Finite Group Databasebeta

← #36365560

Subgroups — #36365560

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#26012147C211yesmaximal, characteristic, complemented
2#26012141C211yesmaximal, characteristic, complemented
3#26011849C211yesmaximal, characteristic, complemented
4#26011828C211yesmaximal, characteristic, complemented
5#26011802C211yesmaximal, characteristic, complemented
6#26011785C211yesmaximal, characteristic, complemented
7#26011736C211yesmaximal, characteristic, complemented
8#26011704C211yesmaximal, characteristic, complemented
9#26011633C211yesmaximal, characteristic, complemented
10#26011629C211yesmaximal, characteristic, complemented
11#26011588C211yesmaximal, characteristic, complemented
12#26011562C211yesmaximal, characteristic, complemented
13#26011489C211yesmaximal, characteristic, complemented
14C2×#19149604C211yesmaximal, characteristic, complemented
15#466×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—21no—
17C2×#14583825C2211yescharacteristic
18C22×Dic346C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21C2×#14583809C2211yescharacteristic
22#19149678C2211yescharacteristic
23#19149666C2211yescharacteristic
24#19149665—21no—
25#19149661—21no—
26#19149660—21no—
27#19149659C2211yescharacteristic
28#19149656—21no—
29#19149656—21no—
30#19149651—21no—
31#19149650C2211yescharacteristic
32#19149644C2211yescharacteristic
33D4×Dic173—21no—
34D4×Dic173C2211yescharacteristic
35C4×C22⋊D173C2211yescharacteristic
36C2×C4⋊Dic173C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149608C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic
52C2×D8×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54#121×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61#14583825—44no—
62C2×Dic346—21no—
63C2×Dic346—22no—
64C2×Dic346—21no—
65C2×Dic346—41no—
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173D411yescharacteristic
71C22×Dic173—21no—
72C22×Dic173—21no—
73C22×Dic173—21no—
74C22×Dic173—21no—
75C2×C4×D173—21no—
76C2×C4×D173C2311yescharacteristic
77#14583814—22no—
78#14583814—41no—
79#14583809—41no—
80C2×#13063436C2311yescharacteristic
81C2×#13063436C2311yescharacteristic
82C4⋊Dic173—21no—
83C4⋊Dic173—21no—
84C4⋊Dic173C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173D411yescharacteristic
95#14583803—21no—
96#14583803D411yescharacteristic
97C4×Dic173—21no—
98C4×Dic173C2311yescharacteristic
99C2×D4×C173—21no—
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104D8×C173—41no—
105C2×C8×C173—41no—
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—42no—
117Dic346—42no—
118Dic346#4112yescomplemented
119C22⋊D173—22no—
120C22⋊D173—42no—
121C22⋊D173—42no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—42no—
127C2×Dic173—42no—
128C2×Dic173—42no—
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—
140C23×C173C2×D411yescharacteristic
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—42no—
146D4×C173—42no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2411yescharacteristic
153C8×C173—21no—
154C8×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×C173AGL(1,ℤ/8)12yes—
164C22×C173C22×D411yescharacteristic
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—21no—
168C4×C173—21no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173C2×#4111yescharacteristic
172C4×C173H(5,𝔽2)11yescharacteristic
173Dic173—21no—
174Dic173—22no—
175Dic173—21no—
176Dic173—21no—
177Dic173—41no—
178Dic173—41no—
179Dic173—41no—
180C2×C173—22no—
181C2×C173—81no—
182C2×C173—81no—
183C2×C173#54611yescharacteristic
184C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C173#57811yescharacteristic
186D173—41no—
187C173#330811yescharacteristic, complemented, Sylow
