Finite Group Databasebeta

← #36365554

Subgroups — #36365554

Order 22144.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012145C211yesmaximal, characteristic, complemented
2#26011850C211yesmaximal, characteristic, complemented
3#26011824C211yesmaximal, characteristic, complemented
4#26011802C211yesmaximal, characteristic, complemented
5#26011786C211yesmaximal, characteristic, complemented
6#26011739C211yesmaximal, characteristic, complemented
7#26011725C211yesmaximal, characteristic, complemented
8#26011703C211yesmaximal, characteristic, complemented
9#26011636C211yesmaximal, characteristic, complemented
10#26011630C211yesmaximal, characteristic, complemented
11#26011589C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011492C211yesmaximal, characteristic, complemented
14C2×#19149603C211yesmaximal, characteristic, complemented
15#548×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—21no—
17C2×#14583813C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583810C2211yescharacteristic
21C22×#13063436C2211yescharacteristic
22#19149679C2211yescharacteristic
23#19149663—21no—
24#19149663—21no—
25#19149659—21no—
26#19149653C2211yescharacteristic
27#19149653C2211yescharacteristic
28C4×D692C2211yescharacteristic
29C2×C4⋊Dic173C2211yescharacteristic
30C2×C4⋊Dic173C2211yescharacteristic
31#19149612C2211yescharacteristic
32#19149612C2211yescharacteristic
33#19149612C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35#19149603C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149603C2211yescharacteristic
39#19149603C2211yescharacteristic
40#19149603C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149600C2211yescharacteristic
43#19149600C2211yescharacteristic
44#19149599C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47#19149590C2211yescharacteristic
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
59#14583825—44no—
60C2×Dic346—21no—
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C22×Dic173—21no—
65C22×Dic173—21no—
66C2×C4×D173—21no—
67#14583813—22no—
68#14583813—41no—
69#14583811—41no—
70#14583810—41no—
71C2×#13063436—41no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583803—21no—
83#14583803—21no—
84#14583803—21no—
85#41×C173—22no—
86#41×C173—41no—
87C2×Q8×C173—21no—
88C2×Q8×C173—22no—
89C2×Q8×C173—41no—
90C2×Q8×C173C2311yescharacteristic
91C2×Q8×C173C2311yescharacteristic
92C2×D4×C173—21no—
93C2×D4×C173—21no—
94C2×D4×C173D411yescharacteristic
95C2×D4×C173C2311yescharacteristic
96C22×C4×C173—21no—
97C22×C4×C173—21no—
98C22×C4×C173C2311yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173—21no—
102C4⋊C4×C173D411yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22⋊C4×C173D411yescharacteristic
109C22⋊C4×C173D411yescharacteristic
110C42×C173—21no—
111C42×C173C2311yescharacteristic
112C22×D173—21no—
113Dic346—42no—
114D692—21no—
115D692—21no—
116C22⋊D173—42no—
117C22⋊D173—42no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—42no—
122C2×Dic173—42no—
123C4×D173—42no—
124#13063436—21no—
125#13063436—21no—
126#13063436—21no—
127#13063436—21no—
128C23×C173—21no—
129C23×C173C2×D411yescharacteristic
130C23×C173C2×D411yescharacteristic
131Q8×C173—21no—
132Q8×C173—21no—
133Q8×C173—21no—
134Q8×C173—42no—
135Q8×C173#4112yescomplemented
136D4×C173—22no—
137D4×C173—21no—
138D4×C173—21no—
139D4×C173—42no—
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—21no—
147C2×C4×C173—21no—
148C2×C4×C173—42no—
149C2×C4×C173—42no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173AGL(1,ℤ/8)12yes—
164C22×C173C22×D411yescharacteristic
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C4×C173—21no—
169C4×C173—21no—
170C4×C173—22no—
171C4×C173—21no—
172C4×C173—21no—
173C4×C173—41no—
174C4×C173—41no—
175C4×C173C2×#4111yescharacteristic
176C4×C173H(5,𝔽2)11yescharacteristic
177Dic173—41no—
178Dic173—41no—
179Dic173—41no—
180C2×C173—22no—
181C2×C173—41no—
182C2×C173—81no—
183C2×C173#54611yescharacteristic
184C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C173#57811yescharacteristic
186D173—81no—
