Finite Group Databasebeta

← #36364433

Subgroups — #36364433

Order 22144.

410 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012065C211yesmaximal, characteristic, complemented
2#26012057C211yesmaximal, characteristic
3#26011831C211yesmaximal, characteristic, complemented
4#26011830C211yesmaximal, characteristic, complemented
5#26011816C211yesmaximal, characteristic, complemented
6#26011812C211yesmaximal, characteristic, complemented
7#26011740C211yesmaximal, characteristic, complemented
8#26011734C211yesmaximal, characteristic, complemented
9#26011723C211yesmaximal, characteristic, complemented
10#26011717C211yesmaximal, characteristic, complemented
11#26011640C211yesmaximal, characteristic, complemented
12#26011640C211yesmaximal, characteristic, complemented
13#26011630C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#448×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×#14583809C2211yescharacteristic, complemented
21C2×C8×D173C2211yescharacteristic, complemented
22#19149677—21no—
23#19149675—21no—
24#19149668—21no—
25#19149665—21no—
26#19149662—21no—
27#19149662—21no—
28#19149655C2211yescharacteristic
29#19149655C2211yescharacteristic
30#19149655C2211yescharacteristic, complemented
31#19149655C2211yescharacteristic, complemented
32#19149654C2211yescharacteristic
33#19149653C2211yescharacteristic
34#19149650C2211yescharacteristic
35#19149649—21no—
36#19149647—21no—
37#19149638C2211yescharacteristic, complemented
38#19149637C2211yescharacteristic
39#19149613C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C22×Q8×C173C2211yescharacteristic, complemented
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—41no—
61#14583826—41no—
62C2×Dic346D411yescharacteristic
63C2×D692—21no—
64C2×D692D411yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173D411yescharacteristic
70C2×C22⋊D173D411yescharacteristic
71D4×D173—41no—
72D4×D173—41no—
73C2×C4×D173C2311yescharacteristic
74C2×C4×D173C2311yescharacteristic
75#14583813—41no—
76#14583813—41no—
77#14583809—41no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C8×D173—41no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97C4×Dic173—21no—
98C4×Dic173—21no—
99C2×Q8×C173—21no—
100C2×Q8×C173—41no—
101C2×Q8×C173—41no—
102C2×Q8×C173C2311yescharacteristic
103C2×Q8×C173C2311yescharacteristic
104C2×D4×C173—21no—
105C2×D4×C173C2311yescharacteristic
106C22×C4×C173—21no—
107C22×C4×C173C2311yescharacteristic
108SD16×C173—41no—
109SD16×C173—41no—
110C2×C8×C173C2311yescharacteristic
111C2×C8×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173—42no—
116C22×D173—81no—
117C22×D173—81no—
118C22×D173C2×D411yescharacteristic
119C22×D173C2×D411yescharacteristic
120Dic346—41no—
121D692—21no—
122D692—21no—
123D692—41no—
124D692—81no—
125C22⋊D173—41no—
126C22⋊D173—41no—
127C22⋊D173—42no—
128C22⋊D173—81no—
129C22⋊D173—81no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173C2×D411yescharacteristic
134C2×Dic173C2×D411yescharacteristic
135C4×D173—21no—
136C4×D173—21no—
137C4×D173—21no—
138C4×D173—21no—
139#13063436—21no—
140#13063436—21no—
141C23×C173—21no—
142C23×C173C2×D411yescharacteristic
143Q8×C173—21no—
144Q8×C173—21no—
145Q8×C173—21no—
146Q8×C173—21no—
147Q8×C173—42no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—81no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—42no—
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2411yescharacteristic
159C8×C173—21no—
160C8×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173C22×D411yescharacteristic
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—21no—
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—84no—
178C4×C173—21no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173H(5,𝔽2)11yescharacteristic
183C4×C173H(5,𝔽2)11yescharacteristic
184Dic173—21no—
185Dic173—21no—
186Dic173—41no—
187Dic173—41no—
188Dic173—41no—
189C2×C173—41no—
190C2×C173—81no—
191C2×C173#53411yescharacteristic
192C2×C173#57711yescharacteristic
193C2×C173#57711yescharacteristic
194D173—41no—
195D173—41no—
196D173—81no—
197C173#320011yescharacteristic, complemented, Sylow
198#3200—1731nomaximal, Sylow
199#536—1731no—
200#535—1731no—
201#485—1731no—
202#485—1731no—
203#482—1731no—
204#482—1731no—
205#468—1731no—
206#468—1731no—
207#465—1731no—
208#465—1731no—
209#450—1731no—
210#450—1731no—
211#448—1731no—
