Finite Group Databasebeta

← #36366163

Subgroups — #36366163

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#26012187C211yesmaximal, characteristic, complemented
2#26011890C211yesmaximal, characteristic, complemented
3#26011815C211yesmaximal, characteristic, complemented
4#26011812C211yesmaximal, characteristic, complemented
5#26011804C211yesmaximal, characteristic, complemented
6#26011802C211yesmaximal, characteristic, complemented
7#26011791C211yesmaximal, characteristic, complemented
8#26011788C211yesmaximal, characteristic, complemented
9#26011742C211yesmaximal, characteristic, complemented
10#26011742C211yesmaximal, characteristic, complemented
11C4×#14583811C211yesmaximal, characteristic, complemented
12#26011569C211yesmaximal, characteristic, complemented
13#26011534C211yesmaximal, characteristic, complemented
14#26011531C211yesmaximal, characteristic, complemented
15#553×C173C211yesmaximal, characteristic, complemented
16C2×#14583811C2211yescharacteristic
17C2×#14583810C2211yescharacteristic
18C2×#14583810C2211yescharacteristic
19#19149674—21no—
20#19149671C2211yescharacteristic
21#19149661—21no—
22#19149653C2211yescharacteristic
23#19149653C2211yescharacteristic
24#19149651—21no—
25C4×D692C2211yescharacteristic
26#19149639—21no—
27#19149638C2211yescharacteristic
28C2×C4⋊Dic173C2211yescharacteristic
29#19149612C2211yescharacteristic
30#19149612C2211yescharacteristic
31#19149612C2211yescharacteristic
32#19149612C2211yescharacteristic
33#19149612C2211yescharacteristic
34#19149610C2211yescharacteristic
35#19149610C2211yescharacteristic
36#19149610C2211yescharacteristic
37#19149610C2211yescharacteristic
38C4⋊#13063436C2211yescharacteristic
39#19149600C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44#19149599C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47C4×#13063436C2211yescharacteristic
48#126×C173—21no—
49#125×C173C2211yescharacteristic
50#123×C173—21no—
51#121×C173—21no—
52#121×C173C2211yescharacteristic
53#121×C173C2211yescharacteristic
54C4×D4×C173—21no—
55C4×D4×C173C2211yescharacteristic
56C4×D4×C173C2211yescharacteristic
57#117×C173C2211yescharacteristic
58C2×C4⋊C4×C173C2211yescharacteristic
59C2×D692C2311yescharacteristic
60C2×C22⋊D173—21no—
61C2×C22⋊D173—21no—
62C22×Dic173—21no—
63C2×C4×D173—21no—
64#14583811—22no—
65#14583810—22no—
66#14583810—22no—
67C2×#13063436C2311yescharacteristic
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C4⋊Dic173—21no—
72C4⋊Dic173C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75#14583805—21no—
76#14583805—21no—
77C22⋊Dic173—21no—
78#14583803—21no—
79#14583803—21no—
80#14583803—21no—
81C4×Dic173—21no—
82C2×D4×C173—21no—
83C2×D4×C173—21no—
84C2×D4×C173D411yescharacteristic
85C2×D4×C173C2311yescharacteristic
86C22×C4×C173—21no—
87C22×C4×C173—21no—
88C22×C4×C173C2311yescharacteristic
89C22×C4×C173C2311yescharacteristic
90C4⋊C4×C173—21no—
91C4⋊C4×C173—21no—
92C4⋊C4×C173—21no—
93C4⋊C4×C173D411yescharacteristic
94C4⋊C4×C173C2311yescharacteristic
95C4⋊C4×C173C2311yescharacteristic
96C4⋊C4×C173C2311yescharacteristic
97C22⋊C4×C173—21no—
98C22⋊C4×C173—21no—
99C22⋊C4×C173—21no—
100C22⋊C4×C173—21no—
101C22⋊C4×C173D411yescharacteristic
102C22⋊C4×C173D411yescharacteristic
103C42×C173—21no—
104C42×C173C2311yescharacteristic
105C22×D173—21no—
106D692—21no—
107D692—21no—
108C22⋊D173—42no—
109C22⋊D173—81no—
110C2×Dic173—21no—
111C2×Dic173—21no—
112C2×Dic173—21no—
113C2×Dic173—81no—
114C4×D173—81no—
115#13063436—21no—
116#13063436—21no—
117#13063436—21no—
118#13063436#4112yes—
119C23×C173—21no—
120C23×C173C2×D411yescharacteristic
121C23×C173C2×D411yescharacteristic
122D4×C173—21no—
123D4×C173—21no—
124D4×C173—41no—
125D4×C173—42no—
126D4×C173—81no—
127C2×C4×C173—21no—
128C2×C4×C173—21no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—81no—
135C2×C4×C173—81no—
136C2×C4×C173C2×D411yescharacteristic
137C2×C4×C173C2×D411yescharacteristic
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2411yescharacteristic
141C22×C173—21no—
142C22×C173—21no—
143C22×C173—21no—
144C22×C173—21no—
145C22×C173—21no—
146C22×C173—21no—
