Finite Group Databasebeta

← #36363825

Subgroups — #36363825

Order 22144.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012112C211yesmaximal, characteristic, complemented
2#26011826C211yesmaximal, characteristic, complemented
3#26011804C211yesmaximal, characteristic, complemented
4#26011786C211yesmaximal, characteristic, complemented
5#26011771C211yesmaximal, characteristic, complemented
6#26011713C211yesmaximal, characteristic, complemented
7#26011706C211yesmaximal, characteristic, complemented
8#26011653C211yesmaximal, characteristic, complemented
9#26011636C211yesmaximal, characteristic
10#26011591C211yesmaximal, characteristic, complemented
11#26011589C211yesmaximal, characteristic, complemented
12C4×#14583816C211yesmaximal, characteristic, complemented
13C4×#14583811C211yesmaximal, characteristic, complemented
14#26011552C211yesmaximal, characteristic, complemented
15#520×C173C211yesmaximal, characteristic, complemented
16C2×#14583816C2211yescharacteristic
17C2×#14583814C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20#19149679C2211yescharacteristic
21#19149674—21no—
22#19149667C2211yescharacteristic
23#19149665—21no—
24#19149661—21no—
25#19149658C2211yescharacteristic
26#19149657—21no—
27#19149653C2211yescharacteristic
28C4×Dic346C2211yescharacteristic
29C4×D692C2211yescharacteristic
30#19149612C2211yescharacteristic
31#19149612C2211yescharacteristic
32#19149612C2211yescharacteristic
33#19149612C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35C4⋊#13063436C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149602C2211yescharacteristic
39#19149602C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46#19149590C2211yescharacteristic
47C4×#13063436C2211yescharacteristic
48C2×#41×C173C2211yescharacteristic
49C4×Q8×C173C2211yescharacteristic
50C4×Q8×C173C2211yescharacteristic
51C4×D4×C173—21no—
52C4×D4×C173—21no—
53C4×D4×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55#117×C173—21no—
56#117×C173C2211yescharacteristic
57C2×C4⋊C4×C173—21no—
58C2×C4⋊C4×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C4×D173—21no—
64#14583816—22no—
65#14583814—22no—
66#14583813—22no—
67#14583811—22no—
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C4⋊Dic173—21no—
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75#14583805—21no—
76#14583805—21no—
77C22⋊Dic173—21no—
78#14583803—21no—
79#14583803—21no—
80#14583803—21no—
81#14583803—21no—
82C4×Dic173—21no—
83#41×C173—22no—
84#41×C173—22no—
85C2×Q8×C173C2311yescharacteristic
86C2×D4×C173—21no—
87C2×D4×C173C2311yescharacteristic
88C22×C4×C173—21no—
89C22×C4×C173—21no—
90C22×C4×C173—21no—
91C22×C4×C173—21no—
92C22×C4×C173C2311yescharacteristic
93C4⋊C4×C173—21no—
94C4⋊C4×C173—21no—
95C4⋊C4×C173—21no—
96C4⋊C4×C173D411yescharacteristic
97C4⋊C4×C173D411yescharacteristic
98C4⋊C4×C173C2311yescharacteristic
99C4⋊C4×C173C2311yescharacteristic
100C22⋊C4×C173—21no—
101C22⋊C4×C173—21no—
102C22⋊C4×C173D411yescharacteristic
103C22⋊C4×C173D411yescharacteristic
104C42×C173—21no—
105C42×C173—21no—
106C42×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109Dic346—21no—
110Dic346—21no—
111Dic346—81no—
112D692—21no—
113D692—21no—
114C22⋊D173—81no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C4×D173—81no—
119#13063436—21no—
120#13063436—21no—
121#13063436—21no—
122#13063436—21no—
123C23×C173—21no—
124C23×C173C2×D411yescharacteristic
125Q8×C173—21no—
126Q8×C173#4112yes—
127D4×C173—24no—
128D4×C173—21no—
129D4×C173#4112yes—
130C2×C4×C173—24no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—81no—
137C2×C4×C173—81no—
138C2×C4×C173—81no—
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2411yescharacteristic
145C22×C173—21no—
146C22×C173—22no—
147C22×C173—21no—
148C22×C173—21no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173C22×D411yescharacteristic
152C2×D173—41no—
153C2×D173—41no—
154C2×D173—41no—
155C4×C173—21no—
156C4×C173—22no—
157C4×C173—21no—
158C4×C173—21no—
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173C2×#4111yescharacteristic
164C4×C173C2×#4111yescharacteristic
165Dic173—41no—
