Finite Group Databasebeta

← #36364968

Subgroups — #36364968

Order 22144.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012112C211yesmaximal, characteristic, complemented
2#26011795C211yesmaximal, characteristic, complemented
3#26011788C211yesmaximal, characteristic, complemented
4#26011781C211yesmaximal, characteristic, complemented
5#26011771C211yesmaximal, characteristic, complemented
6#26011742C211yesmaximal, characteristic, complemented
7#26011735C211yesmaximal, characteristic, complemented
8#26011725C211yesmaximal, characteristic, complemented
9#26011718C211yesmaximal, characteristic, complemented
10#26011706C211yesmaximal, characteristic, complemented
11#26011706C211yesmaximal, characteristic, complemented
12#26011688C211yesmaximal, characteristic, complemented
13#26011683C211yesmaximal, characteristic, complemented
14#26011518C211yesmaximal, characteristic, complemented
15#540×C173C211yesmaximal, characteristic, complemented
16C2×#14583814C2211yescharacteristic
17C2×#14583814C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583810C2211yescharacteristic
21C22×#13063436C2211yescharacteristic
22#19149679C2211yescharacteristic
23#19149674—21no—
24#19149667C2211yescharacteristic
25#19149665—21no—
26#19149661—21no—
27#19149658C2211yescharacteristic
28#19149657—21no—
29#19149653C2211yescharacteristic
30C4×Dic346C2211yescharacteristic
31C4×D692C2211yescharacteristic
32#19149612C2211yescharacteristic
33#19149612C2211yescharacteristic
34#19149610C2211yescharacteristic
35#19149610C2211yescharacteristic
36C4⋊#13063436C2211yescharacteristic
37C4⋊#13063436C2211yescharacteristic
38C4⋊#13063436C2211yescharacteristic
39C4⋊#13063436C2211yescharacteristic
40#19149602C2211yescharacteristic
41#19149602C2211yescharacteristic
42#19149600C2211yescharacteristic
43#19149600C2211yescharacteristic
44#19149600C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic
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
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C4×D173—21no—
64#14583814—41no—
65#14583814—41no—
66#14583811—41no—
67#14583811—41no—
68#14583810—41no—
69C2×#13063436—41no—
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C4⋊Dic173—21no—
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78#14583805—21no—
79C22⋊Dic173—21no—
80#14583803—21no—
81#14583803—21no—
82#14583803—21no—
83#14583803—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—21no—
111Dic346—81no—
112D692—21no—
113D692—21no—
114C22⋊D173—81no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C4×D173—81no—
119#13063436—21no—
120#13063436—21no—
121#13063436—21no—
122#13063436—21no—
123C23×C173—21no—
124C23×C173—21no—
125C23×C173C2×D411yescharacteristic
126D4×C173—21no—
127D4×C173—21no—
128D4×C173—21no—
129D4×C173—21no—
130D4×C173—81no—
131D4×C173—81no—
132D4×C173—81no—
133D4×C173—81no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—81no—
139C2×C4×C173—81no—
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C22×C173—21no—
147C22×C173—21no—
148C22×C173—21no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173C22×D411yescharacteristic
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—21no—
162C4×C173—21no—
163C4×C173—21no—
164C4×C173—41no—
165C4×C173—41no—
166C4×C173H(5,𝔽2)11yescharacteristic
167C4×C1732−1+411yescharacteristic
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171C2×C173—41no—
172C2×C173—81no—
173C2×C173—81no—
174C2×C173#53711yescharacteristic
175C2×C173#57611yescharacteristic
176C2×C173#57711yescharacteristic
177D173—81no—
178C173#325711yescharacteristic, complemented, Sylow
179#3257—1731nomaximal, Sylow
180#542—1731no—
181#540D17311yescharacteristic, complemented
182#479—1731no—
183#478—1731no—