187C173#330811yescharacteristic, complemented, Sylow
188#3308—1731nomaximal, Sylow
189#548D17311yescharacteristic, complemented
190#547—1731no—
191#489—1731no—
192#484—1731no—
193#480—1731no—
194#477—1731no—
195#468—1731no—
196#466—1731no—
197#463—1731no—
198#449—1731no—
199#448—1731no—
200#440—1731no—
201#435—1731no—
202#417—1731no—
203C2×#103—1731no—
204C2×#41—3461no—
205C2×#41C2×D17311yescharacteristic
206C22×Q8C2×D17311yescharacteristic
207C2×SD16—1731no—
208C2×SD16—1731no—
209C2×D8—1731no—
210C2×M16—1731no—
211C22×C8—1731no—
212C4⋊Q8—1731no—
213#124—21no—
214#124C2×D17311yescharacteristic
215#123—3461no—
216#123—3461no—
217C22⋊Q8—21no—
218C22⋊Q8—3461no—
219C22⋊Q8C2×D17311yescharacteristic
220#121—21no—
221#121—1731no—
222#121—1731no—
223#121C2×D17311yescharacteristic
224C4×Q8C2×D17311yescharacteristic
225C4×D4—21no—
226C4×D4—1731no—
227C4×D4C2×D17311yescharacteristic
228C2×C4⋊C4—1731no—
229C2×C4⋊C4—1731no—
230#107—1731no—
231#107—1731no—
232#107—1731no—
233C4⋊C8—1731no—
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#103—1731no—
238#103—1731no—
239#103—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244C22⋊C8—1731no—
245C22⋊C8—1731no—
246#97—1731no—
247#41—22no—
248#41—41no—
249#41—6924no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—3461no—
254C2×Q8C22×D17311yescharacteristic
255C2×Q8C22×D17311yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—1731no—
261C2×D4C22⋊D17311yescharacteristic
262C2×D4C22×D17311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4C22×D17311yescharacteristic
269C22×C4C22×D17311yescharacteristic
270SD16—3462no—
271SD16—6921no—
272D8—6921no—
273M16—6921no—
274C2×C8—1731no—
275C2×C8—1731no—
276C2×C8—1731no—
277C2×C8—1731no—
278C2×C8—6921no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—1731no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4C22⋊D17311yescharacteristic
288C4⋊C4C22×D17311yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4C22⋊D17311yescharacteristic
297C22⋊C4C22⋊D17311yescharacteristic
298C42—21no—
299C42C22×D17311yescharacteristic
300C23—21no—
301C23—3461no—
302C23C2×C22⋊D17311yescharacteristic
303C23C2×C22⋊D17311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—6922no—
309Q8#1458382512yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—3461no—
316D4—3461no—
317D4—6922no—
318D4—6922no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—6922no—
332C2×C4—6922no—
333C2×C4—6922no—
334C2×C4C2×C22⋊D17311yescharacteristic
335C2×C4C2×C22⋊D17311yescharacteristic
336C2×C4C2×C22⋊D17311yescharacteristic
337C2×C4C2×C22⋊D17311yescharacteristic
338C2×C4C23×D17311yescharacteristic
339C8—3461no—
340C8—3461no—
341C8—3461no—
342C8—3461no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—6921no—
352C22—6921no—
353C22—6921no—
354C22#1914970412yes—
355C22C22×C22⋊D17311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—6921no—
364C4—6921no—
365C4—6921no—
366C4C2×#1458382511yescharacteristic
367C4#1914972411yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—13841no—
372C2#2601213511yescharacteristic
373C2C2×#1914970411yescharacteristic
374C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#548×C173C2
socleC22×C173C22×D4
hypercenter#548D173
hypocenterC173#3308
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. #26012145C2semidirect
2. #26011850C2semidirect
3. #26011824C2semidirect
4. #26011802C2semidirect
5. #26011786C2semidirect
6. #26011739C2semidirect
7. #26011725C2semidirect
8. #26011703C2semidirect
9. #26011636C2semidirect
10. #26011630C2semidirect
11. #26011589C2semidirect
12. #26011563C2semidirect
13. #26011492C2semidirect
14. C2×#19149603C2semidirect
15. #548×C173C2semidirect
135. Q8×C173#41semidirect
187. C173#3308semidirect
189. #548D173semidirect
309. Q8#14583825semidirect

Maximal subgroups

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

subgroupindex
#260121452
#260118502
#260118242
#260118022
#260117862
#260117392
#260117252
#260117032
#260116362
#260116302
#260115892
#260115632
#260114922
C2×#191496032
#548×C1732
#3308173