Finite Group Databasebeta

← #36368920

Subgroups — #36368920

Order 22144.

216 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 60 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
2#26012494C211yesmaximal, characteristic, complemented
3Q8⋊#13063443C212yesmaximal, complemented
4#26012339C212yesmaximal, complemented
5#26012027C211yesmaximal, characteristic
6C2×#14583835—21no—
7#19149746—21no—
8C4⋊#13063443—21no—
9C4⋊#13063443C2212yes—
10C22⋊#13063443—21no—
11C4×#13063443C2212yes—
12#19149674—21no—
13#19149670C411yescharacteristic
14#19149668C411yescharacteristic
15#19149663—21no—
16#19149653C411yescharacteristic
17Q8×Dic173C2211yescharacteristic
18#19149641C2211yescharacteristic
19#19149639—21no—
20#19149637—21no—
21C2×C4×Dic173C2211yescharacteristic
22C22⋊Q8×C173C411yescharacteristic
23#14583835—42no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C2×D692C2×C411yescharacteristic
27C2×C22⋊D173—21no—
28C22×Dic173D411yescharacteristic
29C22×Dic173D411yescharacteristic
30C2×C4×D173—21no—
31C4⋊Dic173—21no—
32C4⋊Dic173—21no—
33C4⋊Dic173C2×C411yescharacteristic
34#14583805—21no—
35#14583803—21no—
36#14583803—21no—
37#14583803—21no—
38#14583803—21no—
39C4×Dic173—21no—
40C4×Dic173C2×C411yescharacteristic
41C4×Dic173D411yescharacteristic
42C4×Dic173D411yescharacteristic
43C4×Dic173C2311yescharacteristic
44C2×Q8×C173C2×C411yescharacteristic
45C22×C4×C173C2×C411yescharacteristic
46C4⋊C4×C173—21no—
47C4⋊C4×C173C2×C411yescharacteristic
48C22⋊C4×C173—21no—
49#13063443—42no—
50#13063443—42no—
51C22×D173—21no—
52D692—22no—
53C22⋊D173—42no—
54C2×Dic173—21no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C2×Dic173—41no—
58C2×Dic173C22⋊C411yescharacteristic
59C2×Dic173C22⋊C411yescharacteristic
60C2×Dic173C2×D411yescharacteristic
61C2×Dic173C2×D411yescharacteristic
62C4×D173—41no—
63C4×D173—41no—
64C23×C173C22⋊C411yescharacteristic
65Q8×C173—22no—
66C2×C4×C173—22no—
67C2×C4×C173—21no—
68C2×C4×C173—21no—
69C2×C4×C173C22⋊C411yescharacteristic
70C2×C4×C173C22×C411yescharacteristic
71C22×C173—21no—
72C22×C173—21no—
73C22×C173—21no—
74C22×C173C2×C22⋊C411yescharacteristic
75C2×D173—41no—
76C2×D173—41no—
77C2×D173—41no—
78C4×C173—21no—
79C4×C173—41no—
80C4×C173—41no—
81C4×C173#13112yes—
82Dic173—21no—
83Dic173—21no—
84Dic173—21no—
85Dic173—41no—
86Dic173AGL(1,ℤ/8)11yescharacteristic
87Dic173#13711yescharacteristic
88C2×C173—41no—
89C2×C173#40811yescharacteristic
90C2×C173#41711yescharacteristic
91C2×C173#42111yescharacteristic
92D173—81no—
93C173#149511yescharacteristic, complemented, Sylow
94#1495—1731nomaximal, Sylow
95#529—1731no—
96#431—1731no—
97#423—1731no—
98Q8⋊C8—1732no—
99D4⋊C8—1732no—
100C2×M16—3461no—
101#126—3461no—
102#125—1731no—
103#124—1731no—
104#123—3461no—
105C22⋊Q8#1062192411yescharacteristic
106#121—1731no—
107C4×Q8—1731no—
108C4×D4—3461no—
109C4×D4—1731no—
110#117—3461no—
111C2×C42—1731no—
112C4⋊C8—3461no—
113C4⋊C8—3461no—
114C4⋊C8—1732no—
115C22⋊C8—3461no—
116C4×C8—1732no—
117C2×Q8C2×#1062192411yescharacteristic
118C2×D4—3461no—
119C2×D4—1731no—
120C22×C4—3461no—
121C22×C4—1731no—
122C22×C4—1731no—
123C22×C4C2×#1062192411yescharacteristic
124M16—6922no—
125C2×C8—3462no—
126C2×C8—3462no—
127C4⋊C4—21no—
128C4⋊C4—3461no—
129C4⋊C4—3461no—
130C4⋊C4—3461no—
131C4⋊C4—1731no—
132C4⋊C4C2×#1062192411yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—3461no—
135C22⋊C4—3461no—
136C22⋊C4—3461no—
137C22⋊C4—3461no—
138C42—3461no—
139C42—1731no—
140C42—1731no—
141C42—1731no—
142C42—1731no—
143C23—3461no—
144C23C22⋊#1062192411yescharacteristic
145Q8—22no—
146D4—3462no—
147D4—6922no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—3461no—
152C2×C4—3461no—
153C2×C4—3461no—
154C2×C4—1731no—
155C2×C4—1731no—
156C2×C4—1731no—
157C2×C4—1731no—
158C2×C4—6921no—
159C2×C4—6921no—
160C2×C4—6921no—
161C2×C4C22⋊#1062192411yescharacteristic
162C2×C4C22×#1062192411yescharacteristic
163C8—6922no—
164C8—6922no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—6921no—
169C22—6921no—
170C22—6921no—
171C22C2×C22⋊#1062192411yescharacteristic
172C4—21no—
173C4—41no—
174C4—41no—
175C4—3461no—
176C4—3461no—
177C4—3461no—
178C4—1731no—
179C4—1731no—
180C4—6921no—
181C4#1914976612yes—
182C2—41no—
183C2—13841no—
184C2#2601246411yescharacteristic
185C2#2601248211yescharacteristic
186C2#2601249011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22⋊Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radical#19149653C4
cosocleC4×Dic173C23
hypercenterC22⋊Q8#10621924
hypocenterC173#1495
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #26012507C2semidirect
2. #26012494C2semidirect
3. Q8⋊#13063443C2semidirect
4. #26012339C2semidirect
93. C173#1495semidirect

Maximal subgroups

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

subgroupindex
#260125072
#260124942
Q8⋊#130634432
#260123392
#260120272
#1495173