no—
111#121—3461no—
112#121—1731no—
113#121#1062192411yescharacteristic
114C4×D4—3461no—
115C4×D4—1731no—
116C4×D4—1731no—
117C2×C42—1731no—
118C4⋊C8—1732no—
119C22⋊C8—3461no—
120#97—3461no—
121C4×C8—3461no—
122C4×C8—1732no—
123#41—6924no—
124C2×Q8—3461no—
125C2×D4—21no—
126C2×D4—3461no—
127C2×D4—3461no—
128C2×D4—3461no—
129C2×D4—1731no—
130C2×D4C2×#1062192411yescharacteristic
131C22×C4—3461no—
132C22×C4—3461no—
133C22×C4—1731no—
134C22×C4—1731no—
135C22×C4C2×#1062192411yescharacteristic
136M16—3464no—
137C2×C8—3462no—
138C2×C8—3462no—
139C4⋊C4—3461no—
140C4⋊C4—1731no—
141C4⋊C4C2×#1062192411yescharacteristic
142C22⋊C4—21no—
143C22⋊C4—3461no—
144C22⋊C4—3461no—
145C22⋊C4—3461no—
146C42—1731no—
147C42—1731no—
148C42—1731no—
149C42—1731no—
150C23—21no—
151C23—3461no—
152C23C22⋊#1062192411yescharacteristic
153Q8—6922no—
154D4—22no—
155D4—42no—
156D4—3462no—
157D4—6922no—
158D4—6922no—
159D4—6922no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—3462no—
163C2×C4—3462no—
164C2×C4—3461no—
165C2×C4—1731no—
166C2×C4—1731no—
167C2×C4—1731no—
168C2×C4—1731no—
169C2×C4—6922no—
170C2×C4—6922no—
171C2×C4C22⋊#1062192411yescharacteristic
172C2×C4C22×#1062192411yescharacteristic
173C8—3464no—
174C8—6922no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—6921no—
182C22—6921no—
183C22—6921no—
184C22C2×C22⋊#1062192411yescharacteristic
185C4—21no—
186C4—41no—
187C4—3461no—
188C4—3461no—
189C4—1732no—
190C4—1732no—
191C4—6921no—
192C4#1914976612yes—
193C2—41no—
194C2—81no—
195C2—13841no—
196C2#2601246411yescharacteristic
197C2#2601248311yescharacteristic
198C2D173⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#121×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012036C2
cosocleC4×Dic173C23
hypercenter#121#10621924
hypocenterC173#1484
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #26012507C2semidirect
2. C4×#14583835C2semidirect
3. #26012339C2semidirect
4. D4⋊#13063443C2semidirect
99. C173#1484semidirect

Maximal subgroups

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

subgroupindex
#260125072
C4×#145838352
#260123392
D4⋊#130634432
#260120362
#1484173