Finite Group Databasebeta

← #36362403

Subgroups — #36362403

Order 22144.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149708C211yesmaximal, characteristic
2#26011970C211yesmaximal, characteristic
3#26011967C211yesmaximal, characteristic, complemented
4#26011599C211yesmaximal, characteristic, complemented
5#26011595C211yesmaximal, characteristic, complemented
6#26011588C211yesmaximal, characteristic, complemented
7#26011585C211yesmaximal, characteristic, complemented
8C4×#14583816C211yesmaximal, characteristic, complemented
9SD16×Dic173C211yesmaximal, characteristic, complemented
10C4×#14583814C211yesmaximal, characteristic, complemented
11#26011532C211yesmaximal, characteristic, complemented
12#26011484C211yesmaximal, characteristic
13#103×D173C211yesmaximal, characteristic, complemented
14#26011448C211yesmaximal, characteristic, complemented
15#440×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18#19149708C418yescomplemented
19C2×Dic692C2211yescharacteristic
20C2×#14583816C2211yescharacteristic
21C2×#14583814C2211yescharacteristic
22C2×#14583809C2211yescharacteristic
23C2×C8×D173C2211yescharacteristic
24Q8×Dic173C2211yescharacteristic
25C4×Dic346—21no—
26C4×Dic346C2211yescharacteristic
27C4×Dic346C2211yescharacteristic
28#19149644—21no—
29#19149644C2211yescharacteristic
30#19149644C2211yescharacteristic
31D4×Dic173C2211yescharacteristic
32C4×C22⋊D173—21no—
33#19149639—21no—
34#19149637—21no—
35#19149637C2211yescharacteristic
36#19149636—21no—
37#19149636C2211yescharacteristic
38#19149635—21no—
39C2×#14583805—21no—
40C4⋊C4×D173C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149611C2211yescharacteristic
43#19149610C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149589C2211yescharacteristic
51C4×#13063436C2211yescharacteristic
52C8×Dic173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#97×C173C2211yescharacteristic
60#14583825—22no—
61#14583825C2×C414yes—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65Q8×D173—22no—
66Q8×D173C2×C414yescomplemented
67C2×C22⋊D173—21no—
68C22×Dic173—21no—
69C22×Dic173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73Dic692C2×C414yescomplemented
74#14583816C2×C414yescomplemented
75#14583814C2×C414yescomplemented
76#14583809C2×C414yescomplemented
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C8×D173—22no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805D411yescharacteristic
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91#14583803—21no—
92#14583803D411yescharacteristic
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173—21no—
98C4×Dic173—21no—
99C4×Dic173—21no—
100C4×Dic173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173—21no—
104SD16×C173C2×C414yescomplemented
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C42×C173—21no—
112C42×C173C2311yescharacteristic
113C22×D173C2×D411yescharacteristic
114Dic346—21no—
115Dic346—21no—
116Dic346—24no—
117Dic346C22×C412yes—
118Dic346C22×C412yes—
119C22⋊D173—24no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—24no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—81no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—24no—
130C4×D173—81no—
131C4×D173C22×C412yes—
132C4×D173#4112yes—
133#13063436—21no—
134#13063436C22×C412yes—
135C23×C173—21no—
136Q8×C173—21no—
137Q8×C173C22×C412yes—
138D4×C173—21no—
139D4×C173C22×C412yes—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—81no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173C22×C412yes—
149C22×C173—22no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173C22×D411yescharacteristic
153C2×D173—21no—
154C2×D173—21no—
155C2×D173C4×D412yes—
156C4×C173—22no—
157C4×C173—21no—
158C4×C173—21no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173C23×C411yescharacteristic
162C4×C173C2×#4111yescharacteristic
163Dic173—22no—
164Dic173—22no—
165Dic173—21no—
166Dic173—21no—
167Dic173—41no—
168Dic173—41no—
169Dic173C4×D412yes—
170C2×C173—42no—
171C2×C173C2×C4×D411yescharacteristic
172C2×C173#57711yescharacteristic
173C2×C173#57811yescharacteristic
174D173—22no—
175C173#294711yescharacteristic, complemented, Sylow
176#2947—1731nomaximal, Sylow
177C2×#137—1731no—
