Finite Group Databasebeta

← #36366142

Subgroups — #36366142

Order 22144.

374 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012212C211yesmaximal, characteristic, complemented
2#26012141C211yesmaximal, characteristic
3#26011895C211yesmaximal, characteristic, complemented
4#26011825C211yesmaximal, characteristic, complemented
5#26011824C211yesmaximal, characteristic, complemented
6#26011814C211yesmaximal, characteristic, complemented
7#26011812C211yesmaximal, characteristic, complemented
8#26011774C211yesmaximal, characteristic, complemented
9#26011769C211yesmaximal, characteristic, complemented
10#26011734C211yesmaximal, characteristic, complemented
11SD16×Dic173C211yesmaximal, characteristic, complemented
12#26011567C211yesmaximal, characteristic, complemented
13#26011533C211yesmaximal, characteristic, complemented
14#26011532C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—21no—
17C2×#14583809C2211yescharacteristic
18#19149678C2211yescharacteristic
19#19149678C2211yescharacteristic
20#19149674—21no—
21#19149672—21no—
22#19149665—21no—
23#19149661—21no—
24#19149660—21no—
25#19149656—21no—
26#19149655C2211yescharacteristic
27#19149650C2211yescharacteristic
28Q8×Dic173C2211yescharacteristic
29#19149644—21no—
30D4×Dic173C2211yescharacteristic
31C4×C22⋊D173C2211yescharacteristic
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#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—44no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×C22⋊D173—21no—
63C2×C22⋊D173D411yescharacteristic
64C22×Dic173—21no—
65C22×Dic173—21no—
66C2×C4×D173—21no—
67C2×C4×D173C2311yescharacteristic
68#14583809—22no—
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C4⋊Dic173—21no—
72C4⋊Dic173—21no—
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805D411yescharacteristic
82C22⋊Dic173—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85C22⋊Dic173D411yescharacteristic
86#14583803—21no—
87#14583803—21no—
88#14583803D411yescharacteristic
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173C2311yescharacteristic
93C2×Q8×C173C2311yescharacteristic
94C2×D4×C173—21no—
95C2×D4×C173C2311yescharacteristic
96C22×C4×C173C2311yescharacteristic
97SD16×C173—22no—
98M16×C173—22no—
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22×D173C2×D411yescharacteristic
107Dic346—42no—
108Dic346—42no—
109C22⋊D173—22no—
110C22⋊D173—42no—
111C2×Dic173—21no—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—42no—
116C2×Dic173—42no—
117C2×Dic173C2×D411yescharacteristic
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120C4×D173—21no—
121C4×D173—21no—
122C4×D173—42no—
123#13063436—21no—
124#13063436—21no—
125C23×C173—21no—
126C23×C173C2×D411yescharacteristic
127Q8×C173—21no—
128Q8×C173—21no—
129D4×C173—21no—
130D4×C173—21no—
131D4×C173—42no—
132C2×C4×C173—21no—
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#4112yescomplemented
141C22×C173—21no—
142C22×C173—21no—
143C22×C173—21no—
144C22×C173—41no—
145C22×C173—41no—
146C22×C173—41no—
147C22×C173C22×D411yescharacteristic
148C2×D173—21no—
149C2×D173—21no—
150C2×D173—21no—
151C4×C173—21no—
152C4×C173—41no—
153C4×C173—41no—
154C4×C173—41no—
155C4×C173C2×#4111yescharacteristic
156C4×C1732−1+411yescharacteristic
157Dic173—21no—
158Dic173—21no—
159Dic173—41no—
160Dic173—41no—
161Dic173—41no—
162Dic173—41no—
163Dic173AGL(1,ℤ/8)12yes—
164C2×C173—41no—
165C2×C173—81no—
166C2×C173#54711yescharacteristic
167C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C173#57811yescharacteristic
169D173—41no—
170C173#335311yescharacteristic, complemented, Sylow
171#3353—1731nomaximal, Sylow
172#556—1731no—
173#547—1731no—
174#500—1731no—
175#484—1731no—
176#484—1731no—
177#482—1731no—
178#482—1731no—
179#475—1731no—
180#474—1731no—
181#468—1731no—
