Finite Group Databasebeta

← #36369545

Subgroups — #36369545

Order 22144.

376 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 72 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×D173⋊M16C211yesmaximal, characteristic, complemented
2C2×#19149744C212yesmaximal, complemented
3C4×C22⋊#10621924C211yesmaximal, characteristic, complemented
4#26012361C212yesmaximal, complemented
5#26012004C211yesmaximal, characteristic
6C2×#14583836C2212yes—
7D173⋊M16—24no—
8C2×#14583835—21no—
9C2×D173⋊C8—21no—
10C2×C22⋊#10621924—21no—
11C2×C4×#10621924C2212yes—
12#19149744—24no—
13(C2×#10621924)⋊C4—21no—
14C2×#14583827C2211yescharacteristic
15C22×C4×D173C2211yescharacteristic
16#19149664—21no—
17#19149655C411yescharacteristic, complemented
18#19149654C411yescharacteristic, complemented
19#19149653C411yescharacteristic, complemented
20#19149649—21no—
21#19149641—21no—
22#19149639—21no—
23#19149637C2211yescharacteristic
24C22⋊Q8×C173C411yescharacteristic, complemented
25C22×#10621924—22no—
26#14583836—22no—
27#14583836D414yescomplemented
28#14583835—24no—
29D173⋊C8—24no—
30C2×#13063443—22no—
31C22⋊#10621924—24no—
32C4×#10621924—22no—
33C4×#10621924—22no—
34C23×D173D411yescharacteristic
35#14583827—22no—
36#14583827—22no—
37C2×D692—21no—
38C2×D692—21no—
39C2×D692C2×C411yescharacteristic
40C2×C22⋊D173—21no—
41C22×Dic173D411yescharacteristic
42C2×C4×D173—22no—
43C2×C4×D173—22no—
44C2×C4×D173—21no—
45C2×C4×D173C2×C411yescharacteristic
46C2×C4×D173D411yescharacteristic
47C2×C4×D173D411yescharacteristic
48C2×C4×D173C2311yescharacteristic
49C4⋊Dic173C2×C411yescharacteristic
50#14583805—21no—
51#14583803—21no—
52#14583803—21no—
53#14583803—21no—
54#14583803—21no—
55C4×Dic173—21no—
56C2×Q8×C173C2×C411yescharacteristic
57C22×C4×C173C2×C411yescharacteristic
58C4⋊C4×C173—21no—
59C4⋊C4×C173C2×C411yescharacteristic
60C22⋊C4×C173—21no—
61C2×#10621924—24no—
62C2×#10621924—42no—
63C2×#10621924—42no—
64#13063443—24no—
65C22×D173—22no—
66C22×D173—22no—
67C22×D173—21no—
68C22×D173—41no—
69C22×D173C22⋊C411yescharacteristic
70C22×D173C2×D411yescharacteristic
71D692—22no—
72D692—42no—
73D692—42no—
74C22⋊D173—42no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173C22⋊C411yescharacteristic
79C2×Dic173C2×D411yescharacteristic
80C4×D173—22no—
81C4×D173—22no—
82C4×D173—21no—
83C4×D173—21no—
84C4×D173—41no—
85C4×D173—41no—
86C4×D173C2×D412yes—
87C4×D173#4112yes—
88C23×C173C22⋊C411yescharacteristic
89Q8×C173—22no—
90C2×C4×C173—22no—
91C2×C4×C173—21no—
92C2×C4×C173—21no—
93C2×C4×C173C22⋊C411yescharacteristic
94C2×C4×C173C22×C411yescharacteristic
95C22×C173—21no—
96C22×C173—21no—
97C22×C173—21no—
98C22×C173C2×C22⋊C411yescharacteristic
99C2×D173—21no—
100C2×D173—21no—
101C2×D173—21no—
102C2×D173—21no—
103C2×D173—44no—
104C2×D173—41no—
105C2×D173—41no—
106C2×D173—41no—
107C2×D173#10412yescomplemented
108C2×D173#12112yes—
109#10621924—44no—
110C4×C173—21no—
111C4×C173—41no—
112C4×C173—41no—
113C4×C173C4×D412yes—
114Dic173—21no—
115Dic173—41no—
116Dic173#12011yescharacteristic
117Dic173#12311yescharacteristic
