Finite Group Databasebeta

← #36369029

Subgroups — #36369029

Order 22144.

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

normal subgroups only

This group has 216 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#26012511C211yesmaximal, characteristic, complemented
2C4×#14583835C211yesmaximal, characteristic, complemented
3#26012348C212yesmaximal, complemented
4#26012346C212yesmaximal, complemented
5#26012022C211yesmaximal, characteristic
6C2×#14583835—21no—
7#19149746—21no—
8C22⋊#13063443—21no—
9#19149734C2212yes—
10#19149734C2212yes—
11C4×#13063443—21no—
12#19149672—21no—
13#19149670C2211yescharacteristic
14#19149666C2211yescharacteristic
15#19149660—21no—
16#19149658C411yescharacteristic
17#19149644C411yescharacteristic
18D4×Dic173—21no—
19D4×Dic173C411yescharacteristic
20#19149635—21no—
21C2×C4×Dic173C2211yescharacteristic
22#121×C173C411yescharacteristic
23#14583835—24no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C2×Dic346C2×C411yescharacteristic
27C22×Dic173—21no—
28C22×Dic173D411yescharacteristic
29C22×Dic173D411yescharacteristic
30C4⋊Dic173—21no—
31C4⋊Dic173C2×C411yescharacteristic
32#14583805—21no—
33#14583805—21no—
34#14583805—21no—
35C22⋊Dic173—21no—
36C22⋊Dic173—21no—
37C22⋊Dic173—21no—
38C22⋊Dic173—21no—
39C4×Dic173—21no—
40C4×Dic173C2×C411yescharacteristic
41C4×Dic173D411yescharacteristic
42C4×Dic173D411yescharacteristic
43C4×Dic173C2311yescharacteristic
44C2×D4×C173—21no—
45C2×D4×C173C2×C411yescharacteristic
46C22×C4×C173C2×C411yescharacteristic
47C4⋊C4×C173C2×C411yescharacteristic
48C22⋊C4×C173—21no—
49#13063443—24no—
50#13063443—42no—
51Dic346—41no—
52C2×Dic173—22no—
53C2×Dic173—21no—
54C2×Dic173—21no—
55C2×Dic173—41no—
56C2×Dic173—42no—
57C2×Dic173C22⋊C411yescharacteristic
58C2×Dic173C22⋊C411yescharacteristic
59C2×Dic173C2×D411yescharacteristic
60C2×Dic173C2×D411yescharacteristic
61C23×C173—21no—
62C23×C173C22⋊C411yescharacteristic
63D4×C173—41no—
64D4×C173—42no—
65C2×C4×C173—21no—
66C2×C4×C173—41no—
67C2×C4×C173C22⋊C411yescharacteristic
68C2×C4×C173C22×C411yescharacteristic
69C22×C173—21no—
70C22×C173—21no—
71C22×C173—21no—
72C22×C173—41no—
73C22×C173—41no—
74C22×C173—41no—
75C22×C173C2×C22⋊C411yescharacteristic
76C4×C173—21no—
77C4×C173—21no—
78C4×C173—41no—
79Dic173—21no—
80Dic173—21no—
81Dic173—21no—
82Dic173—41no—
83Dic173—41no—
84Dic173#10412yes—
85C2×C173—41no—
86C2×C173—81no—
87C2×C173#41311yescharacteristic
88C2×C173C2×#10411yescharacteristic
89C2×C173#42111yescharacteristic
90C173#152911yescharacteristic, complemented, Sylow
91#1529—1731nomaximal, Sylow
92#529—1731no—
93#432—1731no—
94C4×M16—1731no—
95#330—1732no—
96#329—1732no—
97C2×M16—3461no—
98#126—3461no—
99#125—1731no—
100#124—1731no—
101#123—3461no—
102C22⋊Q8—1731no—
103#121#1062192411yescharacteristic
104C4×Q8—1731no—
105C4×D4—3461no—
106C4×D4—1731no—
107#117—3461no—
108C2×C42—1731no—
109C4⋊C8—3461no—
110C22⋊C8—3461no—
111#97—1732no—
112#97—1732no—
113C4×C8—3461no—
114C2×Q8—1731no—
115C2×D4—21no—
116C2×D4C2×#1062192411yescharacteristic
117C22×C4—3461no—
118C22×C4—1731no—
119C22×C4—1731no—
120C22×C4C2×#1062192411yescharacteristic
121M16—3464no—
122C2×C8—3462no—
123C2×C8—3462no—
124C4⋊C4—3461no—
125C4⋊C4—3461no—
126C4⋊C4—3461no—
127C4⋊C4—3461no—
128C4⋊C4—1731no—
129C4⋊C4C2×#1062192411yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—3461no—
132C22⋊C4—3461no—
133C22⋊C4—3461no—
134C22⋊C4—3461no—
135C42—3461no—
136C42—1731no—
137C42—1731no—
138C42—1731no—
139C42—1731no—
140C23—21no—
141C23C22⋊#1062192411yescharacteristic
142Q8—6921no—
143D4—41no—
144D4—42no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—3462no—
148C2×C4—3461no—
149C2×C4—3461no—
150C2×C4—1731no—
151C2×C4—1731no—
152C2×C4—1731no—
153C2×C4—1731no—
154C2×C4—6921no—
155C2×C4—6922no—
156C2×C4C22⋊#1062192411yescharacteristic
157C2×C4C22×#1062192411yescharacteristic
158C8—3464no—
159C8—6922no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—41no—
164C22—41no—
165C22—41no—
166C22C2×C22⋊#1062192411yescharacteristic
167C4—21no—
168C4—21no—
169C4—41no—
170C4—3461no—
171C4—3461no—
172C4—3461no—
173C4—1732no—
174C4—6921no—
175C4—6921no—
176C2—41no—
177C2—81no—
178C2#2601247811yescharacteristic
179C2D173⋊#10411yescharacteristic
180C2#2601248911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting, prime radical#121×C173C4
socleC22×C173C2×C22⋊C4
cosocleC4×Dic173C23
hypercenter#121#10621924
hypocenterC173#1529
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #26012511C2semidirect
2. C4×#14583835C2semidirect
3. #26012348C2semidirect
4. #26012346C2semidirect
90. C173#1529semidirect

Maximal subgroups

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

subgroupindex
#260125112
C4×#145838352
#260123482
#260123462
#260120222
#1529173