Finite Group Databasebeta

← #36363747

Subgroups — #36363747

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#26011832C211yesmaximal, characteristic, complemented
3#26011815C211yesmaximal, characteristic, complemented
4#26011781C211yesmaximal, characteristic, complemented
5#26011771C211yesmaximal, characteristic, complemented
6#26011706C211yesmaximal, characteristic, complemented
7#26011706C211yesmaximal, characteristic, complemented
8#26011642C211yesmaximal, characteristic, complemented
9#26011625C211yesmaximal, characteristic
10#26011601C211yesmaximal, characteristic, complemented
11#26011591C211yesmaximal, characteristic, complemented
12C4×#14583814C211yesmaximal, characteristic, complemented
13C4×#14583811C211yesmaximal, characteristic, complemented
14#26011552C211yesmaximal, characteristic, complemented
15#518×C173C211yesmaximal, characteristic, complemented
16C2×#14583814C2211yescharacteristic
17C2×#14583814C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20#19149679C2211yescharacteristic, complemented
21#19149674—21no—
22#19149667C2211yescharacteristic, complemented
23#19149665—21no—
24#19149661—21no—
25#19149658C2211yescharacteristic
26#19149657—21no—
27#19149653C2211yescharacteristic
28C4×Dic346C2211yescharacteristic, complemented
29C4×D692C2211yescharacteristic, complemented
30#19149612C2211yescharacteristic, complemented
31#19149612C2211yescharacteristic, complemented
32#19149610C2211yescharacteristic, complemented
33#19149610C2211yescharacteristic, complemented
34C4⋊#13063436C2211yescharacteristic, complemented
35C4⋊#13063436C2211yescharacteristic, complemented
36#19149602C2211yescharacteristic, complemented
37#19149602C2211yescharacteristic, complemented
38#19149600C2211yescharacteristic
39#19149600C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149599C2211yescharacteristic, complemented
43#19149599C2211yescharacteristic, complemented
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46#19149590C2211yescharacteristic, complemented
47C4×#13063436C2211yescharacteristic, complemented
48C22×D4×C173C2211yescharacteristic
49C4×D4×C173—21no—
50C4×D4×C173—21no—
51C4×D4×C173C2211yescharacteristic
52C4×D4×C173C2211yescharacteristic
53C4×D4×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55#117×C173C2211yescharacteristic
56#117×C173C2211yescharacteristic
57C2×C22⋊C4×C173—21no—
58C2×C22⋊C4×C173—21no—
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C4×D173—21no—
64#14583814—22no—
65#14583814—22no—
66#14583811—22no—
67#14583811—22no—
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C4⋊Dic173—21no—
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75#14583805—21no—
76#14583805—21no—
77C22⋊Dic173—21no—
78#14583803—21no—
79#14583803—21no—
80#14583803—21no—
81#14583803—21no—
82C4×Dic173—21no—
83C24×C173—21no—
84C2×D4×C173—22no—
85C2×D4×C173—22no—
86C2×D4×C173—21no—
87C2×D4×C173C2311yescharacteristic
88C2×D4×C173C2311yescharacteristic
89C22×C4×C173—21no—
90C22×C4×C173—21no—
91C22×C4×C173—21no—
92C22×C4×C173—21no—
93C22×C4×C173C2311yescharacteristic
94C4⋊C4×C173D411yescharacteristic
95C4⋊C4×C173D411yescharacteristic
96C4⋊C4×C173C2311yescharacteristic
97C4⋊C4×C173C2311yescharacteristic
98C22⋊C4×C173—21no—
99C22⋊C4×C173—21no—
100C22⋊C4×C173—21no—
101C22⋊C4×C173—21no—
102C22⋊C4×C173—21no—
103C22⋊C4×C173D411yescharacteristic
104C22⋊C4×C173D411yescharacteristic
105C42×C173—21no—
106C42×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109Dic346—21no—
110Dic346—21no—
111Dic346—81no—
112D692—21no—
113D692—21no—
114C22⋊D173—81no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C4×D173—81no—
119#13063436—21no—
120#13063436—21no—
121#13063436—21no—
122#13063436—21no—
123C23×C173—24no—
124C23×C173—21no—
125C23×C173—21no—
126C23×C173—81no—
127C23×C173—81no—
128C23×C173C2×D411yescharacteristic
129D4×C173—24no—
130D4×C173—21no—
131D4×C173—21no—
132D4×C173#4112yes—
133D4×C173#4112yes—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—81no—
139C2×C4×C173—81no—
140C2×C4×C173—81no—
141C2×C4×C173—81no—
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2411yescharacteristic
148C22×C173—21no—
149C22×C173—22no—
150C22×C173—22no—
151C22×C173—21no—
152C22×C173—21no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—84no—
158C22×C173C22×D411yescharacteristic
159C2×D173—41no—
160C2×D173—41no—
