Finite Group Databasebeta

← #36363795

Subgroups — #36363795

Order 22144.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012114C211yesmaximal, characteristic
2#26011968C211yesmaximal, characteristic, complemented
3#26011815C211yesmaximal, characteristic
4#26011785C211yesmaximal, characteristic, complemented
5#26011775C211yesmaximal, characteristic, complemented
6#26011716C211yesmaximal, characteristic, complemented
7#26011716C211yesmaximal, characteristic, complemented
8#26011646C211yesmaximal, characteristic, complemented
9#26011632C211yesmaximal, characteristic
10#26011595C211yesmaximal, characteristic, complemented
11#26011590C211yesmaximal, characteristic, complemented
12#26011583C211yesmaximal, characteristic, complemented
13#26011578C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#485×C173C211yesmaximal, characteristic, complemented
16C22×Dic346C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149674—21no—
23#19149668C2211yescharacteristic
24#19149665—21no—
25#19149661—21no—
26#19149659—21no—
27#19149653C2211yescharacteristic
28Q8×Dic173C2211yescharacteristic
29C4×Dic346—21no—
30C4×Dic346—21no—
31#19149644C2211yescharacteristic
32#19149644C2211yescharacteristic
33#19149644C2211yescharacteristic
34#19149644C2211yescharacteristic
35#19149641C2211yescharacteristic
36#19149638C2211yescharacteristic
37#19149635—21no—
38#19149635—21no—
39#19149612C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×Dic346—22no—
60C2×Dic346—22no—
61C2×Dic346—21no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C22×Dic173—21no—
68C2×C4×D173—21no—
69#14583816—22no—
70#14583816—22no—
71#14583815—22no—
72#14583815—22no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C4⋊Dic173—21no—
76C4⋊Dic173D411yescharacteristic
77C4⋊Dic173D411yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84C22⋊Dic173D411yescharacteristic
85C22⋊Dic173D411yescharacteristic
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98C22×C4×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C42×C173—21no—
108C22×D173—21no—
109Dic346—24no—
110Dic346—21no—
111Dic346—21no—
112Dic346—81no—
113Dic346#4112yes—
114Dic346#4112yes—
115D692—21no—
116D692—21no—
117C22⋊D173—81no—
118C2×Dic173—24no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C4×D173—81no—
127#13063436—21no—
128#13063436—21no—
129C23×C173C2×D411yescharacteristic
130Q8×C173—21no—
131Q8×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2411yescharacteristic
140C8×C173—21no—
141C8×C173—21no—
142C22×C173—21no—
143C22×C173—21no—
144C22×C173—21no—
145C22×C173C22×D411yescharacteristic
146C2×D173—41no—
147C2×D173—41no—
148C2×D173—41no—
149C4×C173—21no—
150C4×C173—41no—
151C4×C173—41no—
152C4×C173—41no—
153C4×C173—41no—
154C4×C173C2×#4111yescharacteristic
155C4×C173C2×#4111yescharacteristic
156Dic173—22no—
157Dic173—22no—
158Dic173—21no—
159Dic173—21no—
160Dic173—21no—
161Dic173—21no—
162Dic173—41no—
163C2×C173—41no—
164C2×C173#52511yescharacteristic
165C2×C173#57711yescharacteristic
166C2×C173#57811yescharacteristic
167D173—81no—
168C173#312111yescharacteristic, complemented, Sylow
169#3121—1731nomaximal, Sylow
170#542—1731no—
171#519—1731no—
172#485D17311yescharacteristic, complemented
173#482—1731no—
174#477—1731no—
175#475—1731no—
176#464—1731no—
177#464—1731no—
178#451—1731no—
179#448—1731no—
180#441—1731no—
181#440—1731no—
182C4×Q16—1731no—
183C4×SD16—1731no—
184#433—1731no—
185C22×Q8—1731no—
186C2×Q16—1731no—
187C2×Q16—1731no—
188C2×SD16—1731no—
189C2×SD16—1731no—
