Finite Group Databasebeta

← #36364669

Subgroups — #36364669

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#26012106C211yesmaximal, characteristic, complemented
2#26012061C211yesmaximal, characteristic, complemented
3#26011882C211yesmaximal, characteristic, complemented
4#26011874C211yesmaximal, characteristic, complemented
5#26011849C211yesmaximal, characteristic, complemented
6#26011840C211yesmaximal, characteristic, complemented
7#26011710C211yesmaximal, characteristic, complemented
8#26011702C211yesmaximal, characteristic, complemented
9#26011696C211yesmaximal, characteristic, complemented
10#26011654C211yesmaximal, characteristic, complemented
11#26011648C211yesmaximal, characteristic, complemented
12#26011633C211yesmaximal, characteristic
13#26011628C211yesmaximal, characteristic, complemented
14#26011551C211yesmaximal, characteristic, complemented
15#463×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×Dic692C2211yescharacteristic
20C2×#14583816C2211yescharacteristic
21C2×#14583814C2211yescharacteristic
22C2×#14583814C2211yescharacteristic
23C2×#14583814C2211yescharacteristic
24#19149679C2211yescharacteristic
25#19149678C2211yescharacteristic
26#19149677C2211yescharacteristic
27#19149672—21no—
28#19149666C2211yescharacteristic
29#19149666C2211yescharacteristic
30#19149665—21no—
31#19149662—21no—
32#19149661—21no—
33#19149661—21no—
34#19149659C2211yescharacteristic
35#19149658—21no—
36#19149656—21no—
37#19149656—21no—
38#19149650C2211yescharacteristic
39C4×Dic346C2211yescharacteristic
40#19149636C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C2×D8×C173C2211yescharacteristic
54#124×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—41no—
60#14583825—41no—
61C2×Dic346—21no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66Q8×D173—41no—
67Q8×D173—41no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C22×Dic173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73Dic692—41no—
74Dic692—41no—
75#14583816—41no—
76#14583814—41no—
77#14583814—41no—
78#14583814—41no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C4⋊Dic173—21no—
82C4⋊Dic173—21no—
83C4⋊Dic173—21no—
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90#14583805D411yescharacteristic
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96#14583803—21no—
97#14583803D411yescharacteristic
98#14583803D411yescharacteristic
99C4×Dic173—21no—
100C4×Dic173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173—21no—
105SD16×C173—41no—
106D8×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C42×C173C2311yescharacteristic
114C22×D173C2×D411yescharacteristic
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346—21no—
120Dic346—21no—
121Dic346—42no—
122Dic346—81no—
123C22⋊D173—42no—
124C22⋊D173—81no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—42no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—42no—
136#13063436—21no—
137#13063436—21no—
138C23×C173—21no—
139C23×C173—21no—
140Q8×C173—21no—
141Q8×C173—21no—
142D4×C173—21no—
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—81no—
149C2×C4×C173C2×D411yescharacteristic
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C8×C173—21no—
153C8×C173—21no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173C22×D411yescharacteristic
161C2×D173—21no—
162C2×D173—21no—
163C2×D173—21no—
164C4×C173—21no—
165C4×C173—21no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173H(5,𝔽2)11yescharacteristic
169C4×C173H(5,𝔽2)11yescharacteristic
170Dic173—21no—
171Dic173—21no—
172Dic173—21no—
173Dic173—41no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—81no—
178C2×C173—81no—
179C2×C173#53511yescharacteristic
180C2×C173#57711yescharacteristic
181C2×C173#57811yescharacteristic
182D173—41no—
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#463D17311yescharacteristic, complemented
193#462—1731no—
194#461—1731no—
195#452—1731no—
196#451—1731no—
197#449—1731no—
198#448—1731no—
199#433—1731no—
200C2×#41—1731no—
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#124C2×D17311yescharacteristic
218#123—3461no—
219#123—3461no—
220#123—3461no—
221C22⋊Q8—3461no—
222C22⋊Q8—3461no—
223C22⋊Q8—3461no—
224C22⋊Q8—1731no—
225#121—1731no—
226C4×Q8—1731no—
227C4×D4C2×D17311yescharacteristic
228#117—1731no—
229C4⋊C8—1731no—
230C4⋊C8C2×D17311yescharacteristic
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⋊C8—1731no—
241#97—1731no—
242C4×C8—1731no—
243#41—6921no—
244#41—6921no—
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—3461no—
254C2×D4—3461no—
255C2×D4C22×D17311yescharacteristic
256C2×D4C22×D17311yescharacteristic
257C22×C4—21no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—1731no—
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—21no—
286C22⋊C4—21no—
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—
297C42C22×D17311yescharacteristic
298C23—21no—
299C23—21no—
300C23—1731no—
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—13841no—
316D4—6922no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—1731no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—6922no—
330C2×C4—6922no—
331C2×C4D4×D17311yescharacteristic
332C2×C4D4×D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3461no—
337C8—3461no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—3461no—
345C22—3461no—
346C22—3461no—
347C22C2×D4×D17311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—3461no—
353C4—3461no—
354C4—3461no—
355C4—6921no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4#1914972411yescharacteristic
360C4#1914972511yescharacteristic
361C2—81no—
362C2—81no—
363C2—6921no—
364C2#2601205711yescharacteristic
365C2#2601230111yescharacteristic
366C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#463×C173C2
socleC22×C173C22×D4
prime radical#26011633C2
hypercenter#463D173
hypocenterC173#3225
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #26012106C2semidirect
2. #26012061C2semidirect
3. #26011882C2semidirect
4. #26011874C2semidirect
5. #26011849C2semidirect
6. #26011840C2semidirect
7. #26011710C2semidirect
8. #26011702C2semidirect
9. #26011696C2semidirect
10. #26011654C2semidirect
11. #26011648C2semidirect
13. #26011628C2semidirect
14. #26011551C2semidirect
15. #463×C173C2semidirect
183. C173#3225semidirect
192. #463D173semidirect

Maximal subgroups

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

subgroupindex
#260121062
#260120612
#260118822
#260118742
#260118492
#260118402
#260117102
#260117022
#260116962
#260116542
#260116482
#260116332
#260116282
#260115512
#463×C1732
#3225173