Finite Group Databasebeta

← #36364534

Subgroups — #36364534

Order 22144.

400 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012040C211yesmaximal, characteristic, complemented
2#26012020C211yesmaximal, characteristic, complemented
3#26011873C211yesmaximal, characteristic, complemented
4#26011846C211yesmaximal, characteristic, complemented
5#26011845C211yesmaximal, characteristic, complemented
6#26011699C211yesmaximal, characteristic, complemented
7#26011694C211yesmaximal, characteristic, complemented
8#26011689C211yesmaximal, characteristic, complemented
9#26011684C211yesmaximal, characteristic, complemented
10#26011652C211yesmaximal, characteristic, complemented
11#26011650C211yesmaximal, characteristic, complemented
12#26011635C211yesmaximal, characteristic, complemented
13#26011633C211yesmaximal, characteristic, complemented
14#26011542C211yesmaximal, characteristic, complemented
15#495×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22#19149677C2211yescharacteristic
23#19149675C2211yescharacteristic
24#19149672—21no—
25#19149671C2211yescharacteristic
26#19149667C2211yescharacteristic
27#19149661—21no—
28#19149655C2211yescharacteristic
29#19149652—21no—
30#19149652—21no—
31#19149650C2211yescharacteristic
32Q8×Dic173C2211yescharacteristic
33D4×Dic173C2211yescharacteristic
34#19149639—21no—
35#19149639—21no—
36#19149639—21no—
37#19149635—21no—
38C42×D173C2211yescharacteristic
39#19149608C2211yescharacteristic
40#19149608C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51C4×#13063436C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic
57#124×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59#14583827—41no—
60#14583827—41no—
61#14583825—41no—
62#14583825—41no—
63C2×Dic346C2311yescharacteristic
64C2×D692—21no—
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C22×Dic173—21no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73C2×C4×D173D411yescharacteristic
74C2×C4×D173D411yescharacteristic
75C2×C4×D173C2311yescharacteristic
76#14583816—41no—
77#14583814—41no—
78#14583813—41no—
79#14583811—41no—
80C2×#13063436C2311yescharacteristic
81C2×#13063436C2311yescharacteristic
82C4⋊Dic173—21no—
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95C4×Dic173—21no—
96C4×Dic173D411yescharacteristic
97C4×Dic173D411yescharacteristic
98C4×Dic173C2311yescharacteristic
99C2×Q8×C173C2311yescharacteristic
100C2×Q8×C173C2311yescharacteristic
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103Q16×C173—41no—
104SD16×C173—41no—
105SD16×C173—41no—
106D8×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173C2×D411yescharacteristic
116Dic346—21no—
117Dic346—21no—
118D692—21no—
119D692—21no—
120D692—42no—
121C22⋊D173—42no—
122C22⋊D173—42no—
123C22⋊D173—42no—
124C22⋊D173—42no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—42no—
129C2×Dic173—42no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—22no—
136C4×D173—41no—
137C4×D173—42no—
138#13063436—21no—
139#13063436—21no—
140C23×C173—21no—
141C23×C173—21no—
142Q8×C173—21no—
143Q8×C173—21no—
144Q8×C173—21no—
145Q8×C173—21no—
146D4×C173—21no—
147D4×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C8×C173—21no—
156C8×C173—21no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173C22×D411yescharacteristic
164C2×D173—21no—
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C4×C173—21no—
171C4×C173—21no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173H(5,𝔽2)11yescharacteristic
175C4×C173H(5,𝔽2)11yescharacteristic
176Dic173—21no—
177Dic173—21no—
178Dic173—41no—
179Dic173—41no—
180Dic173—41no—
181Dic173#13512yes—
182C2×C173—81no—
183C2×C173—81no—
184C2×C173#53511yescharacteristic
185C2×C173C2×#13511yescharacteristic
186C2×C173#57511yescharacteristic
187D173—41no—
188D173—81no—
189C173#321211yescharacteristic, complemented, Sylow
190#3212—1731nomaximal, Sylow
191#532—1731no—
192#529—1731no—
193#495—1731no—
194#495D17311yescharacteristic, complemented
195#488—1731no—
196#488—1731no—
