Finite Group Databasebeta

← #36363763

Subgroups — #36363763

Order 22144.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012094C211yesmaximal, characteristic, complemented
2#26011974C211yesmaximal, characteristic, complemented
3#26011810C211yesmaximal, characteristic, complemented
4#26011802C211yesmaximal, characteristic, complemented
5#26011796C211yesmaximal, characteristic, complemented
6#26011793C211yesmaximal, characteristic, complemented
7#26011689C211yesmaximal, characteristic, complemented
8#26011684C211yesmaximal, characteristic, complemented
9#26011636C211yesmaximal, characteristic, complemented
10#26011603C211yesmaximal, characteristic, complemented
11#26011594C211yesmaximal, characteristic, complemented
12SD16×Dic173C211yesmaximal, characteristic, complemented
13#26011571C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C2×#14583813C2211yescharacteristic
17C2×#14583811C2211yescharacteristic
18#19149675C2211yescharacteristic
19#19149670C2211yescharacteristic
20#19149663—21no—
21#19149661—21no—
22#19149653C2211yescharacteristic
23#19149651—21no—
24#19149650—21no—
25Q8×Dic173C2211yescharacteristic
26Q8×Dic173C2211yescharacteristic
27D4×Dic173—21no—
28D4×Dic173—21no—
29D4×Dic173C2211yescharacteristic
30D4×Dic173C2211yescharacteristic
31D4×Dic173C2211yescharacteristic
32#19149641C2211yescharacteristic
33#19149638—21no—
34#19149638C2211yescharacteristic
35C2×C4⋊Dic173—21no—
36C2×C4⋊Dic173C2211yescharacteristic
37#19149612C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
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×D692C2311yescharacteristic
60C2×C22⋊D173—21no—
61C2×C22⋊D173—21no—
62C2×C22⋊D173—21no—
63C22×Dic173—21no—
64C22×Dic173—21no—
65C22×Dic173—21no—
66C22×Dic173—21no—
67C2×C4×D173—21no—
68#14583813—22no—
69#14583811—22no—
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C4⋊Dic173—21no—
73C4⋊Dic173—21no—
74C4⋊Dic173—21no—
75C4⋊Dic173—21no—
76C4⋊Dic173D411yescharacteristic
77C4⋊Dic173D411yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82C22⋊Dic173—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85C22⋊Dic173D411yescharacteristic
86C22⋊Dic173D411yescharacteristic
87#14583803—21no—
88#14583803—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173C2311yescharacteristic
92C4×Dic173C2311yescharacteristic
93#41×C173—22no—
94#41×C173—22no—
95C2×Q8×C173C2311yescharacteristic
96C2×D4×C173—21no—
97C2×D4×C173—21no—
98C2×D4×C173C2311yescharacteristic
99C2×D4×C173C2311yescharacteristic
100C22×C4×C173—21no—
101C22×C4×C173C2311yescharacteristic
102SD16×C173—22no—
103D8×C173—22no—
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C22⋊C4×C173—21no—
108C22×D173—21no—
109D692—21no—
110D692—21no—
111C22⋊D173—81no—
112C22⋊D173—81no—
113C22⋊D173—81no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—81no—
118C2×Dic173—81no—
119C2×Dic173—81no—
120C2×Dic173—81no—
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C4×D173—81no—
126#13063436—21no—
127#13063436—21no—
128C23×C173—21no—
129C23×C173—21no—
130C23×C173C2×D411yescharacteristic
131Q8×C173—21no—
132Q8×C173#4112yes—
133D4×C173—24no—
134D4×C173—21no—
135D4×C173—21no—
136D4×C173—21no—
137D4×C173—81no—
138D4×C173#4112yes—
139C2×C4×C173—24no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—21no—
147C8×C173—21no—
148C22×C173—21no—
149C22×C173—22no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173C22×D411yescharacteristic
158C2×D173—41no—
159C2×D173—41no—
160C2×D173—41no—
161C4×C173—21no—
162C4×C173—22no—
163C4×C173—41no—
164C4×C173C2×#4111yescharacteristic
165C4×C173C2×#4111yescharacteristic
166Dic173—21no—
167Dic173—21no—
168Dic173—21no—
169Dic173—21no—
170Dic173—41no—
171Dic173—41no—
172Dic173—41no—
173C2×C173—41no—
174C2×C173—42no—
175C2×C173—81no—
176C2×C173#52511yescharacteristic
177C2×C173#57611yescharacteristic
178C2×C173#57711yescharacteristic