147C22×C173—41no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173C22×D411yescharacteristic
151C2×D173—41no—
152C2×D173—41no—
153C2×D173—41no—
154C4×C173—21no—
155C4×C173—21no—
156C4×C173—21no—
157C4×C173—21no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173C2×#4111yescharacteristic
162C4×C1732−1+411yescharacteristic
163Dic173—41no—
164Dic173—41no—
165Dic173—41no—
166C2×C173—41no—
167C2×C173—41no—
168C2×C173—81no—
169C2×C173#54711yescharacteristic
170C2×C173#57511yescharacteristic
171C2×C173#57611yescharacteristic
172D173—81no—
173C173#335511yescharacteristic, complemented, Sylow
174#3355—1731nomaximal, Sylow
175#553—1731no—
176#553D17311yescharacteristic, complemented
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#469—1731no—
186C4×D8—1731no—
187#436—1731no—
188#428—1731no—
189#428—1731no—
190C2×D8—1731no—
191C2×M16—1731no—
192C2×M16—1731no—
193#126—21no—
194#126—3461no—
195#125—1731no—
196#125C2×D17311yescharacteristic
197#123—21no—
198#123—3461no—
199#121—21no—
200#121—3461no—
201#121—1731no—
202#121—1731no—
203#121C2×D17311yescharacteristic
204#121C2×D17311yescharacteristic
205C4×D4—21no—
206C4×D4—3461no—
207C4×D4—1731no—
208C4×D4C2×D17311yescharacteristic
209C4×D4C2×D17311yescharacteristic
210#117—1731no—
211#117C2×D17311yescharacteristic
212C2×C4⋊C4—1731no—
213C2×C4⋊C4C2×D17311yescharacteristic
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#106—1731no—
223C4⋊C8—1731no—
224#102—1731no—
225#102—1731no—
226#102—1731no—
227#102—1731no—
228#102—1731no—
229#102—1731no—
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—21no—
241C22×C4—21no—
242C22×C4—3461no—
243C22×C4—3461no—
244C22×C4C22×D17311yescharacteristic
245C22×C4C22×D17311yescharacteristic
246D8—3462no—
247M16—3462no—
248M16—3462no—
249C2×C8—1731no—
250C2×C8—1731no—
251C2×C8—1731no—
252C2×C8—1731no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—1731no—
260C4⋊C4—1731no—
261C4⋊C4—1731no—
262C4⋊C4C22⋊D17311yescharacteristic
263C4⋊C4C22×D17311yescharacteristic
264C4⋊C4C22×D17311yescharacteristic
265C4⋊C4C22×D17311yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—3461no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4C22⋊D17311yescharacteristic
275C22⋊C4C22⋊D17311yescharacteristic
276C42—21no—
277C42—3461no—
278C42C22×D17311yescharacteristic
279C23—21no—
280C23—3461no—
281C23C2×C22⋊D17311yescharacteristic
282C23C2×C22⋊D17311yescharacteristic
283D4—21no—
284D4—21no—
285D4—41no—
286D4—42no—
287D4—81no—
288D4—3461no—
289D4—3461no—
290D4—13841no—
291D4—6922no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—3461no—
302C2×C4—3461no—
303C2×C4—3461no—
304C2×C4—13841no—
305C2×C4—13841no—
306C2×C4C2×C22⋊D17311yescharacteristic
307C2×C4C2×C22⋊D17311yescharacteristic
308C2×C4C2×C22⋊D17311yescharacteristic
309C2×C4C2×C22⋊D17311yescharacteristic
310C2×C4C23×D17311yescharacteristic
311C8—3461no—
312C8—3461no—
313C8—3461no—
314C8—1732no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—6921no—
325C22—6921no—
326C22—6921no—
327C22C22×C22⋊D17311yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—6921no—
336C4—6921no—
337C4—6921no—
338C4#41×D17311yescharacteristic
339C4#1914972611yescharacteristic
340C2—41no—
341C2—41no—
342C2—81no—
343C2—13841no—
344C2#2601214011yescharacteristic
345C2#2601228911yescharacteristic
346C2#2601229411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#553×C173C2
socleC22×C173C22×D4
hypercenter#553D173
hypocenterC173#3355
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #26012187C2semidirect
2. #26011890C2semidirect
3. #26011815C2semidirect
4. #26011812C2semidirect
5. #26011804C2semidirect
6. #26011802C2semidirect
7. #26011791C2semidirect
8. #26011788C2semidirect
9. #26011742C2semidirect
10. #26011742C2semidirect
11. C4×#14583811C2semidirect
12. #26011569C2semidirect
13. #26011534C2semidirect
14. #26011531C2semidirect
15. #553×C173C2semidirect
173. C173#3355semidirect
176. #553D173semidirect

Maximal subgroups

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

subgroupindex
#260121872
#260118902
#260118152
#260118122
#260118042
#260118022
#260117912
#260117882
#260117422
C4×#145838112
#260115692
#260115342
#260115312
#553×C1732
#3355173