Finite Group Databasebeta

← #36352129

Subgroups — #36352129

Order 22144.

340 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 71 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#121×D173C211yesmaximal, characteristic
2C22⋊C8×D173C211yesmaximal, characteristic, complemented
3#26011350C211yesmaximal, characteristic, complemented
4#26011226C211yesmaximal, characteristic, complemented
5#26011225C211yesmaximal, characteristic, complemented
6#26011224C211yesmaximal, characteristic, complemented
7#327×C173C211yesmaximal, characteristic, complemented
8C2×D4×D173—21no—
9C2×D4×D173C411yescharacteristic, complemented
10C22×C4×D173C2211yescharacteristic
11C2×C8×D173—21no—
12#19149654C411yescharacteristic, complemented
13#19149653C2211yescharacteristic
14#19149652—21no—
15#19149651—21no—
16#19149650C411yescharacteristic, complemented
17C2×#14583805—21no—
18C4⋊C4×D173C411yescharacteristic, complemented
19C2×#14583803—21no—
20C22⋊C4×D173—21no—
21C22⋊#13063436C2211yescharacteristic, complemented
22#19149591—21no—
23#19149586C2211yescharacteristic, complemented
24#121×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic, complemented
26#95×C173C2211yescharacteristic, complemented
27C23×D173—21no—
28C23×D173D411yescharacteristic
29C2×D692—21no—
30C2×D692C2×C411yescharacteristic
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34D4×D173—42no—
35D4×D173—42no—
36D4×D173—42no—
37C22×Dic173—21no—
38C22×Dic173D411yescharacteristic
39C2×C4×D173—21no—
40C2×C4×D173—42no—
41C2×C4×D173C2×C411yescharacteristic
42C2×C4×D173C2×C411yescharacteristic
43C2×C4×D173D411yescharacteristic
44C2×C4×D173D411yescharacteristic
45C2×#13063436—21no—
46C8×D173—42no—
47C4⋊Dic173C2×C411yescharacteristic
48#14583805—21no—
49#14583805—42no—
50C22⋊Dic173—21no—
51#14583803—21no—
52#14583803—21no—
53#14583803—42no—
54C2×D4×C173—21no—
55C2×D4×C173C2×C411yescharacteristic
56C22×C4×C173—21no—
57C22×C4×C173C2311yescharacteristic
58C2×C8×C173—21no—
59C4⋊C4×C173C2×C411yescharacteristic
60C22⋊C4×C173—21no—
61C22×D173—21no—
62C22×D173—41no—
63C22×D173—41no—
64C22×D173—42no—
65C22×D173—42no—
66C22×D173—42no—
67C22×D173C22⋊C411yescharacteristic
68C22×D173C22⋊C411yescharacteristic
69C22×D173D812yes—
70C22×D173SD1612yes—
71D692—41no—
72D692—42no—
73C22⋊D173—41no—
74C22⋊D173—41no—
75C22⋊D173—42no—
76C22⋊D173—42no—
77C2×Dic173—22no—
78C2×Dic173—21no—
79C2×Dic173—41no—
80C2×Dic173—41no—
81C2×Dic173—41no—
82C2×Dic173C22⋊C411yescharacteristic
83C2×Dic173C22⋊C411yescharacteristic
84C4×D173—22no—
85C4×D173—22no—
86C4×D173—41no—
87C4×D173—41no—
88C4×D173—42no—
89#13063436—41no—
90C23×C173—21no—
91C23×C173C2×D411yescharacteristic
92D4×C173—41no—
93D4×C173—41no—
94D4×C173—41no—
95C2×C4×C173—21no—
96C2×C4×C173—41no—
97C2×C4×C173—41no—
98C2×C4×C173—41no—
99C2×C4×C173—41no—
100C2×C4×C173C22×C411yescharacteristic
101C2×C4×C173C2×D411yescharacteristic
102C8×C173—41no—
103C22×C173—21no—
104C22×C173—21no—
105C22×C173—41no—
106C22×C173—41no—
107C22×C173—41no—
108C22×C173C2×C22⋊C411yescharacteristic
109C22×C173C2×D811yescharacteristic
110C22×C173C2×SD1611yescharacteristic
111C2×D173—21no—
112C2×D173—21no—
113C2×D173—21no—
114C2×D173—21no—
115C2×D173—42no—
116C2×D173—42no—
117C2×D173—41no—
118C2×D173—41no—
119C2×D173—41no—
120C2×D173—84no—
121C2×D173#10212yes—
122C2×D173#10212yes—
123C4×C173—21no—
124C4×C173—21no—
125C4×C173—41no—
126C4×C173—81no—
127Dic173—21no—
128Dic173—41no—
129Dic173—81no—
130Dic173#9912yes—
131C2×C173—21no—
132C2×C173—21no—
133C2×C173—81no—
134C2×C173C2×#9911yescharacteristic
135C2×C173C2×#10211yescharacteristic