166Dic173—41no—
167Dic173—41no—
168C2×C173—41no—
169C2×C173—42no—
170C2×C173#52511yescharacteristic
171C2×C173#57611yescharacteristic
172C2×C173#57811yescharacteristic
173D173—81no—
174C173#312411yescharacteristic, complemented, Sylow
175#3124—1731nomaximal, Sylow
176#542—1731no—
177#520D17311yescharacteristic, complemented
178#484—1731no—
179#480—1731no—
180#477—1731no—
181#474—1731no—
182#464—1731no—
183#463—1731no—
184#452—1731no—
185#449—1731no—
186#440—1731no—
187#440—1731no—
188C4×Q16—1731no—
189C4×D8—1731no—
190#433—1731no—
191C2×#41C2×D17311yescharacteristic
192C2×Q16—1731no—
193C2×SD16—1731no—
194C2×SD16—1731no—
195C2×D8—1731no—
196C4⋊Q8—1731no—
197#126—3461no—
198#124—3461no—
199#124—1731no—
200#123—3461no—
201C22⋊Q8—3461no—
202C22⋊Q8—1731no—
203#121—1731no—
204C4×Q8—1731no—
205C4×Q8C2×D17311yescharacteristic
206C4×Q8C2×D17311yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—1731no—
210C4×D4C2×D17311yescharacteristic
211C4×D4C2×D17311yescharacteristic
212#117—21no—
213#117C2×D17311yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D17311yescharacteristic
216#107—1731no—
217#107—1731no—
218#107—1731no—
219#107—1731no—
220C4⋊C8—1731no—
221C4⋊C8—1731no—
222#103—1731no—
223#103—1731no—
224#103—1731no—
225#103—1731no—
226#102—1731no—
227#102—1731no—
228#102—1731no—
229#102—1731no—
230C22⋊C8—1731no—
231C22⋊C8—1731no—
232#97—1731no—
233C4×C8—1731no—
234#41—22no—
235#41—22no—
236C2×Q8—3461no—
237C2×Q8—1731no—
238C2×Q8C22×D17311yescharacteristic
239C2×D4—21no—
240C2×D4—3461no—
241C2×D4—1731no—
242C2×D4C22×D17311yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3461no—
248C22×C4C22×D17311yescharacteristic
249Q16—3462no—
250SD16—3462no—
251SD16—3462no—
252D8—3462no—
253C2×C8—1731no—
254C2×C8—1731no—
255C2×C8—1731no—
256C2×C8—1731no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—1731no—
264C4⋊C4—1731no—
265C4⋊C4C22⋊D17311yescharacteristic
266C4⋊C4C22⋊D17311yescharacteristic
267C4⋊C4C22×D17311yescharacteristic
268C4⋊C4C22×D17311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4C22⋊D17311yescharacteristic
277C22⋊C4C22⋊D17311yescharacteristic
278C42—21no—
279C42—21no—
280C42—3461no—
281C42C22×D17311yescharacteristic
282C42C22×D17311yescharacteristic
283C23—21no—
284C23—3461no—
285C23C2×C22⋊D17311yescharacteristic
286Q8—21no—
287Q8—3461no—
288Q8—3461no—
289Q8—13841no—
290Q8#1458382612yes—
291D4—24no—
292D4—21no—
293D4—3461no—
294D4—3461no—
295D4—13841no—
296D4#1458382612yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—13841no—
310C2×C4C2×C22⋊D17311yescharacteristic
311C2×C4C2×C22⋊D17311yescharacteristic
312C2×C4C2×C22⋊D17311yescharacteristic
313C2×C4C2×C22⋊D17311yescharacteristic
314C2×C4C2×C22⋊D17311yescharacteristic
315C2×C4C23×D17311yescharacteristic
316C8—3461no—
317C8—3461no—
318C8—3461no—
319C8—3461no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—6921no—
327C22—6921no—
328C22—6921no—
329C22C22×C22⋊D17311yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—6921no—
339C4—6921no—
340C4—6921no—
341C4C2×#1458382611yescharacteristic
342C4C2×#1458382611yescharacteristic
343C2—41no—
344C2—42no—
345C2—13841no—
346C2#2601200211yescharacteristic
347C2#2601229411yescharacteristic
348C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#520×C173C2
socleC22×C173C22×D4
prime radical#26011636C2
hypercenter#520D173
hypocenterC173#3124
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #26012112C2semidirect
2. #26011826C2semidirect
3. #26011804C2semidirect
4. #26011786C2semidirect
5. #26011771C2semidirect
6. #26011713C2semidirect
7. #26011706C2semidirect
8. #26011653C2semidirect
10. #26011591C2semidirect
11. #26011589C2semidirect
12. C4×#14583816C2semidirect
13. C4×#14583811C2semidirect
14. #26011552C2semidirect
15. #520×C173C2semidirect
174. C173#3124semidirect
177. #520D173semidirect

Maximal subgroups

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

subgroupindex
#260121122
#260118262
#260118042
#260117862
#260117712
#260117132
#260117062
#260116532
#260116362
#260115912
#260115892
C4×#145838162
C4×#145838112
#260115522
#520×C1732
#3124173