Finite Group Databasebeta

← #36364622

Subgroups — #36364622

Order 22144.

388 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#26012094C211yesmaximal, characteristic, complemented
2#26012062C211yesmaximal, characteristic, complemented
3#26011879C211yesmaximal, characteristic, complemented
4#26011862C211yesmaximal, characteristic, complemented
5#26011858C211yesmaximal, characteristic, complemented
6#26011844C211yesmaximal, characteristic, complemented
7#26011705C211yesmaximal, characteristic, complemented
8#26011704C211yesmaximal, characteristic, complemented
9#26011695C211yesmaximal, characteristic, complemented
10#26011694C211yesmaximal, characteristic, complemented
11#26011636C211yesmaximal, characteristic, complemented
12#26011632C211yesmaximal, characteristic, complemented
13#26011629C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C22×Dic346C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21C2×#14583813C2211yescharacteristic
22C2×#14583811C2211yescharacteristic
23#19149677C2211yescharacteristic
24#19149677C2211yescharacteristic
25#19149675C2211yescharacteristic
26#19149670C2211yescharacteristic
27#19149666C2211yescharacteristic
28#19149666C2211yescharacteristic
29#19149663—21no—
30#19149661—21no—
31#19149660—21no—
32#19149660—21no—
33#19149658—21no—
34#19149658—21no—
35#19149653C2211yescharacteristic
36#19149651—21no—
37#19149650—21no—
38D4×Dic173C2211yescharacteristic
39#19149641C2211yescharacteristic
40#19149638C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346—41no—
61C2×Dic346—41no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C22×Dic173—21no—
69C22×Dic173—21no—
70C2×C4×D173—21no—
71#14583815—41no—
72#14583815—41no—
73#14583814—41no—
74#14583814—41no—
75#14583813—41no—
76#14583811—41no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173D411yescharacteristic
81C4⋊Dic173D411yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173D411yescharacteristic
94C22⋊Dic173D411yescharacteristic
95#14583803—21no—
96#14583803—21no—
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
99#41×C173—41no—
100#41×C173—41no—
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173—21no—
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C2×D4×C173C2311yescharacteristic
106C22×C4×C173—21no—
107C22×C4×C173C2311yescharacteristic
108SD16×C173—41no—
109D8×C173—41no—
110C2×C8×C173C2311yescharacteristic
111C2×C8×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346—42no—
120D692—21no—
121D692—21no—
122C22⋊D173—81no—
123C22⋊D173—81no—
124C22⋊D173—81no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—42no—
129C2×Dic173—81no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—81no—
135#13063436—21no—
136#13063436—21no—
137C23×C173—21no—
138C23×C173—21no—
139C23×C173C2×D411yescharacteristic
140Q8×C173—21no—
141Q8×C173—21no—
142D4×C173—21no—
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—21no—
146D4×C173—42no—
147D4×C173—81no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—42no—
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C8×C173—21no—
156C8×C173—21no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173C22×D411yescharacteristic
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C4×C173—21no—
171C4×C173—41no—
172C4×C173—41no—
173C4×C173H(5,𝔽2)11yescharacteristic
174C4×C173H(5,𝔽2)11yescharacteristic
175Dic173—21no—
176Dic173—21no—
177Dic173—21no—
178Dic173—21no—
179Dic173—41no—
180Dic173—41no—
181Dic173—41no—
182C2×C173—41no—
183C2×C173—81no—
184C2×C173—81no—
185C2×C173#53511yescharacteristic
186C2×C173#57711yescharacteristic
187C2×C173#57711yescharacteristic
188D173—81no—
189C173#322111yescharacteristic, complemented, Sylow
