Finite Group Databasebeta

← #36352479

Subgroups — #36352479

Order 22144.

368 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 51 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#26012049C211yesmaximal, characteristic, complemented
2#26011625C211yesmaximal, characteristic, complemented
3#26011399C211yesmaximal, characteristic, complemented
4#26011226C211yesmaximal, characteristic, complemented
5#26011225C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#447×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173—21no—
9C2×D4×D173—21no—
10C2×#14583811—21no—
11#19149664—21no—
12#19149661—21no—
13#19149653C2211yescharacteristic
14#19149651—21no—
15#19149649—21no—
16#19149647—21no—
17C2×C22⋊Dic173—21no—
18C2×#14583803C2211yescharacteristic
19#19149600—21no—
20C22⋊#13063436C2211yescharacteristic, complemented
21#19149586C2211yescharacteristic, complemented
22C22×D4×C173C2211yescharacteristic, complemented
23C2×D8×C173—21no—
24#121×C173C2211yescharacteristic
25#102×C173—21no—
26C22⋊C8×C173C2211yescharacteristic, complemented
27C23×D173—21no—
28C23×D173D411yescharacteristic
29C2×D692—21no—
30C2×D692D411yescharacteristic, complemented
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C2×C22⋊D173—42no—
35C2×C22⋊D173—42no—
36C2×C22⋊D173—42no—
37D4×D173—42no—
38D4×D173—42no—
39C22×Dic173—21no—
40C22×Dic173D411yescharacteristic
41C2×C4×D173—21no—
42#14583811—42no—
43C2×#13063436—21no—
44C4⋊Dic173D411yescharacteristic, complemented
45#14583805—21no—
46C22⋊Dic173—21no—
47C22⋊Dic173—42no—
48#14583803—21no—
49#14583803—21no—
50#14583803—21no—
51#14583803—21no—
52C24×C173—21no—
53C2×D4×C173—21no—
54C2×D4×C173—21no—
55C2×D4×C173—21no—
56C2×D4×C173—42no—
57C2×D4×C173D411yescharacteristic, complemented
58C22×C4×C173C2311yescharacteristic
59D8×C173—42no—
60C2×C8×C173—21no—
61C4⋊C4×C173D411yescharacteristic, complemented
62C22⋊C4×C173—21no—
63C22×D173—21no—
64C22×D173—21no—
65C22×D173—21no—
66C22×D173—42no—
67C22×D173—41no—
68C22×D173—41no—
69C22×D173—81no—
70C22×D173—81no—
71C22×D173D812yescomplemented
72D692—41no—
73D692—42no—
74C22⋊D173—42no—
75C22⋊D173—48no—
76C22⋊D173—41no—
77C22⋊D173—41no—
78C22⋊D173—81no—
79C2×Dic173—22no—
80C2×Dic173—21no—
81C2×Dic173—21no—
82C2×Dic173—41no—
83C2×Dic173—41no—
84C2×Dic173—41no—
85C4×D173—42no—
86#13063436—41no—
87C23×C173—22no—
88C23×C173—21no—
89C23×C173—41no—
90C23×C173—41no—
91C23×C173—41no—
92C23×C173—41no—
93C23×C173—41no—
94C23×C173—41no—
95C23×C173C2×D411yescharacteristic
96D4×C173—22no—
97D4×C173—22no—
98D4×C173—41no—
99D4×C173—41no—
100D4×C173—41no—
101D4×C173—41no—
102D4×C173—41no—
103C2×C4×C173—21no—
104C2×C4×C173—41no—
105C2×C4×C173C2×D411yescharacteristic
106C2×C4×C173C2×D411yescharacteristic
107C8×C173—41no—
108C22×C173—21no—
109C22×C173—21no—
110C22×C173—42no—
111C22×C173—42no—
112C22×C173—41no—
113C22×C173—41no—
114C22×C173—42no—
115C22×C173—41no—
116C22×C173—81no—
117C22×C173—81no—
118C22×C173—81no—
119C22×C173—81no—
120C22×C173#12011yescharacteristic
121C22×C173C2×D811yescharacteristic
122C22×C173AGL(1,ℤ/8)11yescharacteristic
123C2×D173—22no—
124C2×D173—41no—
125C2×D173—41no—
126C2×D173—42no—
127C2×D173—41no—
128C2×D173—41no—
129C2×D173—41no—
130C2×D173—81no—
131C2×D173—84no—
132C4×C173—21no—
133C4×C173—21no—
134C4×C173—41no—
135Dic173—42no—
136Dic173—41no—
137Dic173—41no—
138C2×C173—21no—
139C2×C173—21no—
140C2×C173—81no—
141C2×C173—82no—
142C2×C173#44711yescharacteristic
143C2×C173#45311yescharacteristic
144C2×C173B(4,𝔽2)11yescharacteristic
145D173—42no—
