Finite Group Databasebeta

← #36364672

Subgroups — #36364672

Order 22144.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012112C211yesmaximal, characteristic, complemented
2#26011883C211yesmaximal, characteristic, complemented
3#26011873C211yesmaximal, characteristic, complemented
4#26011850C211yesmaximal, characteristic, complemented
5#26011841C211yesmaximal, characteristic, complemented
6#26011710C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011698C211yesmaximal, characteristic, complemented
9#26011693C211yesmaximal, characteristic, complemented
10#26011653C211yesmaximal, characteristic, complemented
11#26011647C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic
13#26011630C211yesmaximal, characteristic, complemented
14#26011552C211yesmaximal, characteristic, complemented
15#536×C173C211yesmaximal, characteristic, complemented
16C2×#14583816C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×#14583813C2211yescharacteristic
22C2×#14583813C2211yescharacteristic
23C2×#14583811C2211yescharacteristic
24#19149679C2211yescharacteristic
25#19149674—21no—
26#19149667C2211yescharacteristic
27#19149665—21no—
28#19149661—21no—
29#19149658C2211yescharacteristic
30#19149657—21no—
31#19149653C2211yescharacteristic
32C4×Dic346C2211yescharacteristic
33C4×D692C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35C4⋊#13063436C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149603C2211yescharacteristic
39#19149603C2211yescharacteristic
40#19149603C2211yescharacteristic
41#19149603C2211yescharacteristic
42#19149600C2211yescharacteristic
43#19149600C2211yescharacteristic
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46#19149590C2211yescharacteristic
47C4×#13063436C2211yescharacteristic
48C2×#41×C173C2211yescharacteristic
49C22×Q8×C173C2211yescharacteristic
50C4⋊Q8×C173C2211yescharacteristic
51C4⋊Q8×C173C2211yescharacteristic
52#124×C173C2211yescharacteristic
53#124×C173C2211yescharacteristic
54#123×C173—21no—
55#123×C173—21no—
56C22⋊Q8×C173—21no—
57C22⋊Q8×C173—21no—
58#117×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C4×D173—21no—
64#14583816—41no—
65#14583816—41no—
66#14583816—41no—
67#14583814—41no—
68#14583813—41no—
69#14583813—41no—
70#14583813—41no—
71#14583811—41no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81C22⋊Dic173—21no—
82#14583803—21no—
83#14583803—21no—
84#14583803—21no—
85#14583803—21no—
86C4×Dic173—21no—
87#41×C173—41no—
88#41×C173—41no—
89C2×Q8×C173—21no—
90C2×Q8×C173—41no—
91C2×Q8×C173—41no—
92C2×Q8×C173C2311yescharacteristic
93C2×Q8×C173C2311yescharacteristic
94C2×Q8×C173C2311yescharacteristic
95C2×D4×C173—21no—
96C2×D4×C173C2311yescharacteristic
97C22×C4×C173—21no—
98C22×C4×C173—21no—
99C22×C4×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173—21no—
104C4⋊C4×C173D411yescharacteristic
105C4⋊C4×C173D411yescharacteristic
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C22⋊C4×C173—21no—
110C22⋊C4×C173D411yescharacteristic
111C22⋊C4×C173D411yescharacteristic
112C42×C173C2311yescharacteristic
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—81no—
118D692—21no—
119D692—21no—
120C22⋊D173—81no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C4×D173—81no—
125#13063436—21no—
126#13063436—21no—
127#13063436—21no—
128#13063436—21no—
129C23×C173—21no—
130C23×C173C2×D411yescharacteristic
131Q8×C173—21no—
132Q8×C173—21no—
133Q8×C173—21no—
134Q8×C173—21no—
135Q8×C173—21no—
136Q8×C173—21no—
137Q8×C173—42no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—42no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—42no—
147C2×C4×C173—42no—
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2×D411yescharacteristic
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2411yescharacteristic
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173C22×D411yescharacteristic
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C4×C173—21no—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173—21no—
168C4×C173—21no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173H(5,𝔽2)11yescharacteristic
