Finite Group Databasebeta

← #36366207

Subgroups — #36366207

Order 22144.

350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012210C211yesmaximal, characteristic, complemented
2#26012200C211yesmaximal, characteristic, complemented
3#26011845C211yesmaximal, characteristic, complemented
4#26011844C211yesmaximal, characteristic, complemented
5#26011840C211yesmaximal, characteristic, complemented
6#26011800C211yesmaximal, characteristic, complemented
7#26011799C211yesmaximal, characteristic, complemented
8#26011779C211yesmaximal, characteristic, complemented
9#26011704C211yesmaximal, characteristic
10#26011694C211yesmaximal, characteristic, complemented
11#26011615C211yesmaximal, characteristic, complemented
12SD16×Dic173C211yesmaximal, characteristic, complemented
13C4×#14583814C211yesmaximal, characteristic, complemented
14D8×Dic173C211yesmaximal, characteristic
15#463×C173C211yesmaximal, characteristic
16C2×#14583814C2211yescharacteristic
17#19149678C2211yescharacteristic
18#19149677C2211yescharacteristic
19#19149672—21no—
20#19149672—21no—
21#19149671C2211yescharacteristic
22#19149670C2211yescharacteristic
23#19149666C2211yescharacteristic
24#19149658—21no—
25#19149656—21no—
26#19149656—21no—
27#19149656—21no—
28Q8×Dic173C2211yescharacteristic
29C4×Dic346C2211yescharacteristic
30C4×Dic346C2211yescharacteristic
31D4×Dic173C2211yescharacteristic
32D4×Dic173C2211yescharacteristic
33D4×Dic173C2211yescharacteristic
34#19149635—21no—
35C2×#14583805—21no—
36#19149613C2211yescharacteristic
37#19149611C2211yescharacteristic
38#19149610C2211yescharacteristic
39#19149608C2211yescharacteristic
40#19149608C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C4×#13063436C2211yescharacteristic
50C8×Dic173C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C2×D8×C173C2211yescharacteristic
54#124×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346D411yescharacteristic
61C2×Dic346C2311yescharacteristic
62C22×Dic173—21no—
63C22×Dic173—21no—
64C22×Dic173—21no—
65#14583814—22no—
66C2×#13063436C2311yescharacteristic
67C2×#13063436C2311yescharacteristic
68C4⋊Dic173—21no—
69C4⋊Dic173—21no—
70C4⋊Dic173D411yescharacteristic
71C4⋊Dic173C2311yescharacteristic
72C4⋊Dic173C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74#14583805—21no—
75#14583805—21no—
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805D411yescharacteristic
82#14583805D411yescharacteristic
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173C2311yescharacteristic
91C4×Dic173C2311yescharacteristic
92C2×Q8×C173C2311yescharacteristic
93C2×D4×C173C2311yescharacteristic
94C2×D4×C173C2311yescharacteristic
95C22×C4×C173—21no—
96SD16×C173—22no—
97D8×C173—22no—
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C4⋊C4×C173C2311yescharacteristic
101C22⋊C4×C173—21no—
102C22⋊C4×C173—21no—
103C22⋊C4×C173—21no—
104C42×C173C2311yescharacteristic
105Dic346—22no—
106Dic346—21no—
107Dic346—21no—
108Dic346—81no—
109C2×Dic173—21no—
110C2×Dic173—21no—
111C2×Dic173—21no—
112C2×Dic173—21no—
113C2×Dic173—42no—
114C2×Dic173—42no—
115C2×Dic173—81no—
116C2×Dic173C2×D411yescharacteristic
117C2×Dic173C2×D411yescharacteristic
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120#13063436—21no—
121#13063436—21no—
122C23×C173—21no—
123C23×C173—21no—
124Q8×C173—21no—
125Q8×C173—21no—
126D4×C173—21no—
127D4×C173—21no—
128D4×C173—21no—
129D4×C173#4112yes—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—81no—
133C2×C4×C173C2×D411yescharacteristic
134C2×C4×C173C2×D411yescharacteristic
135C2×C4×C173C2411yescharacteristic
136C8×C173—21no—
137C8×C173—21no—
138C22×C173—22no—
139C22×C173—41no—
140C22×C173—41no—
141C22×C173—41no—
142C22×C173—41no—
143C22×C173—41no—
144C22×C173C22×D411yescharacteristic
145C4×C173—21no—
146C4×C173—21no—
147C4×C173—41no—
148C4×C173—41no—
149C4×C173C2×#4111yescharacteristic
150C4×C1732−1+411yescharacteristic
151Dic173—21no—
152Dic173—21no—
153Dic173—21no—
154Dic173—41no—
155Dic173—41no—
156Dic173—41no—
157Dic173—41no—
158Dic173#13512yes—
