Finite Group Databasebeta

← #36369050

Subgroups — #36369050

Order 22144.

200 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 54 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#26012512C211yesmaximal, characteristic, complemented
2#26012511C211yesmaximal, characteristic, complemented
3#26012348C212yesmaximal, complemented
4#26012347C212yesmaximal, complemented
5#26012027C211yesmaximal, characteristic
6#19149746—21no—
7C4⋊#13063443—21no—
8C22⋊#13063443—21no—
9C22⋊#13063443—21no—
10#19149734C2212yes—
11#19149734C2212yes—
12#19149674—21no—
13#19149670C2211yescharacteristic
14#19149668C2211yescharacteristic
15#19149663—21no—
16#19149653C411yescharacteristic
17Q8×Dic173C411yescharacteristic
18#19149641C411yescharacteristic
19#19149639—21no—
20#19149637—21no—
21C2×C4×Dic173C2211yescharacteristic
22C22⋊Q8×C173C411yescharacteristic
23C2×#13063443—22no—
24C2×#13063443—22no—
25C2×D692C2×C411yescharacteristic
26C2×C22⋊D173—21no—
27C22×Dic173D411yescharacteristic
28C22×Dic173D411yescharacteristic
29C2×C4×D173—21no—
30C4⋊Dic173—21no—
31C4⋊Dic173—21no—
32C4⋊Dic173C2×C411yescharacteristic
33#14583805—21no—
34#14583803—21no—
35#14583803—21no—
36#14583803—21no—
37#14583803—21no—
38C4×Dic173—21no—
39C4×Dic173C2×C411yescharacteristic
40C4×Dic173D411yescharacteristic
41C4×Dic173D411yescharacteristic
42C4×Dic173C2311yescharacteristic
43C2×Q8×C173C2×C411yescharacteristic
44C22×C4×C173C2×C411yescharacteristic
45C4⋊C4×C173—21no—
46C4⋊C4×C173C2×C411yescharacteristic
47C22⋊C4×C173—21no—
48#13063443—42no—
49#13063443—42no—
50C22×D173—21no—
51D692—41no—
52C22⋊D173—42no—
53C2×Dic173—21no—
54C2×Dic173—41no—
55C2×Dic173—41no—
56C2×Dic173C22⋊C411yescharacteristic
57C2×Dic173C22⋊C411yescharacteristic
58C2×Dic173C2×D411yescharacteristic
59C2×Dic173C2×D411yescharacteristic
60C4×D173—42no—
61C23×C173C22⋊C411yescharacteristic
62Q8×C173—41no—
63C2×C4×C173—21no—
64C2×C4×C173—21no—
65C2×C4×C173—41no—
66C2×C4×C173C22⋊C411yescharacteristic
67C2×C4×C173C22×C411yescharacteristic
68C22×C173—21no—
69C22×C173—21no—
70C22×C173—21no—
71C22×C173C2×C22⋊C411yescharacteristic
72C2×D173—41no—
73C2×D173—41no—
74C2×D173—41no—
75C4×C173—21no—
76C4×C173—21no—
77C4×C173—41no—
78C4×C173—41no—
79Dic173—21no—
80Dic173—21no—
81Dic173—21no—
82Dic173—21no—
83Dic173—41no—
84C2×C173—41no—
85C2×C173#41311yescharacteristic
86C2×C173#42111yescharacteristic
87C2×C173#42111yescharacteristic
88D173—81no—
89C173#153511yescharacteristic, complemented, Sylow
90#1535—1731nomaximal, Sylow
91#529—1731no—
92#432—1731no—
93#432—1731no—
94#330—1732no—
95#329—1732no—
96#126—3461no—
97#125—1731no—
98#124—1731no—
99#123—3461no—
100C22⋊Q8#1062192411yescharacteristic
101#121—1731no—
102C4×Q8—1731no—
103C4×D4—3461no—
104C4×D4—1731no—
105#117—3461no—
106C2×C42—1731no—
107C4⋊C8—3461no—
108C4⋊C8—3461no—
109C22⋊C8—3461no—
110C22⋊C8—3461no—
111#97—1732no—
112#97—1732no—
113C2×Q8C2×#1062192411yescharacteristic
114C2×D4—3461no—
115C2×D4—1731no—
116C22×C4—3461no—
117C22×C4—1731no—
118C22×C4—1731no—
119C22×C4C2×#1062192411yescharacteristic
120C2×C8—3462no—
121C2×C8—3462no—
122C4⋊C4—21no—
123C4⋊C4—3461no—
124C4⋊C4—3461no—
125C4⋊C4—3461no—
126C4⋊C4—1731no—
127C4⋊C4C2×#1062192411yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—3461no—
130C22⋊C4—3461no—
131C22⋊C4—3461no—
132C22⋊C4—3461no—
133C42—3461no—
134C42—1731no—
135C42—1731no—
136C42—1731no—
137C42—1731no—
138C23—3461no—
139C23C22⋊#1062192411yescharacteristic
140Q8—41no—
141D4—6921no—
142D4—6922no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—3461no—
147C2×C4—1731no—
148C2×C4—1731no—
149C2×C4—1731no—
150C2×C4—1731no—
151C2×C4—6922no—
152C2×C4—6921no—
153C2×C4—6921no—
154C2×C4C22⋊#1062192411yescharacteristic
155C2×C4C22×#1062192411yescharacteristic
156C8—6922no—
157C8—6922no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—6921no—
162C22—6921no—
163C22—6921no—
164C22C2×C22⋊#1062192411yescharacteristic
165C4—21no—
166C4—21no—
167C4—41no—
168C4—41no—
169C4—3461no—
170C4—3461no—
171C4—3461no—
172C4—3461no—
173C4—6921no—
174C2—41no—
175C2—13841no—
176C2#2601247811yescharacteristic
177C2#2601248811yescharacteristic
178C2#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#1535
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #26012512C2semidirect
2. #26012511C2semidirect
3. #26012348C2semidirect
4. #26012347C2semidirect
89. C173#1535semidirect

Maximal subgroups

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

subgroupindex
#260125122
#260125112
#260123482
#260123472
#260120272
#1535173