Finite Group Databasebeta

← #36365450

Subgroups — #36365450

Order 22144.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012181C211yesmaximal, characteristic, complemented
2#26011845C211yesmaximal, characteristic, complemented
3#26011829C211yesmaximal, characteristic, complemented
4#26011802C211yesmaximal, characteristic, complemented
5#26011798C211yesmaximal, characteristic, complemented
6#26011730C211yesmaximal, characteristic, complemented
7#26011718C211yesmaximal, characteristic, complemented
8#26011710C211yesmaximal, characteristic, complemented
9#26011653C211yesmaximal, characteristic, complemented
10#26011647C211yesmaximal, characteristic, complemented
11C4×#14583816C211yesmaximal, characteristic, complemented
12D4×#13063436C211yesmaximal, characteristic, complemented
13#26011487C211yesmaximal, characteristic, complemented
14C2×#19149603C211yesmaximal, characteristic, complemented
15#548×C173C211yesmaximal, characteristic
16C2×#14583816C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C22×#13063436C2211yescharacteristic
20C22×#13063436C2211yescharacteristic
21#19149672—21no—
22#19149671C2211yescharacteristic
23#19149669—21no—
24#19149660—21no—
25#19149658C2211yescharacteristic
26#19149658C2211yescharacteristic
27C4×Dic346C2211yescharacteristic
28#19149638C2211yescharacteristic
29#19149635—21no—
30C2×C4⋊Dic173C2211yescharacteristic
31#19149612C2211yescharacteristic
32#19149612C2211yescharacteristic
33#19149610C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35#19149603C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149603C2211yescharacteristic
39#19149603C2211yescharacteristic
40#19149603C2211yescharacteristic
41#19149602C2211yescharacteristic
42#19149600C2211yescharacteristic
43#19149600C2211yescharacteristic
44#19149600C2211yescharacteristic
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×Dic346C2311yescharacteristic
60C22×Dic173—21no—
61#14583816—22no—
62#14583816—41no—
63#14583814—41no—
64C2×#13063436—41no—
65C2×#13063436—41no—
66C2×#13063436C2311yescharacteristic
67C2×#13063436C2311yescharacteristic
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C4⋊Dic173—21no—
71C4⋊Dic173C2311yescharacteristic
72C4⋊Dic173C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74#14583805—21no—
75#14583805—21no—
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79C22⋊Dic173—21no—
80C22⋊Dic173—21no—
81C22⋊Dic173—21no—
82C4×Dic173—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
111Dic346—21no—
112Dic346—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—42no—
118#13063436—21no—
119#13063436—21no—
120#13063436—21no—
121#13063436—21no—
122C23×C173—21no—
123C23×C173C2×D411yescharacteristic
124C23×C173C2×D411yescharacteristic
125Q8×C173—21no—
126Q8×C173—21no—
127Q8×C173—21no—
128Q8×C173—42no—
129Q8×C173#4112yes—
130D4×C173—22no—
131D4×C173—21no—
132D4×C173—21no—
133D4×C173—42no—
134D4×C173—42no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—42no—
143C2×C4×C173—42no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2411yescharacteristic
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173#13512yes—
158C22×C173C22×D411yescharacteristic
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—22no—
162C4×C173—21no—
163C4×C173—21no—
164C4×C173—41no—
165C4×C173—41no—
166C4×C173C2×#4111yescharacteristic
167C4×C173H(5,𝔽2)11yescharacteristic
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172C2×C173—22no—
173C2×C173—41no—
174C2×C173—81no—
175C2×C173#54611yescharacteristic
176C2×C173C2×#13511yescharacteristic
177C2×C173#57811yescharacteristic
178C173#330011yescharacteristic, complemented, Sylow
