Finite Group Databasebeta

← #36366164

Subgroups — #36366164

Order 22144.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012193C211yesmaximal, characteristic, complemented
2#26012188C211yesmaximal, characteristic, complemented
3#26011891C211yesmaximal, characteristic, complemented
4#26011814C211yesmaximal, characteristic, complemented
5#26011812C211yesmaximal, characteristic, complemented
6#26011803C211yesmaximal, characteristic, complemented
7#26011802C211yesmaximal, characteristic, complemented
8#26011793C211yesmaximal, characteristic, complemented
9#26011789C211yesmaximal, characteristic, complemented
10#26011741C211yesmaximal, characteristic, complemented
11D8×Dic173C211yesmaximal, characteristic, complemented
12#26011567C211yesmaximal, characteristic, complemented
13#26011533C211yesmaximal, characteristic, complemented
14#26011530C211yesmaximal, characteristic, complemented
15#469×C173C211yesmaximal, characteristic, complemented
16C2×#14583809C2211yescharacteristic
17#19149672—21no—
18#19149672—21no—
19#19149670C2211yescharacteristic
20#19149670C2211yescharacteristic
21#19149663—21no—
22#19149660—21no—
23#19149651—21no—
24#19149650—21no—
25#19149650C2211yescharacteristic
26#19149650C2211yescharacteristic
27D4×Dic173—21no—
28D4×Dic173C2211yescharacteristic
29D4×Dic173C2211yescharacteristic
30C4×C22⋊D173C2211yescharacteristic
31#19149639—21no—
32#19149638C2211yescharacteristic
33#19149637C2211yescharacteristic
34C2×C4⋊Dic173C2211yescharacteristic
35C4⋊C4×D173C2211yescharacteristic
36#19149613C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149611C2211yescharacteristic
43#19149610C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×D8×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic
54#121×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#106×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C2×C22⋊D173—21no—
60C2×C22⋊D173—21no—
61C2×C22⋊D173D411yescharacteristic
62C22×Dic173—21no—
63C22×Dic173—21no—
64C22×Dic173—21no—
65C2×C4×D173—21no—
66C2×C4×D173C2311yescharacteristic
67#14583809—22no—
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C4⋊Dic173—21no—
71C4⋊Dic173—21no—
72C4⋊Dic173—21no—
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805D411yescharacteristic
81C22⋊Dic173—21no—
82C22⋊Dic173—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173D411yescharacteristic
87#14583803—21no—
88#14583803D411yescharacteristic
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173C2311yescharacteristic
92C2×D4×C173—21no—
93C2×D4×C173—21no—
94C2×D4×C173C2311yescharacteristic
95C2×D4×C173C2311yescharacteristic
96C22×C4×C173C2311yescharacteristic
97D8×C173—22no—
98M16×C173—22no—
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C22×D173C2×D411yescharacteristic
106C22⋊D173—41no—
107C22⋊D173—42no—
108C22⋊D173—81no—
109C2×Dic173—21no—
110C2×Dic173—21no—
111C2×Dic173—21no—
112C2×Dic173—21no—
113C2×Dic173—81no—
114C2×Dic173—81no—
115C2×Dic173—81no—
116C2×Dic173C2×D411yescharacteristic
117C2×Dic173C2×D411yescharacteristic
118C2×Dic173C2×D411yescharacteristic
119C4×D173—21no—
120C4×D173—21no—
121C4×D173—81no—
122#13063436—21no—
123#13063436—21no—
124C23×C173—21no—
125C23×C173—21no—
126C23×C173C2×D411yescharacteristic
127D4×C173—21no—
128D4×C173—21no—
129D4×C173—21no—
130D4×C173—21no—
131D4×C173—42no—
132D4×C173—81no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173C2×D411yescharacteristic
138C2×C4×C173C2411yescharacteristic
139C8×C173—21no—
140C8×C173#4112yes—
141C22×C173—21no—
142C22×C173—21no—
143C22×C173—21no—
144C22×C173—41no—
145C22×C173—41no—
146C22×C173—41no—
147C22×C173—41no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173C22×D411yescharacteristic
151C2×D173—21no—
152C2×D173—21no—
153C2×D173—21no—
154C4×C173—21no—
155C4×C173—41no—
156C4×C173—41no—
157C4×C173C2×#4111yescharacteristic
158C4×C1732−1+411yescharacteristic
159Dic173—21no—
160Dic173—21no—
161Dic173—21no—
162Dic173—41no—
163Dic173—41no—
164Dic173—41no—
165Dic173—41no—
166C2×C173—41no—
167C2×C173—81no—
168C2×C173—81no—
