Finite Group Databasebeta

← #36364253

Subgroups — #36364253

Order 22144.

374 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#26012115C211yesmaximal, characteristic, complemented
2#26012095C211yesmaximal, characteristic, complemented
3#26011897C211yesmaximal, characteristic, complemented
4#26011878C211yesmaximal, characteristic, complemented
5#26011846C211yesmaximal, characteristic, complemented
6#26011743C211yesmaximal, characteristic, complemented
7#26011736C211yesmaximal, characteristic, complemented
8#26011726C211yesmaximal, characteristic, complemented
9#26011719C211yesmaximal, characteristic, complemented
10#26011600C211yesmaximal, characteristic, complemented
11#26011588C211yesmaximal, characteristic, complemented
12#26011582C211yesmaximal, characteristic, complemented
13#26011571C211yesmaximal, characteristic, complemented
14#26011445C211yesmaximal, characteristic, complemented
15#486×C173C211yesmaximal, characteristic, complemented
16C2×#14583814C2211yescharacteristic
17C2×#14583814C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149674—21no—
23#19149671C2211yescharacteristic
24#19149668—21no—
25#19149667C2211yescharacteristic
26#19149664—21no—
27#19149660—21no—
28#19149660—21no—
29#19149659C2211yescharacteristic
30#19149655—21no—
31#19149654C2211yescharacteristic
32#19149651—21no—
33#19149651—21no—
34#19149650C2211yescharacteristic
35#19149650C2211yescharacteristic
36#19149644C2211yescharacteristic
37D4×Dic173C2211yescharacteristic
38D4×Dic173C2211yescharacteristic
39#19149641C2211yescharacteristic
40#19149636C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149612C2211yescharacteristic
43#19149610C2211yescharacteristic
44#19149610C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149590C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C4⋊Q8×C173C2211yescharacteristic
53#127×C173C2211yescharacteristic
54#102×C173C2211yescharacteristic
55#102×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59C2×Dic346C2311yescharacteristic
60C2×D692—21no—
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C22×Dic173—21no—
65C22×Dic173—21no—
66C2×C4×D173—21no—
67C2×C4×D173C2311yescharacteristic
68#14583814—22no—
69#14583814—22no—
70#14583811—22no—
71#14583811—22no—
72#14583809—22no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C8×D173—22no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805D411yescharacteristic
83#14583805D411yescharacteristic
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173—21no—
98C2×D4×C173—21no—
99C2×D4×C173—21no—
100C2×D4×C173C2311yescharacteristic
101C2×D4×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C4⋊C4×C173—21no—
105C4⋊C4×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173C2×D411yescharacteristic
110Dic346—21no—
111Dic346—21no—
112D692—21no—
113D692—21no—
114D692—81no—
115C22⋊D173—81no—
116C22⋊D173—81no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—81no—
121C2×Dic173—81no—
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C4×D173—21no—
126C4×D173—21no—
127C4×D173—81no—
128#13063436#4112yes—
129#13063436#4112yes—
130C23×C173—21no—
131C23×C173—21no—
132Q8×C173—81no—
133D4×C173—21no—
134D4×C173—21no—
135D4×C173—21no—
136D4×C173—21no—
137D4×C173—81no—
138D4×C173—81no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2411yescharacteristic
144C8×C173—21no—
145C8×C173—21no—
146C22×C173—41no—
147C22×C173—41no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173C22×D411yescharacteristic
153C2×D173—21no—
154C2×D173—21no—
155C2×D173—21no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173C2×#4111yescharacteristic
164C4×C173C2×#4111yescharacteristic
165Dic173—21no—
166Dic173—21no—
167Dic173—21no—
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171C2×C173—81no—
172C2×C173—81no—
173C2×C173#53211yescharacteristic
174C2×C173#57611yescharacteristic
175C2×C173#57711yescharacteristic
176D173—41no—
177D173—81no—
178C173#317911yescharacteristic, complemented, Sylow
