Finite Group Databasebeta

← #36363792

Subgroups — #36363792

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#26012112C211yesmaximal, characteristic
2#26011832C211yesmaximal, characteristic, complemented
3#26011815C211yesmaximal, characteristic
4#26011786C211yesmaximal, characteristic, complemented
5#26011776C211yesmaximal, characteristic, complemented
6#26011713C211yesmaximal, characteristic, complemented
7#26011713C211yesmaximal, characteristic, complemented
8#26011647C211yesmaximal, characteristic, complemented
9#26011630C211yesmaximal, characteristic
10#26011596C211yesmaximal, characteristic, complemented
11#26011589C211yesmaximal, characteristic, complemented
12C4×#14583816C211yesmaximal, characteristic, complemented
13C4×#14583813C211yesmaximal, characteristic, complemented
14#26011552C211yesmaximal, characteristic, complemented
15#519×C173C211yesmaximal, characteristic, complemented
16C2×#14583816C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20#19149679C2211yescharacteristic
21#19149674—21no—
22#19149667C2211yescharacteristic
23#19149665—21no—
24#19149661—21no—
25#19149658C2211yescharacteristic
26#19149657—21no—
27#19149653C2211yescharacteristic
28C4×Dic346C2211yescharacteristic
29C4×D692C2211yescharacteristic
30#19149612C2211yescharacteristic
31#19149612C2211yescharacteristic
32#19149610C2211yescharacteristic
33#19149610C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35C4⋊#13063436C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149603C2211yescharacteristic
39#19149603C2211yescharacteristic
40#19149602C2211yescharacteristic
41#19149602C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46#19149590C2211yescharacteristic
47C4×#13063436C2211yescharacteristic
48C22×Q8×C173C2211yescharacteristic
49C4×Q8×C173—21no—
50C4×Q8×C173—21no—
51C4×Q8×C173C2211yescharacteristic
52C4×Q8×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic
55#117×C173—21no—
56#117×C173—21no—
57#117×C173C2211yescharacteristic
58#117×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C4×D173—21no—
64#14583816—22no—
65#14583816—22no—
66#14583813—22no—
67#14583813—22no—
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C4⋊Dic173—21no—
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75#14583805—21no—
76#14583805—21no—
77C22⋊Dic173—21no—
78#14583803—21no—
79#14583803—21no—
80#14583803—21no—
81#14583803—21no—
82C4×Dic173—21no—
83C2×Q8×C173—22no—
84C2×Q8×C173—22no—
85C2×Q8×C173—21no—
86C2×Q8×C173C2311yescharacteristic
87C2×Q8×C173C2311yescharacteristic
88C22×C4×C173—21no—
89C22×C4×C173C2311yescharacteristic
90C4⋊C4×C173—21no—
91C4⋊C4×C173—21no—
92C4⋊C4×C173—21no—
93C4⋊C4×C173—21no—
94C4⋊C4×C173D411yescharacteristic
95C4⋊C4×C173D411yescharacteristic
96C4⋊C4×C173C2311yescharacteristic
97C4⋊C4×C173C2311yescharacteristic
98C22⋊C4×C173—21no—
99C22⋊C4×C173D411yescharacteristic
100C22⋊C4×C173D411yescharacteristic
101C42×C173—21no—
102C42×C173—21no—
103C42×C173—21no—
104C42×C173—21no—
105C42×C173—21no—
106C42×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109Dic346—21no—
110Dic346—21no—
111Dic346—81no—
112D692—21no—
113D692—21no—
114C22⋊D173—81no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C4×D173—81no—
119#13063436—21no—
120#13063436—21no—
121#13063436—21no—
122#13063436—21no—
123C23×C173C2×D411yescharacteristic
124Q8×C173—24no—
125Q8×C173—21no—
126Q8×C173—21no—
127Q8×C173#4112yes—
128Q8×C173#4112yes—
129C2×C4×C173—24no—
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×C173C2×D411yescharacteristic
137C2×C4×C173C2×D411yescharacteristic
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2411yescharacteristic
142C22×C173—21no—
143C22×C173—21no—
144C22×C173—21no—
145C22×C173C22×D411yescharacteristic
146C2×D173—41no—
147C2×D173—41no—
148C2×D173—41no—
149C4×C173—22no—
150C4×C173—22no—
151C4×C173—21no—
152C4×C173—21no—
