Finite Group Databasebeta

← #36363752

Subgroups — #36363752

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#26012115C211yesmaximal, characteristic, complemented
2#26011963C211yesmaximal, characteristic, complemented
3#26011835C211yesmaximal, characteristic, complemented
4#26011817C211yesmaximal, characteristic, complemented
5#26011768C211yesmaximal, characteristic, complemented
6#26011707C211yesmaximal, characteristic, complemented
7#26011706C211yesmaximal, characteristic, complemented
8#26011643C211yesmaximal, characteristic, complemented
9#26011626C211yesmaximal, characteristic
10#26011602C211yesmaximal, characteristic, complemented
11#26011588C211yesmaximal, characteristic, complemented
12#26011578C211yesmaximal, characteristic, complemented
13#26011571C211yesmaximal, characteristic, complemented
14#26011551C211yesmaximal, characteristic, complemented
15#476×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×D1384C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21#19149674—21no—
22#19149668—21no—
23#19149667C2211yescharacteristic, complemented
24#19149664—21no—
25#19149659C2211yescharacteristic
26#19149655—21no—
27#19149654C2211yescharacteristic
28#19149644C2211yescharacteristic, complemented
29C4×D692C2211yescharacteristic
30D4×Dic173C2211yescharacteristic
31#19149641C2211yescharacteristic
32#19149641C2211yescharacteristic, complemented
33C4×C22⋊D173—21no—
34#19149639—21no—
35#19149637C2211yescharacteristic
36#19149636C2211yescharacteristic
37C2×#14583803—21no—
38C22⋊C4×D173—21no—
39#19149612C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41C4⋊#13063436C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149602C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic, complemented
45#19149599C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic, complemented
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C4⋊Q8×C173C2211yescharacteristic, complemented
53C4×D4×C173C2211yescharacteristic
54#106×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56C4⋊C8×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C23×D173—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692—21no—
62C2×D692C2311yescharacteristic
63C2×D692C2311yescharacteristic
64C2×C22⋊D173—21no—
65D4×D173—22no—
66D4×D173—22no—
67C22×Dic173—21no—
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173C2311yescharacteristic
72#14583815—22no—
73#14583814—22no—
74D1384—22no—
75#14583811—22no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805D411yescharacteristic
82#14583805D411yescharacteristic
83C22⋊Dic173—21no—
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803D411yescharacteristic
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173—21no—
98C2×D4×C173C2311yescharacteristic
99C22×C4×C173—21no—
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—24no—
110C22×D173—21no—
111C22×D173—81no—
112C22×D173—81no—
113C22×D173C2×D411yescharacteristic
114Dic346—21no—
115Dic346—21no—
116D692—21no—
117D692—21no—
118D692—21no—
119D692—81no—
120D692#4112yes—
121C22⋊D173—24no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—81no—
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C4×D173—21no—
129C4×D173—21no—
130C4×D173—81no—
131C4×D173—81no—
132C4×D173—81no—
133#13063436—21no—
134#13063436—21no—
135C23×C173—21no—
136Q8×C173—81no—
137D4×C173—21no—
138D4×C173#4112yes—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—81no—
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—21no—
147C8×C173—21no—
148C22×C173—22no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173C22×D411yescharacteristic
152C2×D173—21no—
153C2×D173—22no—
154C2×D173—21no—
155C2×D173—21no—
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—41no—
165C4×C173—41no—
166C4×C173—41no—
167C4×C173C2×#4111yescharacteristic
168C4×C173C2×#4111yescharacteristic
169Dic173—21no—
170Dic173—21no—
171Dic173—21no—
172Dic173—41no—
173Dic173—41no—
