Finite Group Databasebeta

← #36364151

Subgroups — #36364151

Order 22144.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012036C211yesmaximal, characteristic, complemented
2#26012027C211yesmaximal, characteristic, complemented
3#26011899C211yesmaximal, characteristic, complemented
4#26011877C211yesmaximal, characteristic, complemented
5#26011854C211yesmaximal, characteristic, complemented
6#26011852C211yesmaximal, characteristic, complemented
7#26011744C211yesmaximal, characteristic, complemented
8#26011742C211yesmaximal, characteristic, complemented
9#26011739C211yesmaximal, characteristic, complemented
10SD16×Dic173C211yesmaximal, characteristic, complemented
11#26011576C211yesmaximal, characteristic, complemented
12#26011573C211yesmaximal, characteristic, complemented
13#26011571C211yesmaximal, characteristic, complemented
14M16×Dic173C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—21no—
17C2×#14583813C2211yescharacteristic
18C2×D1384C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583810C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149675C2211yescharacteristic
23#19149674—21no—
24#19149670C2211yescharacteristic
25#19149668C2211yescharacteristic
26#19149665—21no—
27#19149663—21no—
28#19149653C2211yescharacteristic
29#19149653C2211yescharacteristic
30#19149651—21no—
31Q8×Dic173C2211yescharacteristic
32D4×Dic173C2211yescharacteristic
33#19149641C2211yescharacteristic
34#19149641C2211yescharacteristic
35#19149639—21no—
36#19149639—21no—
37#19149637—21no—
38C2×C4×Dic173C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149603C2211yescharacteristic
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149589C2211yescharacteristic
49#19149589C2211yescharacteristic
50C8×Dic173C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—44no—
60C2×Dic346—21no—
61C2×D692C2311yescharacteristic
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C22×Dic173—21no—
68C22×Dic173D411yescharacteristic
69C22×Dic173D411yescharacteristic
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72#14583813—22no—
73D1384—22no—
74#14583811—22no—
75#14583810—22no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C4⋊Dic173—21no—
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84C22⋊Dic173—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91C4×Dic173—21no—
92C4×Dic173D411yescharacteristic
93C4×Dic173D411yescharacteristic
94C4×Dic173C2311yescharacteristic
95C4×Dic173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173—21no—
98C2×D4×C173C2311yescharacteristic
99C22×C4×C173C2311yescharacteristic
100SD16×C173—22no—
101M16×C173—22no—
102C2×C8×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C4⋊C4×C173—21no—
105C4⋊C4×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C22⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C22×D173—21no—
110C22×D173—21no—
111Dic346—42no—
112D692—21no—
113D692—21no—
114D692—21no—
115D692—21no—
116C22⋊D173—42no—
117C22⋊D173—42no—
118C22⋊D173—42no—
119C22⋊D173—42no—
120C2×Dic173—22no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—41no—
124C2×Dic173—42no—
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—42no—
130C4×D173—42no—
131#13063436—21no—
132#13063436#4112yescomplemented
133C23×C173—21no—
134C23×C173C2×D411yescharacteristic
135Q8×C173—21no—
136Q8×C173—21no—
137D4×C173—21no—
138D4×C173—21no—
139D4×C173—42no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173#4112yescomplemented
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173C22×D411yescharacteristic
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C2×D173—41no—
160C2×D173—41no—
161C2×D173—41no—
162C4×C173—21no—
163C4×C173—41no—
164C4×C173—41no—
165C4×C173—41no—
166C4×C173C2×#4111yescharacteristic
167C4×C173C2×#4111yescharacteristic
168Dic173—21no—
169Dic173—21no—
170Dic173—21no—
171Dic173—41no—
172Dic173—41no—
173Dic173AGL(1,ℤ/8)12yes—
174C2×C173—41no—
175C2×C173—81no—
176C2×C173#53211yescharacteristic
177C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
178C2×C173#57511yescharacteristic
179D173—81no—
180D173—81no—
