Finite Group Databasebeta

← #36363753

Subgroups — #36363753

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#26012113C211yesmaximal, characteristic, complemented
2#26011963C211yesmaximal, characteristic, complemented
3#26011831C211yesmaximal, characteristic, complemented
4#26011817C211yesmaximal, characteristic, complemented
5#26011780C211yesmaximal, characteristic, complemented
6#26011769C211yesmaximal, characteristic, complemented
7#26011709C211yesmaximal, characteristic, complemented
8#26011708C211yesmaximal, characteristic, complemented
9#26011640C211yesmaximal, characteristic, complemented
10#26011626C211yesmaximal, characteristic
11#26011601C211yesmaximal, characteristic, complemented
12#26011589C211yesmaximal, characteristic, complemented
13C4×D1384C211yesmaximal, characteristic, complemented
14#26011550C211yesmaximal, characteristic, complemented
15C4×SD16×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173C2211yescharacteristic
17C2×#14583813C2211yescharacteristic
18C2×D1384C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20#19149678C2211yescharacteristic, complemented
21#19149673—21no—
22#19149668C2211yescharacteristic, complemented
23#19149667—21no—
24#19149664—21no—
25#19149657—21no—
26#19149655C2211yescharacteristic
27#19149654C2211yescharacteristic
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#19149612C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic, complemented
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic, complemented
54C4×D4×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic, complemented
56#103×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C2×Dic346—21no—
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#14583813—22no—
73D1384—22no—
74#14583811—22no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
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×Dic173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99SD16×C173—22no—
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—24no—
110C22×D173—21no—
111C22×D173—81no—
112C22×D173—81no—
113C22×D173C2×D411yescharacteristic
114Dic346—81no—
115D692—21no—
116D692—21no—
117D692—21no—
118D692—81no—
119D692#4112yes—
120C22⋊D173—24no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—81no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C4×D173—21no—
128C4×D173—21no—
129C4×D173—81no—
130C4×D173—81no—
131C4×D173—81no—
132#13063436—21no—
133#13063436—21no—
134C23×C173—21no—
135Q8×C173—21no—
136Q8×C173—21no—
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#476—1731no—
188#474—1731no—
189#463—1731no—
190#463—1731no—
191#450—1731no—
192#447—1731no—
193#442—1731no—
194#440—1731no—
195C4×SD16D17311yescharacteristic, complemented
196C4×D8—1731no—
197#433—1731no—
198C22×D4—1731no—
199C2×SD16—1731no—
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⋊Q8—1731no—
210#121—1731no—
211C4×Q8C2×D17311yescharacteristic, complemented
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#106—1731no—
226#106C2×D17311yescharacteristic, complemented
227C4⋊C8—1731no—
228C4⋊C8—1731no—
229#103—1731no—
230#103C2×D17311yescharacteristic, complemented
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235#102—1731no—
236#102C2×D17311yescharacteristic
237C22⋊C8—1731no—
238C22⋊C8—1731no—
239#97—1731no—
240C4×C8C2×D17311yescharacteristic, complemented
241C24—3461no—
242C2×Q8—3461no—
243C2×Q8C22×D17311yescharacteristic
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—22no—
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⋊C4—1731no—
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—21no—
287C42—3461no—
288C42—1731no—
289C42C22×D17311yescharacteristic
290C23—21no—
291C23—3464no—
292C23—3461no—
293C23—3461no—
294C23—13841no—
295C23—13841no—
296C23—1731no—
297Q8—21no—
298Q8—21no—
299Q8—13841no—
300D4—21no—
301D4—3464no—
302D4—3461no—
303D4—3461no—
304D4—3461no—
305D4—13841no—
306D4—1732no—
307D4#1458382612yes—
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×#1458382611yescharacteristic
356C2—42no—
357C2—13841no—
358C2—6921no—
359C2—6922no—
360C2#2601200011yescharacteristic
361C2#2601229311yescharacteristic
362C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×SD16×C173C2
socleC22×C173C22×D4
prime radical#26011626C2
hypercenterC4×SD16D173
hypocenterC173#3118
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #26012113C2semidirect
2. #26011963C2semidirect
3. #26011831C2semidirect
4. #26011817C2semidirect
5. #26011780C2semidirect
6. #26011769C2semidirect
7. #26011709C2semidirect
8. #26011708C2semidirect
9. #26011640C2semidirect
11. #26011601C2semidirect
12. #26011589C2semidirect
13. C4×D1384C2semidirect
14. #26011550C2semidirect
15. C4×SD16×C173C2semidirect
20. #19149678C22semidirect
22. #19149668C22semidirect
29. C4×D692C22semidirect
38. #19149613C22semidirect
39. #19149612C22semidirect
40. #19149610C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
43. #19149603C22semidirect
45. #19149601C22semidirect
48. #19149599C22semidirect
51. #19149590C22semidirect
53. C4×Q8×C173C22semidirect
55. #106×C173C22semidirect
56. #103×C173C22semidirect
58. C4×C8×C173C22semidirect
181. C173#3118semidirect
195. C4×SD16D173semidirect
211. C4×Q8C2×D173semidirect
226. #106C2×D173semidirect
230. #103C2×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
#260121132
#260119632
#260118312
#260118172
#260117802
#260117692
#260117092
#260117082
#260116402
#260116262
#260116012
#260115892
C4×D13842
#260115502
C4×SD16×C1732
#3118173