169C2×C173#54711yescharacteristic
170C2×C173#57511yescharacteristic
171C2×C173#57611yescharacteristic
172D173—41no—
173C173#335511yescharacteristic, complemented, Sylow
174#3355—1731nomaximal, Sylow
175#553—1731no—
176#553—1731no—
177#499—1731no—
178#482—1731no—
179#482—1731no—
180#480—1731no—
181#480—1731no—
182#478—1731no—
183#478—1731no—
184#469—1731no—
185#469D17311yescharacteristic, complemented
186C4×D8—1731no—
187#436—1731no—
188#428—1731no—
189#428—1731no—
190C2×D8C2×D17311yescharacteristic
191C2×M16—1731no—
192C2×M16C2×D17311yescharacteristic
193#126—3461no—
194#126—3461no—
195#125—1731no—
196#125—1731no—
197#123—3461no—
198#123—3461no—
199#121—3461no—
200#121—3461no—
201#121—1731no—
202#121—1731no—
203#121C2×D17311yescharacteristic
204#121C2×D17311yescharacteristic
205C4×D4—3461no—
206C4×D4—3461no—
207C4×D4—1731no—
208C4×D4—1731no—
209C4×D4—1731no—
210#117—1731no—
211#117—1731no—
212C2×C4⋊C4—1731no—
213C2×C4⋊C4—1731no—
214#107—1731no—
215#107—1731no—
216#107—1731no—
217#107—1731no—
218#107—1731no—
219#106—1731no—
220#106—1731no—
221#106—1731no—
222#106C2×D17311yescharacteristic
223C4⋊C8—1731no—
224#102—1731no—
225#102—1731no—
226#102—1731no—
227#102—1731no—
228#102C2×D17311yescharacteristic
229#102C2×D17311yescharacteristic
230C22⋊C8—1731no—
231C22⋊C8—1731no—
232C4×C8—1731no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—3461no—
236C2×D4—3461no—
237C2×D4—1731no—
238C2×D4C22×D17311yescharacteristic
239C2×D4C22×D17311yescharacteristic
240C22×C4—3461no—
241C22×C4—3461no—
242C22×C4—3461no—
243C22×C4—3461no—
244C22×C4—1731no—
245C22×C4C22×D17311yescharacteristic
246D8—22no—
247M16—22no—
248M16—3462no—
249C2×C8—1731no—
250C2×C8—1731no—
251C2×C8C22×D17311yescharacteristic
252C2×C8C22×D17311yescharacteristic
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—1731no—
260C4⋊C4—1731no—
261C4⋊C4—1731no—
262C4⋊C4—1731no—
263C4⋊C4—1731no—
264C4⋊C4C22×D17311yescharacteristic
265C4⋊C4C22×D17311yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—3461no—
269C22⋊C4—3461no—
270C22⋊C4—3461no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—1731no—
275C22⋊C4—1731no—
276C42—3461no—
277C42—3461no—
278C42—1731no—
279C23—21no—
280C23—21no—
281C23—1731no—
282C23D4×D17311yescharacteristic
283D4—21no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—42no—
288D4—81no—
289D4—13841no—
290D4—6921no—
291D4—6922no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—3461no—
301C2×C4—3461no—
302C2×C4—13841no—
303C2×C4—13841no—
304C2×C4—13841no—
305C2×C4—13841no—
306C2×C4—1731no—
307C2×C4—1731no—
308C2×C4—1731no—
309C2×C4D4×D17311yescharacteristic
310C2×C4C23×D17311yescharacteristic
311C8—21no—
312C8—3461no—
313C8—3461no—
314C8#1458382512yes—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—3461no—
325C22—3461no—
326C22—3461no—
327C22C2×D4×D17311yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—3461no—
332C4—3461no—
333C4—3461no—
334C4—6921no—
335C4—6921no—
336C4—6921no—
337C4—6921no—
338C4C2×#1458382511yescharacteristic
339C4#1914972611yescharacteristic
340C2—41no—
341C2—81no—
342C2—81no—
343C2—6921no—
344C2#2601214111yescharacteristic
345C2#2601228711yescharacteristic
346C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#469×C173C2
socleC22×C173C22×D4
hypercenter#469D173
hypocenterC173#3355
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #26012193C2semidirect
2. #26012188C2semidirect
3. #26011891C2semidirect
4. #26011814C2semidirect
5. #26011812C2semidirect
6. #26011803C2semidirect
7. #26011802C2semidirect
8. #26011793C2semidirect
9. #26011789C2semidirect
10. #26011741C2semidirect
11. D8×Dic173C2semidirect
12. #26011567C2semidirect
13. #26011533C2semidirect
14. #26011530C2semidirect
15. #469×C173C2semidirect
173. C173#3355semidirect
185. #469D173semidirect

Maximal subgroups

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

subgroupindex
#260121932
#260121882
#260118912
#260118142
#260118122
#260118032
#260118022
#260117932
#260117892
#260117412
D8×Dic1732
#260115672
#260115332
#260115302
#469×C1732
#3355173