179#3300—1731nomaximal, Sylow
180#552—1731no—
181#548D17311yescharacteristic
182#488—1731no—
183#485—1731no—
184#480—1731no—
185#479—1731no—
186#467—1731no—
187#465—1731no—
188#464—1731no—
189#452—1731no—
190#451—1731no—
191C4×Q16—1731no—
192C8×D4—1731no—
193#416—1731no—
194C2×#103—1731no—
195C2×#41C2×D17311yescharacteristic
196C22×Q8C2×D17311yescharacteristic
197C2×Q16—1731no—
198C2×Q16—1731no—
199C2×SD16—1731no—
200C22×C8—1731no—
201C22×C8—1731no—
202#126—3461no—
203#125—3461no—
204#125—1731no—
205#124—21no—
206#124C2×D17311yescharacteristic
207#123—3461no—
208C22⋊Q8—21no—
209C22⋊Q8—1731no—
210C22⋊Q8—1731no—
211C22⋊Q8C2×D17311yescharacteristic
212#121—21no—
213#121C2×D17311yescharacteristic
214C4×Q8—1731no—
215C4×Q8C2×D17311yescharacteristic
216C4×D4—21no—
217C4×D4C2×D17311yescharacteristic
218#117—3461no—
219#117—1731no—
220C2×C4⋊C4—1731no—
221#107—1731no—
222#107—1731no—
223#106—1731no—
224C4⋊C8—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103—1731no—
231#103—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235C22⋊C8—1731no—
236C22⋊C8—1731no—
237C4×C8—1731no—
238#41—22no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—1731no—
244C2×Q8C22×D17311yescharacteristic
245C2×Q8C22×D17311yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22⋊D17311yescharacteristic
249C2×D4C22×D17311yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—3461no—
253C22×C4C22×D17311yescharacteristic
254C22×C4C22×D17311yescharacteristic
255Q16—3462no—
256Q16—6921no—
257SD16—6921no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8—1731no—
261C2×C8—1731no—
262C2×C8—6921no—
263C2×C8—6921no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—3461no—
272C4⋊C4—1731no—
273C4⋊C4—1731no—
274C4⋊C4—1731no—
275C4⋊C4C22⋊D17311yescharacteristic
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4C22⋊D17311yescharacteristic
285C22⋊C4C22⋊D17311yescharacteristic
286C42—21no—
287C42—3461no—
288C42—3461no—
289C42C22×D17311yescharacteristic
290C23—21no—
291C23C2×C22⋊D17311yescharacteristic
292C23C2×C22⋊D17311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—3461no—
298Q8—3461no—
299Q8#1458382512yes—
300D4—22no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—6922no—
319C2×C4C2×C22⋊D17311yescharacteristic
320C2×C4C2×C22⋊D17311yescharacteristic
321C2×C4C2×C22⋊D17311yescharacteristic
322C2×C4C2×C22⋊D17311yescharacteristic
323C2×C4C23×D17311yescharacteristic
324C8—3461no—
325C8—3461no—
326C8—3461no—
327C8—3461no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1914969912yes—
337C22C22×C22⋊D17311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4C2×#1458382511yescharacteristic
350C4#1914972411yescharacteristic
351C2—22no—
352C2—41no—
353C2—81no—
354C2#2601213511yescharacteristic
355C2C2×#1914969911yescharacteristic
356C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#548×C173C2
socleC22×C173C22×D4
hypercenter#548D173
hypocenterC173#3300
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. #26012181C2semidirect
2. #26011845C2semidirect
3. #26011829C2semidirect
4. #26011802C2semidirect
5. #26011798C2semidirect
6. #26011730C2semidirect
7. #26011718C2semidirect
8. #26011710C2semidirect
9. #26011653C2semidirect
10. #26011647C2semidirect
11. C4×#14583816C2semidirect
12. D4×#13063436C2semidirect
13. #26011487C2semidirect
14. C2×#19149603C2semidirect
178. C173#3300semidirect

Maximal subgroups

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

subgroupindex
#260121812
#260118452
#260118292
#260118022
#260117982
#260117302
#260117182
#260117102
#260116532
#260116472
C4×#145838162
D4×#130634362
#260114872
C2×#191496032
#548×C1732
#3300173