Finite Group Databasebeta

← #36364461

Subgroups — #36364461

Order 22144.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012071C211yesmaximal, characteristic, complemented
2#26012070C211yesmaximal, characteristic, complemented
3#26011828C211yesmaximal, characteristic, complemented
4#26011825C211yesmaximal, characteristic, complemented
5#26011806C211yesmaximal, characteristic, complemented
6#26011803C211yesmaximal, characteristic, complemented
7#26011737C211yesmaximal, characteristic, complemented
8#26011730C211yesmaximal, characteristic, complemented
9#26011725C211yesmaximal, characteristic, complemented
10#26011656C211yesmaximal, characteristic, complemented
11#26011650C211yesmaximal, characteristic, complemented
12#26011637C211yesmaximal, characteristic, complemented
13#26011635C211yesmaximal, characteristic, complemented
14#26011459C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×#14583825C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149659C2211yescharacteristic
24#19149658C2211yescharacteristic
25#19149657—21no—
26#19149656—21no—
27#19149655C2211yescharacteristic
28#19149654—21no—
29#19149654C2211yescharacteristic
30#19149653C2211yescharacteristic
31#19149652—21no—
32#19149652—21no—
33#19149652—21no—
34#19149651—21no—
35#19149650—21no—
36#19149650C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583827—41no—
60#14583827—41no—
61#14583826—41no—
62#14583826—41no—
63#14583825—41no—
64#14583825—41no—
65C2×Dic346D411yescharacteristic
66C2×Dic346C2311yescharacteristic
67C2×D692—21no—
68C2×D692—21no—
69C2×D692D411yescharacteristic
70C2×D692C2311yescharacteristic
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173D411yescharacteristic
75C2×C22⋊D173D411yescharacteristic
76C22×Dic173—21no—
77C2×C4×D173—21no—
78C2×C4×D173—21no—
79C2×C4×D173—21no—
80C2×C4×D173C2311yescharacteristic
81C2×C4×D173C2311yescharacteristic
82#14583816—41no—
83#14583815—41no—
84#14583811—41no—
85C2×#13063436—41no—
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C4⋊Dic173—21no—
89C4⋊Dic173C2311yescharacteristic
90C4⋊Dic173C2311yescharacteristic
91#14583805—21no—
92#14583805—21no—
93#14583805—21no—
94#14583805—21no—
95C22⋊Dic173—21no—
96C22⋊Dic173—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173—21no—
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105SD16×C173—41no—
106M16×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173—21no—
110C4⋊C4×C173C2311yescharacteristic
111C4⋊C4×C173C2311yescharacteristic
112C22⋊C4×C173—21no—
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173C2×D411yescharacteristic
116C22×D173C2×D411yescharacteristic
117Dic346—21no—
118Dic346—21no—
119Dic346—41no—
120D692—21no—
121D692—21no—
122D692—41no—
123D692—42no—
124D692—81no—
125C22⋊D173—41no—
126C22⋊D173—41no—
127C22⋊D173—42no—
128C22⋊D173—81no—
129C22⋊D173—81no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—42no—
134C2×Dic173C2×D411yescharacteristic
135C2×Dic173C2×D411yescharacteristic
136C4×D173—21no—
137C4×D173—21no—
138C4×D173—21no—
139C4×D173—21no—
140C4×D173—42no—
141C4×D173—81no—
142C4×D173—81no—
143#13063436—21no—
144#13063436—21no—
145C23×C173—21no—
146C23×C173C2×D411yescharacteristic
147Q8×C173—21no—
148Q8×C173—21no—
149D4×C173—21no—
150D4×C173—21no—
151D4×C173—81no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2411yescharacteristic
159C8×C173—21no—
160C8×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173C22×D411yescharacteristic
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—21no—
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C4×C173—21no—
178C4×C173—41no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173H(5,𝔽2)11yescharacteristic
182C4×C173H(5,𝔽2)11yescharacteristic
183Dic173—21no—
184Dic173—21no—
185Dic173—41no—
186Dic173—41no—
187Dic173—41no—
188C2×C173—41no—
189C2×C173—81no—
190C2×C173#53411yescharacteristic
191C2×C173#57611yescharacteristic
192C2×C173#57811yescharacteristic
193D173—41no—
194D173—41no—
195D173—81no—
196C173#320311yescharacteristic, complemented, Sylow
197#3203—1731nomaximal, Sylow
198#537—1731no—
199#537—1731no—