118C2×C173—41no—
119C2×C173#38611yescharacteristic
120C2×C173C2×#10411yescharacteristic
121C2×C173#42111yescharacteristic
122D173—22no—
123D173—22no—
124D173—81no—
125C173#187111yescharacteristic, complemented, Sylow
126#1871—1731nomaximal, Sylow
127C22×M16—1731no—
128#525—1731no—
129C2×#104—1732no—
130C4×C22⋊C4—1731no—
131#339—1732no—
132C2×#41—1731no—
133C23×C4—1731no—
134C2×M16—3461no—
135C2×M16—3464no—
136C2×M16—1732no—
137C22×C8—3461no—
138#123—3461no—
139C22⋊Q8—1731no—
140C22⋊Q8#1062192411yescharacteristic, complemented
141#121—1731no—
142#121—1731no—
143#120—3461no—
144C4×D4—3461no—
145C4×D4—3461no—
146#117—1731no—
147C2×C22⋊C4—3461no—
148C2×C42—1732no—
149#104—3464no—
150#95—3461no—
151C24—1731no—
152#41—3462no—
153#41—3462no—
154C2×Q8C2×#1062192411yescharacteristic
155C2×D4—3461no—
156C2×D4—3461no—
157C2×D4—3461no—
158C2×D4—1731no—
159C22×C4—3462no—
160C22×C4—3462no—
161C22×C4—3461no—
162C22×C4—3462no—
163C22×C4—1731no—
164C22×C4—1731no—
165C22×C4—1731no—
166C22×C4—1731no—
167C22×C4—1731no—
168C22×C4C2×#1062192411yescharacteristic
169M16—3464no—
170M16—3462no—
171M16—1734no—
172C2×C8—3464no—
173C2×C8—3462no—
174C4⋊C4—21no—
175C4⋊C4—3461no—
176C4⋊C4—1731no—
177C4⋊C4C2×#1062192411yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—3461no—
180C22⋊C4—3461no—
181C22⋊C4—3461no—
182C22⋊C4—3464no—
183C22⋊C4—3461no—
184C42—3461no—
185C42—3462no—
186C42—3462no—
187C23—3462no—
188C23—3462no—
189C23—3461no—
190C23—1731no—
191C23—1731no—
192C23—6921no—
193C23C22⋊#1062192411yescharacteristic
194Q8—22no—
195D4—3462no—
196D4—6922no—
197D4—6922no—
198D4—6922no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—3462no—
203C2×C4—3461no—
204C2×C4—3461no—
205C2×C4—3461no—
206C2×C4—3461no—
207C2×C4—3462no—
208C2×C4—3464no—
209C2×C4—3461no—
210C2×C4—1731no—
211C2×C4—1731no—
212C2×C4—1732no—
213C2×C4—1732no—
214C2×C4—6921no—
215C2×C4—6921no—
216C2×C4—6922no—
217C2×C4—6922no—
218C2×C4C22⋊#1062192411yescharacteristic
219C2×C4C22×#1062192411yescharacteristic
220C8—3464no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—3461no—
225C22—3461no—
226C22—3461no—
227C22—3461no—
228C22—1732no—
229C22—1732no—
230C22—6924no—
231C22—6921no—
232C22—6921no—
233C22—6921no—
234C22C2×C22⋊#1062192411yescharacteristic
235C4—21no—
236C4—41no—
237C4—41no—
238C4—3461no—
239C4—1731no—
240C4—1731no—
241C4—6924no—
242C4—6921no—
243C4D4×#1062192412yes—
244C2—41no—
245C2—3462no—
246C2—3462no—
247C2—13841no—
248C2#2601243911yescharacteristic
249C2D173⋊#10411yescharacteristic
250C2#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#1871
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×D173⋊M16C2semidirect
2. C2×#19149744C2semidirect
3. C4×C22⋊#10621924C2semidirect
4. #26012361C2semidirect
17. #19149655C4semidirect
18. #19149654C4semidirect
19. #19149653C4semidirect
24. C22⋊Q8×C173C4semidirect
27. #14583836D4semidirect
107. C2×D173#104semidirect
125. C173#1871semidirect
140. C22⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
C2×D173⋊M162
C2×#191497442
C4×C22⋊#106219242
#260123612
#260120042
#1871173