Finite Group Databasebeta

← #36368941

Subgroups — #36368941

Order 22144.

304 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C2×#19149736C211yesmaximal, characteristic, complemented
2C4×C22⋊#10621924C211yesmaximal, characteristic, complemented
3#26012345C212yesmaximal, complemented
4#26012343C212yesmaximal, complemented
5#26012004C211yesmaximal, characteristic
6C22×#13063443—21no—
7C2×C22⋊#10621924—21no—
8C2×C4×#10621924—21no—
9#19149736—21no—
10#19149736C2212yes—
11(C2×#10621924)⋊C4C2212yes—
12C2×#14583827C411yescharacteristic, complemented
13C22×C4×D173C2211yescharacteristic
14#19149664—21no—
15#19149655C2211yescharacteristic
16#19149654C2211yescharacteristic
17#19149653C411yescharacteristic, complemented
18#19149649—21no—
19#19149641—21no—
20#19149639—21no—
21#19149637C411yescharacteristic, complemented
22C22⋊Q8×C173C411yescharacteristic, complemented
23C22×#10621924—22no—
24C2×#13063443—22no—
25C2×#13063443—42no—
26C22⋊#10621924—24no—
27C4×#10621924—42no—
28C23×D173D411yescharacteristic
29#14583827—42no—
30C2×D692—21no—
31C2×D692—21no—
32C2×D692C2×C411yescharacteristic
33C2×C22⋊D173—21no—
34C22×Dic173D411yescharacteristic
35C2×C4×D173—21no—
36C2×C4×D173—42no—
37C2×C4×D173C2×C411yescharacteristic
38C2×C4×D173D411yescharacteristic
39C2×C4×D173D411yescharacteristic
40C2×C4×D173C2311yescharacteristic
41C4⋊Dic173C2×C411yescharacteristic
42#14583805—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—21no—
47C4×Dic173—21no—
48C2×Q8×C173C2×C411yescharacteristic
49C22×C4×C173C2×C411yescharacteristic
50C4⋊C4×C173—21no—
51C4⋊C4×C173C2×C411yescharacteristic
52C22⋊C4×C173—21no—
53C2×#10621924—24no—
54C2×#10621924—42no—
55C2×#10621924—42no—
56#13063443—42no—
57C22×D173—22no—
58C22×D173—22no—
59C22×D173—21no—
60C22×D173—41no—
61C22×D173C22⋊C411yescharacteristic
62C22×D173C2×D411yescharacteristic
63D692—41no—
64D692—42no—
65D692—42no—
66C22⋊D173—42no—
67C2×Dic173—22no—
68C2×Dic173—21no—
69C2×Dic173C22⋊C411yescharacteristic
70C2×Dic173C2×D411yescharacteristic
71C4×D173—22no—
72C4×D173—22no—
73C4×D173—41no—
74C4×D173—41no—
75C4×D173—42no—
76C23×C173C22⋊C411yescharacteristic
77Q8×C173—41no—
78C2×C4×C173—21no—
79C2×C4×C173—21no—
80C2×C4×C173—41no—
81C2×C4×C173C22⋊C411yescharacteristic
82C2×C4×C173C22×C411yescharacteristic
83C22×C173—21no—
84C22×C173—21no—
85C22×C173#10412yes—
86C22×C173C2×C22⋊C411yescharacteristic
87C2×D173—21no—
88C2×D173—21no—
89C2×D173—21no—
90C2×D173—21no—
91C2×D173—21no—
92C2×D173—44no—
93C2×D173—41no—
94C2×D173—41no—
95C2×D173—41no—
96C2×D173#13512yes—
97#10621924—44no—
98C4×C173—21no—
99C4×C173—21no—
100C4×C173—41no—
101C4×C173—41no—
102Dic173—21no—
103Dic173—41no—
104Dic173#9912yes—
105C2×C173—22no—
106C2×C173C2×#9911yescharacteristic
107C2×C173#41611yescharacteristic
108C2×C173C2×#10411yescharacteristic
