Finite Group Databasebeta

← #36366182

Subgroups — #36366182

Order 22144.

354 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012205C211yesmaximal, characteristic
2#26012196C211yesmaximal, characteristic, complemented
3#26011895C211yesmaximal, characteristic, complemented
4#26011834C211yesmaximal, characteristic, complemented
5#26011833C211yesmaximal, characteristic, complemented
6#26011803C211yesmaximal, characteristic, complemented
7#26011785C211yesmaximal, characteristic, complemented
8#26011779C211yesmaximal, characteristic, complemented
9#26011736C211yesmaximal, characteristic, complemented
10#26011735C211yesmaximal, characteristic, complemented
11#26011582C211yesmaximal, characteristic, complemented
12#26011566C211yesmaximal, characteristic, complemented
13#26011532C211yesmaximal, characteristic, complemented
14#26011531C211yesmaximal, characteristic, complemented
15#480×C173C211yesmaximal, characteristic, complemented
16C2×#14583814C2211yescharacteristic
17C2×#14583810C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19#19149678C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149669—21no—
22#19149666—21no—
23#19149665—21no—
24#19149659C2211yescharacteristic
25#19149658—21no—
26#19149658C2211yescharacteristic
27#19149657—21no—
28#19149651—21no—
29#19149650C2211yescharacteristic
30#19149644C2211yescharacteristic
31D4×Dic173C2211yescharacteristic
32C4×C22⋊D173—21no—
33C4×C22⋊D173C2211yescharacteristic
34#19149638C2211yescharacteristic
35#19149637C2211yescharacteristic
36#19149635—21no—
37C4⋊C4×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149611C2211yescharacteristic
43#19149610C2211yescharacteristic
44#19149610C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52#121×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346—21no—
61C2×Dic346C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173D411yescharacteristic
64C22×Dic173—21no—
65C2×C4×D173—21no—
66C2×C4×D173C2311yescharacteristic
67#14583814—22no—
68#14583810—22no—
69#14583809—22no—
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C4⋊Dic173—21no—
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805D411yescharacteristic
82C22⋊Dic173—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173D411yescharacteristic
87#14583803—21no—
88#14583803—21no—
89#14583803D411yescharacteristic
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173C2311yescharacteristic
94C2×D4×C173—21no—
95C2×D4×C173C2311yescharacteristic
96C22×C4×C173—21no—
97C22×C4×C173C2311yescharacteristic
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C22⋊C4×C173—21no—
105C22×D173C2×D411yescharacteristic
106Dic346—21no—
107Dic346—21no—
108Dic346—81no—
109Dic346—81no—
110C22⋊D173—22no—
111C22⋊D173—81no—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—42no—
117C2×Dic173C2×D411yescharacteristic
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120C4×D173—21no—
121C4×D173—21no—
122C4×D173—81no—
123#13063436—21no—
124#13063436#4112yes—
125C23×C173—21no—
126C23×C173C2×D411yescharacteristic
127D4×C173—21no—
128D4×C173—21no—
129D4×C173—81no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—81no—
136C2×C4×C173C2×D411yescharacteristic
137C2×C4×C173C2411yescharacteristic
138C8×C173—21no—
139C8×C173—21no—
140C22×C173—21no—
141C22×C173—21no—
142C22×C173—21no—
143C22×C173—41no—
144C22×C173—41no—
145C22×C173—41no—
146C22×C173C22×D411yescharacteristic
147C2×D173—21no—
148C2×D173—21no—
149C2×D173—21no—
150C4×C173—21no—
151C4×C173—41no—
152C4×C173—41no—
153C4×C173—41no—
154C4×C173C2×#4111yescharacteristic
155C4×C1732−1+411yescharacteristic
156Dic173—21no—
157Dic173—21no—
158Dic173—41no—
159Dic173—41no—
160Dic173—41no—
161Dic173—41no—
162Dic173#13712yes—
163C2×C173—41no—
164C2×C173—81no—
165C2×C173#54711yescharacteristic
166C2×C173C2×#13711yescharacteristic
