Finite Group Databasebeta

← #36363749

Subgroups — #36363749

Order 22144.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012112C211yesmaximal, characteristic, complemented
2#26011965C211yesmaximal, characteristic, complemented
3#26011833C211yesmaximal, characteristic, complemented
4#26011816C211yesmaximal, characteristic, complemented
5#26011782C211yesmaximal, characteristic, complemented
6#26011772C211yesmaximal, characteristic, complemented
7#26011707C211yesmaximal, characteristic, complemented
8#26011707C211yesmaximal, characteristic, complemented
9#26011644C211yesmaximal, characteristic, complemented
10#26011627C211yesmaximal, characteristic
11#26011604C211yesmaximal, characteristic, complemented
12#26011593C211yesmaximal, characteristic, complemented
13C4×#14583815C211yesmaximal, characteristic, complemented
14C4×D1384C211yesmaximal, characteristic, complemented
15#433×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×D1384C2211yescharacteristic
20C2×D1384C2211yescharacteristic
21#19149679C2211yescharacteristic, complemented
22#19149674—21no—
23#19149667C2211yescharacteristic, complemented
24#19149665—21no—
25#19149661—21no—
26#19149658C2211yescharacteristic
27#19149657—21no—
28#19149653C2211yescharacteristic
29C4×Dic346C2211yescharacteristic, complemented
30C4×D692C2211yescharacteristic
31C4×D692C2211yescharacteristic
32C4×D692C2211yescharacteristic
33C4×D692C2211yescharacteristic
34C4×D692C2211yescharacteristic, complemented
35#19149641—21no—
36#19149641—21no—
37#19149638C2211yescharacteristic
38C22⋊C4×D173—21no—
39C22⋊C4×D173—21no—
40#19149613C2211yescharacteristic, complemented
41#19149613C2211yescharacteristic, complemented
42#19149611C2211yescharacteristic, complemented
43#19149611C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149604C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic, complemented
52#117×C173C2211yescharacteristic
53C4⋊C8×C173C2211yescharacteristic, complemented
54C4⋊C8×C173C2211yescharacteristic, complemented
55C22⋊C8×C173C2211yescharacteristic
56C22⋊C8×C173C2211yescharacteristic
57#97×C173C2211yescharacteristic, complemented
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C2×Dic346—21no—
61C2×Dic346C2311yescharacteristic
62C2×D692—22no—
63C2×D692—22no—
64C2×D692—21no—
65C2×D692C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74#14583815—22no—
75#14583815—22no—
76D1384—22no—
77D1384—22no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C22×C4×C173C2311yescharacteristic
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173D411yescharacteristic
103C4⋊C4×C173D411yescharacteristic
104C22⋊C4×C173D411yescharacteristic
105C22⋊C4×C173D411yescharacteristic
106C42×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—21no—
110C22×D173—24no—
111C22×D173—21no—
112C22×D173—81no—
113C22×D173—81no—
114Dic346—21no—
115Dic346—21no—
116Dic346—81no—
117D692—24no—
118D692—21no—
119D692—21no—
120D692—21no—
121D692—21no—
122D692#4112yes—
123D692#4112yes—
124C22⋊D173—81no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C4×D173—81no—
131C4×D173—81no—
132C4×D173—81no—
133C4×D173—81no—
134C4×D173—81no—
135C23×C173C2×D411yescharacteristic
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2411yescharacteristic
144C8×C173—21no—
145C8×C173—21no—
146C8×C173—21no—
147C8×C173—21no—
148C22×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173C22×D411yescharacteristic
152C2×D173—22no—
153C2×D173—22no—
154C2×D173—41no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C2×D173—41no—
160C2×D173—41no—
161C2×D173—84no—
162C4×C173—21no—
163C4×C173—21no—
164C4×C173—21no—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173C2×#4111yescharacteristic
168C4×C173C2×#4111yescharacteristic
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172Dic173—41no—
173Dic173—41no—
174C2×C173—41no—
175C2×C173#52511yescharacteristic
176C2×C173#57611yescharacteristic
177C2×C173#57711yescharacteristic
178D173—42no—
179D173—42no—
180D173—81no—
181C173#311811yescharacteristic, complemented, Sylow
182#3118—1731nomaximal, Sylow
183#542—1731no—
184#518—1731no—
185#485—1731no—
186#482—1731no—
187#476—1731no—
188#474—1731no—
189#463—1731no—
190#463—1731no—
191#450—1731no—
192#447—1731no—
193#442—1731no—
194#440—1731no—
195C4×SD16—1731no—
196C4×D8—1731no—
197#433D17311yescharacteristic, complemented
