Finite Group Databasebeta

← #36352118

Subgroups — #36352118

Order 22144.

284 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 63 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#26012004C211yesmaximal, characteristic
2C22⋊C8×D173C211yesmaximal, characteristic, complemented
3#26011353C211yesmaximal, characteristic, complemented
4#26011228C211yesmaximal, characteristic, complemented
5#26011226C211yesmaximal, characteristic, complemented
6#26011224C211yesmaximal, characteristic, complemented
7#328×C173C211yesmaximal, characteristic, complemented
8C2×#14583827C411yescharacteristic, complemented
9C22×C4×D173C2211yescharacteristic
10C2×C8×D173—21no—
11#19149664—21no—
12#19149655C411yescharacteristic, complemented
13#19149654C411yescharacteristic, complemented
14#19149653C2211yescharacteristic
15#19149649—21no—
16#19149641—21no—
17#19149639—21no—
18#19149637C411yescharacteristic, complemented
19C2×#14583803—21no—
20C22⋊#13063436C2211yescharacteristic
21#19149591—21no—
22#19149586—21no—
23#19149586C2211yescharacteristic
24C22⋊Q8×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28#14583827—42no—
29C2×D692—21no—
30C2×D692—21no—
31C2×D692C2×C411yescharacteristic
32C2×C22⋊D173—21no—
33C22×Dic173—21no—
34C22×Dic173D411yescharacteristic
35C2×C4×D173—21no—
36C2×C4×D173—42no—
37C2×C4×D173C2×C411yescharacteristic
38C2×C4×D173C2×C411yescharacteristic
39C2×C4×D173D411yescharacteristic
40C2×C4×D173D411yescharacteristic
41C2×#13063436—21no—
42C8×D173—42no—
43C4⋊Dic173C2×C411yescharacteristic
44#14583805—21no—
45#14583803—21no—
46#14583803—21no—
47#14583803—21no—
48#14583803—21no—
49#14583803—42no—
50C4×Dic173—21no—
51C2×Q8×C173C2×C411yescharacteristic
52C22×C4×C173—21no—
53C22×C4×C173C2311yescharacteristic
54C2×C8×C173—21no—
55C4⋊C4×C173—21no—
56C4⋊C4×C173C2×C411yescharacteristic
57C22⋊C4×C173—21no—
58C22×D173—21no—
59C22×D173—21no—
60C22×D173—22no—
61C22×D173—21no—
62C22×D173—41no—
63C22×D173C22⋊C411yescharacteristic
64C22×D173C22⋊C411yescharacteristic
65D692—41no—
66D692—42no—
67D692—42no—
68C22⋊D173—41no—
69C22⋊D173—41no—
70C2×Dic173—21no—
71C2×Dic173—41no—
72C2×Dic173—41no—
73C2×Dic173—41no—
74C2×Dic173—41no—
75C2×Dic173C22⋊C411yescharacteristic
76C2×Dic173C22⋊C411yescharacteristic
77C4×D173—22no—
78C4×D173—22no—
79C4×D173—41no—
80C4×D173—41no—
81C4×D173—42no—
82#13063436—41no—
83C23×C173C2×D411yescharacteristic
84Q8×C173—41no—
85C2×C4×C173—21no—
86C2×C4×C173—21no—
87C2×C4×C173—41no—
88C2×C4×C173—41no—
89C2×C4×C173—41no—
90C2×C4×C173—41no—
91C2×C4×C173C22×C411yescharacteristic
92C2×C4×C173C2×D411yescharacteristic
93C8×C173—41no—
94C22×C173—21no—
95C22×C173—21no—
96C22×C173C2×C22⋊C411yescharacteristic
97C22×C173#13511yescharacteristic
98C22×C173#13511yescharacteristic
99C2×D173—21no—
100C2×D173—21no—
101C2×D173—21no—
102C2×D173—21no—
103C2×D173—21no—
104C2×D173—42no—
105C2×D173—42no—
106C2×D173—41no—
107C2×D173—41no—
108C2×D173—41no—
109C2×D173#9912yes—
110C4×C173—21no—
111C4×C173—21no—
112C4×C173—41no—
113C4×C173—41no—
114C4×C173—81no—
115Dic173—21no—
116Dic173—21no—
117Dic173—41no—
118Dic173—81no—
119C2×C173—21no—
120C2×C173—21no—