179D173—81no—
180C173#311911yescharacteristic, complemented, Sylow
181#3119—1731nomaximal, Sylow
182#540—1731no—
183#520—1731no—
184#481—1731no—
185#480—1731no—
186#479—1731no—
187#478—1731no—
188#460—1731no—
189#459—1731no—
190#449—1731no—
191#449D17311yescharacteristic, complemented
192#442—1731no—
193#440—1731no—
194C4×SD16—1731no—
195C4×D8—1731no—
196#433—1731no—
197C2×#41C2×D17311yescharacteristic
198C2×SD16—1731no—
199C2×SD16C2×D17311yescharacteristic
200C2×D8—1731no—
201C2×D8C2×D17311yescharacteristic
202#127—1731no—
203#125—1731no—
204#123—3461no—
205#123—3461no—
206#121—3461no—
207#121—3461no—
208#121—1731no—
209#121C2×D17311yescharacteristic
210C4×Q8—1731no—
211C4×Q8—1731no—
212C4×D4—3461no—
213C4×D4—3461no—
214C4×D4—1731no—
215C4×D4—1731no—
216C4×D4—1731no—
217C4×D4—1731no—
218#117—3461no—
219#117—1731no—
220C2×C4⋊C4—3461no—
221C2×C4⋊C4—1731no—
222#107—1731no—
223#107—1731no—
224#106—1731no—
225#106—1731no—
226C4⋊C8—1731no—
227C4⋊C8—1731no—
228#103—1731no—
229#103C2×D17311yescharacteristic
230#102—1731no—
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235#102C2×D17311yescharacteristic
236C22⋊C8—1731no—
237C22⋊C8C2×D17311yescharacteristic
238#97—1731no—
239C4×C8—1731no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D17311yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—3461no—
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4C22×D17311yescharacteristic
250C2×D4C22×D17311yescharacteristic
251C22×C4—21no—
252C22×C4—3461no—
253C22×C4—3461no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4—3461no—
257C22×C4C22×D17311yescharacteristic
258SD16—22no—
259SD16—3462no—
260D8—22no—
261D8—3462no—
262C2×C8—1731no—
263C2×C8—1731no—
264C2×C8C22×D17311yescharacteristic
265C2×C8C22×D17311yescharacteristic
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4—1731no—
274C4⋊C4—1731no—
275C4⋊C4—1731no—
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—1731no—
284C22⋊C4—1731no—
285C42—3461no—
286C42—3461no—
287C42—1731no—
288C42—1731no—
289C23—21no—
290C23—21no—
291C23—3461no—
292C23D4×D17311yescharacteristic
293Q8—21no—
294Q8#1458382512yes—
295D4—24no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—3461no—
301D4—3461no—
302D4—13841no—
303D4—13841no—
304D4—13841no—
305D4#1458382512yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—13841no—
315C2×C4—13841no—
316C2×C4—13841no—
317C2×C4—13841no—
318C2×C4—13841no—
319C2×C4—1731no—
320C2×C4—1731no—
321C2×C4—1731no—
322C2×C4—1731no—
323C2×C4D4×D17311yescharacteristic
324C2×C4C23×D17311yescharacteristic
325C8—21no—
326C8—21no—
327C8—3461no—
328C8—3461no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—6921no—
339C22—6921no—
340C22—6921no—
341C22C2×D4×D17311yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—3461no—
346C4—3461no—
347C4—3461no—
348C4—3461no—
349C4—6921no—
350C4—6921no—
351C4—6921no—
352C4C2×#1458382511yescharacteristic
353C4C2×#1458382511yescharacteristic
354C2—41no—
355C2—42no—
356C2—81no—
357C2—13841no—
358C2#2601199711yescharacteristic
359C2#2601229511yescharacteristic
360C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3119
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. #26011974C2semidirect
3. #26011810C2semidirect
4. #26011802C2semidirect
5. #26011796C2semidirect
6. #26011793C2semidirect
7. #26011689C2semidirect
8. #26011684C2semidirect
9. #26011636C2semidirect
10. #26011603C2semidirect
11. #26011594C2semidirect
12. SD16×Dic173C2semidirect
13. #26011571C2semidirect
14. #26011549C2semidirect
15. #449×C173C2semidirect
180. C173#3119semidirect
191. #449D173semidirect

Maximal subgroups

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

subgroupindex
#260120942
#260119742
#260118102
#260118022
#260117962
#260117932
#260116892
#260116842
#260116362
#260116032
#260115942
SD16×Dic1732
#260115712
#260115492
#449×C1732
#3119173