200#484—1731no—
201#484—1731no—
202#480—1731no—
203#480—1731no—
204#468—1731no—
205#468D17311yescharacteristic, complemented
206#467—1731no—
207#466—1731no—
208#452—1731no—
209#452—1731no—
210#449—1731no—
211#449—1731no—
212#408—1731no—
213C2×#41—1731no—
214C2×#41—1731no—
215C2×#41—1731no—
216C2×Q16—1731no—
217C2×SD16—1731no—
218C2×SD16C2×D17311yescharacteristic
219C2×D8—1731no—
220C2×M16C2×D17311yescharacteristic
221C22×C8—1731no—
222C22⋊Q8—3461no—
223C22⋊Q8—3461no—
224C22⋊Q8—1731no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227C22⋊Q8C2×D17311yescharacteristic
228#121—3461no—
229#121—3461no—
230#121—3461no—
231#121—3461no—
232#121—3461no—
233#121—3461no—
234#121—1731no—
235#121—1731no—
236#121—1731no—
237#121C2×D17311yescharacteristic
238C2×C4⋊C4—1731no—
239C2×C4⋊C4—1731no—
240#107—1731no—
241#107—1731no—
242#107—1731no—
243#107C2×D17311yescharacteristic
244#103—1731no—
245#103—1731no—
246#103—1731no—
247#103C2×D17311yescharacteristic
248#102—1731no—
249#102—1731no—
250#102—1731no—
251#102C2×D17311yescharacteristic
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254C22⋊C8—1731no—
255C22⋊C8—1731no—
256#41—6921no—
257#41—6921no—
258#41—6921no—
259#41—6921no—
260#41—6921no—
261#41—6921no—
262C2×Q8—1731no—
263C2×Q8—1731no—
264C2×Q8C22×D17311yescharacteristic
265C2×D4—21no—
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—1731no—
272C2×D4—1731no—
273C2×D4—1731no—
274C2×D4—1731no—
275C2×D4C22×D17311yescharacteristic
276C22×C4—3461no—
277C22×C4—3461no—
278C22×C4—3461no—
279C22×C4—3461no—
280C22×C4—1731no—
281C22×C4—1731no—
282C22×C4C22×D17311yescharacteristic
283Q16—6921no—
284SD16—41no—
285SD16—6921no—
286D8—6921no—
287M16—41no—
288C2×C8—1731no—
289C2×C8—1731no—
290C2×C8—6921no—
291C2×C8C22×D17311yescharacteristic
292C2×C8C22×D17311yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—3461no—
297C4⋊C4—3461no—
298C4⋊C4—3461no—
299C4⋊C4—1731no—
300C4⋊C4—1731no—
301C4⋊C4C22×D17311yescharacteristic
302C4⋊C4C22×D17311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C23—21no—
312C23—3461no—
313C23—1731no—
314C23—1731no—
315C23D4×D17311yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—3461no—
319Q8—3461no—
320Q8—6921no—
321D4—21no—
322D4—21no—
323D4—81no—
324D4—3461no—
325D4—3461no—
326D4—13841no—
327D4—13841no—
328D4—13841no—
329D4—6921no—
330D4—6922no—
331D4—6921no—
332D4—6921no—
333D4—6922no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—3461no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—13841no—
347C2×C4—13841no—
348C2×C4—1731no—
349C2×C4—1731no—
350C2×C4—6922no—
351C2×C4—6922no—
352C2×C4D4×D17311yescharacteristic
353C2×C4C23×D17311yescharacteristic
354C8—21no—
355C8—21no—
356C8—3461no—
357C8—3461no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—3461no—
365C22—3461no—
366C22—3461no—
367C22—3461no—
368C22—3461no—
369C22—3461no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22C2×D4×D17311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—3461no—
379C4—3461no—
380C4—6921no—
381C4—6921no—
382C4—6921no—
383C4#1914972411yescharacteristic
384C4#1914972511yescharacteristic
385C2—41no—
386C2—81no—
387C2—13841no—
388C2—6921no—
389C2—6921no—
390C2#2601204911yescharacteristic
391C2#2601229511yescharacteristic
392C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
hypercenter#468D173
hypocenterC173#3203
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #26012071C2semidirect
2. #26012070C2semidirect
3. #26011828C2semidirect
4. #26011825C2semidirect
5. #26011806C2semidirect
6. #26011803C2semidirect
7. #26011737C2semidirect
8. #26011730C2semidirect
9. #26011725C2semidirect
10. #26011656C2semidirect
11. #26011650C2semidirect
12. #26011637C2semidirect
13. #26011635C2semidirect
14. #26011459C2semidirect
15. #468×C173C2semidirect
196. C173#3203semidirect
205. #468D173semidirect

Maximal subgroups

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

subgroupindex
#260120712
#260120702
#260118282
#260118252
#260118062
#260118032
#260117372
#260117302
#260117252
#260116562
#260116502
#260116372
#260116352
#260114592
#468×C1732
#3203173