Finite Group Databasebeta

← #36364623

Subgroups — #36364623

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#26012092C211yesmaximal, characteristic, complemented
2#26012066C211yesmaximal, characteristic, complemented
3#26011877C211yesmaximal, characteristic, complemented
4#26011862C211yesmaximal, characteristic, complemented
5#26011858C211yesmaximal, characteristic, complemented
6#26011847C211yesmaximal, characteristic, complemented
7#26011707C211yesmaximal, characteristic, complemented
8#26011706C211yesmaximal, characteristic, complemented
9#26011693C211yesmaximal, characteristic, complemented
10#26011639C211yesmaximal, characteristic, complemented
11#26011635C211yesmaximal, characteristic, complemented
12#26011631C211yesmaximal, characteristic, complemented
13#26011628C211yesmaximal, characteristic, complemented
14#26011551C211yesmaximal, characteristic, complemented
15#461×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×#14583813C2211yescharacteristic
22C2×D1384C2211yescharacteristic
23C2×#14583811C2211yescharacteristic
24#19149677C2211yescharacteristic
25#19149675C2211yescharacteristic
26#19149670C2211yescharacteristic
27#19149668C2211yescharacteristic
28#19149667C2211yescharacteristic
29#19149664—21no—
30#19149664—21no—
31#19149662—21no—
32#19149661—21no—
33#19149657—21no—
34#19149657—21no—
35#19149654C2211yescharacteristic
36#19149653—21no—
37#19149652—21no—
38#19149650C2211yescharacteristic
39C4×D692C2211yescharacteristic
40#19149636C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C4⋊Q8×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59#14583827—41no—
60#14583827—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63Q8×D173—41no—
64Q8×D173—41no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76#14583815—41no—
77#14583814—41no—
78#14583813—41no—
79#14583813—41no—
80D1384—41no—
81#14583811—41no—
82C2×#13063436C2311yescharacteristic
83C2×#13063436C2311yescharacteristic
84C4⋊Dic173—21no—
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90#14583805D411yescharacteristic
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803D411yescharacteristic
99#14583803D411yescharacteristic
100C4×Dic173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173—21no—
105SD16×C173—41no—
106SD16×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173—21no—
110C4⋊C4×C173—21no—
111C4⋊C4×C173C2311yescharacteristic
112C22⋊C4×C173—21no—
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173—21no—
116C22×D173C2×D411yescharacteristic
117Dic346—21no—
118Dic346—21no—
119Dic346—42no—
120D692—21no—
121D692—21no—
122D692—21no—
123D692—21no—
124D692—42no—
125D692—81no—
126C22⋊D173—81no—
127C22⋊D173—81no—
128C22⋊D173—81no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—21no—
135C4×D173—21no—
136C4×D173—42no—
137C4×D173—42no—
138C4×D173—81no—
139#13063436—21no—
140#13063436—21no—
141C23×C173—21no—
142Q8×C173—21no—
143Q8×C173—21no—
144Q8×C173—21no—
145Q8×C173—21no—
146D4×C173—21no—
147D4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—81no—
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×C173C22×D411yescharacteristic
161C2×D173—21no—
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C4×C173—21no—
171C4×C173—21no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173—41no—
175C4×C173H(5,𝔽2)11yescharacteristic
176C4×C173H(5,𝔽2)11yescharacteristic
177Dic173—21no—
178Dic173—21no—
179Dic173—21no—
180Dic173—41no—
181Dic173—41no—
182C2×C173—81no—
183C2×C173#53511yescharacteristic
184C2×C173#57711yescharacteristic
185C2×C173#57711yescharacteristic
186D173—41no—
187D173—81no—
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#461D17311yescharacteristic, complemented
201#449—1731no—
202#449—1731no—
203#448—1731no—
204#448—1731no—
205#433—1731no—
206C2×#41—1731no—
207C22×Q8—1731no—
208C2×SD16—1731no—
209C2×SD16—1731no—
210C2×SD16—1731no—
211C2×SD16—1731no—
212C2×SD16C2×D17311yescharacteristic
213C2×SD16C2×D17311yescharacteristic
214C2×D8—1731no—
215C2×D8—1731no—
216C4⋊Q8—1731no—
217C4⋊Q8C2×D17311yescharacteristic
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#121—1731no—
232C4×D4—1731no—
233C4×D4C2×D17311yescharacteristic
234#117—1731no—
235C4⋊C8—1731no—
236C4⋊C8C2×D17311yescharacteristic
237#103—1731no—
238#103—1731no—
239#103—1731no—
240#103—1731no—
241#103C2×D17311yescharacteristic
242#103C2×D17311yescharacteristic
243#102—1731no—
244#102—1731no—
245C22⋊C8—1731no—
246C22⋊C8—1731no—
247#97—1731no—
248C4×C8—1731no—
249#41—6921no—
250#41—6921no—
251C2×Q8—3461no—
252C2×Q8—1731no—
253C2×Q8—6921no—
254C2×Q8—6921no—
255C2×Q8C22×D17311yescharacteristic
256C2×Q8C22×D17311yescharacteristic
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—1731no—
263C2×D4—1731no—
264C2×D4C22×D17311yescharacteristic
265C22×C4—21no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4—1731no—
270SD16—41no—
271SD16—41no—
272SD16—6921no—
273SD16—6921no—
274SD16—6921no—
275SD16—6921no—
276D8—6921no—
277D8—6921no—
278C2×C8—1731no—
279C2×C8—1731no—
280C2×C8C22×D17311yescharacteristic
281C2×C8C22×D17311yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—21no—
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—
303C42C22×D17311yescharacteristic
304C23—21no—
305C23—3461no—
306C23—3461no—
307C23—1731no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—3461no—
313Q8—3461no—
314Q8—6922no—
315D4—21no—
316D4—21no—
317D4—3461no—
318D4—3461no—
319D4—3461no—
320D4—3461no—
321D4—13841no—
322D4—13841no—
323D4—13841no—
324D4—13841no—
325D4—6922no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—81no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—13841no—
335C2×C4—1731no—
336C2×C4—1731no—
337C2×C4—1731no—
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—3461no—
351C22—3461no—
352C22—3461no—
353C22—6921no—
354C22—6921no—
355C22—6921no—
356C22—6921no—
357C22—6921no—
358C22—6921no—
359C22C2×D4×D17311yescharacteristic
360C4—21no—
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—3461no—
366C4—3461no—
367C4—3461no—
368C4—6921no—
369C4—6921no—
370C4#1914972411yescharacteristic
371C4#1914972511yescharacteristic
372C2—81no—
373C2—13841no—
374C2—13841no—
375C2—6921no—
376C2#2601205711yescharacteristic
377C2#2601229711yescharacteristic
378C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#461×C173C2
socleC22×C173C22×D4
hypercenter#461D173
hypocenterC173#3221
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #26012092C2semidirect
2. #26012066C2semidirect
3. #26011877C2semidirect
4. #26011862C2semidirect
5. #26011858C2semidirect
6. #26011847C2semidirect
7. #26011707C2semidirect
8. #26011706C2semidirect
9. #26011693C2semidirect
10. #26011639C2semidirect
11. #26011635C2semidirect
12. #26011631C2semidirect
13. #26011628C2semidirect
14. #26011551C2semidirect
15. #461×C173C2semidirect
189. C173#3221semidirect
200. #461D173semidirect

Maximal subgroups

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

subgroupindex
#260120922
#260120662
#260118772
#260118622
#260118582
#260118472
#260117072
#260117062
#260116932
#260116392
#260116352
#260116312
#260116282
#260115512
#461×C1732
#3221173