Finite Group Databasebeta

← #36366126

Subgroups — #36366126

Order 22144.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012189C211yesmaximal, characteristic
2#26012175C211yesmaximal, characteristic, complemented
3#26011891C211yesmaximal, characteristic, complemented
4#26011830C211yesmaximal, characteristic, complemented
5#26011816C211yesmaximal, characteristic, complemented
6#26011811C211yesmaximal, characteristic, complemented
7#26011800C211yesmaximal, characteristic, complemented
8#26011790C211yesmaximal, characteristic, complemented
9#26011740C211yesmaximal, characteristic, complemented
10#26011739C211yesmaximal, characteristic, complemented
11#26011576C211yesmaximal, characteristic, complemented
12#26011566C211yesmaximal, characteristic, complemented
13#26011531C211yesmaximal, characteristic, complemented
14#26011530C211yesmaximal, characteristic, complemented
15#483×C173C211yesmaximal, characteristic, complemented
16C2×#14583813C2211yescharacteristic
17C2×#14583810C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19#19149674—21no—
20#19149672—21no—
21#19149670C2211yescharacteristic
22#19149670C2211yescharacteristic
23#19149669—21no—
24#19149664—21no—
25#19149661—21no—
26#19149655C2211yescharacteristic
27#19149654C2211yescharacteristic
28#19149653—21no—
29#19149653C2211yescharacteristic
30Q8×Dic173C2211yescharacteristic
31#19149641C2211yescharacteristic
32C4×C22⋊D173—21no—
33C4×C22⋊D173C2211yescharacteristic
34#19149638C2211yescharacteristic
35#19149637—21no—
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#106×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×D692—21no—
60C2×D692C2311yescharacteristic
61C2×C22⋊D173—21no—
62C2×C22⋊D173D411yescharacteristic
63C2×C4×D173—21no—
64C2×C4×D173—21no—
65C2×C4×D173C2311yescharacteristic
66#14583813—22no—
67#14583810—22no—
68#14583809—22no—
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C4⋊Dic173—21no—
72C4⋊Dic173—21no—
73C4⋊Dic173—21no—
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805D411yescharacteristic
82C22⋊Dic173—21no—
83C22⋊Dic173D411yescharacteristic
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803D411yescharacteristic
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173C2311yescharacteristic
94C2×Q8×C173C2311yescharacteristic
95C22×C4×C173—21no—
96C22×C4×C173C2311yescharacteristic
97C2×C8×C173C2311yescharacteristic
98C2×C8×C173C2311yescharacteristic
99C4⋊C4×C173—21no—
100C4⋊C4×C173—21no—
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C22⋊C4×C173—21no—
105C22×D173—21no—
106C22×D173C2×D411yescharacteristic
107D692—21no—
108D692—21no—
109D692—81no—
110C22⋊D173—41no—
111C22⋊D173—42no—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173C2×D411yescharacteristic
116C2×Dic173C2×D411yescharacteristic
117C2×Dic173C2×D411yescharacteristic
118C4×D173—21no—
119C4×D173—21no—
120C4×D173—42no—
121C4×D173—81no—
122#13063436—21no—
123#13063436#4112yes—
124C23×C173C2×D411yescharacteristic
125Q8×C173—21no—
126Q8×C173—21no—
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—81no—
134C2×C4×C173C2×D411yescharacteristic
135C2×C4×C173C2411yescharacteristic
136C8×C173—21no—
137C8×C173—21no—
138C22×C173—21no—
139C22×C173—21no—
140C22×C173—21no—
141C22×C173C22×D411yescharacteristic
142C2×D173—21no—
143C2×D173—21no—
144C2×D173—21no—
145C2×D173—41no—
146C2×D173—41no—
147C2×D173—41no—
148C4×C173—21no—
149C4×C173—41no—
150C4×C173—41no—
151C4×C173—41no—
152C4×C173—41no—
153C4×C173C2×#4111yescharacteristic
154C4×C1732−1+411yescharacteristic
155Dic173—21no—
156Dic173—21no—
157Dic173—21no—
158Dic173—41no—
159Dic173—41no—
160Dic173—41no—
161C2×C173—41no—
162C2×C173#54711yescharacteristic
163C2×C173#57511yescharacteristic
164C2×C173#57711yescharacteristic
165D173—41no—
166D173—81no—
167C173#335111yescharacteristic, complemented, Sylow
168#3351—1731nomaximal, Sylow
169#553—1731no—
170#552—1731no—
171#499—1731no—
