Finite Group Databasebeta

← #36363803

Subgroups — #36363803

Order 22144.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012092C211yesmaximal, characteristic, complemented
2#26011963C211yesmaximal, characteristic, complemented
3#26011818C211yesmaximal, characteristic, complemented
4#26011813C211yesmaximal, characteristic, complemented
5#26011794C211yesmaximal, characteristic, complemented
6#26011789C211yesmaximal, characteristic, complemented
7#26011685C211yesmaximal, characteristic, complemented
8#26011684C211yesmaximal, characteristic, complemented
9#26011626C211yesmaximal, characteristic, complemented
10#26011623C211yesmaximal, characteristic, complemented
11#26011601C211yesmaximal, characteristic, complemented
12#26011601C211yesmaximal, characteristic, complemented
13C4×D1384C211yesmaximal, characteristic, complemented
14#26011550C211yesmaximal, characteristic, complemented
15C4×D8×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173C2211yescharacteristic
17C2×D1384C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20#19149675C2211yescharacteristic, complemented
21#19149670C2211yescharacteristic, complemented
22#19149662—21no—
23#19149661—21no—
24#19149654C2211yescharacteristic
25#19149653—21no—
26#19149652—21no—
27#19149650C2211yescharacteristic
28C4×D692C2211yescharacteristic
29C4×D692C2211yescharacteristic, complemented
30D4×Dic173C2211yescharacteristic
31#19149641C2211yescharacteristic
32C4×C22⋊D173—21no—
33#19149639—21no—
34#19149637C2211yescharacteristic
35#19149636C2211yescharacteristic
36C2×#14583803—21no—
37C22⋊C4×D173—21no—
38#19149613C2211yescharacteristic, complemented
39#19149610C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic, complemented
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic, complemented
52C2×D8×C173C2211yescharacteristic
53C4×D4×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic, complemented
55#107×C173C2211yescharacteristic, complemented
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic, complemented
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C2×D692—21no—
61C2×D692C2311yescharacteristic
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67D4×D173—22no—
68D4×D173—22no—
69C22×Dic173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74D1384—22no—
75#14583811—22no—
76#14583811—22no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805D411yescharacteristic
84#14583805D411yescharacteristic
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803D411yescharacteristic
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C2×D4×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99C22×C4×C173—21no—
100D8×C173—22no—
101C2×C8×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—24no—
110C22×D173—21no—
111C22×D173—81no—
112C22×D173—81no—
113C22×D173C2×D411yescharacteristic
114D692—21no—
115D692—21no—
116D692—21no—
117D692—81no—
118D692#4112yes—
119C22⋊D173—24no—
120C22⋊D173—81no—
121C22⋊D173—81no—
122C22⋊D173—81no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—81no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—81no—
132C4×D173—81no—
133C4×D173—81no—
134#13063436—21no—
135#13063436—21no—
136C23×C173—21no—
137C23×C173—21no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—21no—
141D4×C173#4112yes—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—81no—
145C2×C4×C173—81no—
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2411yescharacteristic
149C8×C173—21no—
150C8×C173—21no—
151C22×C173—22no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173C22×D411yescharacteristic
158C2×D173—21no—
159C2×D173—22no—
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—84no—
168C4×C173—21no—
169C4×C173—21no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173C2×#4111yescharacteristic
173C4×C173C2×#4111yescharacteristic
174Dic173—21no—
175Dic173—21no—
176Dic173—21no—
177Dic173—41no—
178Dic173—41no—
179C2×C173—42no—
180C2×C173—81no—
181C2×C173#52511yescharacteristic
182C2×C173#57611yescharacteristic
183C2×C173#57611yescharacteristic
184D173—41no—
185D173—42no—
186D173—81no—
187C173#312211yescharacteristic, complemented, Sylow
188#3122—1731nomaximal, Sylow
189#540—1731no—
190#518—1731no—
191#482—1731no—
192#482—1731no—
193#478—1731no—
194#478—1731no—
195#459—1731no—
196#459—1731no—
197#447—1731no—
198#447—1731no—