146D173—81no—
147C173#159511yescharacteristic, complemented, Sylow
148#1595—1731nomaximal, Sylow
149#534—1731no—
150#447—1731no—
151#447D17311yescharacteristic, complemented
152#392—1731no—
153#327—1731no—
154#327—1731no—
155#323—1731no—
156C22×D4—3461no—
157C22×D4—3461no—
158C22×D4C2×D17311yescharacteristic, complemented
159C2×D8—21no—
160C2×D8—3461no—
161#123—3461no—
162#123—3461no—
163#121—3461no—
164#121—1731no—
165#121C2×D17311yescharacteristic
166#120—3461no—
167#120—3461no—
168C2×C22⋊C4—3461no—
169C2×C22⋊C4—1731no—
170#102—21no—
171#102—3461no—
172C22⋊C8—1731no—
173C22⋊C8C2×D17311yescharacteristic, complemented
174#95—1731no—
175C24—21no—
176C24—3461no—
177C24—1731no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—42no—
182C2×D4—3461no—
183C2×D4—3461no—
184C2×D4—3461no—
185C2×D4—3461no—
186C2×D4—1731no—
187C2×D4—6922no—
188C2×D4—6922no—
189C2×D4—6922no—
190C2×D4—6922no—
191C2×D4—6922no—
192C2×D4C22⋊D17311yescharacteristic, complemented
193C22×C4—3461no—
194C22×C4—3461no—
195C22×C4—1731no—
196C22×C4C22×D17311yescharacteristic
197D8—42no—
198D8—6922no—
199C2×C8—21no—
200C2×C8—3461no—
201C4⋊C4—3461no—
202C4⋊C4—1731no—
203C4⋊C4C22⋊D17311yescharacteristic, complemented
204C22⋊C4—21no—
205C22⋊C4—3461no—
206C22⋊C4—3461no—
207C22⋊C4—3461no—
208C22⋊C4—3461no—
209C22⋊C4—3461no—
210C22⋊C4—6922no—
211C23—22no—
212C23—21no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—3461no—
220C23—3461no—
221C23—3461no—
222C23—13841no—
223C23—13841no—
224C23—1732no—
225C23—6921no—
226C23—6921no—
227C23—6922no—
228C23D4×D17311yescharacteristic
229D4—22no—
230D4—22no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—13841no—
237D4—6921no—
238D4—6922no—
239D4—6922no—
240D4—6928no—
241D4—6921no—
242D4—6921no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—3462no—
246C2×C4—3461no—
247C2×C4—3461no—
248C2×C4—6922no—
249C2×C4—6921no—
250C2×C4—6921no—
251C2×C4—6921no—
252C2×C4D4×D17311yescharacteristic
253C2×C4C2×C22⋊D17311yescharacteristic
254C8—41no—
255C8—6921no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—42no—
260C22—41no—
261C22—42no—
262C22—41no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22—81no—
268C22—3462no—
269C22—13841no—
270C22—13844no—
271C22—6922no—
272C22—6921no—
273C22—6921no—
274C22—6921no—
275C22—6921no—
276C22—6921no—
277C22#1914964711yescharacteristic
278C22D8×D17311yescharacteristic
279C22#1914970311yescharacteristic
280C4—21no—
281C4—21no—
282C4—41no—
283C4—6922no—
284C4—6921no—
285C4—6921no—
286C2—21no—
287C2—21no—
288C2—82no—
289C2—81no—
290C2—13841no—
291C2—6922no—
292C2#2601162311yescharacteristic
293C2#2601165711yescharacteristic
294C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#447×C173C2
socleC22×C173#120
hypercenter#447D173
hypocenterC173#1595
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #26012049C2semidirect
2. #26011625C2semidirect
3. #26011399C2semidirect
4. #26011226C2semidirect
5. #26011225C2semidirect
6. #26011214C2semidirect
7. #447×C173C2semidirect
20. C22⋊#13063436C22semidirect
21. #19149586C22semidirect
22. C22×D4×C173C22semidirect
26. C22⋊C8×C173C22semidirect
30. C2×D692D4semidirect
44. C4⋊Dic173D4semidirect
57. C2×D4×C173D4semidirect
61. C4⋊C4×C173D4semidirect
71. C22×D173D8semidirect
147. C173#1595semidirect
151. #447D173semidirect
158. C22×D4C2×D173semidirect
173. C22⋊C8C2×D173semidirect
192. C2×D4C22⋊D173semidirect
203. C4⋊C4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260120492
#260116252
#260113992
#260112262
#260112252
#260112142
#447×C1732
#1595173