136C2×C173#42111yescharacteristic
137D173—22no—
138D173—22no—
139D173—81no—
140C173#151911yescharacteristic, complemented, Sylow
141#1519—1731nomaximal, Sylow
142C2×#121—1731no—
143C2×C22⋊C8—1731no—
144#380—1731no—
145#327—1731no—
146#327—1731no—
147#327—1731no—
148#327D17311yescharacteristic, complemented
149C22×D4—3461no—
150C22×D4—1731no—
151C23×C4—1731no—
152C22×C8—3461no—
153#121—3461no—
154#121—3461no—
155#121—1731no—
156#121—1731no—
157#121—1731no—
158#121C2×D17311yescharacteristic
159C2×C4⋊C4—3461no—
160C2×C4⋊C4—1731no—
161C2×C22⋊C4—3461no—
162C2×C22⋊C4—3461no—
163C22⋊C8—3461no—
164C22⋊C8—1731no—
165C22⋊C8C2×D17311yescharacteristic, complemented
166#95—1731no—
167#95C2×D17311yescharacteristic, complemented
168C24—3461no—
169C24—1731no—
170C2×D4—21no—
171C2×D4—3461no—
172C2×D4—3461no—
173C2×D4—3461no—
174C2×D4—3461no—
175C2×D4—1731no—
176C2×D4—6922no—
177C2×D4—6922no—
178C2×D4—6922no—
179C2×D4C4×D17311yescharacteristic
180C22×C4—21no—
181C22×C4—3461no—
182C22×C4—3461no—
183C22×C4—1731no—
184C22×C4—1731no—
185C22×C4—1731no—
186C22×C4—1731no—
187C22×C4—1731no—
188C22×C4—6922no—
189C22×C4C22×D17311yescharacteristic
190C2×C8—21no—
191C2×C8—3461no—
192C2×C8—6922no—
193C4⋊C4—3461no—
194C4⋊C4—1731no—
195C4⋊C4—6922no—
196C4⋊C4C4×D17311yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—3461no—
199C22⋊C4—3461no—
200C22⋊C4—3461no—
201C22⋊C4—6922no—
202C23—21no—
203C23—3461no—
204C23—1731no—
205C23—1731no—
206C23—1732no—
207C23—1732no—
208C23—6921no—
209C23—6921no—
210C23—6922no—
211C23—6922no—
212C23—6922no—
213C23D4×D17311yescharacteristic
214D4—41no—
215D4—41no—
216D4—41no—
217D4—6921no—
218D4—6922no—
219D4—6922no—
220D4—6922no—
221D4—6921no—
222D4—6921no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—3462no—
229C2×C4—3462no—
230C2×C4—3462no—
231C2×C4—3461no—
232C2×C4—1731no—
233C2×C4—1731no—
234C2×C4—6921no—
235C2×C4—6921no—
236C2×C4—6922no—
237C2×C4—6921no—
238C2×C4—6921no—
239C2×C4—6921no—
240C2×C4C2×C4×D17311yescharacteristic
241C2×C4D4×D17311yescharacteristic
242C8—41no—
243C8—6921no—
244C22—21no—
245C22—21no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—3461no—
250C22—3461no—
251C22—3461no—
252C22—3461no—
253C22—13844no—
254C22—1732no—
255C22—1732no—
256C22—6922no—
257C22—6922no—
258C22—6921no—
259C22—6921no—
260C22—6921no—
261C22C22⋊C4×D17311yescharacteristic
262C22D8×D17311yescharacteristic
263C22SD16×D17311yescharacteristic
264C4—21no—
265C4—21no—
266C4—41no—
267C4—81no—
268C4—3461no—
269C4—13841no—
270C4—1732no—
271C4—6921no—
272C2—21no—
273C2—21no—
274C2—81no—
275C2—3462no—
276C2—3462no—
277C2—13841no—
278C2#99×D17311yescharacteristic
279C2#102×D17311yescharacteristic
280C2#2601150611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#327×C173C2
socleC22×C173C2×C22⋊C4
prime radical#121×D173C2
cosocleC22×C4×C173C23
hypercenter#327D173
hypocenterC173#1519
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D173C2semidirect
3. #26011350C2semidirect
4. #26011226C2semidirect
5. #26011225C2semidirect
6. #26011224C2semidirect
7. #327×C173C2semidirect
9. C2×D4×D173C4semidirect
12. #19149654C4semidirect
16. #19149650C4semidirect
18. C4⋊C4×D173C4semidirect
21. C22⋊#13063436C22semidirect
23. #19149586C22semidirect
25. C22⋊C8×C173C22semidirect
26. #95×C173C22semidirect
140. C173#1519semidirect
148. #327D173semidirect
165. C22⋊C8C2×D173semidirect
167. #95C2×D173semidirect

Maximal subgroups

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

subgroupindex
#121×D1732
C22⋊C8×D1732
#260113502
#260112262
#260112252
#260112242
#327×C1732
#1519173