Finite Group Databasebeta

← #36366140

Subgroups — #36366140

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#26012142C211yesmaximal, characteristic
3#26011895C211yesmaximal, characteristic, complemented
4#26011825C211yesmaximal, characteristic, complemented
5#26011818C211yesmaximal, characteristic, complemented
6#26011816C211yesmaximal, characteristic, complemented
7#26011774C211yesmaximal, characteristic, complemented
8#26011770C211yesmaximal, characteristic, complemented
9#26011740C211yesmaximal, characteristic, complemented
10#26011739C211yesmaximal, characteristic, complemented
11#26011576C211yesmaximal, characteristic, complemented
12#26011566C211yesmaximal, characteristic, complemented
13#26011532C211yesmaximal, characteristic, complemented
14#26011531C211yesmaximal, characteristic, complemented
15#484×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C2×#14583813C2211yescharacteristic
18C2×#14583810C2211yescharacteristic
19C2×#14583809C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149674—21no—
23#19149672—21no—
24#19149665—21no—
25#19149664—21no—
26#19149662—21no—
27#19149657—21no—
28#19149655C2211yescharacteristic
29#19149654C2211yescharacteristic
30#19149653C2211yescharacteristic
31Q8×Dic173C2211yescharacteristic
32#19149644—21no—
33#19149641C2211yescharacteristic
34C4×C22⋊D173C2211yescharacteristic
35#19149638C2211yescharacteristic
36#19149637C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149611C2211yescharacteristic
43#19149610C2211yescharacteristic
44#19149610C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—44no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×D692—21no—
63C2×D692C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173D411yescharacteristic
66C2×C4×D173—21no—
67C2×C4×D173—21no—
68C2×C4×D173C2311yescharacteristic
69#14583813—22no—
70#14583810—22no—
71#14583809—22no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C4⋊Dic173—21no—
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805D411yescharacteristic
83C22⋊Dic173—21no—
84C22⋊Dic173D411yescharacteristic
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803D411yescharacteristic
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173C2311yescharacteristic
95C2×Q8×C173C2311yescharacteristic
96C22×C4×C173—21no—
97C22×C4×C173C2311yescharacteristic
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173—21no—
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22×D173—21no—
107C22×D173C2×D411yescharacteristic
108Dic346—42no—
109Dic346—42no—
110D692—21no—
111D692—21no—
112D692—42no—
113C22⋊D173—22no—
114C22⋊D173—42no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C4×D173—21no—
122C4×D173—21no—
123C4×D173—42no—
124C4×D173—42no—
125#13063436—21no—
126#13063436#4112yescomplemented
127C23×C173C2×D411yescharacteristic
128Q8×C173—21no—
129Q8×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—42no—
137C2×C4×C173C2×D411yescharacteristic
138C2×C4×C173C2411yescharacteristic
139C8×C173—21no—
140C8×C173—21no—
141C22×C173—21no—
142C22×C173—21no—
143C22×C173—21no—
144C22×C173C22×D411yescharacteristic
145C2×D173—21no—
146C2×D173—21no—
147C2×D173—21no—
148C2×D173—41no—
149C2×D173—41no—
150C2×D173—41no—
151C4×C173—21no—
152C4×C173—41no—
153C4×C173—41no—
154C4×C173—41no—
155C4×C173—41no—
156C4×C173C2×#4111yescharacteristic
157C4×C1732−1+411yescharacteristic
158Dic173—21no—
159Dic173—21no—
160Dic173—41no—
161Dic173—41no—
162Dic173—41no—
163Dic173AGL(1,ℤ/8)12yes—
164C2×C173—41no—
165C2×C173#54711yescharacteristic
166C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C173#57811yescharacteristic
168D173—41no—