181C173#317111yescharacteristic, complemented, Sylow
182#3171—1731nomaximal, Sylow
183#532—1731no—
184#529—1731no—
185#501—1731no—
186#496—1731no—
187#490—1731no—
188#489—1731no—
189#469—1731no—
190#469—1731no—
191#468—1731no—
192#468D17311yescharacteristic, complemented
193C4×SD16—1731no—
194C4×SD16—1731no—
195C4×D8—1731no—
196C4×D8—1731no—
197C4×M16—1731no—
198C2×#41—3461no—
199C2×SD16—1731no—
200C2×SD16C2×D17311yescharacteristic
201C2×D8—1731no—
202C2×D8—1731no—
203C2×M16—1731no—
204C2×M16C2×D17311yescharacteristic
205C4⋊Q8—1731no—
206#127—1731no—
207#126—3461no—
208#125—1731no—
209#124—3461no—
210#124—1731no—
211#123—3461no—
212C22⋊Q8C2×D17311yescharacteristic
213#121—3461no—
214#121—1731no—
215#121—1731no—
216#121C2×D17311yescharacteristic
217C4×Q8—1731no—
218C4×D4—3461no—
219C4×D4—3461no—
220C4×D4—1731no—
221C4×D4—1731no—
222C4×D4—1731no—
223#117—3461no—
224C2×C42—1731no—
225#107—1731no—
226#107C2×D17311yescharacteristic
227#106—1731no—
228#106—1731no—
229#103—1731no—
230#103C2×D17311yescharacteristic
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235#102—1731no—
236#102C2×D17311yescharacteristic
237#97—1731no—
238#97—1731no—
239C4×C8—1731no—
240C4×C8—1731no—
241#41—6924no—
242C2×Q8—3461no—
243C2×Q8C22×D17311yescharacteristic
244C2×D4—21no—
245C2×D4—3461no—
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—3461no—
249C2×D4—1731no—
250C2×D4—1731no—
251C2×D4C22×D17311yescharacteristic
252C22×C4—3461no—
253C22×C4—3461no—
254C22×C4—3461no—
255C22×C4—1731no—
256C22×C4—1731no—
257C22×C4C22×D17311yescharacteristic
258SD16—22no—
259SD16—3462no—
260D8—3462no—
261D8—3462no—
262M16—22no—
263M16—3462no—
264C2×C8—1731no—
265C2×C8—1731no—
266C2×C8C22×D17311yescharacteristic
267C2×C8C22×D17311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—3461no—
272C4⋊C4—3461no—
273C4⋊C4—1731no—
274C4⋊C4—1731no—
275C4⋊C4C22×D17311yescharacteristic
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C42—3461no—
287C42—1731no—
288C42—1731no—
289C42—1731no—
290C42—1731no—
291C23—21no—
292C23—3461no—
293C23—3461no—
294C23D4×D17311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—6922no—
298D4—21no—
299D4—21no—
300D4—42no—
301D4—3461no—
302D4—3461no—
303D4—3461no—
304D4—3461no—
305D4—6922no—
306D4—6922no—
307D4—6922no—
308D4—6922no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—3462no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—1731no—
318C2×C4—1731no—
319C2×C4—1731no—
320C2×C4—1731no—
321C2×C4—6922no—
322C2×C4—6922no—
323C2×C4—6921no—
324C2×C4—6922no—
325C2×C4D4×D17311yescharacteristic
326C2×C4C23×D17311yescharacteristic
327C8—21no—
328C8—3461no—
329C8—1732no—
330C8#1458382512yescomplemented
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—6921no—
338C22—6921no—
339C22—6921no—
340C22—6921no—
341C22—6921no—
342C22—6921no—
343C22C2×D4×D17311yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—3461no—
349C4—3461no—
350C4—3461no—
351C4—1732no—
352C4—6921no—
353C4—6921no—
354C4#41×D17311yescharacteristic
355C4C2×#1458382511yescharacteristic
356C2—41no—
357C2—81no—
358C2—13841no—
359C2—13841no—
360C2#2601203611yescharacteristic
361C2AGL(1,ℤ/8)×D17311yescharacteristic
362C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
hypercenter#468D173
hypocenterC173#3171
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #26012036C2semidirect
2. #26012027C2semidirect
3. #26011899C2semidirect
4. #26011877C2semidirect
5. #26011854C2semidirect
6. #26011852C2semidirect
7. #26011744C2semidirect
8. #26011742C2semidirect
9. #26011739C2semidirect
10. SD16×Dic173C2semidirect
11. #26011576C2semidirect
12. #26011573C2semidirect
13. #26011571C2semidirect
14. M16×Dic173C2semidirect
15. #468×C173C2semidirect
132. #13063436#41semidirect
148. C8×C173#41semidirect
181. C173#3171semidirect
192. #468D173semidirect
330. C8#14583825semidirect

Maximal subgroups

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

subgroupindex
#260120362
#260120272
#260118992
#260118772
#260118542
#260118522
#260117442
#260117422
#260117392
SD16×Dic1732
#260115762
#260115732
#260115712
M16×Dic1732
#468×C1732
#3171173