Finite Group Databasebeta

← #33239324

Subgroups — #33239324

Order 18688.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989025C211yesmaximal, characteristic
2#22618805C211yesmaximal, characteristic, complemented
3#22618782C211yesmaximal, characteristic, complemented
4#22611806C212yesmaximal, complemented
5#22611794C212yesmaximal, complemented
6#22611782C212yesmaximal, complemented
7#22611777C212yesmaximal, complemented
8#22611636C211yesmaximal, characteristic, complemented
9C2×#17988387C211yesmaximal, characteristic, complemented
10#22611530C211yesmaximal, characteristic, complemented
11C73×#2159C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13C22×D584C2211yescharacteristic
14#17989068—21no—
15#17989061—21no—
16#17989025C418yescomplemented
17C2×#14556536C2212yes—
18C2×D1168C2212yes—
19C22×#11924003C2211yescharacteristic
20#17988698—22no—
21#17988686—22no—
22#17988681C2212yescomplemented
23#17988669C2212yes—
24#17988669C2212yes—
25#17988669C2212yescomplemented
26#17988591—21no—
27#17988586—21no—
28#17988542C2211yescharacteristic
29#17988390C2211yescharacteristic, complemented
30#17988390C2211yescharacteristic, complemented
31#17988387C2212yes—
32#17988387C2212yes—
33#17988387C2211yescharacteristic, complemented
34#17988387C2211yescharacteristic, complemented
35#17988362C2212yescomplemented
36C16×Dic73C2212yescomplemented
37C2×M32×C73C2211yescharacteristic
38#427×C73C2211yescharacteristic
39C2×#107×C73C2211yescharacteristic
40#367×C73C2212yescomplemented
41#366×C73C2212yescomplemented
42C2×#11924028—21no—
43C22×D292—21no—
44#14556613—22no—
45#14556613C2×C414yescomplemented
46C2×Dic292C2311yescharacteristic
47C2×#11924017—21no—
48C2×D584—21no—
49C2×D584—22no—
50C2×D584C2×C414yescomplemented
51C2×D584C2312yes—
52C2×D584C2311yescharacteristic
53C2×#11924012D411yescharacteristic
54#14556556—22no—
55C4×D292—41no—
56C4×D292—41no—
57#14556553—22no—
58#14556553—22no—
59#14556553—22no—
60C4×C22⋊D73—42no—
61#14556550D411yescharacteristic
62#14556549—41no—
63C4⋊C4×D73—41no—
64C22⋊C4×D73—42no—
65#14556536C2×C418yescomplemented
66D1168C2×C418yescomplemented
67C2×#11924003—22no—
68C2×#11924003C2311yescharacteristic
69C2×#11924003C2311yescharacteristic
70#14556514—22no—
71#14556511—22no—
72#14556511—22no—
73#14556511—22no—
74#14556501D412yes—
75C8×Dic73C2312yes—
76C22×C8×C73C2311yescharacteristic
77#117×C73—21no—
78C2×C4⋊C4×C73—21no—
79M32×C73C2×C414yescomplemented
80C2×C16×C73C2312yes—
81#107×C73—21no—
82#107×C73C2312yes—
83#107×C73C2311yescharacteristic
84#107×C73C2311yescharacteristic
85#106×C73—21no—
86C23×D73—41no—
87#11924028—24no—
88#11924028—42no—
89C2×Dic146—21no—
90C2×D292—24no—
91C2×D292—22no—
92C2×D292—21no—
93C2×D292—41no—
94C2×D292—42no—
95C2×C22⋊D73—41no—
96C2×C4×D73—41no—
97C2×C4×D73—42no—
98C2×C4×D73—42no—
99C2×C4×D73—42no—
100Dic292—21no—
101Dic292C22×C412yes—
102#11924017—24no—
103D584—24no—
104D584—22no—
105D584—21no—
106D584C22×C412yes—
107D584C22×C414yes—
108#11924012—22no—