172#485—1731no—
173#483D17311yescharacteristic, complemented
174#482—1731no—
175#481—1731no—
176#479—1731no—
177#478—1731no—
178#468—1731no—
179#468—1731no—
180C4×SD16—1731no—
181#436—1731no—
182#428—1731no—
183#428—1731no—
184C2×SD16—1731no—
185C2×M16—1731no—
186C2×M16—1731no—
187#126—3461no—
188#126—3461no—
189#125—3461no—
190#125—1731no—
191#125—1731no—
192#123—3461no—
193#123—3461no—
194C22⋊Q8—1731no—
195C22⋊Q8C2×D17311yescharacteristic
196#121—3461no—
197#121—1731no—
198#121—1731no—
199C4×Q8—1731no—
200C4×D4—3461no—
201C4×D4—1731no—
202C4×D4—1731no—
203#117—3461no—
204#117—1731no—
205#117—1731no—
206C2×C4⋊C4—1731no—
207C2×C4⋊C4C2×D17311yescharacteristic
208#107—1731no—
209#107—1731no—
210#107—1731no—
211#107—1731no—
212#106—1731no—
213#106—1731no—
214#106—1731no—
215#106C2×D17311yescharacteristic
216#106C2×D17311yescharacteristic
217C4⋊C8—1731no—
218#103—1731no—
219#103C2×D17311yescharacteristic
220#103C2×D17311yescharacteristic
221#102—1731no—
222#102—1731no—
223#102—1731no—
224C22⋊C8—1731no—
225C22⋊C8C2×D17311yescharacteristic
226C4×C8—1731no—
227C2×Q8C22×D17311yescharacteristic
228C2×D4—3461no—
229C2×D4—3461no—
230C2×D4—1731no—
231C2×D4—1731no—
232C22×C4—21no—
233C22×C4—3461no—
234C22×C4—3461no—
235C22×C4—1731no—
236C22×C4C22×D17311yescharacteristic
237SD16—3462no—
238M16—3462no—
239M16—3462no—
240C2×C8—1731no—
241C2×C8—1731no—
242C2×C8C22×D17311yescharacteristic
243C2×C8C22×D17311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—3461no—
247C4⋊C4—3461no—
248C4⋊C4—3461no—
249C4⋊C4—3461no—
250C4⋊C4—3461no—
251C4⋊C4—3461no—
252C4⋊C4—3461no—
253C4⋊C4—1731no—
254C4⋊C4—1731no—
255C4⋊C4—1731no—
256C4⋊C4—1731no—
257C4⋊C4C22×D17311yescharacteristic
258C4⋊C4C22×D17311yescharacteristic
259C4⋊C4C22×D17311yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—3461no—
262C22⋊C4—3461no—
263C22⋊C4—3461no—
264C22⋊C4—3461no—
265C22⋊C4—3461no—
266C22⋊C4—3461no—
267C22⋊C4—1731no—
268C22⋊C4—1731no—
269C42—3461no—
270C42—3461no—
271C42—3461no—
272C42—1731no—
273C23—3461no—
274C23—1731no—
275C23D4×D17311yescharacteristic
276Q8—21no—
277Q8—21no—
278D4—3461no—
279D4—3461no—
280D4—13841no—
281D4—6921no—
282D4—6922no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—81no—
290C2×C4—3461no—
291C2×C4—3461no—
292C2×C4—3461no—
293C2×C4—3461no—
294C2×C4—3461no—
295C2×C4—13841no—
296C2×C4—1731no—
297C2×C4—1731no—
298C2×C4—1731no—
299C2×C4—6922no—
300C2×C4D4×D17311yescharacteristic
301C2×C4C23×D17311yescharacteristic
302C8—21no—
303C8—21no—
304C8—3461no—
305C8—1732no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—3461no—
310C22—3461no—
311C22—3461no—
312C22—6921no—
313C22—6921no—
314C22—6921no—
315C22C2×D4×D17311yescharacteristic
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—3461no—
322C4—3461no—
323C4—3461no—
324C4—6921no—
325C4—6921no—
326C4—6921no—
327C4#41×D17311yescharacteristic
328C4#1914972611yescharacteristic
329C2—41no—
330C2—13841no—
331C2—6921no—
332C2#2601213911yescharacteristic
333C2#2601229011yescharacteristic
334C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#483×C173C2
socleC22×C173C22×D4
prime radical#26012189C2
hypercenter#483D173
hypocenterC173#3351
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. #26012175C2semidirect
3. #26011891C2semidirect
4. #26011830C2semidirect
5. #26011816C2semidirect
6. #26011811C2semidirect
7. #26011800C2semidirect
8. #26011790C2semidirect
9. #26011740C2semidirect
10. #26011739C2semidirect
11. #26011576C2semidirect
12. #26011566C2semidirect
13. #26011531C2semidirect
14. #26011530C2semidirect
15. #483×C173C2semidirect
167. C173#3351semidirect
173. #483D173semidirect

Maximal subgroups

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

subgroupindex
#260121892
#260121752
#260118912
#260118302
#260118162
#260118112
#260118002
#260117902
#260117402
#260117392
#260115762
#260115662
#260115312
#260115302
#483×C1732
#3351173