Finite Group Databasebeta

← #36365487

Subgroups — #36365487

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#26012204C211yesmaximal, characteristic, complemented
2#26012140C211yesmaximal, characteristic, complemented
3#26011840C211yesmaximal, characteristic, complemented
4#26011832C211yesmaximal, characteristic, complemented
5#26011830C211yesmaximal, characteristic, complemented
6#26011779C211yesmaximal, characteristic, complemented
7#26011723C211yesmaximal, characteristic, complemented
8#26011720C211yesmaximal, characteristic, complemented
9#26011695C211yesmaximal, characteristic, complemented
10#26011637C211yesmaximal, characteristic, complemented
11#26011636C211yesmaximal, characteristic, complemented
12#26011571C211yesmaximal, characteristic, complemented
13C8×C22⋊D173C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#416×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149677C2211yescharacteristic
23#19149667—21no—
24#19149665—21no—
25#19149662—21no—
26#19149660—21no—
27#19149658C2211yescharacteristic
28#19149655C2211yescharacteristic
29#19149654C2211yescharacteristic
30#19149653C2211yescharacteristic
31#19149652—21no—
32#19149651—21no—
33D4×Dic173C2211yescharacteristic
34#19149641C2211yescharacteristic
35C4×C22⋊D173C2211yescharacteristic
36#19149637C2211yescharacteristic
37#19149636—21no—
38C2×#14583805—21no—
39#19149612C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54#117×C173C2211yescharacteristic
55#103×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×D692—21no—
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173D411yescharacteristic
68C22×Dic173—21no—
69C22×Dic173—21no—
70C2×C4×D173—21no—
71C2×C4×D173C2311yescharacteristic
72#14583815—41no—
73#14583813—41no—
74#14583811—22no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C8×D173—41no—
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805D411yescharacteristic
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173D411yescharacteristic
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97#41×C173—22no—
98#41×C173—41no—
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C22×C4×C173C2311yescharacteristic
104C2×C8×C173—41no—
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C42×C173—21no—
111C22×D173—21no—
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—21no—
115Dic346—81no—
116D692—21no—
117D692—21no—
118D692—81no—
119C22⋊D173—22no—
120C22⋊D173—42no—
121C22⋊D173—81no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—42no—
126C2×Dic173—42no—
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C4×D173—21no—
131C4×D173—21no—
132C4×D173—42no—
133#13063436—21no—
134#13063436—21no—
135C23×C173—21no—
136C23×C173C2×D411yescharacteristic
137Q8×C173—21no—
138Q8×C173—21no—
139D4×C173—21no—
140D4×C173—42no—
141D4×C173#4112yes—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—42no—
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—21no—
153C22×C173—22no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173C22×D411yescharacteristic
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C4×C173—21no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173C2×#4111yescharacteristic
170C4×C173H(5,𝔽2)11yescharacteristic
171Dic173—21no—
172Dic173—21no—
173Dic173—41no—
174Dic173—41no—
175Dic173—41no—
176Dic173#13512yes—
177C2×C173—41no—
178C2×C173—42no—
179C2×C173#54611yescharacteristic
180C2×C173C2×#13511yescharacteristic
181C2×C173#57711yescharacteristic
182D173—41no—
183D173—81no—
184C173#330311yescharacteristic, complemented, Sylow
185#3303—1731nomaximal, Sylow
186#555—1731no—
187#547—1731no—
188#487—1731no—
189#485—1731no—
190#485—1731no—
191#476—1731no—
192#465—1731no—
193#465—1731no—
194#461—1731no—
195#449—1731no—
196#449—1731no—
197C4×D8—1731no—
198C8×D4—1731no—
199#416—1731no—
200#416D17311yescharacteristic, complemented
201C2×#41—1731no—
202C2×#41C2×D17311yescharacteristic
203C2×SD16—1731no—
204C2×SD16—1731no—
205C2×D8—1731no—
206C22×C8—1731no—
207C22×C8C2×D17311yescharacteristic
208C4⋊Q8—1731no—
209C4⋊Q8—1731no—
210#124—3461no—
211#124—3461no—
212#123—3461no—
213#123—3461no—
214C22⋊Q8—1731no—
215C22⋊Q8—1731no—
216#121—3461no—
217#121—3461no—
218#121—1731no—
219#121—1731no—
220C4×D4—1731no—
221C4×D4—1731no—
222C4×D4—1731no—
223#117—3461no—
224#117—1731no—
225#117C2×D17311yescharacteristic
226C2×C4⋊C4—3461no—
227#107—1731no—
228#106—1731no—
229#106—1731no—
230C4⋊C8—1731no—
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103C2×D17311yescharacteristic
236#103C2×D17311yescharacteristic
237#102—1731no—
238#102—1731no—
239#102C2×D17311yescharacteristic
240#102C2×D17311yescharacteristic
241C22⋊C8—1731no—
242C22⋊C8—1731no—
243C4×C8—1731no—
244#41—22no—
245#41—41no—
246#41—3462no—
247#41—6921no—
248C2×Q8—3461no—
249C2×Q8—1731no—
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—21no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—1731no—
263C22×C4C22×D17311yescharacteristic
264SD16—6921no—
265SD16—6921no—
266D8—3462no—
267C2×C8—41no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8—6921no—
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—3461no—
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—3461no—
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—21no—
294C42—3461no—
295C42—1731no—
296C23—21no—
297C23—3461no—
298C23—1731no—
299C23D4×D17311yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—3461no—
303Q8—3461no—
304Q8—13841no—
305D4—21no—
306D4—42no—
307D4—3462no—
308D4—3461no—
309D4—3461no—
310D4—13841no—
311D4—13841no—
312D4—6922no—
313D4#1458382612yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—1731no—
326C2×C4—1731no—
327C2×C4—1731no—
328C2×C4—6922no—
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—22no—
341C22—41no—
342C22—41no—
343C22—3461no—
344C22—3461no—
345C22—3461no—
346C22—6921no—
347C22—6921no—
348C22—6921no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—3461no—
355C4—3461no—
356C4—1732no—
357C4—6921no—
358C4—6921no—
359C4—6921no—
360C4C2×#1458382611yescharacteristic
361C4#1914972411yescharacteristic
362C2—41no—
363C2—42no—
364C2—13841no—
365C2—6921no—
366C2#2601213111yescharacteristic
367C2#135×D17311yescharacteristic
368C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#416×C173C2
socleC22×C173C22×D4
hypercenter#416D173
hypocenterC173#3303
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #26012204C2semidirect
2. #26012140C2semidirect
3. #26011840C2semidirect
4. #26011832C2semidirect
5. #26011830C2semidirect
6. #26011779C2semidirect
7. #26011723C2semidirect
8. #26011720C2semidirect
9. #26011695C2semidirect
10. #26011637C2semidirect
11. #26011636C2semidirect
12. #26011571C2semidirect
13. C8×C22⋊D173C2semidirect
14. #26011484C2semidirect
15. #416×C173C2semidirect
184. C173#3303semidirect
200. #416D173semidirect

Maximal subgroups

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

subgroupindex
#260122042
#260121402
#260118402
#260118322
#260118302
#260117792
#260117232
#260117202
#260116952
#260116372
#260116362
#260115712
C8×C22⋊D1732
#260114842
#416×C1732
#3303173