Finite Group Databasebeta

← #36365000

Subgroups — #36365000

Order 22144.

370 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012095C211yesmaximal, characteristic, complemented
2#26012092C211yesmaximal, characteristic, complemented
3#26011793C211yesmaximal, characteristic, complemented
4#26011792C211yesmaximal, characteristic, complemented
5#26011791C211yesmaximal, characteristic, complemented
6#26011789C211yesmaximal, characteristic, complemented
7#26011743C211yesmaximal, characteristic, complemented
8#26011741C211yesmaximal, characteristic, complemented
9#26011726C211yesmaximal, characteristic, complemented
10#26011724C211yesmaximal, characteristic, complemented
11#26011684C211yesmaximal, characteristic, complemented
12#26011684C211yesmaximal, characteristic, complemented
13#26011683C211yesmaximal, characteristic, complemented
14#26011516C211yesmaximal, characteristic, complemented
15#459×C173C211yesmaximal, characteristic, complemented
16C2×#14583811C2211yescharacteristic
17C2×#14583811C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149675C2211yescharacteristic
21#19149671C2211yescharacteristic
22#19149670C2211yescharacteristic
23#19149662—21no—
24#19149661—21no—
25#19149660—21no—
26#19149660—21no—
27#19149654C2211yescharacteristic
28#19149653—21no—
29#19149652—21no—
30#19149651—21no—
31#19149651—21no—
32#19149650C2211yescharacteristic
33#19149650C2211yescharacteristic
34#19149650C2211yescharacteristic
35C4×D692C2211yescharacteristic
36D4×Dic173C2211yescharacteristic
37D4×Dic173C2211yescharacteristic
38#19149636C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149610C2211yescharacteristic
43C4⋊#13063436C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149600C2211yescharacteristic
51#19149600C2211yescharacteristic
52C2×D8×C173C2211yescharacteristic
53C2×D8×C173C2211yescharacteristic
54#127×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C2×D692—21no—
60C2×D692C2311yescharacteristic
61C2×C22⋊D173—21no—
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C22×Dic173—21no—
67C22×Dic173—21no—
68C2×C4×D173—21no—
69C2×C4×D173C2311yescharacteristic
70#14583811—41no—
71#14583811—41no—
72#14583809—41no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C8×D173—41no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84#14583805D411yescharacteristic
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803D411yescharacteristic
93#14583803D411yescharacteristic
94C4×Dic173C2311yescharacteristic
95C2×D4×C173—21no—
96C2×D4×C173—21no—
97C2×D4×C173C2311yescharacteristic
98C2×D4×C173C2311yescharacteristic
99C2×D4×C173C2311yescharacteristic
100C22×C4×C173—21no—
101D8×C173—41no—
102D8×C173—41no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173C2×D411yescharacteristic
110D692—21no—
111D692—21no—
112D692—81no—
113C22⋊D173—81no—
114C22⋊D173—81no—
115C22⋊D173—81no—
116C22⋊D173—81no—
117C22⋊D173—81no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—81no—
122C2×Dic173—81no—
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C4×D173—21no—
127C4×D173—21no—
128C4×D173—81no—
129#13063436—21no—
130#13063436—21no—
131C23×C173—21no—
132C23×C173—21no—
133C23×C173—21no—
134D4×C173—21no—
135D4×C173—21no—
136D4×C173—21no—
137D4×C173—21no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—81no—
141D4×C173—81no—
142C2×C4×C173—21no—
143C2×C4×C173—81no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173C22×D411yescharacteristic
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173H(5,𝔽2)11yescharacteristic
169C4×C1732−1+411yescharacteristic
170Dic173—21no—
171Dic173—21no—
172Dic173—21no—
173Dic173—41no—
174Dic173—41no—
175Dic173—41no—
176C2×C173—81no—
177C2×C173—81no—
178C2×C173—81no—
179C2×C173#53711yescharacteristic
180C2×C173#57611yescharacteristic
181C2×C173#57611yescharacteristic
182D173—41no—
183D173—81no—