174C2×C173—42no—
175C2×C173#52511yescharacteristic
176C2×C173#57611yescharacteristic
177C2×C173#57711yescharacteristic
178D173—41no—
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#476D17311yescharacteristic, complemented
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×D4—1731no—
199C2×SD16—1731no—
200C2×SD16—1731no—
201C2×D8—1731no—
202C2×D8—1731no—
203C4⋊Q8C2×D17311yescharacteristic, complemented
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×D4C2×D17311yescharacteristic
219#117—1731no—
220#117—1731no—
221C2×C22⋊C4—3461no—
222C2×C22⋊C4—3461no—
223#107—1731no—
224#107—1731no—
225#106C2×D17311yescharacteristic, complemented
226#106C2×D17311yescharacteristic, complemented
227C4⋊C8—1731no—
228C4⋊C8C2×D17311yescharacteristic, complemented
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⋊C8—1731no—
239#97—1731no—
240C4×C8—1731no—
241C24—3461no—
242C2×Q8—21no—
243C2×Q8—1731no—
244C2×D4—3461no—
245C2×D4—3462no—
246C2×D4—3462no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4—1731no—
250C2×D4C22×D17311yescharacteristic
251C22×C4—21no—
252C22×C4—3461no—
253C22×C4—3461no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4—1731no—
257SD16—3462no—
258SD16—3462no—
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⋊C4C22×D17311yescharacteristic
272C4⋊C4C22×D17311yescharacteristic
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—
289C42C22×D17311yescharacteristic
290C23—21no—
291C23—3464no—
292C23—3461no—
293C23—3461no—
294C23—13841no—
295C23—13841no—
296C23—1731no—
297Q8—81no—
298Q8—3461no—
299Q8—3461no—
300D4—21no—
301D4—3464no—
302D4—3461no—
303D4—3461no—
304D4—3461no—
305D4—13841no—
306D4—1732no—
307D4#1458382712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—13841no—
317C2×C4—13841no—
318C2×C4—13841no—
319C2×C4—13841no—
320C2×C4—1731no—
321C2×C4—1731no—
322C2×C4—1731no—
323C2×C4D4×D17311yescharacteristic
324C2×C4D4×D17311yescharacteristic
325C2×C4C23×D17311yescharacteristic
326C8—21no—
327C8—21no—
328C8—3461no—
329C8—3461no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—3461no—
334C22—3462no—
335C22—3461no—
336C22—3461no—
337C22—13844no—
338C22—6921no—
339C22—6921no—
340C22—6921no—
341C22—6921no—
342C22—6921no—
343C22C2×D4×D17311yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—3461no—
350C4—3461no—
351C4—3461no—
352C4—6921no—
353C4—6921no—
354C4#41×D17311yescharacteristic
355C4C2×#1458382711yescharacteristic
356C2—42no—
357C2—13841no—
358C2—6921no—
359C2—6922no—
360C2#2601200411yescharacteristic
361C2#2601229511yescharacteristic
362C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#476×C173C2
socleC22×C173C22×D4
prime radical#26011626C2
hypercenter#476D173
hypocenterC173#3118
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #26012115C2semidirect
2. #26011963C2semidirect
3. #26011835C2semidirect
4. #26011817C2semidirect
5. #26011768C2semidirect
6. #26011707C2semidirect
7. #26011706C2semidirect
8. #26011643C2semidirect
10. #26011602C2semidirect
11. #26011588C2semidirect
12. #26011578C2semidirect
13. #26011571C2semidirect
14. #26011551C2semidirect
15. #476×C173C2semidirect
23. #19149667C22semidirect
28. #19149644C22semidirect
32. #19149641C22semidirect
39. #19149612C22semidirect
40. #19149612C22semidirect
41. C4⋊#13063436C22semidirect
42. #19149604C22semidirect
43. #19149602C22semidirect
44. #19149601C22semidirect
47. #19149599C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
52. C4⋊Q8×C173C22semidirect
54. #106×C173C22semidirect
55. #106×C173C22semidirect
56. C4⋊C8×C173C22semidirect
181. C173#3118semidirect
187. #476D173semidirect
203. C4⋊Q8C2×D173semidirect
225. #106C2×D173semidirect
226. #106C2×D173semidirect
228. C4⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121152
#260119632
#260118352
#260118172
#260117682
#260117072
#260117062
#260116432
#260116262
#260116022
#260115882
#260115782
#260115712
#260115512
#476×C1732
#3118173