Finite Group Databasebeta

← #36364668

Subgroups — #36364668

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#26012108C211yesmaximal, characteristic, complemented
2#26012062C211yesmaximal, characteristic, complemented
3#26011882C211yesmaximal, characteristic, complemented
4#26011874C211yesmaximal, characteristic, complemented
5#26011849C211yesmaximal, characteristic, complemented
6#26011840C211yesmaximal, characteristic, complemented
7#26011712C211yesmaximal, characteristic, complemented
8#26011704C211yesmaximal, characteristic, complemented
9#26011700C211yesmaximal, characteristic, complemented
10#26011694C211yesmaximal, characteristic, complemented
11#26011653C211yesmaximal, characteristic, complemented
12#26011649C211yesmaximal, characteristic, complemented
13#26011629C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic
16C22×Dic346C2211yescharacteristic
17C2×Dic692C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21C2×#14583814C2211yescharacteristic
22C2×#14583814C2211yescharacteristic
23#19149678C2211yescharacteristic
24#19149677C2211yescharacteristic
25#19149677C2211yescharacteristic
26#19149672—21no—
27#19149666—21no—
28#19149666C2211yescharacteristic
29#19149666C2211yescharacteristic
30#19149666C2211yescharacteristic
31#19149660—21no—
32#19149660—21no—
33#19149660—21no—
34#19149658—21no—
35#19149658—21no—
36#19149658C2211yescharacteristic
37#19149656—21no—
38#19149644C2211yescharacteristic
39D4×Dic173C2211yescharacteristic
40#19149638C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
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×Dic346—21no—
60C2×Dic346—21no—
61C2×Dic346—41no—
62C2×Dic346—41no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66C22×Dic173—21no—
67C22×Dic173—21no—
68Dic692—41no—
69Dic692—41no—
70#14583816—41no—
71#14583814—41no—
72#14583814—41no—
73#14583814—41no—
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173D411yescharacteristic
77C4⋊Dic173D411yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173D411yescharacteristic
95C22⋊Dic173D411yescharacteristic
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
99#41×C173—41no—
100#41×C173—41no—
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173—21no—
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C2×D4×C173C2311yescharacteristic
106C22×C4×C173—21no—
107C22×C4×C173C2311yescharacteristic
108SD16×C173—41no—
109D8×C173—41no—
110C2×C8×C173C2311yescharacteristic
111C2×C8×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346—21no—
120Dic346—42no—
121Dic346—81no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—42no—
127C2×Dic173—81no—
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132#13063436—21no—
133#13063436—21no—
134C23×C173—21no—
135C23×C173—21no—
136C23×C173C2×D411yescharacteristic
137Q8×C173—21no—
138Q8×C173—21no—
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—42no—
144D4×C173—81no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—42no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C8×C173—21no—
153C8×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173C22×D411yescharacteristic
164C4×C173—21no—
165C4×C173—41no—
166C4×C173—41no—
167C4×C173H(5,𝔽2)11yescharacteristic
168C4×C173H(5,𝔽2)11yescharacteristic
169Dic173—21no—
170Dic173—21no—
171Dic173—21no—
172Dic173—21no—
173Dic173—41no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—41no—
178C2×C173—81no—
179C2×C173—81no—
180C2×C173#53511yescharacteristic
181C2×C173#57711yescharacteristic
182C2×C173#57811yescharacteristic
183C173#322511yescharacteristic, complemented, Sylow
184#3225—1731nomaximal, Sylow
185#542—1731no—
186#536—1731no—
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#449D17311yescharacteristic
198#448—1731no—
199#433—1731no—
200C2×#41C2×D17311yescharacteristic
201C22×Q8—1731no—
202C2×Q16—1731no—
203C2×Q16—1731no—
204C2×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16—1731no—
207C2×SD16—1731no—
208C2×SD16C2×D17311yescharacteristic
209C2×D8C2×D17311yescharacteristic
210C4⋊Q8—1731no—
211C4⋊Q8—1731no—
212C4⋊Q8—1731no—
213#126—3461no—
214#124—3461no—
215#124—1731no—
216#124—1731no—
217#124—1731no—
218#123—3461no—
219#123—3461no—
220#123—3461no—
221C22⋊Q8—3461no—
222C22⋊Q8—3461no—
223C22⋊Q8—3461no—
224C22⋊Q8—1731no—
225#121C2×D17311yescharacteristic
226C4×Q8—1731no—
227C4×D4—1731no—
228#117—1731no—
229C4⋊C8—1731no—
230C4⋊C8—1731no—
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#103C2×D17311yescharacteristic
237#102—1731no—
238#102C2×D17311yescharacteristic
239C22⋊C8—1731no—
240C22⋊C8C2×D17311yescharacteristic
241#97—1731no—
242C4×C8—1731no—
243#41—41no—
244#41—41no—
245C2×Q8—3461no—
246C2×Q8—3461no—
247C2×Q8—1731no—
248C2×Q8—1731no—
249C2×Q8—1731no—
250C2×Q8—6921no—
251C2×Q8—6921no—
252C2×Q8C22×D17311yescharacteristic
253C2×D4—21no—
254C2×D4—21no—
255C2×D4C22×D17311yescharacteristic
256C2×D4C22×D17311yescharacteristic
257C22×C4—21no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4C22×D17311yescharacteristic
261Q16—6921no—
262Q16—6921no—
263Q16—6921no—
264SD16—41no—
265SD16—6921no—
266SD16—6921no—
267SD16—6921no—
268D8—41no—
269C2×C8—1731no—
270C2×C8—1731no—
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4—1731no—
283C4⋊C4C22×D17311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—1731no—
294C22⋊C4—1731no—
295C42—3461no—
296C42—1731no—
297C42—1731no—
298C23—21no—
299C23—21no—
300C23D4×D17311yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—3461no—
304Q8—3461no—
305Q8—3461no—
306Q8—3461no—
307Q8—3461no—
308Q8—3461no—
309Q8—13841no—
310Q8—6922no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—81no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—13841no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—1731no—
330C2×C4—1731no—
331C2×C4—6922no—
332C2×C4D4×D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3461no—
337C8—3461no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22C2×D4×D17311yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—3461no—
352C4—3461no—
353C4—3461no—
354C4—3461no—
355C4—6921no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4#1914972411yescharacteristic
360C4#1914972411yescharacteristic
361C2—41no—
362C2—81no—
363C2—81no—
364C2#2601205611yescharacteristic
365C2#2601230111yescharacteristic
366C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3225
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. #26012108C2semidirect
2. #26012062C2semidirect
3. #26011882C2semidirect
4. #26011874C2semidirect
5. #26011849C2semidirect
6. #26011840C2semidirect
7. #26011712C2semidirect
8. #26011704C2semidirect
9. #26011700C2semidirect
10. #26011694C2semidirect
11. #26011653C2semidirect
12. #26011649C2semidirect
13. #26011629C2semidirect
14. #26011549C2semidirect
183. C173#3225semidirect

Maximal subgroups

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

subgroupindex
#260121082
#260120622
#260118822
#260118742
#260118492
#260118402
#260117122
#260117042
#260117002
#260116942
#260116532
#260116492
#260116292
#260115492
#449×C1732
#3225173