Finite Group Databasebeta

← #36363756

Subgroups — #36363756

Order 22144.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26011971C211yesmaximal, characteristic, complemented
2#26011807C211yesmaximal, characteristic, complemented
3#26011802C211yesmaximal, characteristic, complemented
4#26011795C211yesmaximal, characteristic, complemented
5#26011788C211yesmaximal, characteristic, complemented
6#26011688C211yesmaximal, characteristic, complemented
7#26011683C211yesmaximal, characteristic, complemented
8#26011634C211yesmaximal, characteristic, complemented
9#26011634C211yesmaximal, characteristic, complemented
10#26011601C211yesmaximal, characteristic, complemented
11#26011591C211yesmaximal, characteristic, complemented
12C4×#14583814C211yesmaximal, characteristic, complemented
13C4×#14583811C211yesmaximal, characteristic, complemented
14#26011552C211yesmaximal, characteristic, complemented
15#540×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583814C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C4×Dic346C2211yescharacteristic
22C4×Dic346C2211yescharacteristic
23C4×D692C2211yescharacteristic
24C4×D692C2211yescharacteristic
25#19149639—21no—
26#19149639—21no—
27#19149637—21no—
28C2×C4⋊Dic173C2211yescharacteristic
29C4⋊C4×D173—21no—
30#19149612C2211yescharacteristic
31#19149612C2211yescharacteristic
32#19149610C2211yescharacteristic
33#19149610C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35C4⋊#13063436C2211yescharacteristic
36#19149602C2211yescharacteristic
37#19149602C2211yescharacteristic
38#19149600C2211yescharacteristic
39#19149600C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46#19149590C2211yescharacteristic
47C4×#13063436C2211yescharacteristic
48#127×C173C2211yescharacteristic
49#125×C173C2211yescharacteristic
50#123×C173—21no—
51#123×C173—21no—
52#121×C173—21no—
53#121×C173—21no—
54#121×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56C4×D4×C173C2211yescharacteristic
57C4×D4×C173C2211yescharacteristic
58#117×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—22no—
61C2×Dic346C2311yescharacteristic
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C22×Dic173—21no—
65C2×C4×D173—21no—
66C2×C4×D173—21no—
67C2×C4×D173—21no—
68#14583814—22no—
69#14583814—22no—
70#14583811—22no—
71#14583811—22no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583803—21no—
82#14583803—21no—
83C4×Dic173—21no—
84C4×Dic173—21no—
85C2×D4×C173—21no—
86C2×D4×C173—21no—
87C2×D4×C173—21no—
88C2×D4×C173—21no—
89C2×D4×C173C2311yescharacteristic
90C2×D4×C173C2311yescharacteristic
91C22×C4×C173—21no—
92C22×C4×C173—21no—
93C22×C4×C173C2311yescharacteristic
94C4⋊C4×C173—21no—
95C4⋊C4×C173—21no—
96C4⋊C4×C173D411yescharacteristic
97C4⋊C4×C173D411yescharacteristic
98C4⋊C4×C173C2311yescharacteristic
99C4⋊C4×C173C2311yescharacteristic
100C22⋊C4×C173—21no—
101C22⋊C4×C173—21no—
102C22⋊C4×C173—21no—
103C22⋊C4×C173—21no—
104C22⋊C4×C173D411yescharacteristic
105C22⋊C4×C173D411yescharacteristic
106C42×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109Dic346—21no—
110Dic346#4112yes—
111D692—21no—
112D692#4112yes—
113C22⋊D173—24no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—81no—
118C4×D173—24no—
119C4×D173—81no—
120C4×D173—81no—
121#13063436—21no—
122#13063436—21no—
123#13063436—21no—
124#13063436—21no—
125C23×C173—21no—
126C23×C173—21no—
127C23×C173C2×D411yescharacteristic
128D4×C173—21no—
129D4×C173—21no—
130D4×C173—21no—
131D4×C173—21no—
132D4×C173—81no—
133D4×C173—81no—
134D4×C173—81no—
135D4×C173—81no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—81no—
141C2×C4×C173—81no—
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2411yescharacteristic
148C22×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173C22×D411yescharacteristic
158C2×D173—22no—
159C2×D173—41no—
160C2×D173—41no—
161C4×C173—21no—
162C4×C173—21no—
163C4×C173—21no—
164C4×C173—21no—
165C4×C173—21no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173C2×#4111yescharacteristic
169C4×C173C2×#4111yescharacteristic
170Dic173—22no—
171Dic173—41no—
172Dic173—41no—
173C2×C173—41no—