109D173—22no—
110D173—41no—
111D173—81no—
112C173#150111yescharacteristic, complemented, Sylow
113#1501—1731nomaximal, Sylow
114#525—1731no—
115C2×C22⋊C8—1731no—
116C4×C22⋊C4—1731no—
117#328—1732no—
118#327—1732no—
119C2×#41—1731no—
120C23×C4—1731no—
121C22×C8—3461no—
122#123—3461no—
123C22⋊Q8—1731no—
124C22⋊Q8#1062192411yescharacteristic, complemented
125#121—1731no—
126#121—1731no—
127#120—3461no—
128C4×D4—3461no—
129C4×D4—3461no—
130#117—1731no—
131C2×C22⋊C4—3461no—
132C2×C42—3461no—
133C22⋊C8—3461no—
134C22⋊C8—1732no—
135#95—1732no—
136C24—1731no—
137#41—6922no—
138C2×Q8C2×#1062192411yescharacteristic
139C2×D4—3461no—
140C2×D4—3461no—
141C2×D4—3461no—
142C2×D4—1731no—
143C22×C4—3461no—
144C22×C4—3462no—
145C22×C4—1731no—
146C22×C4—1731no—
147C22×C4—1731no—
148C22×C4—1731no—
149C22×C4—1731no—
150C22×C4—6922no—
151C22×C4C2×#1062192411yescharacteristic
152C2×C8—3462no—
153C2×C8—6922no—
154C4⋊C4—21no—
155C4⋊C4—3461no—
156C4⋊C4—1731no—
157C4⋊C4C2×#1062192411yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—3461no—
160C22⋊C4—3461no—
161C22⋊C4—3461no—
162C22⋊C4—3464no—
163C22⋊C4—3461no—
164C42—3461no—
165C42—6922no—
166C23—3462no—
167C23—3462no—
168C23—3461no—
169C23—1731no—
170C23—1731no—
171C23—6921no—
172C23C22⋊#1062192411yescharacteristic
173Q8—41no—
174D4—6921no—
175D4—6922no—
176D4—6922no—
177D4—6922no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—3462no—
182C2×C4—3462no—
183C2×C4—3462no—
184C2×C4—3464no—
185C2×C4—3461no—
186C2×C4—1731no—
187C2×C4—1731no—
188C2×C4—6921no—
189C2×C4—6921no—
190C2×C4—6922no—
191C2×C4—6922no—
192C2×C4—6922no—
193C2×C4C22⋊#1062192411yescharacteristic
194C2×C4C22×#1062192411yescharacteristic
195C8—6922no—
196C22—21no—
197C22—21no—
198C22—3461no—
199C22—3461no—
200C22—3461no—
201C22—3461no—
202C22—3461no—
203C22—1732no—
204C22—6924no—
205C22—6921no—
206C22—6921no—
207C22—6921no—
208C22#1914974412yes—
209C22C2×C22⋊#1062192411yescharacteristic
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—3461no—
215C4—1732no—
216C4—6924no—
217C4—6921no—
218C2—22no—
219C2—3462no—
220C2—13841no—
221C2—6921no—
222C2D173⋊#9911yescharacteristic
223C2#2601248111yescharacteristic
224C2C2×#1914974411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22⋊Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012004C2
cosocleC2×C4×D173C23
hypercenterC22⋊Q8#10621924
hypocenterC173#1501
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. C2×#19149736C2semidirect
2. C4×C22⋊#10621924C2semidirect
3. #26012345C2semidirect
4. #26012343C2semidirect
12. C2×#14583827C4semidirect
17. #19149653C4semidirect
21. #19149637C4semidirect
22. C22⋊Q8×C173C4semidirect
112. C173#1501semidirect
124. C22⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
C2×#191497362
C4×C22⋊#106219242
#260123452
#260123432
#260120042
#1501173