Finite Group Databasebeta

← #36365465

Subgroups — #36365465

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#26012187C211yesmaximal, characteristic, complemented
2#26011845C211yesmaximal, characteristic, complemented
3#26011829C211yesmaximal, characteristic, complemented
4#26011807C211yesmaximal, characteristic, complemented
5#26011798C211yesmaximal, characteristic, complemented
6#26011725C211yesmaximal, characteristic, complemented
7#26011722C211yesmaximal, characteristic, complemented
8#26011703C211yesmaximal, characteristic, complemented
9#26011636C211yesmaximal, characteristic, complemented
10#26011630C211yesmaximal, characteristic, complemented
11C4×#14583813C211yesmaximal, characteristic, complemented
12D4×#13063436C211yesmaximal, characteristic, complemented
13#26011487C211yesmaximal, characteristic, complemented
14C2×#19149603C211yesmaximal, characteristic, complemented
15#548×C173C211yesmaximal, characteristic, complemented
16C2×#14583813C2211yescharacteristic
17C2×#14583813C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C22×#13063436C2211yescharacteristic
20C22×#13063436C2211yescharacteristic
21#19149674—21no—
22#19149671C2211yescharacteristic
23#19149661—21no—
24#19149653C2211yescharacteristic
25#19149653C2211yescharacteristic
26#19149651—21no—
27C4×D692C2211yescharacteristic
28#19149639—21no—
29#19149638C2211yescharacteristic
30C2×C4⋊Dic173C2211yescharacteristic
31#19149612C2211yescharacteristic
32#19149610C2211yescharacteristic
33#19149610C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35#19149603C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149603C2211yescharacteristic
39#19149603C2211yescharacteristic
40#19149603C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149600C2211yescharacteristic
43#19149600C2211yescharacteristic
44#19149599C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47C4×#13063436C2211yescharacteristic
48C2×#41×C173C2211yescharacteristic
49C22×Q8×C173C2211yescharacteristic
50#124×C173—21no—
51#124×C173C2211yescharacteristic
52C22⋊Q8×C173—21no—
53C22⋊Q8×C173C2211yescharacteristic
54#121×C173—21no—
55#121×C173C2211yescharacteristic
56C4×Q8×C173C2211yescharacteristic
57C4×D4×C173—21no—
58C4×D4×C173C2211yescharacteristic
59C2×D692C2311yescharacteristic
60C2×C22⋊D173—21no—
61C2×C22⋊D173—21no—
62C22×Dic173—21no—
63C2×C4×D173—21no—
64#14583813—22no—
65#14583813—41no—
66#14583811—41no—
67C2×#13063436—41no—
68C2×#13063436—41no—
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C2×#13063436C2311yescharacteristic
73C4⋊Dic173—21no—
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78#14583805—21no—
79C22⋊Dic173—21no—
80#14583803—21no—
81#14583803—21no—
82#14583803—21no—
83C4×Dic173—21no—
84#41×C173—22no—
85#41×C173—41no—
86C2×Q8×C173—21no—
87C2×Q8×C173—22no—
88C2×Q8×C173—41no—
89C2×Q8×C173C2311yescharacteristic
90C2×Q8×C173C2311yescharacteristic
91C2×D4×C173—21no—
92C2×D4×C173—21no—
93C2×D4×C173D411yescharacteristic
94C2×D4×C173C2311yescharacteristic
95C22×C4×C173—21no—
96C22×C4×C173—21no—
97C22×C4×C173C2311yescharacteristic
98C22×C4×C173C2311yescharacteristic
99C4⋊C4×C173—21no—
100C4⋊C4×C173—21no—
101C4⋊C4×C173D411yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173D411yescharacteristic
108C22⋊C4×C173D411yescharacteristic
109C42×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112D692—21no—
113D692—21no—
114C22⋊D173—42no—
115C22⋊D173—81no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—81no—
120C4×D173—81no—
121#13063436—21no—
122#13063436—21no—
123#13063436—21no—
124#13063436—21no—
125C23×C173—21no—
126C23×C173C2×D411yescharacteristic
127C23×C173C2×D411yescharacteristic
128Q8×C173—21no—
129Q8×C173—21no—
130Q8×C173—21no—
131Q8×C173—42no—
132Q8×C173#4112yes—
133D4×C173—22no—
134D4×C173—21no—
135D4×C173—21no—
136D4×C173—42no—
137D4×C173—42no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—42no—
146C2×C4×C173—42no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2×D411yescharacteristic
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173#13512yes—
161C22×C173C22×D411yescharacteristic
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173—22no—
