Finite Group Databasebeta

← #36365458

Subgroups — #36365458

Order 22144.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012188C211yesmaximal, characteristic, complemented
2#26012148C211yesmaximal, characteristic, complemented
3#26011844C211yesmaximal, characteristic, complemented
4#26011833C211yesmaximal, characteristic, complemented
5#26011808C211yesmaximal, characteristic, complemented
6#26011800C211yesmaximal, characteristic, complemented
7#26011724C211yesmaximal, characteristic, complemented
8#26011718C211yesmaximal, characteristic, complemented
9#26011704C211yesmaximal, characteristic, complemented
10#26011628C211yesmaximal, characteristic, complemented
11SD16×Dic173C211yesmaximal, characteristic, complemented
12#26011554C211yesmaximal, characteristic, complemented
13#26011487C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×#14583814C2211yescharacteristic
18C22×#13063436C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149672—21no—
21#19149670C2211yescharacteristic
22#19149666—21no—
23#19149666C2211yescharacteristic
24#19149660—21no—
25#19149658—21no—
26#19149658C2211yescharacteristic
27#19149650—21no—
28#19149650—21no—
29#19149650C2211yescharacteristic
30#19149650C2211yescharacteristic
31Q8×Dic173C2211yescharacteristic
32D4×Dic173C2211yescharacteristic
33C4×C22⋊D173—21no—
34C4×C22⋊D173C2211yescharacteristic
35#19149639—21no—
36#19149638C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149591C2211yescharacteristic
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×Dic346C2311yescharacteristic
61Q8×D173—22no—
62Q8×D173—41no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173D411yescharacteristic
66C22×Dic173—21no—
67C2×C4×D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173C2311yescharacteristic
70#14583814—41no—
71C2×#13063436—41no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C8×D173—41no—
75C4⋊Dic173—21no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173D411yescharacteristic
92#14583803D411yescharacteristic
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96#41×C173—22no—
97#41×C173—41no—
98C2×Q8×C173C2311yescharacteristic
99C2×D4×C173—21no—
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173—21no—
104C22×C4×C173C2311yescharacteristic
105SD16×C173—22no—
106D8×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114Dic346—42no—
115C22⋊D173—22no—
116C22⋊D173—42no—
117C22⋊D173—42no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—81no—
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C4×D173—21no—
127C4×D173—21no—
128C4×D173—42no—
129C4×D173—81no—
130#13063436—21no—
131#13063436—21no—
132C23×C173—21no—
133C23×C173—21no—
134C23×C173C2×D411yescharacteristic
135Q8×C173—21no—
136Q8×C173#4112yes—
137D4×C173—21no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—42no—
142D4×C173—81no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—42no—
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173C22×D411yescharacteristic
162C2×D173—21no—
163C2×D173—21no—
164C2×D173#13512yes—
165C4×C173—21no—
166C4×C173—22no—
167C4×C173—41no—
168C4×C173C2×#4111yescharacteristic
169C4×C173H(5,𝔽2)11yescharacteristic
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#54611yescharacteristic
181C2×C173C2×#13511yescharacteristic
182C2×C173#57711yescharacteristic
183D173—22no—
184C173#330111yescharacteristic, complemented, Sylow
185#3301—1731nomaximal, Sylow
186#553—1731no—
187#548—1731no—
188#488—1731no—
189#485—1731no—
190#481—1731no—
191#479—1731no—
192#466—1731no—
193#465—1731no—
194#463—1731no—
195#449D17311yescharacteristic, complemented
196#448—1731no—
197C4×SD16—1731no—
198C8×D4—1731no—
199#416—1731no—
200C2×#103—1731no—
201C2×#41C2×D17311yescharacteristic
202C22×Q8—1731no—
203C2×SD16—1731no—
204C2×SD16C2×D17311yescharacteristic
205C2×D8C2×D17311yescharacteristic
206C22×C8—1731no—
207C22×C8—1731no—
208#126—3461no—
209#125—1731no—
210#124—3461no—
211#124—1731no—
212#123—3461no—
213C22⋊Q8—3461no—
214C22⋊Q8—1731no—
215#121—3461no—
216#121—3461no—
217#121—1731no—
218#121—1731no—
219#121C2×D17311yescharacteristic
220C4×Q8—1731no—
221C4×D4—3461no—
222C4×D4—3461no—
223C4×D4—1731no—
224C4×D4—1731no—
225#117—1731no—
226C2×C4⋊C4—1731no—
227#107—1731no—
228#106—1731no—
229#106—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#102—1731no—
239#102—1731no—
240#102C2×D17311yescharacteristic
241C22⋊C8—1731no—
242C22⋊C8C2×D17311yescharacteristic
243C4×C8—1731no—
244#41—22no—
245#41—41no—
246C2×Q8—3461no—
247C2×Q8—3462no—
248C2×Q8—1731no—
249C2×Q8—6921no—
250C2×Q8C22×D17311yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4C22×D17311yescharacteristic
257C2×D4C22×D17311yescharacteristic
258C22×C4—21no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—1731no—
263C22×C4C22×D17311yescharacteristic
264SD16—22no—
265SD16—6921no—
266D8—41no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8—6921no—
270C2×C8—6921no—
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—1731no—
279C4⋊C4—1731no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4C22×D17311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—1731no—
292C22⋊C4—1731no—
293C42—3461no—
294C42—3461no—
295C42—1731no—
296C23—21no—
297C23—21no—
298C23—1731no—
299C23D4×D17311yescharacteristic
300Q8—21no—
301Q8—3461no—
302Q8—3461no—
303Q8—6922no—
304Q8#1458382512yes—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—42no—
310D4—81no—
311D4—3462no—
312D4—6922no—
313D4—6922no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—13841no—
326C2×C4—13841no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—1731no—
330C2×C4—6922no—
331C2×C4D4×D17311yescharacteristic
332C2×C4C23×D17311yescharacteristic
333C8—21no—
334C8—21no—
335C8—3461no—
336C8—3461no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3461no—
347C22—3461no—
348C22—1732no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—3461no—
354C4—3461no—
355C4—3461no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4—6921no—
360C4C2×#1458382511yescharacteristic
361C4#1914972411yescharacteristic
362C2—41no—
363C2—81no—
364C2—81no—
365C2—3462no—
366C2#2601212811yescharacteristic
367C2#135×D17311yescharacteristic
368C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3301
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. #26012188C2semidirect
2. #26012148C2semidirect
3. #26011844C2semidirect
4. #26011833C2semidirect
5. #26011808C2semidirect
6. #26011800C2semidirect
7. #26011724C2semidirect
8. #26011718C2semidirect
9. #26011704C2semidirect
10. #26011628C2semidirect
11. SD16×Dic173C2semidirect
12. #26011554C2semidirect
13. #26011487C2semidirect
14. #103×D173C2semidirect
15. #449×C173C2semidirect
184. C173#3301semidirect
195. #449D173semidirect

Maximal subgroups

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

subgroupindex
#260121882
#260121482
#260118442
#260118332
#260118082
#260118002
#260117242
#260117182
#260117042
#260116282
SD16×Dic1732
#260115542
#260114872
#103×D1732
#449×C1732
#3301173