198C22×D4—1731no—
199C2×SD16—1731no—
200C2×SD16—1731no—
201C2×D8—1731no—
202C2×D8—1731no—
203C4⋊Q8—1731no—
204#126—3461no—
205#124—3461no—
206#124—1731no—
207#123—3461no—
208C22⋊Q8—3461no—
209C22⋊Q8—1731no—
210#121—1731no—
211C4×Q8—1731no—
212C4×D4—3461no—
213C4×D4—3461no—
214C4×D4—1731no—
215C4×D4—1731no—
216C4×D4—1731no—
217C4×D4—1731no—
218C4×D4—1731no—
219#117—1731no—
220#117C2×D17311yescharacteristic
221C2×C22⋊C4—3461no—
222C2×C22⋊C4—3461no—
223#107—1731no—
224#107—1731no—
225#106—1731no—
226#106—1731no—
227C4⋊C8C2×D17311yescharacteristic, complemented
228C4⋊C8C2×D17311yescharacteristic, complemented
229#103—1731no—
230#103—1731no—
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235#102—1731no—
236#102—1731no—
237C22⋊C8C2×D17311yescharacteristic
238C22⋊C8C2×D17311yescharacteristic
239#97C2×D17311yescharacteristic, complemented
240C4×C8C2×D17311yescharacteristic, complemented
241C24—3461no—
242C2×Q8—3461no—
243C2×Q8—1731no—
244C2×D4—3462no—
245C2×D4—3462no—
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4—1731no—
250C2×D4—1731no—
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4—3461no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4C22×D17311yescharacteristic
257SD16—3462no—
258SD16—3462no—
259D8—3462no—
260D8—3462no—
261C2×C8C22×D17311yescharacteristic
262C2×C8C22×D17311yescharacteristic
263C2×C8C22×D17311yescharacteristic
264C2×C8C22×D17311yescharacteristic
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—1731no—
269C4⋊C4—1731no—
270C4⋊C4—1731no—
271C4⋊C4—1731no—
272C4⋊C4D69211yescharacteristic
273C4⋊C4D69211yescharacteristic
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4D69211yescharacteristic
285C22⋊C4D69211yescharacteristic
286C42—3461no—
287C42—3461no—
288C42C22×D17311yescharacteristic
289C42C22×D17311yescharacteristic
290C23—3464no—
291C23—3461no—
292C23—3461no—
293C23—3461no—
294C23—13841no—
295C23—13841no—
296C23C2×D69211yescharacteristic
297Q8—3461no—
298Q8—3461no—
299Q8—13841no—
300D4—3464no—
301D4—3461no—
302D4—3461no—
303D4—3461no—
304D4—3461no—
305D4—13841no—
306D4—1732no—
307D4—1732no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—13841no—
316C2×C4—13841no—
317C2×C4—13841no—
318C2×C4—13841no—
319C2×C4—13841no—
320C2×C4C2×D69211yescharacteristic
321C2×C4C2×D69211yescharacteristic
322C2×C4C2×D69211yescharacteristic
323C2×C4C2×D69211yescharacteristic
324C2×C4C2×D69211yescharacteristic
325C2×C4C23×D17311yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—3462no—
334C22—3462no—
335C22—13844no—
336C22—6921no—
337C22—6921no—
338C22—6921no—
339C22—6921no—
340C22—6921no—
341C22—6921no—
342C22—6921no—
343C22C22×D69211yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—6921no—
350C4—6921no—
351C4—6921no—
352C4—6921no—
353C4—6921no—
354C4#41×D17311yescharacteristic
355C4#41×D17311yescharacteristic
356C2—41no—
357C2—13841no—
358C2—6922no—
359C2—6922no—
360C2#2601200311yescharacteristic
361C2#2601229611yescharacteristic
362C2#2601230311yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#433×C173C2
socleC22×C173C22×D4
prime radical#26011627C2
hypercenter#433D173
hypocenterC173#3118
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #26012112C2semidirect
2. #26011965C2semidirect
3. #26011833C2semidirect
4. #26011816C2semidirect
5. #26011782C2semidirect
6. #26011772C2semidirect
7. #26011707C2semidirect
8. #26011707C2semidirect
9. #26011644C2semidirect
11. #26011604C2semidirect
12. #26011593C2semidirect
13. C4×#14583815C2semidirect
14. C4×D1384C2semidirect
15. #433×C173C2semidirect
21. #19149679C22semidirect
23. #19149667C22semidirect
29. C4×Dic346C22semidirect
34. C4×D692C22semidirect
40. #19149613C22semidirect
41. #19149613C22semidirect
42. #19149611C22semidirect
43. #19149611C22semidirect
44. #19149604C22semidirect
45. #19149604C22semidirect
50. #19149601C22semidirect
51. #19149601C22semidirect
53. C4⋊C8×C173C22semidirect
54. C4⋊C8×C173C22semidirect
57. #97×C173C22semidirect
58. C4×C8×C173C22semidirect
181. C173#3118semidirect
197. #433D173semidirect
227. C4⋊C8C2×D173semidirect
228. C4⋊C8C2×D173semidirect
239. #97C2×D173semidirect
240. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121122
#260119652
#260118332
#260118162
#260117822
#260117722
#260117072
#260116442
#260116272
#260116042
#260115932
C4×#145838152
C4×D13842
#433×C1732
#3118173