Finite Group Databasebeta

← #36364409

Subgroups — #36364409

Order 22144.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012055C211yesmaximal, characteristic, complemented
2#26011829C211yesmaximal, characteristic, complemented
3#26011812C211yesmaximal, characteristic, complemented
4#26011809C211yesmaximal, characteristic, complemented
5#26011804C211yesmaximal, characteristic, complemented
6#26011742C211yesmaximal, characteristic, complemented
7#26011739C211yesmaximal, characteristic, complemented
8#26011725C211yesmaximal, characteristic, complemented
9#26011722C211yesmaximal, characteristic, complemented
10#26011641C211yesmaximal, characteristic, complemented
11#26011636C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic, complemented
13#26011624C211yesmaximal, characteristic, complemented
14#26011458C211yesmaximal, characteristic, complemented
15#537×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C2×#14583813C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C2×#14583810C2211yescharacteristic, complemented
22C22×#13063436C2211yescharacteristic, complemented
23#19149675—21no—
24#19149668—21no—
25#19149653C2211yescharacteristic
26#19149653C2211yescharacteristic
27#19149653C2211yescharacteristic
28#19149653C2211yescharacteristic
29#19149647—21no—
30#19149647—21no—
31#19149638C2211yescharacteristic
32#19149612C2211yescharacteristic, complemented
33#19149612C2211yescharacteristic, complemented
34#19149610C2211yescharacteristic, complemented
35#19149610C2211yescharacteristic, complemented
36#19149603C2211yescharacteristic, complemented
37#19149603C2211yescharacteristic, complemented
38#19149600C2211yescharacteristic, complemented
39#19149600C2211yescharacteristic, complemented
40#19149599C2211yescharacteristic
41#19149599C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic, complemented
47C22⋊#13063436C2211yescharacteristic, complemented
48C2×#41×C173C2211yescharacteristic
49C2×#41×C173C2211yescharacteristic, complemented
50C22⋊Q8×C173—21no—
51C22⋊Q8×C173C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic, complemented
53#121×C173—21no—
54#121×C173—21no—
55#121×C173—21no—
56#121×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic, complemented
58C2×C4⋊C4×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C2×D692—21no—
61C2×D692—41no—
62C2×D692—41no—
63C2×D692C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69#14583813—41no—
70#14583813—41no—
71#14583811—41no—
72#14583811—41no—
73#14583810—41no—
74C2×#13063436—41no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C22⋊Dic173—21no—
82#14583803—21no—
83#14583803—21no—
84#14583803—21no—
85#14583803—21no—
86C4×Dic173—21no—
87#41×C173—41no—
88#41×C173—41no—
89#41×C173—41no—
90#41×C173—41no—
91C2×Q8×C173D411yescharacteristic
92C2×Q8×C173C2311yescharacteristic
93C2×D4×C173—21no—
94C2×D4×C173—21no—
95C2×D4×C173—21no—
96C2×D4×C173D411yescharacteristic
97C2×D4×C173D411yescharacteristic
98C2×D4×C173D411yescharacteristic
99C2×D4×C173C2311yescharacteristic
100C22×C4×C173—21no—
101C22×C4×C173—21no—
102C22×C4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173—21no—
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—42no—
117C22×D173—81no—
118C22×D173—81no—
119D692—21no—
120D692—21no—
121D692—21no—
122D692—21no—
123D692—42no—
124C22⋊D173—81no—
125C22⋊D173—81no—
126C22⋊D173—81no—
127C22⋊D173—81no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130#13063436—21no—
131#13063436—21no—
132#13063436—21no—
133#13063436—21no—
134C23×C173—21no—
135C23×C173C2×D411yescharacteristic
136C23×C173C2×D411yescharacteristic
137C23×C173C2×D411yescharacteristic
138Q8×C173—21no—
139Q8×C173—21no—
140Q8×C173—41no—
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—41no—
144D4×C173—41no—
145D4×C173—41no—
146D4×C173—42no—
147D4×C173—81no—
148D4×C173—81no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173—42no—
159C2×C4×C173—81no—
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2×D411yescharacteristic
162C2×C4×C173C2×D411yescharacteristic
163C2×C4×C173C2411yescharacteristic
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—21no—
167C22×C173—21no—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—21no—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173C22×D411yescharacteristic
177C2×D173—41no—
178C2×D173—41no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—84no—
184C4×C173—21no—
185C4×C173—21no—
186C4×C173—21no—
187C4×C173—41no—
188C4×C173—41no—
189C4×C173—41no—
190C4×C173H(5,𝔽2)11yescharacteristic
191C4×C173H(5,𝔽2)11yescharacteristic
192Dic173—41no—
193Dic173—41no—
194C2×C173—41no—
195C2×C173—41no—
196C2×C173—41no—
197C2×C173—81no—
198C2×C173#53411yescharacteristic
199C2×C173#57611yescharacteristic
200C2×C173#57711yescharacteristic
201D173—81no—
202D173—81no—
203C173#319811yescharacteristic, complemented, Sylow