190#3221—1731nomaximal, Sylow
191#540—1731no—
192#536—1731no—
193#496—1731no—
194#492—1731no—
195#491—1731no—
196#488—1731no—
197#463—1731no—
198#463—1731no—
199#461—1731no—
200#461—1731no—
201#449—1731no—
202#449D17311yescharacteristic, complemented
203#448—1731no—
204#448—1731no—
205#433—1731no—
206C2×#41C2×D17311yescharacteristic
207C22×Q8—1731no—
208C2×SD16—1731no—
209C2×SD16—1731no—
210C2×SD16—1731no—
211C2×SD16—1731no—
212C2×SD16—1731no—
213C2×SD16C2×D17311yescharacteristic
214C2×D8—1731no—
215C2×D8C2×D17311yescharacteristic
216C4⋊Q8—1731no—
217C4⋊Q8—1731no—
218#127—1731no—
219#125—1731no—
220#124—1731no—
221#124—1731no—
222#123—3461no—
223#123—3461no—
224#123—3461no—
225#123—3461no—
226C22⋊Q8—3461no—
227C22⋊Q8—3461no—
228#121—3461no—
229#121—3461no—
230#121—1731no—
231#121C2×D17311yescharacteristic
232C4×D4—1731no—
233C4×D4—1731no—
234#117—1731no—
235C4⋊C8—1731no—
236C4⋊C8—1731no—
237#103—1731no—
238#103—1731no—
239#103—1731no—
240#103—1731no—
241#103—1731no—
242#103C2×D17311yescharacteristic
243#102—1731no—
244#102C2×D17311yescharacteristic
245C22⋊C8—1731no—
246C22⋊C8C2×D17311yescharacteristic
247#97—1731no—
248C4×C8—1731no—
249#41—41no—
250#41—41no—
251C2×Q8—3461no—
252C2×Q8—1731no—
253C2×Q8—1731no—
254C2×Q8—6921no—
255C2×Q8—6921no—
256C2×Q8C22×D17311yescharacteristic
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—1731no—
263C2×D4C22×D17311yescharacteristic
264C2×D4C22×D17311yescharacteristic
265C22×C4—21no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4C22×D17311yescharacteristic
270SD16—41no—
271SD16—6921no—
272SD16—6921no—
273SD16—6921no—
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—
290C4⋊C4—1731no—
291C4⋊C4C22×D17311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—1731no—
301C22⋊C4—1731no—
302C42—1731no—
303C42—1731no—
304C23—21no—
305C23—21no—
306C23—3461no—
307C23D4×D17311yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—3461no—
311Q8—3461no—
312Q8—3461no—
313Q8—3461no—
314Q8—6922no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—42no—
320D4—81no—
321D4—3461no—
322D4—3461no—
323D4—13841no—
324D4—13841no—
325D4—13841no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—13841no—
335C2×C4—13841no—
336C2×C4—1731no—
337C2×C4—1731no—
338C2×C4—1731no—
339C2×C4—1731no—
340C2×C4—6922no—
341C2×C4D4×D17311yescharacteristic
342C2×C4C23×D17311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3461no—
346C8—3461no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—6921no—
357C22—6921no—
358C22—6921no—
359C22C2×D4×D17311yescharacteristic
360C4—21no—
361C4—41no—
362C4—41no—
363C4—3461no—
364C4—3461no—
365C4—3461no—
366C4—3461no—
367C4—6921no—
368C4—6921no—
369C4—6921no—
370C4#1914972411yescharacteristic
371C4#1914972411yescharacteristic
372C2—41no—
373C2—81no—
374C2—81no—
375C2—13841no—
376C2#2601205611yescharacteristic
377C2#2601229811yescharacteristic
378C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3221
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #26012094C2semidirect
2. #26012062C2semidirect
3. #26011879C2semidirect
4. #26011862C2semidirect
5. #26011858C2semidirect
6. #26011844C2semidirect
7. #26011705C2semidirect
8. #26011704C2semidirect
9. #26011695C2semidirect
10. #26011694C2semidirect
11. #26011636C2semidirect
12. #26011632C2semidirect
13. #26011629C2semidirect
14. #26011549C2semidirect
15. #449×C173C2semidirect
189. C173#3221semidirect
202. #449D173semidirect

Maximal subgroups

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

subgroupindex
#260120942
#260120622
#260118792
#260118622
#260118582
#260118442
#260117052
#260117042
#260116952
#260116942
#260116362
#260116322
#260116292
#260115492
#449×C1732
#3221173