179#3179—1731nomaximal, Sylow
180#542—1731no—
181#540—1731no—
182#501—1731no—
183#496—1731no—
184#488—1731no—
185#486D17311yescharacteristic, complemented
186#469—1731no—
187#468—1731no—
188#466—1731no—
189#465—1731no—
190#442—1731no—
191#440—1731no—
192C4×SD16—1731no—
193C4×D8—1731no—
194#405—1731no—
195C2×SD16—1731no—
196C2×SD16—1731no—
197C2×D8—1731no—
198C2×D8—1731no—
199C2×M16—1731no—
200C22×C8—1731no—
201C4⋊Q8C2×D17311yescharacteristic
202#127C2×D17311yescharacteristic
203#126—3461no—
204#125—1731no—
205#124—3461no—
206#124—1731no—
207#123—3461no—
208#123—3461no—
209#123—3461no—
210C22⋊Q8—3461no—
211C22⋊Q8—1731no—
212#121—3461no—
213#121—3461no—
214#121—1731no—
215#121—1731no—
216#121—1731no—
217C4×Q8—1731no—
218C4×D4—1731no—
219C4×D4—1731no—
220C4×D4—1731no—
221#117—1731no—
222#107—1731no—
223#107—1731no—
224#106—1731no—
225#106—1731no—
226#103—1731no—
227#103—1731no—
228#102—1731no—
229#102—1731no—
230#102C2×D17311yescharacteristic
231#102C2×D17311yescharacteristic
232#102C2×D17311yescharacteristic
233#102C2×D17311yescharacteristic
234#97—1731no—
235#97—1731no—
236C4×C8—1731no—
237C4×C8C2×D17311yescharacteristic
238C2×Q8—21no—
239C2×Q8—1731no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—3461no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—1731no—
246C2×D4C22×D17311yescharacteristic
247C2×D4C22×D17311yescharacteristic
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4—1731no—
252SD16—3462no—
253SD16—3462no—
254D8—3462no—
255D8—3462no—
256M16—3462no—
257C2×C8—3462no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8C22×D17311yescharacteristic
261C2×C8C22×D17311yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—3461no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—1731no—
268C4⋊C4—1731no—
269C4⋊C4—1731no—
270C4⋊C4—1731no—
271C4⋊C4C22×D17311yescharacteristic
272C4⋊C4C22×D17311yescharacteristic
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—1731no—
283C22⋊C4—1731no—
284C42—3461no—
285C42—1731no—
286C42C22×D17311yescharacteristic
287C23—21no—
288C23—21no—
289C23—3461no—
290C23—1731no—
291Q8—81no—
292Q8—3461no—
293Q8—3461no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—81no—
299D4—81no—
300D4—3461no—
301D4—3461no—
302D4—13841no—
303D4—13841no—
304D4—13841no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—13841no—
313C2×C4—13841no—
314C2×C4—13841no—
315C2×C4—1731no—
316C2×C4—1731no—
317C2×C4—1731no—
318C2×C4D4×D17311yescharacteristic
319C2×C4D4×D17311yescharacteristic
320C2×C4C23×D17311yescharacteristic
321C8—21no—
322C8—21no—
323C8—1732no—
324C8—1732no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—3461no—
332C22—3461no—
333C22—3461no—
334C22—6921no—
335C22—6921no—
336C22—6921no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—3461no—
343C4—3461no—
344C4—3461no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4#41×D17311yescharacteristic
349C4#41×D17311yescharacteristic
350C2—81no—
351C2—81no—
352C2—13841no—
353C2—6921no—
354C2#2601204011yescharacteristic
355C2#2601229211yescharacteristic
356C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#486×C173C2
socleC22×C173C22×D4
hypercenter#486D173
hypocenterC173#3179
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #26012115C2semidirect
2. #26012095C2semidirect
3. #26011897C2semidirect
4. #26011878C2semidirect
5. #26011846C2semidirect
6. #26011743C2semidirect
7. #26011736C2semidirect
8. #26011726C2semidirect
9. #26011719C2semidirect
10. #26011600C2semidirect
11. #26011588C2semidirect
12. #26011582C2semidirect
13. #26011571C2semidirect
14. #26011445C2semidirect
15. #486×C173C2semidirect
178. C173#3179semidirect
185. #486D173semidirect

Maximal subgroups

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

subgroupindex
#260121152
#260120952
#260118972
#260118782
#260118462
#260117432
#260117362
#260117262
#260117192
#260116002
#260115882
#260115822
#260115712
#260114452
#486×C1732
#3179173