199#442—1731no—
200#442—1731no—
201C4×D8—1731no—
202C4×D8D17311yescharacteristic, complemented
203#433—1731no—
204C22×D4—1731no—
205C2×D8—1731no—
206C2×D8—1731no—
207C2×D8—1731no—
208C2×D8C2×D17311yescharacteristic
209#127—1731no—
210#125—1731no—
211#123—3461no—
212#123—3461no—
213#121—3461no—
214#121—3461no—
215#121—1731no—
216#121—1731no—
217C4×D4—3461no—
218C4×D4—3461no—
219C4×D4—1731no—
220C4×D4—1731no—
221C4×D4—1731no—
222C4×D4—1731no—
223C4×D4C2×D17311yescharacteristic
224C4×D4C2×D17311yescharacteristic, complemented
225#117—1731no—
226#117—1731no—
227C2×C22⋊C4—3461no—
228C2×C22⋊C4—3461no—
229#107—1731no—
230#107C2×D17311yescharacteristic, complemented
231#106—1731no—
232#106—1731no—
233C4⋊C8—1731no—
234C4⋊C8—1731no—
235#102—1731no—
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102C2×D17311yescharacteristic
242#102C2×D17311yescharacteristic, complemented
243C22⋊C8—1731no—
244C22⋊C8—1731no—
245#97—1731no—
246C4×C8C2×D17311yescharacteristic, complemented
247C24—3461no—
248C2×D4—3461no—
249C2×D4—3462no—
250C2×D4—3462no—
251C2×D4—3461no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4—1731no—
257C2×D4C22×D17311yescharacteristic
258C2×D4C22×D17311yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—3461no—
262C22×C4—3461no—
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4—1731no—
266D8—22no—
267D8—3462no—
268D8—3462no—
269D8—3462no—
270C2×C8—1731no—
271C2×C8—1731no—
272C2×C8C22×D17311yescharacteristic
273C2×C8C22×D17311yescharacteristic
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—1731no—
277C4⋊C4—1731no—
278C4⋊C4—1731no—
279C4⋊C4—1731no—
280C4⋊C4C22×D17311yescharacteristic
281C4⋊C4C22×D17311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—1731no—
292C22⋊C4—1731no—
293C42—3461no—
294C42—1731no—
295C42C22×D17311yescharacteristic
296C23—21no—
297C23—21no—
298C23—3464no—
299C23—3461no—
300C23—3461no—
301C23—13841no—
302C23—13841no—
303C23—1731no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—3464no—
308D4—3461no—
309D4—3461no—
310D4—3461no—
311D4—13841no—
312D4—13841no—
313D4—13841no—
314D4—13841no—
315D4—1732no—
316D4#1458382612yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—13841no—
326C2×C4—13841no—
327C2×C4—13841no—
328C2×C4—13841no—
329C2×C4—1731no—
330C2×C4—1731no—
331C2×C4—1731no—
332C2×C4D4×D17311yescharacteristic
333C2×C4D4×D17311yescharacteristic
334C2×C4C23×D17311yescharacteristic
335C8—21no—
336C8—21no—
337C8—3461no—
338C8—3461no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—3461no—
346C22—3462no—
347C22—3461no—
348C22—3461no—
349C22—13844no—
350C22—6921no—
351C22—6921no—
352C22—6921no—
353C22—6921no—
354C22—6921no—
355C22C2×D4×D17311yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—3461no—
361C4—3461no—
362C4—3461no—
363C4—6921no—
364C4—6921no—
365C4#41×D17311yescharacteristic
366C4C2×#1458382611yescharacteristic
367C2—42no—
368C2—81no—
369C2—13841no—
370C2—6921no—
371C2—6922no—
372C2#2601200011yescharacteristic
373C2#2601229211yescharacteristic
374C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×D8×C173C2
socleC22×C173C22×D4
hypercenterC4×D8D173
hypocenterC173#3122
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #26012092C2semidirect
2. #26011963C2semidirect
3. #26011818C2semidirect
4. #26011813C2semidirect
5. #26011794C2semidirect
6. #26011789C2semidirect
7. #26011685C2semidirect
8. #26011684C2semidirect
9. #26011626C2semidirect
10. #26011623C2semidirect
11. #26011601C2semidirect
12. #26011601C2semidirect
13. C4×D1384C2semidirect
14. #26011550C2semidirect
15. C4×D8×C173C2semidirect
20. #19149675C22semidirect
21. #19149670C22semidirect
29. C4×D692C22semidirect
38. #19149613C22semidirect
39. #19149610C22semidirect
40. #19149610C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
44. #19149601C22semidirect
46. #19149600C22semidirect
48. #19149599C22semidirect
51. #19149590C22semidirect
54. C4×D4×C173C22semidirect
55. #107×C173C22semidirect
57. #102×C173C22semidirect
58. C4×C8×C173C22semidirect
187. C173#3122semidirect
202. C4×D8D173semidirect
224. C4×D4C2×D173semidirect
230. #107C2×D173semidirect
242. #102C2×D173semidirect
246. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120922
#260119632
#260118182
#260118132
#260117942
#260117892
#260116852
#260116842
#260116262
#260116232
#260116012
C4×D13842
#260115502
C4×D8×C1732
#3122173