Finite Group Databasebeta

← #36362401

Subgroups — #36362401

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×#19149711C211yesmaximal, characteristic
2#26011973C211yesmaximal, characteristic
3#26011967C211yesmaximal, characteristic, complemented
4#26011598C211yesmaximal, characteristic, complemented
5#26011596C211yesmaximal, characteristic, complemented
6#26011590C211yesmaximal, characteristic, complemented
7#26011589C211yesmaximal, characteristic, complemented
8#26011583C211yesmaximal, characteristic, complemented
9C4×#14583815C211yesmaximal, characteristic, complemented
10#26011576C211yesmaximal, characteristic, complemented
11#26011532C211yesmaximal, characteristic, complemented
12#26011486C211yesmaximal, characteristic
13#103×D173C211yesmaximal, characteristic, complemented
14#26011445C211yesmaximal, characteristic, complemented
15C4×Q16×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18#19149711C418yescomplemented
19C2×#14583816C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C2×#14583813C2211yescharacteristic
22C2×#14583809C2211yescharacteristic
23C2×C8×D173C2211yescharacteristic
24Q8×Dic173C2211yescharacteristic
25Q8×Dic173C2211yescharacteristic
26C4×Dic346—21no—
27C4×Dic346C2211yescharacteristic
28#19149644—21no—
29#19149644C2211yescharacteristic
30C4×D692—21no—
31C4×D692C2211yescharacteristic
32#19149641—21no—
33#19149641C2211yescharacteristic
34#19149637—21no—
35#19149637C2211yescharacteristic
36#19149636—21no—
37#19149636—21no—
38#19149636C2211yescharacteristic
39C4⋊C4×D173—21no—
40C4⋊C4×D173C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149611C2211yescharacteristic
43#19149610C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149590C2211yescharacteristic
51#19149589C2211yescharacteristic
52C8×Dic173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic
55C4×Q8×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59C4×C8×C173C2211yescharacteristic
60#14583827—22no—
61#14583827C2×C414yes—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64Q8×D173—22no—
65Q8×D173C2×C414yescomplemented
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74#14583816C2×C414yescomplemented
75#14583815C2×C414yescomplemented
76#14583813C2×C414yescomplemented
77#14583809C2×C414yescomplemented
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C8×D173—22no—
81C4⋊Dic173—21no—
82C4⋊Dic173—21no—
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805D411yescharacteristic
89#14583805D411yescharacteristic
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173—21no—
98C4×Dic173—21no—
99C4×Dic173—21no—
100C4×Dic173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103Q16×C173C2×C414yescomplemented
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C4⋊C4×C173—21no—
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C42×C173—21no—
111C42×C173—21no—
112C42×C173C2311yescharacteristic
113C22×D173—21no—
114C22×D173C2×D411yescharacteristic
115Dic346—21no—
116Dic346—24no—
117Dic346C22×C412yes—
118D692—24no—
119D692—21no—
120D692C22×C412yes—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C4×D173—24no—
128C4×D173—24no—
129C4×D173—81no—
130C4×D173—81no—
131C4×D173—81no—
132C4×D173C22×C412yes—
133C4×D173#4112yes—
134#13063436—21no—
135#13063436C22×C412yes—
136Q8×C173—21no—
137Q8×C173—21no—
138Q8×C173C22×C412yes—
139Q8×C173C22×C412yes—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173C22×C412yes—
149C22×C173C22×D411yescharacteristic
150C2×D173—21no—
151C2×D173—21no—
152C2×D173—22no—
153C2×D173—41no—
154C2×D173—41no—
155C2×D173C4×D412yes—
156C4×C173—22no—
157C4×C173—22no—
158C4×C173—21no—
159C4×C173—21no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173C23×C411yescharacteristic
