Finite Group Databasebeta

← #36368990

Subgroups — #36368990

Order 22144.

284 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 58 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
1D173⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
2#26012434C211yesmaximal, characteristic, complemented
3#26012345C212yesmaximal, complemented
4#26012343C212yesmaximal, complemented
5#26012004C211yesmaximal, characteristic
6C2×D173⋊C8—21no—
7C2×C22⋊#10621924—21no—
8#19149736C2212yes—
9C22⋊#13063443—21no—
10(C2×#10621924)⋊C4—21no—
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—
24D173⋊C8—42no—
25C2×#13063443—22no—
26C22⋊#10621924—42no—
27C23×D173D411yescharacteristic
28#14583827—42no—
29C2×D692—21no—
30C2×D692—21no—
31C2×D692C2×C411yescharacteristic
32C2×C22⋊D173—21no—
33C22×Dic173D411yescharacteristic
34C2×C4×D173—21no—
35C2×C4×D173—42no—
36C2×C4×D173C2×C411yescharacteristic
37C2×C4×D173D411yescharacteristic
38C2×C4×D173D411yescharacteristic
39C2×C4×D173C2311yescharacteristic
40C4⋊Dic173C2×C411yescharacteristic
41#14583805—21no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46C4×Dic173—21no—
47C2×Q8×C173C2×C411yescharacteristic
48C22×C4×C173C2×C411yescharacteristic
49C4⋊C4×C173—21no—
50C4⋊C4×C173C2×C411yescharacteristic
51C22⋊C4×C173—21no—
52C2×#10621924—42no—
53C2×#10621924—42no—
54C2×#10621924—42no—
55#13063443—42no—
56C22×D173—22no—
57C22×D173—22no—
58C22×D173—21no—
59C22×D173—41no—
60C22×D173C22⋊C411yescharacteristic
61C22×D173C2×D411yescharacteristic
62D692—41no—
63D692—42no—
64D692—42no—
65C22⋊D173—42no—
66C2×Dic173—21no—
67C2×Dic173—41no—
68C2×Dic173C22⋊C411yescharacteristic
69C2×Dic173C2×D411yescharacteristic
70C4×D173—22no—
71C4×D173—22no—
72C4×D173—41no—
73C4×D173—41no—
74C4×D173—42no—
75C23×C173C22⋊C411yescharacteristic
76Q8×C173—41no—
77C2×C4×C173—21no—
78C2×C4×C173—21no—
79C2×C4×C173—41no—
80C2×C4×C173C22⋊C411yescharacteristic
81C2×C4×C173C22×C411yescharacteristic
82C22×C173—21no—
83C22×C173—21no—
84C22×C173—21no—
85C22×C173C2×C22⋊C411yescharacteristic
86C2×D173—21no—
87C2×D173—21no—
88C2×D173—21no—
89C2×D173—21no—
90C2×D173—44no—
91C2×D173—41no—
92C2×D173—41no—
93C2×D173—41no—
94C2×D173#9912yes—
95C2×D173#13512yes—
96#10621924—82no—
97C4×C173—21no—
98C4×C173—21no—
99C4×C173—41no—
100C4×C173—41no—
101Dic173—21no—
102Dic173—21no—
103Dic173—41no—
104C2×C173—41no—
105C2×C173C2×#9911yescharacteristic
106C2×C173#41611yescharacteristic
107C2×C173#42111yescharacteristic
108D173—22no—
109D173—22no—
110D173—81no—
111C173#151611yescharacteristic, complemented, Sylow
112#1516—1731nomaximal, Sylow
113#525—1731no—
114C2×C22⋊C8—1731no—
115#381—1731no—
116#328—1732no—
117#327—1732no—
118C2×#41—1731no—
119C23×C4—1731no—
120C22×C8—3461no—
121#123—3461no—
122C22⋊Q8—1731no—
123C22⋊Q8#1062192411yescharacteristic, complemented
124#121—1731no—
125#121—1731no—
126#120—3461no—
127C4×D4—3461no—
128C4×D4—3461no—
129#117—1731no—
130C2×C22⋊C4—3461no—
131C22⋊C8—3461no—
132C22⋊C8—1732no—
133#95—3461no—
134#95—1732no—
135C24—1731no—
136#41—6922no—
137C2×Q8C2×#1062192411yescharacteristic
138C2×D4—3461no—
139C2×D4—3461no—
140C2×D4—3461no—
141C2×D4—1731no—
142C22×C4—3461no—
143C22×C4—3462no—
144C22×C4—1731no—
145C22×C4—1731no—
146C22×C4—1731no—
147C22×C4—1731no—
148C22×C4—1731no—
149C22×C4—6922no—
150C22×C4C2×#1062192411yescharacteristic
151C2×C8—3462no—
152C2×C8—6922no—
153C4⋊C4—21no—
154C4⋊C4—3461no—
155C4⋊C4—1731no—
156C4⋊C4C2×#1062192411yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—3461no—
159C22⋊C4—3461no—
160C22⋊C4—3461no—
161C22⋊C4—3461no—
162C22⋊C4—6922no—
163C42—3461no—
164C23—3462no—
165C23—3462no—
166C23—3461no—
167C23—1731no—
168C23—1731no—
169C23—6921no—
170C23C22⋊#1062192411yescharacteristic
171Q8—41no—
172D4—6921no—
173D4—6922no—
174D4—6922no—
175D4—6922no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—3462no—
180C2×C4—3462no—
181C2×C4—3461no—
182C2×C4—1731no—
183C2×C4—1731no—
184C2×C4—6921no—
185C2×C4—6921no—
186C2×C4—6922no—
187C2×C4—6921no—
188C2×C4—6922no—
189C2×C4—6922no—
190C2×C4—6922no—
191C2×C4C22⋊#1062192411yescharacteristic
192C2×C4C22×#1062192411yescharacteristic
193C8—6922no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—3461no—
198C22—3461no—
199C22—3461no—
200C22—3461no—
201C22—1732no—
202C22—1732no—
203C22—6924no—
204C22—6921no—
205C22—6921no—
206C22—6921no—
207C22C2×C22⋊#1062192411yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—3461no—
213C4—3461no—
214C4—13842no—
215C4—6921no—
216C2—41no—
217C2—3462no—
218C2—3462no—
219C2—13841no—
220C2D173⋊#9911yescharacteristic
221C2#2601248111yescharacteristic
222C2#2601249011yescharacteristic
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#1516
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. D173⋊(C22⋊C8)C2semidirect
2. #26012434C2semidirect
3. #26012345C2semidirect
4. #26012343C2semidirect
12. C2×#14583827C4semidirect
17. #19149653C4semidirect
21. #19149637C4semidirect
22. C22⋊Q8×C173C4semidirect
111. C173#1516semidirect
123. C22⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊(C22⋊C8)2
#260124342
#260123452
#260123432
#260120042
#1516173