188#3308—1731nomaximal, Sylow
189#548—1731no—
190#547—1731no—
191#489—1731no—
192#484—1731no—
193#480—1731no—
194#477—1731no—
195#468—1731no—
196#466D17311yescharacteristic, complemented
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×#41—1731no—
206C22×Q8—1731no—
207C2×SD16—1731no—
208C2×SD16—1731no—
209C2×D8C2×D17311yescharacteristic
210C2×M16—1731no—
211C22×C8C2×D17311yescharacteristic
212C4⋊Q8—1731no—
213#124—3461no—
214#124—1731no—
215#123—3461no—
216#123—3461no—
217C22⋊Q8—3461no—
218C22⋊Q8—3461no—
219C22⋊Q8—1731no—
220#121—3461no—
221#121—1731no—
222#121C2×D17311yescharacteristic
223#121C2×D17311yescharacteristic
224C4×Q8—1731no—
225C4×D4—3461no—
226C4×D4—1731no—
227C4×D4—1731no—
228C2×C4⋊C4—1731no—
229C2×C4⋊C4—1731no—
230#107—1731no—
231#107—1731no—
232#107C2×D17311yescharacteristic
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#102C2×D17311yescharacteristic
243#102C2×D17311yescharacteristic
244C22⋊C8—1731no—
245C22⋊C8—1731no—
246#97—1731no—
247#41—3462no—
248#41—6921no—
249#41—6924no—
250C2×Q8—3461no—
251C2×Q8—3462no—
252C2×Q8—3461no—
253C2×Q8—1731no—
254C2×Q8—1731no—
255C2×Q8—6921no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—1731no—
261C2×D4C22×D17311yescharacteristic
262C2×D4C22×D17311yescharacteristic
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—1731no—
269C22×C4C22×D17311yescharacteristic
270SD16—3462no—
271SD16—6921no—
272D8—41no—
273M16—6921no—
274C2×C8—41no—
275C2×C8—1731no—
276C2×C8—1731no—
277C2×C8C22×D17311yescharacteristic
278C2×C8C22×D17311yescharacteristic
279C4⋊C4—3461no—
280C4⋊C4—3461no—
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—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—1731no—
297C22⋊C4—1731no—
298C42—3461no—
299C42—1731no—
300C23—21no—
301C23—21no—
302C23—1731no—
303C23D4×D17311yescharacteristic
304Q8—3461no—
305Q8—3461no—
306Q8—3461no—
307Q8—1732no—
308Q8—6922no—
309Q8—6922no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—42no—
316D4—3462no—
317D4—6922no—
318D4—6922no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—1731no—
330C2×C4—1731no—
331C2×C4—1731no—
332C2×C4—6922no—
333C2×C4—6922no—
334C2×C4—6922no—
335C2×C4—6922no—
336C2×C4—6922no—
337C2×C4D4×D17311yescharacteristic
338C2×C4C23×D17311yescharacteristic
339C8—21no—
340C8—21no—
341C8—3461no—
342C8—3461no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—3461no—
352C22—3461no—
353C22—3461no—
354C22#1914970312yes—
355C22C2×D4×D17311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—3461no—
360C4—3461no—
361C4—3461no—
362C4—3462no—
363C4—6921no—
364C4—6921no—
365C4—6921no—
366C4#41×D17311yescharacteristic
367C4#1914972411yescharacteristic
368C2—22no—
369C2—81no—
370C2—81no—
371C2—6921no—
372C2#2601213011yescharacteristic
373C2C2×#1914970311yescharacteristic
374C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#466×C173C2
socleC22×C173C22×D4
hypercenter#466D173
hypocenterC173#3308
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #26012147C2semidirect
2. #26012141C2semidirect
3. #26011849C2semidirect
4. #26011828C2semidirect
5. #26011802C2semidirect
6. #26011785C2semidirect
7. #26011736C2semidirect
8. #26011704C2semidirect
9. #26011633C2semidirect
10. #26011629C2semidirect
11. #26011588C2semidirect
12. #26011562C2semidirect
13. #26011489C2semidirect
14. C2×#19149604C2semidirect
15. #466×C173C2semidirect
118. Dic346#41semidirect
187. C173#3308semidirect
196. #466D173semidirect

Maximal subgroups

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

subgroupindex
#260121472
#260121412
#260118492
#260118282
#260118022
#260117852
#260117362
#260117042
#260116332
#260116292
#260115882
#260115622
#260114892
C2×#191496042
#466×C1732
#3308173