161C2×D173—41no—
162C4×C173—21no—
163C4×C173—21no—
164C4×C173—21no—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173C2×#4111yescharacteristic
170C4×C173C2×#4111yescharacteristic
171Dic173—41no—
172Dic173—41no—
173Dic173—41no—
174C2×C173—41no—
175C2×C173—42no—
176C2×C173—42no—
177C2×C173#52511yescharacteristic
178C2×C173#57611yescharacteristic
179C2×C173#57711yescharacteristic
180D173—81no—
181C173#311811yescharacteristic, complemented, Sylow
182#3118—1731nomaximal, Sylow
183#542—1731no—
184#518D17311yescharacteristic, complemented
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#433—1731no—
198C22×D4C2×D17311yescharacteristic
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—21no—
213C4×D4—21no—
214C4×D4—1731no—
215C4×D4C2×D17311yescharacteristic
216C4×D4C2×D17311yescharacteristic
217C4×D4C2×D17311yescharacteristic
218C4×D4C2×D17311yescharacteristic
219#117C2×D17311yescharacteristic
220#117C2×D17311yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—1731no—
224#107—1731no—
225#106—1731no—
226#106—1731no—
227C4⋊C8—1731no—
228C4⋊C8—1731no—
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⋊C8—1731no—
238C22⋊C8—1731no—
239#97—1731no—
240C4×C8—1731no—
241C24—21no—
242C2×Q8—3461no—
243C2×Q8—1731no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4C22×D17311yescharacteristic
250C2×D4C22×D17311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3461no—
256C22×C4C22×D17311yescharacteristic
257SD16—3462no—
258SD16—3462no—
259D8—3462no—
260D8—3462no—
261C2×C8—1731no—
262C2×C8—1731no—
263C2×C8—1731no—
264C2×C8—1731no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—1731no—
269C4⋊C4—1731no—
270C4⋊C4C22⋊D17311yescharacteristic
271C4⋊C4C22⋊D17311yescharacteristic
272C4⋊C4C22×D17311yescharacteristic
273C4⋊C4C22×D17311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4C22⋊D17311yescharacteristic
285C22⋊C4C22⋊D17311yescharacteristic
286C42—21no—
287C42—3461no—
288C42C22×D17311yescharacteristic
289C42C22×D17311yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—3461no—
296C23C2×C22⋊D17311yescharacteristic
297Q8—3461no—
298Q8—3461no—
299Q8—13841no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—3461no—
304D4—3461no—
305D4—13841no—
306D4#1458382612yes—
307D4#1458382612yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—13841no—
320C2×C4C2×C22⋊D17311yescharacteristic
321C2×C4C2×C22⋊D17311yescharacteristic
322C2×C4C2×C22⋊D17311yescharacteristic
323C2×C4C2×C22⋊D17311yescharacteristic
324C2×C4C2×C22⋊D17311yescharacteristic
325C2×C4C23×D17311yescharacteristic
326C8—3461no—
327C8—3461no—
328C8—3461no—
329C8—3461no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—6921no—
341C22—6921no—
342C22—6921no—
343C22C22×C22⋊D17311yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—6921no—
352C4—6921no—
353C4—6921no—
354C4C2×#1458382611yescharacteristic
355C4C2×#1458382611yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—13841no—
360C2#2601200211yescharacteristic
361C2#2601229411yescharacteristic
362C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#518×C173C2
socleC22×C173C22×D4
prime radical#26011625C2
hypercenter#518D173
hypocenterC173#3118
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #26012112C2semidirect
2. #26011832C2semidirect
3. #26011815C2semidirect
4. #26011781C2semidirect
5. #26011771C2semidirect
6. #26011706C2semidirect
7. #26011706C2semidirect
8. #26011642C2semidirect
10. #26011601C2semidirect
11. #26011591C2semidirect
12. C4×#14583814C2semidirect
13. C4×#14583811C2semidirect
14. #26011552C2semidirect
15. #518×C173C2semidirect
20. #19149679C22semidirect
22. #19149667C22semidirect
28. C4×Dic346C22semidirect
29. C4×D692C22semidirect
30. #19149612C22semidirect
31. #19149612C22semidirect
32. #19149610C22semidirect
33. #19149610C22semidirect
34. C4⋊#13063436C22semidirect
35. C4⋊#13063436C22semidirect
36. #19149602C22semidirect
37. #19149602C22semidirect
42. #19149599C22semidirect
43. #19149599C22semidirect
46. #19149590C22semidirect
47. C4×#13063436C22semidirect
181. C173#3118semidirect
184. #518D173semidirect

Maximal subgroups

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

subgroupindex
#260121122
#260118322
#260118152
#260117812
#260117712
#260117062
#260116422
#260116252
#260116012
#260115912
C4×#145838142
C4×#145838112
#260115522
#518×C1732
#3118173