184#476—1731no—
185#474—1731no—
186#469—1731no—
187#468—1731no—
188#466—1731no—
189#465—1731no—
190#463—1731no—
191#463—1731no—
192#460—1731no—
193#459—1731no—
194#424—1731no—
195C2×SD16—1731no—
196C2×SD16—1731no—
197C2×D8—1731no—
198C2×D8—1731no—
199C2×M16—1731no—
200C22×C8—1731no—
201C4⋊Q8—1731no—
202#127C2×D17311yescharacteristic
203#126—3461no—
204#125C2×D17311yescharacteristic
205#124—3461no—
206#124—1731no—
207#123—21no—
208#123—21no—
209#123—3461no—
210C22⋊Q8—3461no—
211C22⋊Q8—1731no—
212#121—21no—
213#121—21no—
214#121—1731no—
215#121C2×D17311yescharacteristic
216#121C2×D17311yescharacteristic
217C4×Q8—1731no—
218C4×D4—1731no—
219C4×D4C2×D17311yescharacteristic
220C4×D4C2×D17311yescharacteristic
221#117C2×D17311yescharacteristic
222#107—1731no—
223#107—1731no—
224#106—1731no—
225#106—1731no—
226C4⋊C8—1731no—
227C4⋊C8—1731no—
228C4⋊C8—1731no—
229C4⋊C8—1731no—
230#103—1731no—
231#103—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235#102—1731no—
236#102—1731no—
237#102—1731no—
238C2×Q8—3461no—
239C2×Q8—1731no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—3461no—
245C2×D4—1731no—
246C2×D4C22×D17311yescharacteristic
247C2×D4C22×D17311yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—3461no—
251C22×C4C22×D17311yescharacteristic
252SD16—6921no—
253SD16—6921no—
254D8—6921no—
255D8—6921no—
256M16—6921no—
257C2×C8—1731no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8—1731no—
261C2×C8—6921no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—1731no—
268C4⋊C4—1731no—
269C4⋊C4C22⋊D17311yescharacteristic
270C4⋊C4C22⋊D17311yescharacteristic
271C4⋊C4C22×D17311yescharacteristic
272C4⋊C4C22×D17311yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4C22⋊D17311yescharacteristic
283C22⋊C4C22⋊D17311yescharacteristic
284C42—3461no—
285C42C22×D17311yescharacteristic
286C42C22×D17311yescharacteristic
287C23—21no—
288C23—21no—
289C23—3461no—
290C23C2×C22⋊D17311yescharacteristic
291Q8—3461no—
292Q8—3461no—
293Q8—13841no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—81no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—3461no—
303D4—3461no—
304D4—13841no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—13841no—
315C2×C4C2×C22⋊D17311yescharacteristic
316C2×C4C2×C22⋊D17311yescharacteristic
317C2×C4C2×C22⋊D17311yescharacteristic
318C2×C4C2×C22⋊D17311yescharacteristic
319C2×C4C2×C22⋊D17311yescharacteristic
320C2×C4C23×D17311yescharacteristic
321C8—3461no—
322C8—3461no—
323C8—3461no—
324C8—3461no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—6921no—
335C22—6921no—
336C22—6921no—
337C22C22×C22⋊D17311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4#1914972511yescharacteristic
349C4#1914972611yescharacteristic
350C2—41no—
351C2—81no—
352C2—81no—
353C2—13841no—
354C2#2601206911yescharacteristic
355C2#2601229411yescharacteristic
356C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#540×C173C2
socleC22×C173C22×D4
hypercenter#540D173
hypocenterC173#3257
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. #26012112C2semidirect
2. #26011795C2semidirect
3. #26011788C2semidirect
4. #26011781C2semidirect
5. #26011771C2semidirect
6. #26011742C2semidirect
7. #26011735C2semidirect
8. #26011725C2semidirect
9. #26011718C2semidirect
10. #26011706C2semidirect
11. #26011706C2semidirect
12. #26011688C2semidirect
13. #26011683C2semidirect
14. #26011518C2semidirect
15. #540×C173C2semidirect
178. C173#3257semidirect
181. #540D173semidirect

Maximal subgroups

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

subgroupindex
#260121122
#260117952
#260117882
#260117812
#260117712
#260117422
#260117352
#260117252
#260117182
#260117062
#260116882
#260116832
#260115182
#540×C1732
#3257173