173C4×C173H(5,𝔽2)11yescharacteristic
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—41no—
178C2×C173—81no—
179C2×C173#53511yescharacteristic
180C2×C173#57711yescharacteristic
181C2×C173#57811yescharacteristic
182D173—81no—
183C173#322511yescharacteristic, complemented, Sylow
184#3225—1731nomaximal, Sylow
185#542—1731no—
186#536D17311yescharacteristic, complemented
187#497—1731no—
188#495—1731no—
189#489—1731no—
190#487—1731no—
191#464—1731no—
192#463—1731no—
193#462—1731no—
194#461—1731no—
195#452—1731no—
196#451—1731no—
197#449—1731no—
198#448—1731no—
199#433—1731no—
200C2×#41C2×D17311yescharacteristic
201C22×Q8C2×D17311yescharacteristic
202C2×Q16—1731no—
203C2×Q16—1731no—
204C2×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16—1731no—
207C2×SD16—1731no—
208C2×SD16—1731no—
209C2×D8—1731no—
210C4⋊Q8—1731no—
211C4⋊Q8C2×D17311yescharacteristic
212C4⋊Q8C2×D17311yescharacteristic
213#126—3461no—
214#124—3461no—
215#124—1731no—
216#124C2×D17311yescharacteristic
217#124C2×D17311yescharacteristic
218#123—21no—
219#123—21no—
220#123—3461no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—3461no—
224C22⋊Q8—1731no—
225#121—1731no—
226C4×Q8—1731no—
227C4×D4—1731no—
228#117C2×D17311yescharacteristic
229C4⋊C8—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—
239C22⋊C8—1731no—
240C22⋊C8—1731no—
241#97—1731no—
242C4×C8—1731no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—3461no—
249C2×Q8—1731no—
250C2×Q8C22×D17311yescharacteristic
251C2×Q8C22×D17311yescharacteristic
252C2×Q8C22×D17311yescharacteristic
253C2×D4—21no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4C22×D17311yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—3461no—
260C22×C4C22×D17311yescharacteristic
261Q16—6921no—
262Q16—6921no—
263Q16—6921no—
264SD16—6921no—
265SD16—6921no—
266SD16—6921no—
267SD16—6921no—
268D8—6921no—
269C2×C8—1731no—
270C2×C8—1731no—
271C2×C8—1731no—
272C2×C8—1731no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4C22⋊D17311yescharacteristic
283C4⋊C4C22⋊D17311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4C22⋊D17311yescharacteristic
294C22⋊C4C22⋊D17311yescharacteristic
295C42—3461no—
296C42C22×D17311yescharacteristic
297C42C22×D17311yescharacteristic
298C23—21no—
299C23—3461no—
300C23C2×C22⋊D17311yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—3461no—
309Q8—3461no—
310Q8—13841no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—3461no—
315D4—3461no—
316D4—13841no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—13841no—
328C2×C4C2×C22⋊D17311yescharacteristic
329C2×C4C2×C22⋊D17311yescharacteristic
330C2×C4C2×C22⋊D17311yescharacteristic
331C2×C4C2×C22⋊D17311yescharacteristic
332C2×C4C2×C22⋊D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—3461no—
335C8—3461no—
336C8—3461no—
337C8—3461no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—6921no—
345C22—6921no—
346C22—6921no—
347C22C22×C22⋊D17311yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4#1914972411yescharacteristic
360C4#1914972411yescharacteristic
361C2—41no—
362C2—81no—
363C2—13841no—
364C2#2601205811yescharacteristic
365C2#2601229911yescharacteristic
366C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#536×C173C2
socleC22×C173C22×D4
prime radical#26011636C2
hypercenter#536D173
hypocenterC173#3225
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #26012112C2semidirect
2. #26011883C2semidirect
3. #26011873C2semidirect
4. #26011850C2semidirect
5. #26011841C2semidirect
6. #26011710C2semidirect
7. #26011703C2semidirect
8. #26011698C2semidirect
9. #26011693C2semidirect
10. #26011653C2semidirect
11. #26011647C2semidirect
13. #26011630C2semidirect
14. #26011552C2semidirect
15. #536×C173C2semidirect
183. C173#3225semidirect
186. #536D173semidirect

Maximal subgroups

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

subgroupindex
#260121122
#260118832
#260118732
#260118502
#260118412
#260117102
#260117032
#260116982
#260116932
#260116532
#260116472
#260116362
#260116302
#260115522
#536×C1732
#3225173