163C4×C173C2×#4111yescharacteristic
164Dic173—22no—
165Dic173—21no—
166Dic173—21no—
167Dic173—41no—
168Dic173—41no—
169Dic173C4×D412yes—
170C2×C173C2×C4×D411yescharacteristic
171C2×C173#57711yescharacteristic
172C2×C173#57811yescharacteristic
173D173—22no—
174D173—42no—
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#440—1731no—
184C4×Q16—1731no—
185C4×Q16D17311yescharacteristic, complemented
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×Q16C2×D17311yescharacteristic
197C2×SD16—1731no—
198C2×SD16—1731no—
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×Q8C2×D17311yescharacteristic
208C4×Q8C2×D17311yescharacteristic
209C4×D4—3461no—
210C4×D4—3461no—
211C4×D4—1731no—
212C4×D4—1731no—
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#103C2×D17311yescharacteristic
229#103C2×D17311yescharacteristic
230#102—1731no—
231#102—1731no—
232#97—1731no—
233#97—1731no—
234C4×C8—1731no—
235C4×C8C2×D17311yescharacteristic
236#41—3462no—
237#41—1734no—
238C2×Q8—3461no—
239C2×Q8—3462no—
240C2×Q8—1734no—
241C2×Q8—1731no—
242C2×Q8C22×D17311yescharacteristic
243C2×Q8C22×D17311yescharacteristic
244C2×D4—3461no—
245C2×D4—1731no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4—1731no—
252Q16—1734no—
253Q16C4×D17314yescomplemented
254SD16—1734no—
255SD16—1734no—
256M16—1734no—
257C2×C8—3462no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8C22×D17311yescharacteristic
261C2×C8C22×D17311yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
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—3461no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—1731no—
279C22⋊C4—1731no—
280C42—21no—
281C42—21no—
282C42—3461no—
283C42—3461no—
284C42—3461no—
285C42—3461no—
286C42—3461no—
287C42—1731no—
288C42C22×D17311yescharacteristic
289C23—3461no—
290C23—1731no—
291Q8—21no—
292Q8—21no—
293Q8—3464no—
294Q8—3461no—
295Q8—1732no—
296Q8C2×C4×D17312yes—
297Q8C2×C4×D17312yes—
298D4—3464no—
299D4—3461no—
300D4—1732no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—3464no—
306C2×C4—3464no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—13841no—
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—3461no—
326C22—3461no—
327C22—3462no—
328C22—1732no—
329C22—6921no—
330C22—6921no—
331C22C2×D4×D17311yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—3462no—
339C4—3461no—
340C4—3461no—
341C4—1732no—
342C4—6921no—
343C4—6921no—
344C4C22×C4×D17311yescharacteristic
345C4#41×D17311yescharacteristic
346C2—3462no—
347C2—6922no—
348C2C4×D4×D17311yescharacteristic
349C2#2601229811yescharacteristic
350C2#2601230411yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutatorC4×C173C23×C4
FrattiniC2×C4C23×D173
FittingC4×Q16×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583827C22
cosocleC2×C4×C173C24
hypercenterC4×Q16D173
hypocenterC173#2947
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
3. #26011967C2semidirect
4. #26011598C2semidirect
5. #26011596C2semidirect
6. #26011590C2semidirect
7. #26011589C2semidirect
8. #26011583C2semidirect
9. C4×#14583815C2semidirect
10. #26011576C2semidirect
11. #26011532C2semidirect
13. #103×D173C2semidirect
14. #26011445C2semidirect
15. C4×Q16×C173C2semidirect
18. #19149711C4semidirect
65. Q8×D173C2×C4semidirect
74. #14583816C2×C4semidirect
75. #14583815C2×C4semidirect
76. #14583813C2×C4semidirect
77. #14583809C2×C4semidirect
103. Q16×C173C2×C4semidirect
175. C173#2947semidirect
185. C4×Q16D173semidirect
253. Q16C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497112
#260119732
#260119672
#260115982
#260115962
#260115902
#260115892
#260115832
C4×#145838152
#260115762
#260115322
#260114862
#103×D1732
#260114452
C4×Q16×C1732
#2947173