169D173—81no—
170C173#335311yescharacteristic, complemented, Sylow
171#3353—1731nomaximal, Sylow
172#556—1731no—
173#547—1731no—
174#500—1731no—
175#484—1731no—
176#484D17311yescharacteristic, complemented
177#482—1731no—
178#482—1731no—
179#475—1731no—
180#474—1731no—
181#468—1731no—
182#468—1731no—
183C4×SD16—1731no—
184#436—1731no—
185#428—1731no—
186#428—1731no—
187C2×#41—3461no—
188C2×SD16—1731no—
189C2×M16—1731no—
190C2×M16—1731no—
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#121—1731no—
203C4×Q8—3461no—
204C4×Q8—1731no—
205C4×D4—1731no—
206C4×D4—1731no—
207#117—1731no—
208#117—1731no—
209C2×C4⋊C4—1731no—
210C2×C4⋊C4C2×D17311yescharacteristic
211#107—1731no—
212#107—1731no—
213#107—1731no—
214#107—1731no—
215#107C2×D17311yescharacteristic
216#107C2×D17311yescharacteristic
217#106—1731no—
218#106—1731no—
219#106—1731no—
220C4⋊C8—1731no—
221#103—1731no—
222#103C2×D17311yescharacteristic
223#103C2×D17311yescharacteristic
224#102—1731no—
225#102—1731no—
226#102—1731no—
227C22⋊C8—1731no—
228C22⋊C8C2×D17311yescharacteristic
229C4×C8—1731no—
230#41—6924no—
231C2×Q8—3461no—
232C2×Q8—3461no—
233C2×Q8C22×D17311yescharacteristic
234C2×D4—3461no—
235C2×D4—3461no—
236C2×D4—1731no—
237C2×D4—1731no—
238C22×C4—21no—
239C22×C4—3461no—
240C22×C4—3461no—
241C22×C4—1731no—
242C22×C4C22×D17311yescharacteristic
243SD16—3462no—
244M16—3462no—
245M16—3462no—
246C2×C8—1731no—
247C2×C8—1731no—
248C2×C8C22×D17311yescharacteristic
249C2×C8C22×D17311yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
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⋊C4C22×D17311yescharacteristic
262C4⋊C4C22×D17311yescharacteristic
263C4⋊C4C22×D17311yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—3461no—
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—3461no—
278C23—1731no—
279C23D4×D17311yescharacteristic
280Q8—21no—
281Q8—21no—
282Q8—6922no—
283Q8—6922no—
284D4—3462no—
285D4—3461no—
286D4—3461no—
287D4—6922no—
288D4—6922no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—3461no—
301C2×C4—1731no—
302C2×C4—1731no—
303C2×C4—1731no—
304C2×C4—6922no—
305C2×C4—6922no—
306C2×C4D4×D17311yescharacteristic
307C2×C4C23×D17311yescharacteristic
308C8—21no—
309C8—21no—
310C8—3461no—
311C8—1732no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—3461no—
316C22—3461no—
317C22—3461no—
318C22—6921no—
319C22—6921no—
320C22—6921no—
321C22C2×D4×D17311yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—3461no—
328C4—3461no—
329C4—1732no—
330C4—6921no—
331C4—6921no—
332C4—6921no—
333C4#41×D17311yescharacteristic
334C4#1914972611yescharacteristic
335C2—41no—
336C2—13841no—
337C2—6921no—
338C2#2601213911yescharacteristic
339C2AGL(1,ℤ/8)×D17311yescharacteristic
340C2#2601230411yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#484×C173C2
socleC22×C173C22×D4
prime radical#26012142C2
hypercenter#484D173
hypocenterC173#3353
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #26012212C2semidirect
3. #26011895C2semidirect
4. #26011825C2semidirect
5. #26011818C2semidirect
6. #26011816C2semidirect
7. #26011774C2semidirect
8. #26011770C2semidirect
9. #26011740C2semidirect
10. #26011739C2semidirect
11. #26011576C2semidirect
12. #26011566C2semidirect
13. #26011532C2semidirect
14. #26011531C2semidirect
15. #484×C173C2semidirect
126. #13063436#41semidirect
170. C173#3353semidirect
176. #484D173semidirect

Maximal subgroups

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

subgroupindex
#260122122
#260121422
#260118952
#260118252
#260118182
#260118162
#260117742
#260117702
#260117402
#260117392
#260115762
#260115662
#260115322
#260115312
#484×C1732
#3353173