Finite Group Databasebeta

← #36365400

Subgroups — #36365400

Order 22144.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012181C211yesmaximal, characteristic, complemented
2#26012137C211yesmaximal, characteristic
3#26011846C211yesmaximal, characteristic, complemented
4#26011822C211yesmaximal, characteristic, complemented
5#26011816C211yesmaximal, characteristic, complemented
6#26011792C211yesmaximal, characteristic, complemented
7#26011732C211yesmaximal, characteristic, complemented
8#26011720C211yesmaximal, characteristic, complemented
9#26011714C211yesmaximal, characteristic, complemented
10#26011655C211yesmaximal, characteristic, complemented
11#26011644C211yesmaximal, characteristic, complemented
12C4×#14583815C211yesmaximal, characteristic, complemented
13#26011488C211yesmaximal, characteristic, complemented
14C2×#19149601C211yesmaximal, characteristic, complemented
15C8×D4×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×Dic692C2211yescharacteristic, complemented
19C2×#14583815C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21#19149672—21no—
22#19149671C2211yescharacteristic, complemented
23#19149669—21no—
24#19149667C2211yescharacteristic
25#19149664—21no—
26#19149660—21no—
27#19149658C2211yescharacteristic
28#19149658C2211yescharacteristic
29#19149658C2211yescharacteristic, complemented
30#19149653C2211yescharacteristic
31#19149652—21no—
32#19149647—21no—
33C4×Dic346C2211yescharacteristic, complemented
34C4×D692C2211yescharacteristic
35#19149638C2211yescharacteristic, complemented
36#19149635—21no—
37C2×C4⋊Dic173C2211yescharacteristic
38C22⋊C4×D173—21no—
39#19149613C2211yescharacteristic, complemented
40#19149611C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149604C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic
51#19149601C2211yescharacteristic
52C22×C8×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic, complemented
54C4×D4×C173C2211yescharacteristic
55C4⋊C8×C173C2211yescharacteristic, complemented
56C22⋊C8×C173C2211yescharacteristic
57C22⋊C8×C173C2211yescharacteristic, complemented
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—22no—
61#14583826—41no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C2×D692—22no—
65C2×D692—21no—
66C2×D692—41no—
67C2×D692C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74Dic692—41no—
75#14583815—22no—
76#14583815—41no—
77C4⋊Dic173—21no—
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96C4×Dic173—21no—
97C4×Dic173—21no—
98C2×D4×C173D411yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C2×C8×C173—41no—
102C2×C8×C173—41no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173D411yescharacteristic
108C22⋊C4×C173D411yescharacteristic
109C22⋊C4×C173D411yescharacteristic
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—81no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346—21no—
120D692—21no—
121D692—21no—
122D692—21no—
123D692—42no—
124D692#4112yes—
125C22⋊D173—42no—
126C22⋊D173—81no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—42no—
134C4×D173—42no—
135C4×D173—81no—
136C23×C173C2×D411yescharacteristic
137C23×C173C2×D411yescharacteristic
138D4×C173—22no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2411yescharacteristic
148C8×C173—21no—
149C8×C173—21no—
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173#13512yes—
158C22×C173C22×D411yescharacteristic
159C2×D173—22no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—84no—
166C4×C173—21no—
167C4×C173—21no—
168C4×C173—21no—
169C4×C173—21no—
170C4×C173C2×#4111yescharacteristic
171C4×C173H(5,𝔽2)11yescharacteristic
172Dic173—41no—
173Dic173—41no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178C2×C173—22no—
179C2×C173—41no—
180C2×C173#54611yescharacteristic
181C2×C173C2×#13511yescharacteristic
182C2×C173#57711yescharacteristic
183D173—42no—
184D173—81no—
185C173#329611yescharacteristic, complemented, Sylow
186#3296—1731nomaximal, Sylow
187#552—1731no—
188#546—1731no—
189#488—1731no—
190#483—1731no—
191#482—1731no—
192#478—1731no—
193#467—1731no—
194#465—1731no—
195#464—1731no—
196#452—1731no—
197#450—1731no—
198C4×SD16—1731no—
199C8×D4D17311yescharacteristic, complemented
200#416—1731no—