178#520—1731no—
179#519—1731no—
180#441—1731no—
181#441—1731no—
182#440—1731no—
183#440D17311yescharacteristic, complemented
184C4×Q16—1731no—
185C4×Q16—1731no—
186C4×SD16—1731no—
187C4×SD16—1731no—
188#428—1731no—
189#416—1731no—
190C2×#103—1731no—
191#405—1731no—
192C2×#41—1731no—
193C22×Q8—1731no—
194#137—1738no—
195C2×Q16—1731no—
196C2×Q16—1731no—
197C2×SD16—1731no—
198C2×SD16C2×D17311yescharacteristic
199C2×M16—1731no—
200C22×C8—1731no—
201C4×Q8—3461no—
202C4×Q8—3461no—
203C4×Q8—1731no—
204C4×Q8—1731no—
205C4×Q8—1731no—
206C4×Q8—1731no—
207C4×Q8—1731no—
208C4×Q8C2×D17311yescharacteristic
209C4×D4—3461no—
210C4×D4—3461no—
211C4×D4—1731no—
212C4×D4C2×D17311yescharacteristic
213#117—3461no—
214#117—3461no—
215#117—3461no—
216#117—1731no—
217#117—1731no—
218C2×C4⋊C4—3461no—
219C2×C4⋊C4—1731no—
220#107—1731no—
221#107C2×D17311yescharacteristic
222#106—1731no—
223#106—1731no—
224#103—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103C2×D17311yescharacteristic
230#102—1731no—
231#102C2×D17311yescharacteristic
232#97—1731no—
233#97C2×D17311yescharacteristic
234C4×C8—1731no—
235C4×C8—1731no—
236#41—3462no—
237#41—1734no—
238C2×Q8—3461no—
239C2×Q8—3462no—
240C2×Q8—1734no—
241C2×Q8—1731no—
242C2×Q8—1731no—
243C2×Q8C22×D17311yescharacteristic
244C2×D4—3461no—
245C2×D4C22×D17311yescharacteristic
246C22×C4—21no—
247C22×C4—3461no—
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4—1731no—
252Q16—1734no—
253Q16—1734no—
254SD16—1734no—
255SD16C4×D17314yescomplemented
256M16—1734no—
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—3461no—
268C4⋊C4—3461no—
269C4⋊C4—1731no—
270C4⋊C4—1731no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4C22×D17311yescharacteristic
274C4⋊C4C22×D17311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—1731no—
279C22⋊C4—1731no—
280C42—21no—
281C42—3461no—
282C42—3461no—
283C42—3461no—
284C42—3461no—
285C42—3461no—
286C42—3461no—
287C42—1731no—
288C42C22×D17311yescharacteristic
289C23—21no—
290C23—1731no—
291Q8—21no—
292Q8—3464no—
293Q8—3461no—
294Q8—3461no—
295Q8—1732no—
296Q8—1732no—
297Q8C2×C4×D17312yes—
298D4—21no—
299D4—3464no—
300D4C2×C4×D17312yes—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—81no—
305C2×C4—3464no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—3464no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—13841no—
312C2×C4—13841no—
313C2×C4—1731no—
314C2×C4—1731no—
315C2×C4—1731no—
316C2×C4—1732no—
317C2×C4—1732no—
318C2×C4D4×D17311yescharacteristic
319C2×C4D4×D17311yescharacteristic
320C2×C4C23×D17311yescharacteristic
321C8—21no—
322C8—3461no—
323C8—1732no—
324C8C2×C4×D17312yes—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—3461no—
329C22—3461no—
330C22—1732no—
331C22C2×D4×D17311yescharacteristic
332C4—22no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—3462no—
338C4—3462no—
339C4—3461no—
340C4—3461no—
341C4—1732no—
342C4—6921no—
343C4—6921no—
344C4C22×C4×D17311yescharacteristic
345C4#41×D17311yescharacteristic
346C2—42no—
347C2—3462no—
348C2C4×D4×D17311yescharacteristic
349C2#2601230111yescharacteristic
350C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutatorC4×C173C23×C4
FrattiniC2×C4C23×D173
Fitting#440×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583825C22
cosocleC2×C4×C173C24
hypercenter#440D173
hypocenterC173#2947
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
3. #26011967C2semidirect
4. #26011599C2semidirect
5. #26011595C2semidirect
6. #26011588C2semidirect
7. #26011585C2semidirect
8. C4×#14583816C2semidirect
9. SD16×Dic173C2semidirect
10. C4×#14583814C2semidirect
11. #26011532C2semidirect
13. #103×D173C2semidirect
14. #26011448C2semidirect
15. #440×C173C2semidirect
18. #19149708C4semidirect
66. Q8×D173C2×C4semidirect
73. Dic692C2×C4semidirect
74. #14583816C2×C4semidirect
75. #14583814C2×C4semidirect
76. #14583809C2×C4semidirect
104. SD16×C173C2×C4semidirect
175. C173#2947semidirect
183. #440D173semidirect
255. SD16C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
#260119702
#260119672
#260115992
#260115952
#260115882
#260115852
C4×#145838162
SD16×Dic1732
C4×#145838142
#260115322
#260114842
#103×D1732
#260114482
#440×C1732
#2947173