184C173#325911yescharacteristic, complemented, Sylow
185#3259—1731nomaximal, Sylow
186#540—1731no—
187#540—1731no—
188#478—1731no—
189#478—1731no—
190#478—1731no—
191#478—1731no—
192#469—1731no—
193#469—1731no—
194#466—1731no—
195#466—1731no—
196#459—1731no—
197#459—1731no—
198#459—1731no—
199#459D17311yescharacteristic, complemented
200#424—1731no—
201C2×D8—1731no—
202C2×D8—1731no—
203C2×D8C2×D17311yescharacteristic
204C2×D8C2×D17311yescharacteristic
205C2×M16—1731no—
206C22×C8—1731no—
207#127—1731no—
208#127C2×D17311yescharacteristic
209#125—1731no—
210#125—1731no—
211#123—3461no—
212#123—3461no—
213#123—3461no—
214#123—3461no—
215#121—3461no—
216#121—3461no—
217#121—3461no—
218#121—3461no—
219#121—1731no—
220#121—1731no—
221#121—1731no—
222#121—1731no—
223C4×D4—1731no—
224C4×D4—1731no—
225C4×D4—1731no—
226C4×D4C2×D17311yescharacteristic
227#117—1731no—
228#107—1731no—
229#107—1731no—
230#106—1731no—
231#106—1731no—
232C4⋊C8—1731no—
233C4⋊C8—1731no—
234C4⋊C8—1731no—
235C4⋊C8C2×D17311yescharacteristic
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242#102C2×D17311yescharacteristic
243#102C2×D17311yescharacteristic
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—3461no—
249C2×D4—3461no—
250C2×D4—3461no—
251C2×D4—3461no—
252C2×D4—1731no—
253C2×D4C22×D17311yescharacteristic
254C2×D4C22×D17311yescharacteristic
255C2×D4C22×D17311yescharacteristic
256C22×C4—21no—
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—1731no—
261D8—41no—
262D8—41no—
263D8—6921no—
264D8—6921no—
265M16—6921no—
266C2×C8—1731no—
267C2×C8—1731no—
268C2×C8—6921no—
269C2×C8C22×D17311yescharacteristic
270C2×C8C22×D17311yescharacteristic
271C4⋊C4—3461no—
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—1731no—
276C4⋊C4—1731no—
277C4⋊C4—1731no—
278C4⋊C4—1731no—
279C4⋊C4—1731no—
280C4⋊C4C22×D17311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—1731no—
290C22⋊C4—1731no—
291C42—1731no—
292C42C22×D17311yescharacteristic
293C23—21no—
294C23—21no—
295C23—21no—
296C23—3461no—
297C23—1731no—
298D4—21no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—81no—
305D4—81no—
306D4—3461no—
307D4—3461no—
308D4—13841no—
309D4—13841no—
310D4—13841no—
311D4—13841no—
312D4—13841no—
313D4—13841no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—13841no—
322C2×C4—13841no—
323C2×C4—13841no—
324C2×C4—1731no—
325C2×C4—1731no—
326C2×C4—1731no—
327C2×C4D4×D17311yescharacteristic
328C2×C4D4×D17311yescharacteristic
329C2×C4C23×D17311yescharacteristic
330C8—21no—
331C8—21no—
332C8—3461no—
333C8—3461no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—3461no—
344C22—3461no—
345C22—3461no—
346C22—6921no—
347C22—6921no—
348C22—6921no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—3461no—
354C4—3461no—
355C4—3461no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4#1914972411yescharacteristic
360C4#1914972611yescharacteristic
361C2—81no—
362C2—81no—
363C2—81no—
364C2—13841no—
365C2—6921no—
366C2#2601207011yescharacteristic
367C2#2601229211yescharacteristic
368C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#459×C173C2
socleC22×C173C22×D4
hypercenter#459D173
hypocenterC173#3259
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #26012095C2semidirect
2. #26012092C2semidirect
3. #26011793C2semidirect
4. #26011792C2semidirect
5. #26011791C2semidirect
6. #26011789C2semidirect
7. #26011743C2semidirect
8. #26011741C2semidirect
9. #26011726C2semidirect
10. #26011724C2semidirect
11. #26011684C2semidirect
12. #26011684C2semidirect
13. #26011683C2semidirect
14. #26011516C2semidirect
15. #459×C173C2semidirect
184. C173#3259semidirect
199. #459D173semidirect

Maximal subgroups

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

subgroupindex
#260120952
#260120922
#260117932
#260117922
#260117912
#260117892
#260117432
#260117412
#260117262
#260117242
#260116842
#260116832
#260115162
#459×C1732
#3259173