212#448D17311yescharacteristic, complemented
213#408—1731no—
214C2×#41—1731no—
215C22×Q8C2×D17311yescharacteristic, complemented
216C22×D4—1731no—
217C2×SD16—1731no—
218C2×SD16—1731no—
219C2×SD16C2×D17311yescharacteristic
220C2×SD16C2×D17311yescharacteristic
221C2×M16—1731no—
222C22×C8—1731no—
223C4⋊Q8—3461no—
224#127—3461no—
225#124—3461no—
226#124—3461no—
227#123—3461no—
228#123—3461no—
229C22⋊Q8—1731no—
230C22⋊Q8—1731no—
231C22⋊Q8—1731no—
232C22⋊Q8—1731no—
233#121—1731no—
234#121—1731no—
235#121—1731no—
236#121C2×D17311yescharacteristic
237#120—3461no—
238#120—3461no—
239#117—1731no—
240#117—1731no—
241#107—1731no—
242#107—1731no—
243#106—1731no—
244#106—1731no—
245#103—1731no—
246#103—1731no—
247#103C2×D17311yescharacteristic, complemented
248#103C2×D17311yescharacteristic, complemented
249#102—1731no—
250#102—1731no—
251#102—1731no—
252#102—1731no—
253C22⋊C8—1731no—
254C22⋊C8—1731no—
255C22⋊C8—1731no—
256C22⋊C8C2×D17311yescharacteristic, complemented
257C24—3461no—
258#41—6921no—
259#41—6921no—
260C2×Q8—21no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—1731no—
264C2×Q8C22×D17311yescharacteristic
265C2×Q8C22×D17311yescharacteristic
266C2×D4—21no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—1731no—
272C2×D4—1731no—
273C2×D4—1731no—
274C2×D4—1731no—
275C2×D4—6921no—
276C2×D4—6921no—
277C2×D4C22×D17311yescharacteristic
278C22×C4—21no—
279C22×C4—1731no—
280C22×C4—1731no—
281C22×C4C22×D17311yescharacteristic
282SD16—41no—
283SD16—41no—
284SD16—6921no—
285SD16—6921no—
286M16—6921no—
287C2×C8—1731no—
288C2×C8—1731no—
289C2×C8—6921no—
290C2×C8C22×D17311yescharacteristic
291C2×C8C22×D17311yescharacteristic
292C4⋊C4—3461no—
293C4⋊C4—3461no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—1731no—
297C4⋊C4—1731no—
298C4⋊C4—1731no—
299C4⋊C4C22×D17311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C42—3461no—
311C42—3461no—
312C23—21no—
313C23—3461no—
314C23—13841no—
315C23—13841no—
316C23—1731no—
317C23—1731no—
318C23—6922no—
319C23D4×D17311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—6921no—
326D4—21no—
327D4—21no—
328D4—81no—
329D4—3461no—
330D4—3461no—
331D4—13841no—
332D4—13841no—
333D4—13841no—
334D4—6921no—
335D4—6921no—
336D4—6921no—
337D4—6922no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—3461no—
351C2×C4—1731no—
352C2×C4—1731no—
353C2×C4D4×D17311yescharacteristic
354C2×C4C23×D17311yescharacteristic
355C8—21no—
356C8—21no—
357C8—3461no—
358C8—3461no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—3461no—
366C22—3461no—
367C22—3461no—
368C22—3461no—
369C22—3461no—
370C22—3461no—
371C22—13844no—
372C22—6921no—
373C22—6921no—
374C22—6921no—
375C22C2×D4×D17311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—3461no—
381C4—3461no—
382C4—6921no—
383C4—6921no—
384C4—6921no—
385C4#1914972411yescharacteristic
386C4#1914972411yescharacteristic
387C2—41no—
388C2—81no—
389C2—13841no—
390C2—6921no—
391C2—6921no—
392C2#2601205111yescharacteristic
393C2#2601229711yescharacteristic
394C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#448×C173C2
socleC22×C173C22×D4
prime radical#26012057C2
hypercenter#448D173
hypocenterC173#3200
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #26012065C2semidirect
3. #26011831C2semidirect
4. #26011830C2semidirect
5. #26011816C2semidirect
6. #26011812C2semidirect
7. #26011740C2semidirect
8. #26011734C2semidirect
9. #26011723C2semidirect
10. #26011717C2semidirect
11. #26011640C2semidirect
12. #26011640C2semidirect
13. #26011630C2semidirect
14. #26011456C2semidirect
15. #448×C173C2semidirect
20. C2×#14583809C22semidirect
21. C2×C8×D173C22semidirect
30. #19149655C22semidirect
31. #19149655C22semidirect
37. #19149638C22semidirect
39. #19149613C22semidirect
40. #19149612C22semidirect
41. #19149611C22semidirect
42. #19149610C22semidirect
43. #19149603C22semidirect
44. #19149603C22semidirect
49. C22⋊#13063436C22semidirect
52. C22×Q8×C173C22semidirect
56. #103×C173C22semidirect
57. #103×C173C22semidirect
58. C22⋊C8×C173C22semidirect
197. C173#3200semidirect
212. #448D173semidirect
215. C22×Q8C2×D173semidirect
247. #103C2×D173semidirect
248. #103C2×D173semidirect
256. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120652
#260120572
#260118312
#260118302
#260118162
#260118122
#260117402
#260117342
#260117232
#260117172
#260116402
#260116302
#260114562
#448×C1732
#3200173