153C4×C173—21no—
154C4×C173—21no—
155C4×C173—21no—
156C4×C173—41no—
157C4×C173—41no—
158C4×C173C2×#4111yescharacteristic
159C4×C173C2×#4111yescharacteristic
160Dic173—41no—
161Dic173—41no—
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#519D17311yescharacteristic, complemented
172#485—1731no—
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×Q8C2×D17311yescharacteristic
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⋊Q8—1731no—
197#121—1731no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—1731no—
201C4×Q8C2×D17311yescharacteristic
202C4×Q8C2×D17311yescharacteristic
203C4×Q8C2×D17311yescharacteristic
204C4×Q8C2×D17311yescharacteristic
205C4×D4—1731no—
206#117—21no—
207#117—21no—
208#117C2×D17311yescharacteristic
209#117C2×D17311yescharacteristic
210#107—1731no—
211#107—1731no—
212#106—1731no—
213#106—1731no—
214C4⋊C8—1731no—
215C4⋊C8—1731no—
216#103—1731no—
217#103—1731no—
218#103—1731no—
219#103—1731no—
220#103—1731no—
221#103—1731no—
222#102—1731no—
223#102—1731no—
224C22⋊C8—1731no—
225C22⋊C8—1731no—
226#97—1731no—
227C4×C8—1731no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—3461no—
232C2×Q8—1731no—
233C2×Q8C22×D17311yescharacteristic
234C2×Q8C22×D17311yescharacteristic
235C2×D4—3461no—
236C2×D4—1731no—
237C22×C4—21no—
238C22×C4—3461no—
239C22×C4C22×D17311yescharacteristic
240Q16—3462no—
241Q16—3462no—
242SD16—3462no—
243SD16—3462no—
244C2×C8—1731no—
245C2×C8—1731no—
246C2×C8—1731no—
247C2×C8—1731no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—3461no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—1731no—
256C4⋊C4—1731no—
257C4⋊C4C22⋊D17311yescharacteristic
258C4⋊C4C22⋊D17311yescharacteristic
259C4⋊C4C22×D17311yescharacteristic
260C4⋊C4C22×D17311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—3461no—
263C22⋊C4—3461no—
264C22⋊C4—3461no—
265C22⋊C4—3461no—
266C22⋊C4—3461no—
267C22⋊C4C22⋊D17311yescharacteristic
268C22⋊C4C22⋊D17311yescharacteristic
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—3461no—
275C42C22×D17311yescharacteristic
276C42C22×D17311yescharacteristic
277C23—3461no—
278C23C2×C22⋊D17311yescharacteristic
279Q8—24no—
280Q8—21no—
281Q8—21no—
282Q8—3461no—
283Q8—3461no—
284Q8—13841no—
285Q8#1458382612yes—
286Q8#1458382612yes—
287D4—3461no—
288D4—3461no—
289D4—13841no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—13841no—
301C2×C4C2×C22⋊D17311yescharacteristic
302C2×C4C2×C22⋊D17311yescharacteristic
303C2×C4C2×C22⋊D17311yescharacteristic
304C2×C4C2×C22⋊D17311yescharacteristic
305C2×C4C2×C22⋊D17311yescharacteristic
306C2×C4C23×D17311yescharacteristic
307C8—3461no—
308C8—3461no—
309C8—3461no—
310C8—3461no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—6921no—
315C22—6921no—
316C22—6921no—
317C22C22×C22⋊D17311yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—6921no—
328C4—6921no—
329C4—6921no—
330C4C2×#1458382611yescharacteristic
331C4C2×#1458382611yescharacteristic
332C2—41no—
333C2—13841no—
334C2#2601200211yescharacteristic
335C2#2601229911yescharacteristic
336C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#519×C173C2
socleC22×C173C22×D4
prime radical#19149653C22
hypercenter#519D173
hypocenterC173#3121
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #26011832C2semidirect
4. #26011786C2semidirect
5. #26011776C2semidirect
6. #26011713C2semidirect
7. #26011713C2semidirect
8. #26011647C2semidirect
10. #26011596C2semidirect
11. #26011589C2semidirect
12. C4×#14583816C2semidirect
13. C4×#14583813C2semidirect
14. #26011552C2semidirect
15. #519×C173C2semidirect
168. C173#3121semidirect
171. #519D173semidirect

Maximal subgroups

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

subgroupindex
#260121122
#260118322
#260118152
#260117862
#260117762
#260117132
#260116472
#260116302
#260115962
#260115892
C4×#145838162
C4×#145838132
#260115522
#519×C1732
#3121173