190C4⋊Q8—1731no—
191#126—3461no—
192#124—3461no—
193#124—1731no—
194#123—3461no—
195C22⋊Q8—3461no—
196C22⋊Q8C2×D17311yescharacteristic
197#121—1731no—
198C4×Q8—3461no—
199C4×Q8—3461no—
200C4×Q8—1731no—
201C4×Q8—1731no—
202C4×Q8—1731no—
203C4×Q8—1731no—
204C4×Q8—1731no—
205C4×D4—1731no—
206#117—3461no—
207#117—3461no—
208#117—1731no—
209#117C2×D17311yescharacteristic
210#107—1731no—
211#107C2×D17311yescharacteristic
212#106—1731no—
213#106C2×D17311yescharacteristic
214C4⋊C8—1731no—
215C4⋊C8—1731no—
216#103—1731no—
217#103—1731no—
218#103—1731no—
219#103—1731no—
220#103C2×D17311yescharacteristic
221#103C2×D17311yescharacteristic
222#102—1731no—
223#102—1731no—
224C22⋊C8—1731no—
225C22⋊C8C2×D17311yescharacteristic
226#97—1731no—
227C4×C8—1731no—
228C2×Q8—3462no—
229C2×Q8—3462no—
230C2×Q8—3461no—
231C2×Q8—3461no—
232C2×Q8—1731no—
233C2×Q8—1731no—
234C2×Q8C22×D17311yescharacteristic
235C2×D4—3461no—
236C2×D4—1731no—
237C22×C4—3461no—
238C22×C4—3461no—
239C22×C4C22×D17311yescharacteristic
240Q16—3462no—
241Q16—3462no—
242SD16—3462no—
243SD16—3462no—
244C2×C8—1731no—
245C2×C8—1731no—
246C2×C8C22×D17311yescharacteristic
247C2×C8C22×D17311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—3461no—
250C4⋊C4—3461no—
251C4⋊C4—3461no—
252C4⋊C4—3461no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—1731no—
256C4⋊C4—1731no—
257C4⋊C4—1731no—
258C4⋊C4C22×D17311yescharacteristic
259C4⋊C4C22×D17311yescharacteristic
260C4⋊C4C22×D17311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—3461no—
264C22⋊C4—3461no—
265C22⋊C4—3461no—
266C22⋊C4—3461no—
267C22⋊C4—1731no—
268C22⋊C4—1731no—
269C42—21no—
270C42—3461no—
271C42—3461no—
272C42—3461no—
273C42—3461no—
274C42—3461no—
275C42—1731no—
276C42—1731no—
277C23—3461no—
278C23D4×D17311yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—3464no—
282Q8—3461no—
283Q8—3461no—
284Q8—13841no—
285Q8—1732no—
286Q8—1732no—
287D4—3461no—
288D4—3461no—
289D4—13841no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—3464no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—13841no—
301C2×C4—1731no—
302C2×C4—1731no—
303C2×C4—1731no—
304C2×C4—1731no—
305C2×C4D4×D17311yescharacteristic
306C2×C4C23×D17311yescharacteristic
307C8—21no—
308C8—21no—
309C8—3461no—
310C8—3461no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—6921no—
315C22—6921no—
316C22—6921no—
317C22C2×D4×D17311yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—3462no—
324C4—3462no—
325C4—3461no—
326C4—3461no—
327C4—3461no—
328C4—3461no—
329C4—6921no—
330C4#41×D17311yescharacteristic
331C4#41×D17311yescharacteristic
332C2—41no—
333C2—13841no—
334C2#2601199911yescharacteristic
335C2#2601229811yescharacteristic
336C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#485×C173C2
socleC22×C173C22×D4
prime radical#19149653C22
hypercenter#485D173
hypocenterC173#3121
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #26011968C2semidirect
4. #26011785C2semidirect
5. #26011775C2semidirect
6. #26011716C2semidirect
7. #26011716C2semidirect
8. #26011646C2semidirect
10. #26011595C2semidirect
11. #26011590C2semidirect
12. #26011583C2semidirect
13. #26011578C2semidirect
14. #26011549C2semidirect
15. #485×C173C2semidirect
168. C173#3121semidirect
172. #485D173semidirect

Maximal subgroups

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

subgroupindex
#260121142
#260119682
#260118152
#260117852
#260117752
#260117162
#260116462
#260116322
#260115952
#260115902
#260115832
#260115782
#260115492
#485×C1732
#3121173