Finite Group Databasebeta

← #36368822

Subgroups — #36368822

Order 22144.

300 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 68 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#26012512C212yesmaximal, complemented
2#26012335C212yesmaximal, complemented
3#26012334C212yesmaximal, complemented
4C2×C4×C22⋊D173C211yesmaximal, characteristic
5C4⋊#13063443—22no—
6C22⋊#13063443C2214yes—
7#19149734—22no—
8C22×C22⋊D173C411yescharacteristic
9C22×C4×D173C411yescharacteristic
10C4×C22⋊D173—22no—
11C4×C22⋊D173—22no—
12C2×#14583805C411yescharacteristic
13C2×C22⋊Dic173C2211yescharacteristic
14C2×#14583803C2211yescharacteristic
15C2×C4×Dic173C2211yescharacteristic
16C23×C4×C173C411yescharacteristic
17C2×#13063443—24no—
18C23×D173C2×C411yescharacteristic
19C2×C22⋊D173—22no—
20C2×C22⋊D173—22no—
21C2×C22⋊D173—22no—
22C22×Dic173C2×C411yescharacteristic
23C22×Dic173C2×C411yescharacteristic
24C22×Dic173C2311yescharacteristic
25C2×C4×D173—21no—
26C2×C4×D173—21no—
27C2×C4×D173—42no—
28#14583805—22no—
29C22⋊Dic173—22no—
30#14583803—22no—
31C4×Dic173D412yes—
32C4×Dic173D412yes—
33C24×C173C2×C411yescharacteristic
34C22×C4×C173—21no—
35C22×C4×C173—21no—
36C22×C4×C173—42no—
37C22×C4×C173C2×C411yescharacteristic
38C22×C4×C173C2×C411yescharacteristic
39#13063443—44no—
40C22×D173—21no—
41C22×D173—21no—
42C22×D173—41no—
43C22×D173—41no—
44C22×D173M1612yes—
45C22⋊D173—28no—
46C2×Dic173—21no—
47C2×Dic173—21no—
48C2×Dic173—21no—
49C2×Dic173—41no—
50C2×Dic173—41no—
51C2×Dic173C22⋊C412yes—
52C2×Dic173C2×D412yes—
53C4×D173—42no—
54C4×D173—42no—
55C23×C173—21no—
56C23×C173—21no—
57C23×C173—41no—
58C23×C173—41no—
59C23×C173M1612yes—
60C23×C173C22×C411yescharacteristic
61C2×C4×C173—22no—
62C2×C4×C173—21no—
63C2×C4×C173—42no—
64C2×C4×C173—42no—
65C2×C4×C173—41no—
66C2×C4×C173C22⋊C411yescharacteristic
67C2×C4×C173C22⋊C411yescharacteristic
68C22×C173—21no—
69C22×C173—21no—
70C22×C173—22no—
71C22×C173—42no—
72C22×C173—42no—
73C22×C173—41no—
74C22×C173—41no—
75C22×C173C2×C22⋊C411yescharacteristic
76C22×C173C2×M1611yescharacteristic
77C22×C173C2×M1611yescharacteristic
78C2×D173—22no—
79C2×D173—42no—
80C2×D173—41no—
81C2×D173—41no—
82C2×D173—42no—
83C4×C173—21no—
84C4×C173—41no—
85C4×C173#10012yes—
86Dic173—22no—
87Dic173—22no—
88Dic173—41no—
89C2×C173—21no—
90C2×C173—21no—
91C2×C173—42no—
92C2×C173#40711yescharacteristic
93C2×C173#41011yescharacteristic
94C2×C173C2×#10011yescharacteristic
95D173—42no—
96C173#146611yescharacteristic, complemented, Sylow
97#1466—1731nomaximal, Sylow
98C2×C4×D4—1731no—
99#432—1732no—
100#323—1732no—
101#323—1732no—
102C22×D4—1731no—
103C23×C4—1731no—
104C23×C4#1062192411yescharacteristic
105C4×D4—3462no—
106C4×D4—3462no—
107C2×C4⋊C4—1731no—
108C2×C22⋊C4—1731no—
109C2×C22⋊C4—1731no—
110C2×C42—1731no—
111C4⋊C8—3462no—
112C22⋊C8—1734no—
113#97—3462no—
114C24—1731no—
115C24C2×#1062192411yescharacteristic
116C2×D4—3462no—
117C2×D4—3462no—
118C2×D4—3462no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—42no—
122C22×C4—3461no—
123C22×C4—3461no—
124C22×C4—1731no—
125C22×C4—1731no—
126C22×C4—1731no—
127C22×C4—6922no—
128C22×C4C2×#1062192411yescharacteristic
129C22×C4C2×#1062192411yescharacteristic
130C2×C8—3464no—
131C4⋊C4—3462no—
132C22⋊C4—3462no—
133C22⋊C4—3462no—
134C42—1732no—
135C42—1732no—
136C23—21no—
137C23—21no—
138C23—41no—
139C23—41no—
140C23—3461no—
141C23—3461no—
142C23—1732no—
143C23—6921no—
144C23—6921no—
145C23#1458383612yes—
146C23C22×#1062192411yescharacteristic
147D4—3468no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—3461no—
154C2×C4—3461no—
155C2×C4—3461no—
156C2×C4—1732no—
157C2×C4—1732no—
158C2×C4—6922no—
159C2×C4—6921no—
160C2×C4—6922no—
161C2×C4—6921no—
162C2×C4C22⋊#1062192411yescharacteristic
163C2×C4C22⋊#1062192411yescharacteristic
164C8—6924no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—41no—
171C22—41no—
172C22—3462no—
173C22—6922no—
174C22—6922no—
175C22—6921no—
176C22—6921no—
177C22C2×C22⋊#1062192411yescharacteristic
178C22D173⋊M1611yescharacteristic
179C22C2×#1458383611yescharacteristic
180C4—21no—
181C4—41no—
182C4—3462no—
183C4—3462no—
184C4—6921no—
185C4#1914973912yes—
186C2—21no—
187C2—21no—
188C2—42no—
189C2—6922no—
190C2#2601246211yescharacteristic
191C2#2601247011yescharacteristic
192C2C2×#1914973911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC23×C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×C22⋊D173C4
cosocleC22×Dic173C23
hypercenterC23×C4#10621924
hypocenterC173#1466
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. #26012512C2semidirect
2. #26012335C2semidirect
3. #26012334C2semidirect
96. C173#1466semidirect

Maximal subgroups

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

subgroupindex
#260125122
#260123352
#260123342
C2×C4×C22⋊D1732
#1466173