167C2×C173#57611yescharacteristic
168D173—41no—
169C173#335711yescharacteristic, complemented, Sylow
170#3357—1731nomaximal, Sylow
171#555—1731no—
172#554—1731no—
173#500—1731no—
174#485—1731no—
175#485—1731no—
176#480—1731no—
177#480D17311yescharacteristic, complemented
178#477—1731no—
179#476—1731no—
180#468—1731no—
181#468—1731no—
182C4×SD16—1731no—
183#436—1731no—
184#428—1731no—
185#428—1731no—
186C2×SD16—1731no—
187C2×M16—1731no—
188C2×M16—1731no—
189C4⋊Q8—1731no—
190C4⋊Q8—1731no—
191#125—3461no—
192#124—3461no—
193#124—3461no—
194C22⋊Q8—3461no—
195C22⋊Q8—3461no—
196C22⋊Q8—1731no—
197C22⋊Q8—1731no—
198#121—3461no—
199#121—1731no—
200#121C2×D17311yescharacteristic
201C4×Q8—1731no—
202C4×D4—3461no—
203C4×D4—1731no—
204C4×D4—1731no—
205#117—3461no—
206#117—1731no—
207#117—1731no—
208C2×C4⋊C4—1731no—
209C2×C4⋊C4C2×D17311yescharacteristic
210#107—1731no—
211#107—1731no—
212#107—1731no—
213#107—1731no—
214#107C2×D17311yescharacteristic
215#107C2×D17311yescharacteristic
216#106—1731no—
217#106—1731no—
218#106—1731no—
219C4⋊C8—1731no—
220#103—1731no—
221#103—1731no—
222#103—1731no—
223#102—1731no—
224#102C2×D17311yescharacteristic
225#102C2×D17311yescharacteristic
226C22⋊C8—1731no—
227C22⋊C8C2×D17311yescharacteristic
228C4×C8—1731no—
229C2×Q8—3461no—
230C2×Q8—3461no—
231C2×Q8—1731no—
232C2×D4—21no—
233C2×D4—3461no—
234C2×D4—1731no—
235C2×D4C22×D17311yescharacteristic
236C22×C4—21no—
237C22×C4—3461no—
238C22×C4—3461no—
239C22×C4—1731no—
240C22×C4C22×D17311yescharacteristic
241SD16—3462no—
242M16—3462no—
243M16—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⋊C4—1731no—
259C4⋊C4C22×D17311yescharacteristic
260C4⋊C4C22×D17311yescharacteristic
261C4⋊C4C22×D17311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—3461no—
264C22⋊C4—3461no—
265C22⋊C4—3461no—
266C22⋊C4—3461no—
267C22⋊C4—3461no—
268C22⋊C4—3461no—
269C22⋊C4—1731no—
270C22⋊C4—1731no—
271C42—3461no—
272C42—3461no—
273C42—3461no—
274C42—1731no—
275C23—21no—
276C23—1731no—
277C23D4×D17311yescharacteristic
278Q8—3461no—
279Q8—3461no—
280Q8—13841no—
281Q8—13841no—
282D4—21no—
283D4—21no—
284D4—81no—
285D4—3462no—
286D4—13841no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—81no—
293C2×C4—3461no—
294C2×C4—3461no—
295C2×C4—3461no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—13841no—
300C2×C4—1731no—
301C2×C4—1731no—
302C2×C4—1731no—
303C2×C4—6922no—
304C2×C4D4×D17311yescharacteristic
305C2×C4C23×D17311yescharacteristic
306C8—21no—
307C8—21no—
308C8—3461no—
309C8—1732no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—3461no—
317C22—3461no—
318C22—3461no—
319C22C2×D4×D17311yescharacteristic
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—3461no—
325C4—3461no—
326C4—1732no—
327C4—6921no—
328C4—6921no—
329C4—6921no—
330C4—6921no—
331C4#41×D17311yescharacteristic
332C4#1914972611yescharacteristic
333C2—41no—
334C2—81no—
335C2—6921no—
336C2#2601213911yescharacteristic
337C2#137×D17311yescharacteristic
338C2#2601229211yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#480×C173C2
socleC22×C173C22×D4
prime radical#26012205C2
hypercenter#480D173
hypocenterC173#3357
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #26012196C2semidirect
3. #26011895C2semidirect
4. #26011834C2semidirect
5. #26011833C2semidirect
6. #26011803C2semidirect
7. #26011785C2semidirect
8. #26011779C2semidirect
9. #26011736C2semidirect
10. #26011735C2semidirect
11. #26011582C2semidirect
12. #26011566C2semidirect
13. #26011532C2semidirect
14. #26011531C2semidirect
15. #480×C173C2semidirect
169. C173#3357semidirect
177. #480D173semidirect

Maximal subgroups

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

subgroupindex
#260122052
#260121962
#260118952
#260118342
#260118332
#260118032
#260117852
#260117792
#260117362
#260117352
#260115822
#260115662
#260115322
#260115312
#480×C1732
#3357173