Finite Group Databasebeta

← #36363502

Subgroups — #36363502

Order 22144.

462 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×D173C211yesmaximal, characteristic
2#117×D173C211yesmaximal, characteristic, complemented
3#26011816C211yesmaximal, characteristic, complemented
4#26011815C211yesmaximal, characteristic, complemented
5#26011812C211yesmaximal, characteristic, complemented
6#26011811C211yesmaximal, characteristic, complemented
7#26011808C211yesmaximal, characteristic, complemented
8#26011806C211yesmaximal, characteristic, complemented
9#26011803C211yesmaximal, characteristic, complemented
10#26011532C211yesmaximal, characteristic, complemented
11#26011530C211yesmaximal, characteristic, complemented
12#26011494C211yesmaximal, characteristic, complemented
13#26011494C211yesmaximal, characteristic, complemented
14#26011453C211yesmaximal, characteristic, complemented
15#482×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173—21no—
17C2×D4×D173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19C2×#14583809C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21#19149654C2211yescharacteristic
22#19149653C2211yescharacteristic
23#19149652—21no—
24#19149651—21no—
25#19149650C2211yescharacteristic
26#19149638C2211yescharacteristic, complemented
27#19149637C2211yescharacteristic
28#19149637C2211yescharacteristic
29#19149636—21no—
30#19149635—21no—
31C4⋊C4×D173C2211yescharacteristic
32C4⋊C4×D173C2211yescharacteristic, complemented
33C4⋊C4×D173C2211yescharacteristic, complemented
34C22⋊C4×D173—21no—
35C22⋊C4×D173—21no—
36C42×D173—21no—
37#19149613C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149611C2211yescharacteristic, complemented
41#19149610C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic, complemented
48#19149599C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52#121×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic, complemented
54#107×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56#102×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C23×D173D411yescharacteristic
61C2×D692—21no—
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66D4×D173—22no—
67D4×D173—22no—
68D4×D173—44no—
69C22×Dic173D411yescharacteristic
70C2×C4×D173—22no—
71C2×C4×D173—22no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173D411yescharacteristic
76C2×C4×D173D411yescharacteristic
77C2×C4×D173C2311yescharacteristic
78C2×C4×D173C2311yescharacteristic
79#14583809—22no—
80#14583809—22no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173—21no—
98C2×D4×C173—21no—
99C2×D4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C42×C173—21no—
109C22×D173—22no—
110C22×D173—22no—
111C22×D173—21no—
112C22×D173—41no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—42no—
116C22×D173C2×D411yescharacteristic
117C22×D173C2×D411yescharacteristic
118D692—21no—
119D692—21no—
120D692—42no—
121C22⋊D173—42no—
122C22⋊D173—42no—
123C22⋊D173—42no—
124C2×Dic173—22no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C4×D173—22no—
131C4×D173—21no—
132C4×D173—21no—
133C4×D173—22no—
134C4×D173—42no—
135C4×D173—42no—
136C4×D173—42no—
137C4×D173#4112yes—
138C4×D173#4112yes—
139#13063436—21no—
140#13063436—21no—
141C23×C173—21no—
142C23×C173C2×D411yescharacteristic
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—42no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2411yescharacteristic
153C8×C173—21no—
154C8×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173C22×D411yescharacteristic
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—21no—
165C2×D173—21no—
166C2×D173—44no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—84no—
171C2×D173#12312yes—
172C2×D173AGL(1,ℤ/8)12yes—
173C4×C173—21no—
174C4×C173—41no—
175C4×C173—41no—
176C4×C173—41no—
177C4×C173C2×#4111yescharacteristic
178C4×C173C2×#4111yescharacteristic
179Dic173—21no—
180Dic173—41no—
181Dic173—41no—
182Dic173—41no—
183Dic173#12312yes—
184C2×C173—41no—
185C2×C173—81no—
186C2×C173C2×#12311yescharacteristic
187C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C173#57511yescharacteristic
189D173—22no—
190D173—22no—
191D173—81no—
192C173#309411yescharacteristic, complemented, Sylow
193#3094—1731nomaximal, Sylow
194C2×#121—1731no—
195C2×#117—1731no—
196#482—1731no—
197#482—1731no—
198#482—1731no—
199#482D17311yescharacteristic, complemented
200#481—1731no—
201#481—1731no—
202#480—1731no—
203#480—1731no—
204#428—1731no—
205#428—1731no—
206#418—1731no—
207#418—1731no—
208#407—1731no—
209C22×D4—3461no—
210C22×D4—1731no—
211C23×C4—1731no—
212C2×M16—1731no—
213C2×M16—1731no—