168C4×C173—21no—
169C4×C173—21no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173C2×#4111yescharacteristic
173C4×C173H(5,𝔽2)11yescharacteristic
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—22no—
178C2×C173—41no—
179C2×C173—81no—
180C2×C173#54611yescharacteristic
181C2×C173C2×#13511yescharacteristic
182C2×C173#57711yescharacteristic
183D173—81no—
184C173#330111yescharacteristic, complemented, Sylow
185#3301—1731nomaximal, Sylow
186#553—1731no—
187#548D17311yescharacteristic, complemented
188#488—1731no—
189#485—1731no—
190#481—1731no—
191#479—1731no—
192#466—1731no—
193#465—1731no—
194#463—1731no—
195#449—1731no—
196#448—1731no—
197C4×SD16—1731no—
198C8×D4—1731no—
199#416—1731no—
200C2×#103—1731no—
201C2×#41C2×D17311yescharacteristic
202C22×Q8C2×D17311yescharacteristic
203C2×SD16—1731no—
204C2×SD16—1731no—
205C2×D8—1731no—
206C22×C8—1731no—
207C22×C8—1731no—
208#126—3461no—
209#125—1731no—
210#124—21no—
211#124C2×D17311yescharacteristic
212#123—3461no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D17311yescharacteristic
215#121—21no—
216#121—3461no—
217#121—1731no—
218#121—1731no—
219#121C2×D17311yescharacteristic
220C4×Q8C2×D17311yescharacteristic
221C4×D4—21no—
222C4×D4—3461no—
223C4×D4—1731no—
224C4×D4C2×D17311yescharacteristic
225#117—1731no—
226C2×C4⋊C4—1731no—
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#103—1731no—
236#103—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241C22⋊C8—1731no—
242C22⋊C8—1731no—
243C4×C8—1731no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D17311yescharacteristic
250C2×Q8C22×D17311yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4C22⋊D17311yescharacteristic
257C2×D4C22×D17311yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4C22×D17311yescharacteristic
263C22×C4C22×D17311yescharacteristic
264SD16—3462no—
265SD16—6921no—
266D8—6921no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8—1731no—
271C2×C8—6921no—
272C2×C8—6921no—
273C4⋊C4—21no—
274C4⋊C4—21no—
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—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4C22⋊D17311yescharacteristic
292C22⋊C4C22⋊D17311yescharacteristic
293C42—21no—
294C42—3461no—
295C42C22×D17311yescharacteristic
296C23—21no—
297C23—3461no—
298C23C2×C22⋊D17311yescharacteristic
299C23C2×C22⋊D17311yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1458382512yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—3461no—
311D4—3461no—
312D4—13841no—
313D4—6922no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—13841no—
327C2×C4—13841no—
328C2×C4C2×C22⋊D17311yescharacteristic
329C2×C4C2×C22⋊D17311yescharacteristic
330C2×C4C2×C22⋊D17311yescharacteristic
331C2×C4C2×C22⋊D17311yescharacteristic
332C2×C4C23×D17311yescharacteristic
333C8—3461no—
334C8—3461no—
335C8—3461no—
336C8—3461no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—6921no—
346C22—6921no—
347C22—6921no—
348C22#1914969912yes—
349C22C22×C22⋊D17311yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—6921no—
358C4—6921no—
359C4—6921no—
360C4C2×#1458382511yescharacteristic
361C4#1914972411yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—13841no—
366C2#2601213511yescharacteristic
367C2C2×#1914969911yescharacteristic
368C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#548×C173C2
socleC22×C173C22×D4
hypercenter#548D173
hypocenterC173#3301
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #26012187C2semidirect
2. #26011845C2semidirect
3. #26011829C2semidirect
4. #26011807C2semidirect
5. #26011798C2semidirect
6. #26011725C2semidirect
7. #26011722C2semidirect
8. #26011703C2semidirect
9. #26011636C2semidirect
10. #26011630C2semidirect
11. C4×#14583813C2semidirect
12. D4×#13063436C2semidirect
13. #26011487C2semidirect
14. C2×#19149603C2semidirect
15. #548×C173C2semidirect
184. C173#3301semidirect
187. #548D173semidirect

Maximal subgroups

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

subgroupindex
#260121872
#260118452
#260118292
#260118072
#260117982
#260117252
#260117222
#260117032
#260116362
#260116302
C4×#145838132
D4×#130634362
#260114872
C2×#191496032
#548×C1732
#3301173