159C2×C173—42no—
160C2×C173—81no—
161C2×C173#54811yescharacteristic
162C2×C173C2×#13511yescharacteristic
163C2×C173#57711yescharacteristic
164C173#336011yescharacteristic, complemented, Sylow
165#3360—1731nomaximal, Sylow
166#556—1731no—
167#554—1731no—
168#488—1731no—
169#488—1731no—
170#487—1731no—
171#479—1731no—
172#479—1731no—
173#476—1731no—
174#463—1731no—
175#463D17311yescharacteristic
176#461—1731no—
177C8×Q8—1731no—
178C4×SD16—1731no—
179C4×SD16—1731no—
180C4×D8—1731no—
181C2×SD16—1731no—
182C2×SD16C2×D17311yescharacteristic
183C2×D8C2×D17311yescharacteristic
184C4⋊Q8—1731no—
185C4⋊Q8—1731no—
186#126—3461no—
187#126—3461no—
188#125—1731no—
189#125—1731no—
190#124—1731no—
191#124C2×D17311yescharacteristic
192C22⋊Q8—3461no—
193C22⋊Q8—3461no—
194C22⋊Q8—3461no—
195C22⋊Q8—3461no—
196C4×Q8—1731no—
197C4×Q8—1731no—
198C4×Q8—1731no—
199C4×D4—1731no—
200C4×D4—1731no—
201C4×D4—1731no—
202C4×D4C2×D17311yescharacteristic
203#117—3461no—
204C2×C4⋊C4—3461no—
205#107—1731no—
206#106—1731no—
207#106—1731no—
208C4⋊C8—1731no—
209C4⋊C8—1731no—
210C4⋊C8C2×D17311yescharacteristic
211#103—1731no—
212#103—1731no—
213#103—1731no—
214#103—1731no—
215#103—1731no—
216#103C2×D17311yescharacteristic
217#102—1731no—
218#102—1731no—
219#102—1731no—
220#102C2×D17311yescharacteristic
221C4×C8—1731no—
222C4×C8—1731no—
223C4×C8—1731no—
224C2×Q8—3461no—
225C2×Q8—1731no—
226C2×Q8—1731no—
227C2×Q8C22×D17311yescharacteristic
228C2×D4C22×D17311yescharacteristic
229C2×D4C22×D17311yescharacteristic
230C22×C4—21no—
231C22×C4—3461no—
232C22×C4—3461no—
233C22×C4—3461no—
234SD16—22no—
235SD16—3462no—
236D8—22no—
237C2×C8—1731no—
238C2×C8—1731no—
239C2×C8C22×D17311yescharacteristic
240C2×C8C22×D17311yescharacteristic
241C4⋊C4—3461no—
242C4⋊C4—3461no—
243C4⋊C4—3461no—
244C4⋊C4—3461no—
245C4⋊C4—3461no—
246C4⋊C4—3461no—
247C4⋊C4—3461no—
248C4⋊C4—3461no—
249C4⋊C4—3461no—
250C4⋊C4—1731no—
251C4⋊C4—1731no—
252C4⋊C4—1731no—
253C4⋊C4—1731no—
254C4⋊C4—1731no—
255C4⋊C4—1731no—
256C4⋊C4C22×D17311yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—3461no—
261C22⋊C4—3461no—
262C22⋊C4—3461no—
263C22⋊C4—3461no—
264C22⋊C4—3461no—
265C42—3461no—
266C42—3461no—
267C42—1731no—
268C42—1731no—
269C42C22×D17311yescharacteristic
270C23—21no—
271C23—21no—
272Q8—21no—
273Q8—21no—
274Q8—3462no—
275Q8—3461no—
276Q8—3461no—
277Q8—13841no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4#1458382512yes—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—3461no—
286C2×C4—3461no—
287C2×C4—3461no—
288C2×C4—3461no—
289C2×C4—13841no—
290C2×C4—1731no—
291C2×C4—1731no—
292C2×C4—1731no—
293C2×C4—1731no—
294C2×C4—6922no—
295C2×C4—6922no—
296C2×C4D4×D17311yescharacteristic
297C2×C4D4×D17311yescharacteristic
298C2×C4C23×D17311yescharacteristic
299C8—21no—
300C8—21no—
301C8—3461no—
302C8—3461no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C2×D4×D17311yescharacteristic
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—3461no—
315C4—3461no—
316C4—3461no—
317C4—1732no—
318C4—6921no—
319C4—6921no—
320C4—6921no—
321C4—6921no—
322C4C2×#1458382511yescharacteristic
323C4#1914972611yescharacteristic
324C2—42no—
325C2—81no—
326C2#2601214711yescharacteristic
327C2#135×D17311yescharacteristic
328C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#463×C173C2
socleC22×C173C22×D4
prime radicalC2×D8×C173C22
hypercenter#463D173
hypocenterC173#3360
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #26012210C2semidirect
2. #26012200C2semidirect
3. #26011845C2semidirect
4. #26011844C2semidirect
5. #26011840C2semidirect
6. #26011800C2semidirect
7. #26011799C2semidirect
8. #26011779C2semidirect
10. #26011694C2semidirect
11. #26011615C2semidirect
12. SD16×Dic173C2semidirect
13. C4×#14583814C2semidirect
164. C173#3360semidirect

Maximal subgroups

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

subgroupindex
#260122102
#260122002
#260118452
#260118442
#260118402
#260118002
#260117992
#260117792
#260117042
#260116942
#260116152
SD16×Dic1732
C4×#145838142
D8×Dic1732
#463×C1732
#3360173