Finite Group Databasebeta

← #36364659

Subgroups — #36364659

Order 22144.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012112C211yesmaximal, characteristic, complemented
2#26012060C211yesmaximal, characteristic, complemented
3#26011880C211yesmaximal, characteristic, complemented
4#26011875C211yesmaximal, characteristic, complemented
5#26011851C211yesmaximal, characteristic, complemented
6#26011838C211yesmaximal, characteristic, complemented
7#26011714C211yesmaximal, characteristic, complemented
8#26011707C211yesmaximal, characteristic, complemented
9#26011692C211yesmaximal, characteristic, complemented
10#26011687C211yesmaximal, characteristic, complemented
11#26011655C211yesmaximal, characteristic, complemented
12#26011644C211yesmaximal, characteristic, complemented
13#26011638C211yesmaximal, characteristic, complemented
14#26011627C211yesmaximal, characteristic, complemented
15#433×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×Dic692C2211yescharacteristic, complemented
19C2×#14583815C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C2×#14583815C2211yescharacteristic, complemented
22C2×#14583815C2211yescharacteristic, complemented
23C2×D1384C2211yescharacteristic
24C2×D1384C2211yescharacteristic
25C2×D1384C2211yescharacteristic, complemented
26#19149679C2211yescharacteristic, complemented
27#19149676C2211yescharacteristic
28#19149676C2211yescharacteristic
29#19149674—21no—
30#19149667C2211yescharacteristic
31#19149667C2211yescharacteristic
32#19149667C2211yescharacteristic, complemented
33#19149665—21no—
34#19149661—21no—
35#19149658C2211yescharacteristic
36#19149657—21no—
37#19149654—21no—
38#19149654—21no—
39#19149653C2211yescharacteristic
40#19149649—21no—
41#19149649—21no—
42C4×Dic346C2211yescharacteristic, complemented
43C4×D692C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149604C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic, complemented
52#117×C173C2211yescharacteristic
53C4⋊C8×C173C2211yescharacteristic, complemented
54C4⋊C8×C173C2211yescharacteristic, complemented
55C22⋊C8×C173C2211yescharacteristic
56C22⋊C8×C173C2211yescharacteristic
57#97×C173C2211yescharacteristic, complemented
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—41no—
61#14583826—41no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—21no—
68C2×D692—21no—
69C2×D692—21no—
70C2×D692—41no—
71C2×D692—41no—
72C2×D692C2311yescharacteristic
73C2×D692C2311yescharacteristic
74C2×D692C2311yescharacteristic
75C2×D692C2311yescharacteristic
76C2×C22⋊D173—21no—
77C2×C22⋊D173—21no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80Dic692—41no—
81#14583815—41no—
82#14583815—41no—
83#14583815—41no—
84#14583815—41no—
85D1384—41no—
86D1384—41no—
87D1384—41no—
88C4⋊Dic173—21no—
89C4⋊Dic173C2311yescharacteristic
90C4⋊Dic173C2311yescharacteristic
91#14583805—21no—
92#14583805—21no—
93C22⋊Dic173—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102C4×Dic173—21no—
103C22×C4×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173D411yescharacteristic
109C4⋊C4×C173D411yescharacteristic
110C22⋊C4×C173D411yescharacteristic
111C22⋊C4×C173D411yescharacteristic
112C42×C173C2311yescharacteristic
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—21no—
117C22×D173—21no—
118C22×D173—42no—
119C22×D173—81no—
120C22×D173—81no—
121Dic346—21no—
122Dic346—21no—
123Dic346—21no—
124Dic346—21no—
125Dic346—81no—
126D692—21no—
127D692—21no—
128D692—21no—
129D692—21no—
130D692—21no—
131D692—21no—
132D692—21no—
133D692—21no—
134D692—42no—
135D692—81no—
136D692—81no—
137D692—81no—
138D692—81no—
139C22⋊D173—42no—
140C22⋊D173—81no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—21no—
144C2×Dic173—21no—
145C4×D173—42no—
146C4×D173—81no—
147C23×C173C2×D411yescharacteristic
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2411yescharacteristic
156C8×C173—21no—
157C8×C173—21no—
158C8×C173—21no—
159C8×C173—21no—
160C22×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173C22×D411yescharacteristic
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—84no—
177C4×C173—21no—
178C4×C173—21no—
179C4×C173—21no—
180C4×C173—21no—
181C4×C173—21no—
182C4×C173H(5,𝔽2)11yescharacteristic
183C4×C173H(5,𝔽2)11yescharacteristic
184Dic173—41no—
185Dic173—41no—
186Dic173—41no—
187Dic173—41no—
188C2×C173—41no—
189C2×C173#53511yescharacteristic
190C2×C173#57611yescharacteristic
191C2×C173#57711yescharacteristic
192D173—81no—
193D173—81no—
194D173—81no—
195D173—81no—
196C173#322411yescharacteristic, complemented, Sylow
197#3224—1731nomaximal, Sylow
198#542—1731no—
199#535—1731no—
200#496—1731no—
201#495—1731no—
202#489—1731no—
203#486—1731no—
204#464—1731no—
205#463—1731no—
206#460—1731no—