197#462—1731no—
198#461—1731no—
199#460—1731no—
200#459—1731no—
201#452—1731no—
202#452—1731no—
203#449—1731no—
204#449—1731no—
205#431—1731no—
206C2×#41—1731no—
207C2×#41—1731no—
208C2×Q16—1731no—
209C2×Q16C2×D17311yescharacteristic
210C2×SD16—1731no—
211C2×SD16—1731no—
212C2×SD16C2×D17311yescharacteristic
213C2×SD16C2×D17311yescharacteristic
214C2×D8—1731no—
215C2×D8C2×D17311yescharacteristic
216C4⋊Q8—1731no—
217#127—1731no—
218#126—3461no—
219#125—1731no—
220#124—1731no—
221#124C2×D17311yescharacteristic
222#124C2×D17311yescharacteristic
223#123—3461no—
224C22⋊Q8—1731no—
225#121—3461no—
226#121—3461no—
227#121—1731no—
228C4×Q8—1731no—
229C4×D4—3461no—
230C4×D4—3461no—
231C4×D4—3461no—
232C4×D4—1731no—
233#117—3461no—
234C2×C42—1731no—
235C4⋊C8—1731no—
236C4⋊C8—1731no—
237#103—1731no—
238#103—1731no—
239#103—1731no—
240#103—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245C22⋊C8—1731no—
246C22⋊C8—1731no—
247C4×C8—1731no—
248C4×C8C2×D17311yescharacteristic
249#41—6921no—
250#41—6921no—
251#41—6921no—
252#41—6921no—
253C2×Q8—1731no—
254C2×Q8C22×D17311yescharacteristic
255C2×Q8C22×D17311yescharacteristic
256C2×D4—3461no—
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—1731no—
262C2×D4C22×D17311yescharacteristic
263C2×D4C22×D17311yescharacteristic
264C22×C4—3461no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—1731no—
268C22×C4—1731no—
269C22×C4—1731no—
270Q16—41no—
271Q16—6921no—
272SD16—41no—
273SD16—41no—
274SD16—6921no—
275SD16—6921no—
276D8—41no—
277D8—6921no—
278C2×C8—1731no—
279C2×C8—1731no—
280C2×C8C22×D17311yescharacteristic
281C2×C8C22×D17311yescharacteristic
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—3461no—
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4—1731no—
289C4⋊C4—1731no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C42—3461no—
300C42—1731no—
301C42—1731no—
302C42—1731no—
303C42C22×D17311yescharacteristic
304C23—21no—
305C23—21no—
306C23—3461no—
307C23—1731no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—3461no—
313Q8—3461no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—3461no—
319D4—3461no—
320D4—6922no—
321D4—6922no—
322D4—6922no—
323D4—6922no—
324D4—6922no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—3462no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—1731no—
334C2×C4—1731no—
335C2×C4—1731no—
336C2×C4—6921no—
337C2×C4—6922no—
338C2×C4—6922no—
339C2×C4—6922no—
340C2×C4D4×D17311yescharacteristic
341C2×C4D4×D17311yescharacteristic
342C2×C4C23×D17311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3461no—
346C8—3461no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—3461no—
354C22—3461no—
355C22—3461no—
356C22—6921no—
357C22—6921no—
358C22—6921no—
359C22C2×D4×D17311yescharacteristic
360C4—21no—
361C4—21no—
362C4—41no—
363C4—41no—
364C4—3461no—
365C4—3461no—
366C4—1732no—
367C4—6921no—
368C4—6921no—
369C4—6921no—
370C4#1914972411yescharacteristic
371C4#1914972411yescharacteristic
372C2—81no—
373C2—81no—
374C2—13841no—
375C2—6921no—
376C2#2601205411yescharacteristic
377C2#135×D17311yescharacteristic
378C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#495×C173C2
socleC22×C173C22×D4
hypercenter#495D173
hypocenterC173#3212
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #26012040C2semidirect
2. #26012020C2semidirect
3. #26011873C2semidirect
4. #26011846C2semidirect
5. #26011845C2semidirect
6. #26011699C2semidirect
7. #26011694C2semidirect
8. #26011689C2semidirect
9. #26011684C2semidirect
10. #26011652C2semidirect
11. #26011650C2semidirect
12. #26011635C2semidirect
13. #26011633C2semidirect
14. #26011542C2semidirect
15. #495×C173C2semidirect
189. C173#3212semidirect
194. #495D173semidirect

Maximal subgroups

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

subgroupindex
#260120402
#260120202
#260118732
#260118462
#260118452
#260116992
#260116942
#260116892
#260116842
#260116522
#260116502
#260116352
#260116332
#260115422
#495×C1732
#3212173