Finite Group Databasebeta

← #36365460

Subgroups — #36365460

Order 22144.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012188C211yesmaximal, characteristic, complemented
2#26012151C211yesmaximal, characteristic, complemented
3#26011847C211yesmaximal, characteristic, complemented
4#26011829C211yesmaximal, characteristic, complemented
5#26011811C211yesmaximal, characteristic, complemented
6#26011801C211yesmaximal, characteristic, complemented
7#26011726C211yesmaximal, characteristic, complemented
8#26011708C211yesmaximal, characteristic, complemented
9#26011634C211yesmaximal, characteristic, complemented
10#26011628C211yesmaximal, characteristic, complemented
11#26011582C211yesmaximal, characteristic, complemented
12C8×C22⋊D173C211yesmaximal, characteristic, complemented
13#26011488C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149672—21no—
22#19149670C2211yescharacteristic
23#19149668C2211yescharacteristic
24#19149665—21no—
25#19149660—21no—
26#19149657—21no—
27#19149654C2211yescharacteristic
28#19149652—21no—
29#19149650—21no—
30#19149650C2211yescharacteristic
31#19149644C2211yescharacteristic
32D4×Dic173C2211yescharacteristic
33#19149641—21no—
34C4×C22⋊D173C2211yescharacteristic
35#19149639—21no—
36#19149638C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149610C2211yescharacteristic
40#19149608C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#106×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63Q8×D173—22no—
64Q8×D173—41no—
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173D411yescharacteristic
70C22×Dic173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173C2311yescharacteristic
75#14583814—22no—
76#14583811—41no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C8×D173—41no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C2×Q8×C173C2311yescharacteristic
99C2×D4×C173—21no—
100C2×D4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102SD16×C173—41no—
103C2×C8×C173—41no—
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—42no—
115Dic346#4112yes—
116D692—21no—
117D692—21no—
118D692—42no—
119C22⋊D173—22no—
120C22⋊D173—42no—
121C22⋊D173—42no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—81no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—42no—
132C4×D173—42no—
133C4×D173—81no—
134#13063436—21no—
135#13063436—21no—
136C23×C173—21no—
137C23×C173C2×D411yescharacteristic
138Q8×C173—21no—
139Q8×C173—21no—
140D4×C173—21no—
141D4×C173—21no—
142D4×C173—81no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173C22×D411yescharacteristic
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173#13512yes—
165C4×C173—21no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173C2×#4111yescharacteristic
170C4×C173H(5,𝔽2)11yescharacteristic
171Dic173—21no—
172Dic173—22no—
173Dic173—21no—
174Dic173—21no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—41no—
178C2×C173—81no—
179C2×C173#54611yescharacteristic
180C2×C173C2×#13511yescharacteristic
181C2×C173#57711yescharacteristic
182D173—22no—
183D173—81no—
184C173#330111yescharacteristic, complemented, Sylow
185#3301—1731nomaximal, Sylow
186#553—1731no—
187#548—1731no—
188#488—1731no—
189#485—1731no—
190#481—1731no—
191#479—1731no—
192#466—1731no—
193#465D17311yescharacteristic, complemented
194#463—1731no—
195#449—1731no—
196#448—1731no—
197C4×SD16—1731no—
198C8×D4—1731no—
199#416—1731no—
200C2×#103—1731no—
201C2×#41—1731no—
202C22×Q8—1731no—
203C2×SD16—1731no—
204C2×SD16C2×D17311yescharacteristic
205C2×D8—1731no—
206C22×C8—1731no—
207C22×C8C2×D17311yescharacteristic
208#126—3461no—
209#125—1731no—
210#124—3461no—
211#124—1731no—
212#123—3461no—
213C22⋊Q8—3461no—
214C22⋊Q8C2×D17311yescharacteristic
215#121—3461no—
216#121—3461no—
217#121—1731no—
218#121—1731no—
219#121C2×D17311yescharacteristic
220C4×Q8—1731no—
221C4×D4—3461no—
222C4×D4—3461no—
223C4×D4—1731no—
224C4×D4—1731no—
225#117—1731no—
226C2×C4⋊C4—1731no—
227#107—1731no—
228#106—1731no—
229#106C2×D17311yescharacteristic
230C4⋊C8—1731no—
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#103C2×D17311yescharacteristic
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102C2×D17311yescharacteristic
241C22⋊C8—1731no—
242C22⋊C8—1731no—
243C4×C8—1731no—
244#41—3462no—
245#41—6921no—
246C2×Q8—3461no—
247C2×Q8—3462no—
248C2×Q8—1731no—
249C2×Q8—6921no—
250C2×Q8C22×D17311yescharacteristic
251C2×D4—21no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4—1731no—
257C2×D4C22×D17311yescharacteristic
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—1731no—
263C22×C4C22×D17311yescharacteristic
264SD16—41no—
265SD16—3462no—
266D8—6921no—
267C2×C8—41no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8—6921no—
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—1731no—
279C4⋊C4—1731no—
280C4⋊C4—1731no—
281C4⋊C4C22×D17311yescharacteristic
282C4⋊C4C22×D17311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—1731no—
292C22⋊C4—1731no—
293C42—3461no—
294C42—3461no—
295C42—1731no—
296C23—21no—
297C23—3461no—
298C23—1731no—
299C23D4×D17311yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—3461no—
303Q8—1732no—
304Q8—6922no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—3462no—
309D4—3461no—
310D4—3461no—
311D4—6922no—
312D4—6922no—
313D4—6922no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—13841no—
325C2×C4—13841no—
326C2×C4—1731no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—6922no—
330C2×C4—6922no—
331C2×C4D4×D17311yescharacteristic
332C2×C4C23×D17311yescharacteristic
333C8—21no—
334C8—21no—
335C8—3461no—
336C8—3461no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—3461no—
344C22—3461no—
345C22—1732no—
346C22—6921no—
347C22—6921no—
348C22—6921no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—3461no—
355C4—3461no—
356C4—3461no—
357C4—3462no—
358C4—6921no—
359C4—6921no—
360C4#41×D17311yescharacteristic
361C4#1914972411yescharacteristic
362C2—41no—
363C2—81no—
364C2—3462no—
365C2—13841no—
366C2#2601213011yescharacteristic
367C2#135×D17311yescharacteristic
368C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
hypercenter#465D173
hypocenterC173#3301
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #26012188C2semidirect
2. #26012151C2semidirect
3. #26011847C2semidirect
4. #26011829C2semidirect
5. #26011811C2semidirect
6. #26011801C2semidirect
7. #26011726C2semidirect
8. #26011708C2semidirect
9. #26011634C2semidirect
10. #26011628C2semidirect
11. #26011582C2semidirect
12. C8×C22⋊D173C2semidirect
13. #26011488C2semidirect
14. #103×D173C2semidirect
15. #465×C173C2semidirect
184. C173#3301semidirect
193. #465D173semidirect

Maximal subgroups

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

subgroupindex
#260121882
#260121512
#260118472
#260118292
#260118112
#260118012
#260117262
#260117082
#260116342
#260116282
#260115822
C8×C22⋊D1732
#260114882
#103×D1732
#465×C1732
#3301173