121C2×C173C2×#9911yescharacteristic
122C2×C173#41611yescharacteristic
123C2×C173#42111yescharacteristic
124D173—22no—
125D173—41no—
126D173—81no—
127C173#151611yescharacteristic, complemented, Sylow
128#1516—1731nomaximal, Sylow
129#525—1731no—
130C2×C22⋊C8—1731no—
131#381—1731no—
132#328—1731no—
133#328D17311yescharacteristic, complemented
134#327—1731no—
135#327—1731no—
136C2×#41—1731no—
137C23×C4—1731no—
138C22×C8—3461no—
139#123—3461no—
140C22⋊Q8—1731no—
141C22⋊Q8C2×D17311yescharacteristic
142#121—1731no—
143#121—1731no—
144#120—3461no—
145C4×D4—3461no—
146C4×D4—3461no—
147#117—1731no—
148C2×C22⋊C4—3461no—
149C22⋊C8—3461no—
150C22⋊C8—1731no—
151C22⋊C8C2×D17311yescharacteristic
152#95—3461no—
153#95—1731no—
154#95C2×D17311yescharacteristic
155C24—1731no—
156#41—6922no—
157C2×Q8C4×D17311yescharacteristic
158C2×D4—3461no—
159C2×D4—3461no—
160C2×D4—3461no—
161C2×D4—1731no—
162C22×C4—21no—
163C22×C4—3461no—
164C22×C4—3461no—
165C22×C4—1731no—
166C22×C4—1731no—
167C22×C4—1731no—
168C22×C4—1731no—
169C22×C4—1731no—
170C22×C4—6922no—
171C22×C4C22×D17311yescharacteristic
172C2×C8—21no—
173C2×C8—3461no—
174C2×C8—6922no—
175C4⋊C4—21no—
176C4⋊C4—3461no—
177C4⋊C4—1731no—
178C4⋊C4C4×D17311yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—3461no—
181C22⋊C4—3461no—
182C22⋊C4—3461no—
183C22⋊C4—3461no—
184C22⋊C4—6922no—
185C42—3461no—
186C23—3462no—
187C23—3461no—
188C23—3461no—
189C23—3461no—
190C23—1731no—
191C23—1731no—
192C23—6921no—
193C23D4×D17311yescharacteristic
194Q8—41no—
195D4—6921no—
196D4—6922no—
197D4—6922no—
198D4—6921no—
199D4—6921no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—3462no—
207C2×C4—3462no—
208C2×C4—3461no—
209C2×C4—1731no—
210C2×C4—1731no—
211C2×C4—6921no—
212C2×C4—6921no—
213C2×C4—6922no—
214C2×C4—6921no—
215C2×C4—6921no—
216C2×C4—6921no—
217C2×C4—6921no—
218C2×C4C2×C4×D17311yescharacteristic
219C2×C4D4×D17311yescharacteristic
220C8—41no—
221C8—6921no—
222C22—21no—
223C22—21no—
224C22—3461no—
225C22—3461no—
226C22—3461no—
227C22—3461no—
228C22—3461no—
229C22—1732no—
230C22—6922no—
231C22—6922no—
232C22—6921no—
233C22—6921no—
234C22—6921no—
235C22C22⋊C4×D17311yescharacteristic
236C22#1914970011yescharacteristic
237C22#1914970211yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—81no—
243C4—3461no—
244C4—3461no—
245C4—13841no—
246C4—6921no—
247C2—21no—
248C2—21no—
249C2—3462no—
250C2—13841no—
251C2—6921no—
252C2#99×D17311yescharacteristic
253C2#2601148611yescharacteristic
254C2#2601150611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012004C2
cosocleC22×C4×C173C23
hypercenter#328D173
hypocenterC173#1516
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D173C2semidirect
3. #26011353C2semidirect
4. #26011228C2semidirect
5. #26011226C2semidirect
6. #26011224C2semidirect
7. #328×C173C2semidirect
8. C2×#14583827C4semidirect
12. #19149655C4semidirect
13. #19149654C4semidirect
18. #19149637C4semidirect
127. C173#1516semidirect
133. #328D173semidirect

Maximal subgroups

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

subgroupindex
#260120042
C22⋊C8×D1732
#260113532
#260112282
#260112262
#260112242
#328×C1732
#1516173