207#459—1731no—
208#452—1731no—
209#450—1731no—
210#449—1731no—
211#447—1731no—
212#433D17311yescharacteristic, complemented
213C2×#41—1731no—
214C22×D4—1731no—
215C2×Q16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×SD16—1731no—
219C2×SD16—1731no—
220C2×D8—1731no—
221C2×D8—1731no—
222C2×D8—1731no—
223C4⋊Q8—1731no—
224#127—1731no—
225#127—1731no—
226#126—3461no—
227#124—3461no—
228#124—1731no—
229#124—1731no—
230#124—1731no—
231#123—3461no—
232C22⋊Q8—3461no—
233C22⋊Q8—1731no—
234#121—3461no—
235#121—3461no—
236#121—1731no—
237#120—3461no—
238#120—3461no—
239C4×Q8—1731no—
240C4×D4—1731no—
241#117C2×D17311yescharacteristic
242C4⋊C8C2×D17311yescharacteristic, complemented
243C4⋊C8C2×D17311yescharacteristic, complemented
244#103—1731no—
245#103—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102—1731no—
251#102—1731no—
252C22⋊C8C2×D17311yescharacteristic
253C22⋊C8C2×D17311yescharacteristic
254#97C2×D17311yescharacteristic, complemented
255C4×C8C2×D17311yescharacteristic, complemented
256C24—3461no—
257#41—6921no—
258#41—6921no—
259C2×Q8—3461no—
260C2×Q8—1731no—
261C2×Q8—1731no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—1731no—
270C2×D4—1731no—
271C2×D4—1731no—
272C2×D4—1731no—
273C2×D4—6921no—
274C2×D4—6921no—
275C22×C4—3461no—
276C22×C4—3461no—
277C22×C4C22×D17311yescharacteristic
278Q16—6921no—
279SD16—6921no—
280SD16—6921no—
281SD16—6921no—
282SD16—6921no—
283D8—6921no—
284D8—6921no—
285D8—6921no—
286C2×C8C22×D17311yescharacteristic
287C2×C8C22×D17311yescharacteristic
288C2×C8C22×D17311yescharacteristic
289C2×C8C22×D17311yescharacteristic
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—3461no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C4⋊C4D69211yescharacteristic
296C4⋊C4D69211yescharacteristic
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4D69211yescharacteristic
307C22⋊C4D69211yescharacteristic
308C42—3461no—
309C42C22×D17311yescharacteristic
310C42C22×D17311yescharacteristic
311C23—3461no—
312C23—3461no—
313C23—3461no—
314C23—3461no—
315C23—13841no—
316C23—13841no—
317C23—6922no—
318C23C2×D69211yescharacteristic
319Q8—3461no—
320Q8—3461no—
321Q8—3461no—
322Q8—3461no—
323Q8—13841no—
324D4—3461no—
325D4—3461no—
326D4—3461no—
327D4—3461no—
328D4—3461no—
329D4—3461no—
330D4—3461no—
331D4—3461no—
332D4—13841no—
333D4—13841no—
334D4—13841no—
335D4—13841no—
336D4—13841no—
337D4—6922no—
338D4—6922no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—13841no—
346C2×C4—6922no—
347C2×C4C2×D69211yescharacteristic
348C2×C4C2×D69211yescharacteristic
349C2×C4C2×D69211yescharacteristic
350C2×C4C2×D69211yescharacteristic
351C2×C4C2×D69211yescharacteristic
352C2×C4C23×D17311yescharacteristic
353C8—21no—
354C8—21no—
355C8—21no—
356C8—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—13844no—
361C22—6921no—
362C22—6921no—
363C22—6921no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22—6921no—
368C22—6921no—
369C22—6921no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22C22×D69211yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—6921no—
380C4—6921no—
381C4—6921no—
382C4—6921no—
383C4#1914972511yescharacteristic
384C4#1914972511yescharacteristic
385C2—41no—
386C2—13841no—
387C2—13841no—
388C2—13841no—
389C2—13841no—
390C2#2601206011yescharacteristic
391C2#2601229611yescharacteristic
392C2#2601230311yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#433×C173C2
socleC22×C173C22×D4
hypercenter#433D173
hypocenterC173#3224
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #26012112C2semidirect
2. #26012060C2semidirect
3. #26011880C2semidirect
4. #26011875C2semidirect
5. #26011851C2semidirect
6. #26011838C2semidirect
7. #26011714C2semidirect
8. #26011707C2semidirect
9. #26011692C2semidirect
10. #26011687C2semidirect
11. #26011655C2semidirect
12. #26011644C2semidirect
13. #26011638C2semidirect
14. #26011627C2semidirect
15. #433×C173C2semidirect
18. C2×Dic692C22semidirect
21. C2×#14583815C22semidirect
22. C2×#14583815C22semidirect
25. C2×D1384C22semidirect
26. #19149679C22semidirect
32. #19149667C22semidirect
42. C4×Dic346C22semidirect
43. C4×D692C22semidirect
44. #19149604C22semidirect
45. #19149604C22semidirect
50. #19149601C22semidirect
51. #19149601C22semidirect
53. C4⋊C8×C173C22semidirect
54. C4⋊C8×C173C22semidirect
57. #97×C173C22semidirect
58. C4×C8×C173C22semidirect
196. C173#3224semidirect
212. #433D173semidirect
242. C4⋊C8C2×D173semidirect
243. C4⋊C8C2×D173semidirect
254. #97C2×D173semidirect
255. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121122
#260120602
#260118802
#260118752
#260118512
#260118382
#260117142
#260117072
#260116922
#260116872
#260116552
#260116442
#260116382
#260116272
#433×C1732
#3224173