Finite Group Databasebeta

← #36368829

Subgroups — #36368829

Order 22144.

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

normal subgroups only

This group has 280 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#26012462C211yesmaximal, characteristic, complemented
2D173⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
3#26012336C212yesmaximal, complemented
4#26012334C212yesmaximal, complemented
5C2×#19149661C211yesmaximal, characteristic
6C2×#14583836—21no—
7C2×D173⋊C8—21no—
8#19149736—21no—
9#19149736—21no—
10C22⋊#13063443C2212yes—
11C22⋊#13063443C2212yes—
12C22×C22⋊D173C411yescharacteristic
13C22×C4×D173C2211yescharacteristic
14#19149661—21no—
15#19149661—21no—
16#19149661—21no—
17#19149661—21no—
18C2×C4⋊Dic173C2211yescharacteristic
19C2×#14583805C411yescharacteristic
20C2×C22⋊Dic173C2211yescharacteristic
21C2×#14583803C411yescharacteristic
22C2×C22⋊C4×C173C411yescharacteristic
23#14583836—42no—
24D173⋊C8—42no—
25C2×#13063443—22no—
26C2×#13063443—22no—
27C23×D173C2×C411yescharacteristic
28C2×C22⋊D173—21no—
29C2×C22⋊D173—21no—
30C2×C22⋊D173—42no—
31C22×Dic173C2×C411yescharacteristic
32C22×Dic173C2×C411yescharacteristic
33C22×Dic173C2311yescharacteristic
34C2×C4×D173—42no—
35C2×C4×D173D411yescharacteristic
36C2×C4×D173D411yescharacteristic
37C2×C4×D173D411yescharacteristic
38C2×C4×D173D411yescharacteristic
39C4⋊Dic173—21no—
40C4⋊Dic173—21no—
41#14583805—21no—
42#14583805—21no—
43C22⋊Dic173—21no—
44C22⋊Dic173—21no—
45#14583803—21no—
46#14583803—21no—
47C24×C173C2×C411yescharacteristic
48C22×C4×C173C2×C411yescharacteristic
49C22×C4×C173C2×C411yescharacteristic
50C22⋊C4×C173—21no—
51C22⋊C4×C173—21no—
52#13063443—42no—
53#13063443—42no—
54C22×D173—22no—
55C22×D173—22no—
56C22×D173—41no—
57C22×D173C22⋊C411yescharacteristic
58C22×D173C22⋊C411yescharacteristic
59C22⋊D173—42no—
60C22⋊D173—42no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C2×Dic173—41no—
64C2×Dic173—41no—
65C2×Dic173—41no—
66C2×Dic173C2×D411yescharacteristic
67C2×Dic173C2×D411yescharacteristic
68C4×D173—22no—
69C4×D173—22no—
70C4×D173—41no—
71C4×D173—41no—
72C23×C173—21no—
73C23×C173—22no—
74C23×C173—41no—
75C23×C173—41no—
76C23×C173C22×C411yescharacteristic
77C2×C4×C173—21no—
78C2×C4×C173—22no—
79C2×C4×C173—41no—
80C2×C4×C173C22⋊C411yescharacteristic
81C2×C4×C173C22⋊C411yescharacteristic
82C22×C173—21no—
83C22×C173—21no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173—41no—
87C22×C173—82no—
88C22×C173C2×C22⋊C411yescharacteristic
89C22×C173#13112yes—
90C2×D173—21no—
91C2×D173—21no—
92C2×D173—21no—
93C2×D173—21no—
94C2×D173—21no—
95C2×D173—44no—
96C2×D173#9912yes—
97C4×C173—21no—
98C4×C173—21no—
99C4×C173—41no—
100Dic173—21no—
101Dic173—21no—
102Dic173—41no—
103Dic173—41no—
104C2×C173—22no—
105C2×C173—81no—
106C2×C173#40811yescharacteristic
107C2×C173C2×#9911yescharacteristic
108C2×C173#41311yescharacteristic
109D173—22no—
110D173—41no—
111C173#146811yescharacteristic, complemented, Sylow
112#1468—1731nomaximal, Sylow
113C2×#123—1731no—
114#407—1731no—
115C2×C22⋊C8—1731no—
116#324—1732no—
117#323—1732no—
118C22×D4—1731no—
119C23×C4—1731no—
120C2×M16—3461no—
121C22×C8—3461no—
122#123—3461no—
123#123—3461no—
124#123—3461no—
125#123—3461no—
126C2×C4⋊C4—1731no—
127C2×C4⋊C4—1731no—
128C2×C22⋊C4—1731no—
129C2×C22⋊C4—1731no—
130C2×C22⋊C4#1062192411yescharacteristic
131C22⋊C8—3461no—
132C22⋊C8—3461no—
133C22⋊C8—1732no—
134C22⋊C8—1732no—
135C24—1731no—
136C24C2×#1062192411yescharacteristic
137C2×D4—3461no—
138C2×D4—3461no—
139C2×D4—6922no—
140C22×C4—1731no—
141C22×C4—1731no—
142C22×C4—1731no—
143C22×C4—1731no—
144C22×C4—1731no—
145C22×C4—1731no—
146C22×C4—1731no—
147C22×C4—6922no—
148C22×C4C2×#1062192411yescharacteristic
149C22×C4C2×#1062192411yescharacteristic
150M16—6922no—
151C2×C8—3462no—
152C2×C8—3462no—
153C2×C8—6922no—
154C4⋊C4—3461no—
155C4⋊C4—3461no—
156C4⋊C4—3461no—
157C4⋊C4—3461no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—3461no—
161C22⋊C4—3461no—
162C22⋊C4—3461no—
163C22⋊C4—3461no—
164C23—21no—
165C23—22no—
166C23—41no—
167C23—41no—
168C23—3462no—
169C23—3462no—
170C23—1731no—
171C23—1731no—
172C23—6921no—
173C23C22×#1062192411yescharacteristic
174D4—6922no—
175D4—6922no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—3462no—
180C2×C4—3462no—
181C2×C4—3461no—
182C2×C4—3461no—
183C2×C4—1731no—
184C2×C4—1731no—
185C2×C4—6921no—
186C2×C4—6921no—
187C2×C4—6921no—
188C2×C4—6921no—
189C2×C4—6921no—
190C2×C4C22⋊#1062192411yescharacteristic
191C2×C4C22⋊#1062192411yescharacteristic
192C8—6922no—
193C8—6922no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—82no—
200C22—3461no—
201C22—3461no—
202C22—3461no—
203C22—3461no—
204C22—3461no—
205C22—1732no—
206C22—6924no—
207C22C2×C22⋊#1062192411yescharacteristic
208C22#1914976612yes—
209C4—21no—
210C4—21no—
211C4—41no—
212C4—3461no—
213C4—3461no—
214C4—6921no—
215C4—6921no—
216C2—22no—
217C2—81no—
218C2—3462no—
219C2—6921no—
220C2#2601246511yescharacteristic
221C2D173⋊#9911yescharacteristic
222C2#2601247711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C22⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×C22⋊D173C4
cosocleC22×Dic173C23
hypercenterC2×C22⋊C4#10621924
hypocenterC173#1468
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #26012462C2semidirect
2. D173⋊(C22⋊C8)C2semidirect
3. #26012336C2semidirect
4. #26012334C2semidirect
111. C173#1468semidirect

Maximal subgroups

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

subgroupindex
#260124622
D173⋊(C22⋊C8)2
#260123362
#260123342
C2×#191496612
#1468173