174C2×C173—81no—
175C2×C173—81no—
176C2×C173#52511yescharacteristic
177C2×C173#57611yescharacteristic
178C2×C173#57711yescharacteristic
179D173—42no—
180C173#311911yescharacteristic, complemented, Sylow
181#3119—1731nomaximal, Sylow
182#540D17311yescharacteristic, complemented
183#520—1731no—
184#481—1731no—
185#480—1731no—
186#479—1731no—
187#478—1731no—
188#460—1731no—
189#459—1731no—
190#449—1731no—
191#449—1731no—
192#442—1731no—
193#440—1731no—
194C4×SD16—1731no—
195C4×D8—1731no—
196#433—1731no—
197C2×#41—1731no—
198C2×SD16—1731no—
199C2×SD16—1731no—
200C2×D8—1731no—
201C2×D8—1731no—
202#127C2×D17311yescharacteristic
203#125C2×D17311yescharacteristic
204#123—21no—
205#123—21no—
206#121—21no—
207#121—21no—
208#121C2×D17311yescharacteristic
209#121C2×D17311yescharacteristic
210C4×Q8—1731no—
211C4×Q8—1731no—
212C4×D4—3461no—
213C4×D4—3461no—
214C4×D4—1731no—
215C4×D4—1731no—
216C4×D4C2×D17311yescharacteristic
217C4×D4C2×D17311yescharacteristic
218#117—3461no—
219#117C2×D17311yescharacteristic
220C2×C4⋊C4—3461no—
221C2×C4⋊C4—1731no—
222#107—1731no—
223#107—1731no—
224#106—1731no—
225#106—1731no—
226C4⋊C8—1731no—
227C4⋊C8—1731no—
228#103—1731no—
229#103—1731no—
230#102—1731no—
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235#102—1731no—
236C22⋊C8—1731no—
237C22⋊C8—1731no—
238#97—1731no—
239C4×C8—1731no—
240#41—3462no—
241#41—3462no—
242C2×Q8—1731no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4C22×D17311yescharacteristic
250C2×D4C22×D17311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—3461no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4—3461no—
257C22×C4C22×D17311yescharacteristic
258SD16—3462no—
259SD16—3462no—
260D8—3462no—
261D8—3462no—
262C2×C8—1731no—
263C2×C8—1731no—
264C2×C8—1731no—
265C2×C8—1731no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4C22⋊D17311yescharacteristic
274C4⋊C4C22⋊D17311yescharacteristic
275C4⋊C4C22×D17311yescharacteristic
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4C22⋊D17311yescharacteristic
284C22⋊C4C22⋊D17311yescharacteristic
285C42—3461no—
286C42—3461no—
287C42C22×D17311yescharacteristic
288C42C22×D17311yescharacteristic
289C23—21no—
290C23—21no—
291C23—3461no—
292C23C2×C22⋊D17311yescharacteristic
293Q8—3461no—
294Q8—1732no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—81no—
303D4—3464no—
304D4—3461no—
305D4—1732no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—3464no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—13841no—
317C2×C4—13841no—
318C2×C4—13841no—
319C2×C4C2×C22⋊D17311yescharacteristic
320C2×C4C2×C22⋊D17311yescharacteristic
321C2×C4C2×C22⋊D17311yescharacteristic
322C2×C4C2×C22⋊D17311yescharacteristic
323C2×C4C2×C22⋊D17311yescharacteristic
324C2×C4C23×D17311yescharacteristic
325C8—3461no—
326C8—3461no—
327C8—3461no—
328C8—3461no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—3462no—
339C22—6921no—
340C22—6921no—
341C22C22×C22⋊D17311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—3462no—
350C4—6921no—
351C4—6921no—
352C4#41×D17311yescharacteristic
353C4#41×D17311yescharacteristic
354C2—41no—
355C2—81no—
356C2—81no—
357C2—6922no—
358C2#2601199811yescharacteristic
359C2#2601229411yescharacteristic
360C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#540×C173C2
socleC22×C173C22×D4
hypercenter#540D173
hypocenterC173#3119
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #26011971C2semidirect
2. #26011807C2semidirect
3. #26011802C2semidirect
4. #26011795C2semidirect
5. #26011788C2semidirect
6. #26011688C2semidirect
7. #26011683C2semidirect
8. #26011634C2semidirect
9. #26011634C2semidirect
10. #26011601C2semidirect
11. #26011591C2semidirect
12. C4×#14583814C2semidirect
13. C4×#14583811C2semidirect
14. #26011552C2semidirect
15. #540×C173C2semidirect
180. C173#3119semidirect
182. #540D173semidirect

Maximal subgroups

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

subgroupindex
#260119712
#260118072
#260118022
#260117952
#260117882
#260116882
#260116832
#260116342
#260116012
#260115912
C4×#145838142
C4×#145838112
#260115522
#540×C1732
#3119173