204#3198—1731nomaximal, Sylow
205#537D17311yescharacteristic, complemented
206#535—1731no—
207#485—1731no—
208#482—1731no—
209#481—1731no—
210#480—1731no—
211#469—1731no—
212#468—1731no—
213#466—1731no—
214#465—1731no—
215#450—1731no—
216#449—1731no—
217#449—1731no—
218#447—1731no—
219#408—1731no—
220C2×#41C2×D17311yescharacteristic
221C2×#41C2×D17311yescharacteristic, complemented
222C22×D4—1731no—
223C2×SD16—1731no—
224C2×SD16—1731no—
225C2×D8—1731no—
226C2×D8—1731no—
227C2×M16—1731no—
228C22×C8—1731no—
229#127—3461no—
230#124—3461no—
231C22⋊Q8—21no—
232C22⋊Q8C2×D17311yescharacteristic
233C22⋊Q8C2×D17311yescharacteristic, complemented
234#121—21no—
235#121—21no—
236#121—21no—
237#121—1731no—
238#121—1731no—
239#121—1731no—
240#121—1731no—
241#121C2×D17311yescharacteristic
242#121C2×D17311yescharacteristic, complemented
243#120—3461no—
244#120—3461no—
245#117—1731no—
246C2×C4⋊C4C2×D17311yescharacteristic, complemented
247#107—1731no—
248#107—1731no—
249#106—1731no—
250#106—1731no—
251#103—1731no—
252#103—1731no—
253#102—1731no—
254#102—1731no—
255#102—1731no—
256#102—1731no—
257#102—1731no—
258#102—1731no—
259C22⋊C8—1731no—
260C22⋊C8—1731no—
261C22⋊C8—1731no—
262C22⋊C8—1731no—
263C24—3461no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8C22⋊D17311yescharacteristic
269C2×Q8C22×D17311yescharacteristic
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—3461no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—3461no—
278C2×D4—1731no—
279C2×D4—1731no—
280C2×D4—6921no—
281C2×D4—6921no—
282C2×D4C22⋊D17311yescharacteristic
283C2×D4C22⋊D17311yescharacteristic
284C2×D4C22⋊D17311yescharacteristic
285C2×D4C22×D17311yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4C22×D17311yescharacteristic
289C22×C4C22×D17311yescharacteristic
290C22×C4C22×D17311yescharacteristic
291SD16—6921no—
292SD16—6921no—
293D8—6921no—
294D8—6921no—
295M16—6921no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8—1731no—
299C2×C8—1731no—
300C2×C8—6921no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—1731no—
305C4⋊C4—1731no—
306C4⋊C4C22×D17311yescharacteristic
307C4⋊C4C22×D17311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3461no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C42—3461no—
318C23—21no—
319C23—3461no—
320C23—3461no—
321C23—13841no—
322C23—13841no—
323C23—6922no—
324C23C2×C22⋊D17311yescharacteristic
325C23C2×C22⋊D17311yescharacteristic
326C23C2×C22⋊D17311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—41no—
330D4—21no—
331D4—21no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—42no—
336D4—81no—
337D4—81no—
338D4—3461no—
339D4—3461no—
340D4—3461no—
341D4—3461no—
342D4—13841no—
343D4—13841no—
344D4—13841no—
345D4—13841no—
346D4—6922no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4C2×C22⋊D17311yescharacteristic
361C2×C4C2×C22⋊D17311yescharacteristic
362C2×C4C2×C22⋊D17311yescharacteristic
363C2×C4C23×D17311yescharacteristic
364C8—3461no—
365C8—3461no—
366C8—3461no—
367C8—3461no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—13844no—
381C22—6921no—
382C22—6921no—
383C22—6921no—
384C22—6921no—
385C22—6921no—
386C22—6921no—
387C22C22×C22⋊D17311yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—6921no—
395C4—6921no—
396C4#1914972411yescharacteristic
397C4#1914972411yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—81no—
402C2—13841no—
403C2—13841no—
404C2#2601205011yescharacteristic
405C2#2601229411yescharacteristic
406C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#537×C173C2
socleC22×C173C22×D4
hypercenter#537D173
hypocenterC173#3198
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #26012055C2semidirect
2. #26011829C2semidirect
3. #26011812C2semidirect
4. #26011809C2semidirect
5. #26011804C2semidirect
6. #26011742C2semidirect
7. #26011739C2semidirect
8. #26011725C2semidirect
9. #26011722C2semidirect
10. #26011641C2semidirect
11. #26011636C2semidirect
12. #26011636C2semidirect
13. #26011624C2semidirect
14. #26011458C2semidirect
15. #537×C173C2semidirect
21. C2×#14583810C22semidirect
22. C22×#13063436C22semidirect
32. #19149612C22semidirect
33. #19149612C22semidirect
34. #19149610C22semidirect
35. #19149610C22semidirect
36. #19149603C22semidirect
37. #19149603C22semidirect
38. #19149600C22semidirect
39. #19149600C22semidirect
46. C22⋊#13063436C22semidirect
47. C22⋊#13063436C22semidirect
49. C2×#41×C173C22semidirect
52. C22⋊Q8×C173C22semidirect
57. #121×C173C22semidirect
58. C2×C4⋊C4×C173C22semidirect
203. C173#3198semidirect
205. #537D173semidirect
221. C2×#41C2×D173semidirect
233. C22⋊Q8C2×D173semidirect
242. #121C2×D173semidirect
246. C2×C4⋊C4C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120552
#260118292
#260118122
#260118092
#260118042
#260117422
#260117392
#260117252
#260117222
#260116412
#260116362
#260116242
#260114582
#537×C1732
#3198173