201C2×#102—1731no—
202C2×#41—1731no—
203C22×D4—1731no—
204C2×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16—1731no—
207C22×C8C2×D17311yescharacteristic
208C22×C8C2×D17311yescharacteristic, complemented
209#126—3461no—
210#125—3461no—
211#125—1731no—
212#124—1731no—
213#123—3461no—
214#123—3461no—
215C22⋊Q8—1731no—
216C22⋊Q8—1731no—
217C22⋊Q8—1731no—
218#121—3461no—
219#121—1731no—
220#120—3461no—
221C4×Q8—1731no—
222C4×D4—1731no—
223C4×D4C2×D17311yescharacteristic
224#117—3461no—
225#117—1731no—
226C2×C4⋊C4—1731no—
227C2×C22⋊C4—3461no—
228#107—1731no—
229#106—1731no—
230#106—1731no—
231C4⋊C8C2×D17311yescharacteristic, complemented
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242C22⋊C8C2×D17311yescharacteristic
243C22⋊C8C2×D17311yescharacteristic, complemented
244C4×C8C2×D17311yescharacteristic, complemented
245C24—3461no—
246#41—3462no—
247#41—6921no—
248C2×Q8—1731no—
249C2×Q8—1731no—
250C2×D4—3462no—
251C2×D4—3461no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—1731no—
255C2×D4—1731no—
256C2×D4—6921no—
257C2×D4D69211yescharacteristic
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4C22×D17311yescharacteristic
262C22×C4C22×D17311yescharacteristic
263Q16—6921no—
264SD16—3462no—
265SD16—6921no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D17311yescharacteristic
269C2×C8C22×D17311yescharacteristic
270C2×C8C22×D17311yescharacteristic
271C2×C8C22×D17311yescharacteristic
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—1731no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4—1731no—
283C4⋊C4D69211yescharacteristic
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4D69211yescharacteristic
293C22⋊C4D69211yescharacteristic
294C42—3461no—
295C42—3461no—
296C42C22×D17311yescharacteristic
297C23—3461no—
298C23—3461no—
299C23—13841no—
300C23—6922no—
301C23—6922no—
302C23C2×D69211yescharacteristic
303C23C2×D69211yescharacteristic
304Q8—3461no—
305Q8—3461no—
306Q8—3461no—
307Q8—3461no—
308D4—22no—
309D4—3461no—
310D4—3461no—
311D4—3461no—
312D4—13841no—
313D4—1732no—
314D4—6922no—
315D4—6922no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—13841no—
327C2×C4—6922no—
328C2×C4—6922no—
329C2×C4C2×D69211yescharacteristic
330C2×C4C2×D69211yescharacteristic
331C2×C4C2×D69211yescharacteristic
332C2×C4C2×D69211yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—3462no—
344C22—13844no—
345C22—6921no—
346C22—6921no—
347C22—6921no—
348C22—6921no—
349C22—6921no—
350C22#1914970112yes—
351C22C22×D69211yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4—6921no—
360C4—6921no—
361C4—6921no—
362C4#41×D17311yescharacteristic
363C4#1914972411yescharacteristic
364C2—22no—
365C2—41no—
366C2—13841no—
367C2—6922no—
368C2#2601213611yescharacteristic
369C2C2×#1914970111yescharacteristic
370C2#2601230311yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC8×D4×C173C2
socleC22×C173C22×D4
prime radical#26012137C2
hypercenterC8×D4D173
hypocenterC173#3296
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #26012181C2semidirect
3. #26011846C2semidirect
4. #26011822C2semidirect
5. #26011816C2semidirect
6. #26011792C2semidirect
7. #26011732C2semidirect
8. #26011720C2semidirect
9. #26011714C2semidirect
10. #26011655C2semidirect
11. #26011644C2semidirect
12. C4×#14583815C2semidirect
13. #26011488C2semidirect
14. C2×#19149601C2semidirect
15. C8×D4×C173C2semidirect
18. C2×Dic692C22semidirect
22. #19149671C22semidirect
29. #19149658C22semidirect
33. C4×Dic346C22semidirect
35. #19149638C22semidirect
39. #19149613C22semidirect
40. #19149611C22semidirect
41. #19149611C22semidirect
42. #19149604C22semidirect
43. #19149604C22semidirect
44. #19149604C22semidirect
45. #19149604C22semidirect
53. C22×C8×C173C22semidirect
55. C4⋊C8×C173C22semidirect
57. C22⋊C8×C173C22semidirect
58. C4×C8×C173C22semidirect
185. C173#3296semidirect
199. C8×D4D173semidirect
208. C22×C8C2×D173semidirect
231. C4⋊C8C2×D173semidirect
243. C22⋊C8C2×D173semidirect
244. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121812
#260121372
#260118462
#260118222
#260118162
#260117922
#260117322
#260117202
#260117142
#260116552
#260116442
C4×#145838152
#260114882
C2×#191496012
C8×D4×C1732
#3296173