Finite Group Databasebeta

← #36363748

Subgroups — #36363748

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#26012114C211yesmaximal, characteristic, complemented
2#26011966C211yesmaximal, characteristic, complemented
3#26011829C211yesmaximal, characteristic, complemented
4#26011816C211yesmaximal, characteristic, complemented
5#26011779C211yesmaximal, characteristic, complemented
6#26011769C211yesmaximal, characteristic, complemented
7#26011708C211yesmaximal, characteristic, complemented
8#26011708C211yesmaximal, characteristic, complemented
9#26011625C211yesmaximal, characteristic
10#26011600C211yesmaximal, characteristic, complemented
11#26011592C211yesmaximal, characteristic, complemented
12SD16×Dic173C211yesmaximal, characteristic, complemented
13#26011571C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#450×C173C211yesmaximal, characteristic, complemented
16C2×#14583811C2211yescharacteristic
17C2×#14583811C2211yescharacteristic
18#19149678C2211yescharacteristic, complemented
19#19149674—21no—
20#19149668C2211yescharacteristic, complemented
21#19149665—21no—
22#19149661—21no—
23#19149659—21no—
24#19149653C2211yescharacteristic
25Q8×Dic173C2211yescharacteristic, complemented
26D4×Dic173—21no—
27D4×Dic173—21no—
28D4×Dic173C2211yescharacteristic
29D4×Dic173C2211yescharacteristic
30D4×Dic173C2211yescharacteristic
31D4×Dic173C2211yescharacteristic
32#19149641C2211yescharacteristic, complemented
33#19149638C2211yescharacteristic
34#19149638C2211yescharacteristic
35C2×C22⋊Dic173—21no—
36C2×C22⋊Dic173—21no—
37#19149613C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149611C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×D692C2311yescharacteristic
61C2×C22⋊D173—21no—
62C22×Dic173—21no—
63C22×Dic173—21no—
64C22×Dic173—21no—
65C22×Dic173—21no—
66C2×C4×D173—21no—
67#14583811—22no—
68#14583811—22no—
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C4⋊Dic173—21no—
72C4⋊Dic173D411yescharacteristic
73C4⋊Dic173D411yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78C22⋊Dic173—21no—
79C22⋊Dic173—21no—
80C22⋊Dic173—21no—
81C22⋊Dic173—21no—
82C22⋊Dic173—21no—
83C22⋊Dic173D411yescharacteristic
84C22⋊Dic173D411yescharacteristic
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173C2311yescharacteristic
92C4×Dic173C2311yescharacteristic
93C24×C173—21no—
94C2×Q8×C173C2311yescharacteristic
95C2×D4×C173—22no—
96C2×D4×C173—22no—
97C2×D4×C173—21no—
98C2×D4×C173C2311yescharacteristic
99C2×D4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101SD16×C173—22no—
102SD16×C173—22no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173C2311yescharacteristic
107C22⋊C4×C173—21no—
108C22×D173—21no—
109Dic346—81no—
110D692—21no—
111D692—21no—
112C22⋊D173—81no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—81no—
117C2×Dic173—81no—
118C2×Dic173—81no—
119C2×Dic173—81no—
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C4×D173—81no—
125#13063436—21no—
126#13063436—21no—
127C23×C173—24no—
128C23×C173—21no—
129C23×C173—21no—
130C23×C173—81no—
131C23×C173—81no—
132C23×C173C2×D411yescharacteristic
133Q8×C173—21no—
134Q8×C173—21no—
135D4×C173—24no—
136D4×C173—21no—
137D4×C173—21no—
138D4×C173#4112yes—
139D4×C173#4112yes—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—21no—
147C8×C173—21no—
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—41no—
164C4×C173—41no—
165C4×C173C2×#4111yescharacteristic
166C4×C173C2×#4111yescharacteristic
167Dic173—21no—
168Dic173—21no—
169Dic173—21no—
170Dic173—21no—
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#518—1731no—
185#485—1731no—
186#482—1731no—
187#476—1731no—
188#474—1731no—
189#463—1731no—
190#463—1731no—
191#450D17311yescharacteristic, complemented
192#447—1731no—
193#442—1731no—
194#440—1731no—
195C4×SD16—1731no—
196C4×D8—1731no—
197#433—1731no—
198C22×D4C2×D17311yescharacteristic
199C2×SD16C2×D17311yescharacteristic
200C2×SD16C2×D17311yescharacteristic
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⋊Q8C2×D17311yescharacteristic
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#117—1731no—
221C2×C22⋊C4—3461no—
222C2×C22⋊C4—3461no—
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#102C2×D17311yescharacteristic
236#102C2×D17311yescharacteristic
237C22⋊C8—1731no—
238C22⋊C8C2×D17311yescharacteristic
239#97—1731no—
240C4×C8—1731no—
241C24—21no—
242C2×Q8—3461no—
243C2×Q8C22×D17311yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4C22×D17311yescharacteristic
250C2×D4C22×D17311yescharacteristic
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4—3461no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4C22×D17311yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—3462no—
260D8—3462no—
261C2×C8—1731no—
262C2×C8—1731no—
263C2×C8C22×D17311yescharacteristic
264C2×C8C22×D17311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—1731no—
269C4⋊C4—1731no—
270C4⋊C4—1731no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4C22×D17311yescharacteristic
274C22⋊C4—21no—
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⋊C4—1731no—
285C22⋊C4—1731no—
286C42—3461no—
287C42—3461no—
288C42—1731no—
289C42—1731no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—3461no—
296C23D4×D17311yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—13841no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—3461no—
304D4—3461no—
305D4—13841no—
306D4#1458382512yes—
307D4#1458382512yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
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×C4—1731no—
321C2×C4—1731no—
322C2×C4—1731no—
323C2×C4—1731no—
324C2×C4D4×D17311yescharacteristic
325C2×C4C23×D17311yescharacteristic
326C8—21no—
327C8—21no—
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—
343C22C2×D4×D17311yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—3461no—
348C4—3461no—
349C4—3461no—
350C4—3461no—
351C4—6921no—
352C4—6921no—
353C4—6921no—
354C4C2×#1458382511yescharacteristic
355C4C2×#1458382511yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—13841no—
360C2#2601199711yescharacteristic
361C2#2601229511yescharacteristic
362C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radical#26011625C2
hypercenter#450D173
hypocenterC173#3118
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26012114C2semidirect
2. #26011966C2semidirect
3. #26011829C2semidirect
4. #26011816C2semidirect
5. #26011779C2semidirect
6. #26011769C2semidirect
7. #26011708C2semidirect
8. #26011708C2semidirect
10. #26011600C2semidirect
11. #26011592C2semidirect
12. SD16×Dic173C2semidirect
13. #26011571C2semidirect
14. #26011549C2semidirect
15. #450×C173C2semidirect
18. #19149678C22semidirect
20. #19149668C22semidirect
25. Q8×Dic173C22semidirect
32. #19149641C22semidirect
37. #19149613C22semidirect
38. #19149612C22semidirect
39. #19149611C22semidirect
40. #19149610C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
43. #19149603C22semidirect
44. #19149603C22semidirect
45. #19149601C22semidirect
46. #19149601C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
181. C173#3118semidirect
191. #450D173semidirect

Maximal subgroups

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

subgroupindex
#260121142
#260119662
#260118292
#260118162
#260117792
#260117692
#260117082
#260116252
#260116002
#260115922
SD16×Dic1732
#260115712
#260115492
#450×C1732
#3118173