214#121—3461no—
215#121—3461no—
216#121—1731no—
217#121—1731no—
218#121—1731no—
219#121C2×D17311yescharacteristic
220#117—3461no—
221#117—3461no—
222#117—1731no—
223#117—1731no—
224#117—1731no—
225#117C2×D17311yescharacteristic, complemented
226C2×C4⋊C4—1731no—
227C2×C4⋊C4—1731no—
228C2×C4⋊C4—1731no—
229C2×C22⋊C4—3461no—
230C2×C22⋊C4—3461no—
231C2×C42—3461no—
232#107—1731no—
233#107—1731no—
234#107—1731no—
235#107C2×D17311yescharacteristic, complemented
236#106—1731no—
237#106—1731no—
238#106—1731no—
239#106C2×D17311yescharacteristic, complemented
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102C2×D17311yescharacteristic, complemented
247#102C2×D17311yescharacteristic, complemented
248C22⋊C8—1731no—
249C22⋊C8—1731no—
250C22⋊C8—1731no—
251C22⋊C8C2×D17311yescharacteristic, complemented
252C24—3461no—
253C24—1731no—
254C2×D4—21no—
255C2×D4—3461no—
256C2×D4—3461no—
257C2×D4—3461no—
258C2×D4—3462no—
259C2×D4—3462no—
260C2×D4—3461no—
261C2×D4—1731no—
262C2×D4—6924no—
263C2×D4C22×D17311yescharacteristic
264C22×C4—3462no—
265C22×C4—3462no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4—1731no—
270C22×C4—1731no—
271C22×C4—1731no—
272C22×C4—1731no—
273C22×C4—1731no—
274C22×C4C22×D17311yescharacteristic
275M16—3462no—
276M16—3462no—
277C2×C8—1731no—
278C2×C8—1731no—
279C2×C8C22×D17311yescharacteristic
280C2×C8C22×D17311yescharacteristic
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—1731no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4C22×D17311yescharacteristic
288C4⋊C4C22×D17311yescharacteristic
289C4⋊C4C22×D17311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C42—21no—
299C42—3461no—
300C42—3461no—
301C42—3461no—
302C23—21no—
303C23—3462no—
304C23—3462no—
305C23—3461no—
306C23—1731no—
307C23—1731no—
308C23—6921no—
309C23—6922no—
310C23—6922no—
311C23—6922no—
312C23D4×D17311yescharacteristic
313D4—21no—
314D4—21no—
315D4—42no—
316D4—3461no—
317D4—3461no—
318D4—6922no—
319D4—6922no—
320D4—6922no—
321D4—6922no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—3462no—
328C2×C4—3462no—
329C2×C4—3462no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—1731no—
336C2×C4—1731no—
337C2×C4—1732no—
338C2×C4—1732no—
339C2×C4—6922no—
340C2×C4—6922no—
341C2×C4—6922no—
342C2×C4D4×D17311yescharacteristic
343C2×C4C23×D17311yescharacteristic
344C8—21no—
345C8—21no—
346C8—3461no—
347C8—3461no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—3461no—
355C22—3461no—
356C22—3461no—
357C22—3461no—
358C22—13844no—
359C22—1732no—
360C22—1732no—
361C22—6924no—
362C22—6921no—
363C22—6921no—
364C22—6921no—
365C22C2×D4×D17311yescharacteristic
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—3461no—
371C4—1732no—
372C4—6921no—
373C4—6921no—
374C4—6921no—
375C4#41×D17311yescharacteristic
376C4#41×D17311yescharacteristic
377C2—41no—
378C2—81no—
379C2—3462no—
380C2—3462no—
381C2—13841no—
382C2#123×D17311yescharacteristic
383C2AGL(1,ℤ/8)×D17311yescharacteristic
384C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#482×C173C2
socleC22×C173C22×D4
prime radical#121×D173C2
hypercenter#482D173
hypocenterC173#3094
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #117×D173C2semidirect
3. #26011816C2semidirect
4. #26011815C2semidirect
5. #26011812C2semidirect
6. #26011811C2semidirect
7. #26011808C2semidirect
8. #26011806C2semidirect
9. #26011803C2semidirect
10. #26011532C2semidirect
11. #26011530C2semidirect
12. #26011494C2semidirect
13. #26011494C2semidirect
14. #26011453C2semidirect
15. #482×C173C2semidirect
26. #19149638C22semidirect
32. C4⋊C4×D173C22semidirect
33. C4⋊C4×D173C22semidirect
37. #19149613C22semidirect
38. #19149612C22semidirect
39. #19149612C22semidirect
40. #19149611C22semidirect
41. #19149610C22semidirect
42. #19149610C22semidirect
43. #19149601C22semidirect
44. #19149601C22semidirect
45. #19149600C22semidirect
46. #19149600C22semidirect
47. #19149599C22semidirect
48. #19149599C22semidirect
49. C22⋊#13063436C22semidirect
50. #19149591C22semidirect
51. #19149591C22semidirect
53. #117×C173C22semidirect
54. #107×C173C22semidirect
55. #106×C173C22semidirect
56. #102×C173C22semidirect
57. #102×C173C22semidirect
58. C22⋊C8×C173C22semidirect
192. C173#3094semidirect
199. #482D173semidirect
225. #117C2×D173semidirect
235. #107C2×D173semidirect
239. #106C2×D173semidirect
246. #102C2×D173semidirect
247. #102C2×D173semidirect
251. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#121×D1732
#117×D1732
#260118162
#260118152
#260118122
#260118112
#260118082
#260118062
#260118032
#260115322
#260115302
#260114942
#260114532
#482×C1732
#3094173