182#468D17311yescharacteristic, complemented
183C4×SD16—1731no—
184#436—1731no—
185#428—1731no—
186#428—1731no—
187C2×#41—3461no—
188C2×SD16C2×D17311yescharacteristic
189C2×M16—1731no—
190C2×M16C2×D17311yescharacteristic
191C4⋊Q8—1731no—
192C4⋊Q8—1731no—
193#126—3461no—
194#126—3461no—
195#124—3461no—
196#123—3461no—
197#123—3461no—
198C22⋊Q8—3461no—
199C22⋊Q8—1731no—
200C22⋊Q8C2×D17311yescharacteristic
201#121—1731no—
202#121C2×D17311yescharacteristic
203C4×Q8—3461no—
204C4×Q8—1731no—
205C4×D4—1731no—
206C4×D4—1731no—
207#117—1731no—
208#117—1731no—
209C2×C4⋊C4—1731no—
210C2×C4⋊C4—1731no—
211#107—1731no—
212#107—1731no—
213#107—1731no—
214#107—1731no—
215#107—1731no—
216#107C2×D17311yescharacteristic
217#106—1731no—
218#106—1731no—
219#106—1731no—
220C4⋊C8—1731no—
221#103—1731no—
222#103—1731no—
223#103C2×D17311yescharacteristic
224#102—1731no—
225#102—1731no—
226#102C2×D17311yescharacteristic
227C22⋊C8—1731no—
228C22⋊C8—1731no—
229C4×C8—1731no—
230#41—6924no—
231C2×Q8—3461no—
232C2×Q8—3461no—
233C2×Q8C22×D17311yescharacteristic
234C2×D4—21no—
235C2×D4—3461no—
236C2×D4—1731no—
237C2×D4C22×D17311yescharacteristic
238C22×C4—3461no—
239C22×C4—3461no—
240C22×C4—3461no—
241C22×C4—1731no—
242C22×C4C22×D17311yescharacteristic
243SD16—22no—
244M16—22no—
245M16—3462no—
246C2×C8—1731no—
247C2×C8—1731no—
248C2×C8C22×D17311yescharacteristic
249C2×C8C22×D17311yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—3461no—
252C4⋊C4—3461no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—1731no—
258C4⋊C4—1731no—
259C4⋊C4—1731no—
260C4⋊C4—1731no—
261C4⋊C4—1731no—
262C4⋊C4C22×D17311yescharacteristic
263C4⋊C4C22×D17311yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—3461no—
267C22⋊C4—3461no—
268C22⋊C4—3461no—
269C22⋊C4—3461no—
270C22⋊C4—3461no—
271C22⋊C4—1731no—
272C22⋊C4—1731no—
273C42—3461no—
274C42—3461no—
275C42—3461no—
276C42—1731no—
277C23—21no—
278C23—1731no—
279C23D4×D17311yescharacteristic
280Q8—21no—
281Q8—21no—
282Q8—6922no—
283Q8—6922no—
284D4—21no—
285D4—21no—
286D4—42no—
287D4—3462no—
288D4—6922no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—3461no—
295C2×C4—3461no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—1731no—
301C2×C4—1731no—
302C2×C4—1731no—
303C2×C4—6922no—
304C2×C4—6922no—
305C2×C4—6922no—
306C2×C4D4×D17311yescharacteristic
307C2×C4C23×D17311yescharacteristic
308C8—21no—
309C8—3461no—
310C8—3461no—
311C8#1458382512yescomplemented
312C22—21no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—3461no—
319C22—3461no—
320C22—3461no—
321C22C2×D4×D17311yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—3461no—
327C4—3461no—
328C4—1732no—
329C4—6921no—
330C4—6921no—
331C4—6921no—
332C4—6921no—
333C4C2×#1458382511yescharacteristic
334C4#1914972611yescharacteristic
335C2—41no—
336C2—81no—
337C2—6921no—
338C2#2601214111yescharacteristic
339C2AGL(1,ℤ/8)×D17311yescharacteristic
340C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
prime radical#26012141C2
hypercenter#468D173
hypocenterC173#3353
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #26012212C2semidirect
3. #26011895C2semidirect
4. #26011825C2semidirect
5. #26011824C2semidirect
6. #26011814C2semidirect
7. #26011812C2semidirect
8. #26011774C2semidirect
9. #26011769C2semidirect
10. #26011734C2semidirect
11. SD16×Dic173C2semidirect
12. #26011567C2semidirect
13. #26011533C2semidirect
14. #26011532C2semidirect
15. #468×C173C2semidirect
140. C8×C173#41semidirect
170. C173#3353semidirect
182. #468D173semidirect
311. C8#14583825semidirect

Maximal subgroups

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

subgroupindex
#260122122
#260121412
#260118952
#260118252
#260118242
#260118142
#260118122
#260117742
#260117692
#260117342
SD16×Dic1732
#260115672
#260115332
#260115322
#468×C1732
#3353173