109C2×#10596831C2×D412yes—
110C4⋊Dic73D811yescharacteristic
111C4⋊Dic73D811yescharacteristic
112#11924007—42no—
113C22⋊Dic73D812yes—
114#11924005—42no—
115#11924005—42no—
116#11924005—42no—
117C4×Dic73—42no—
118C4×Dic73C2×D412yes—
119#11924003—21no—
120#11924003—21no—
121C22×C4×C73—41no—
122C22×C4×C73C2×D411yescharacteristic
123C2×C8×C73C22×C412yes—
124C2×C8×C73C2×D411yescharacteristic
125C2×C8×C73#4112yes—
126C2×C8×C73C2411yescharacteristic
127C4⋊C4×C73—21no—
128C4⋊C4×C73—21no—
129C4⋊C4×C73—22no—
130C4⋊C4×C73—41no—
131C22⋊C4×C73—41no—
132C42×C73—41no—
133C16×C73C22×C414yes—
134C23×C73C2×D811yescharacteristic
135C22×D73—41no—
136C22×D73—42no—
137C22×D73—44no—
138C22×D73—161no—
139C22×D73—161no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—42no—
143C2×C4×C73—161no—
144C2×C4×C73C4×D412yes—
145C2×C4×C73C2×D811yescharacteristic
146C2×C4×C73#13512yes—
147C2×C4×C73C22×D411yescharacteristic
148C22⋊D73—44no—
149C2×Dic73—41no—
150C2×Dic73C2×D812yes—
151C2×Dic73C2×D812yes—
152D292—24no—
153D292—22no—
154D292—42no—
155D292—41no—
156D292—44no—
157C4×D73—44no—
158C4×D73—162no—
159C4×D73—162no—
160C4×D73—162no—
161Dic146—22no—
162Dic146—41no—
163C8×C73C4×D412yes—
164C8×C73C23×C411yescharacteristic
165C8×C73C2×#4111yescharacteristic
166#10596831—22no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73C4×D812yes—
170C22×C73C22×D811yescharacteristic
171C2×D73—42no—
172C2×D73—44no—
173C2×D73—81no—
174C2×D73—82no—
175C2×D73—82no—
176C2×D73—81no—
177C2×D73—164no—
178C4×C73—81no—
179C4×C73—82no—
180C4×C73—81no—
181C4×C73C4×D812yes—
182C4×C73C2×C4×D411yescharacteristic
183C4×C73C2×#13511yescharacteristic
184Dic73—22no—
185Dic73—22no—
186Dic73—42no—
187#33454—731nomaximal, Sylow
188C2×C73—22no—
189C2×C73C2×C4×D811yescharacteristic
190C2×C73#341511yescharacteristic
191C2×C73#341611yescharacteristic
192D73—82no—
193D73—84no—
194C2×#509—731no—
195#2951—731no—
196#2948—731no—
197#2177—732no—
198#2175—732no—
199C4×SD32—732no—
200C4×D16—732no—
201#2159D7311yescharacteristic, complemented
202#2138—731no—
203C2×#357—731no—
204#2108—731no—
205C73#3345411yescharacteristic, complemented, Sylow
206C2×#135—731no—
207C22×D8—731no—
208#520—1461no—
209#518—1461no—
210#509—738no—
211C2×SD32—732no—
212C2×D16—732no—
213C2×M32C2×D7311yescharacteristic
214C22×C16—731no—
215#442—1462no—
216#440—1462no—
217C4×Q16—732no—
218C4×D8—732no—
219C4×D8—732no—
220C4×D8—732no—
221#427C2×D7311yescharacteristic
222C2×#107C2×D7311yescharacteristic
223#418—1461no—
224#417—1461no—
225#405—731no—
226#367C2×D7312yescomplemented
227#366C2×D7312yescomplemented
228#358—731no—
229#358—731no—
230#357—732no—
231#357—732no—
232#357—731no—
233#357—731no—
234#346—732no—
235C4×C16—732no—
236C2×#41—1461no—
237C22×D4—1461no—
238#135—734no—
239#135—1462no—
240C2×Q16—731no—
241C2×SD16—1461no—
242C2×D8—734no—
243C2×D8—732no—
244C2×D8—731no—
245C2×D8—1461no—
246C2×D8—1462no—
247C2×M16—731no—
248C22×C8C22×D7311yescharacteristic
249C4×Q8—1462no—
250C4×D4—2921no—
251C4×D4—2921no—
252C4×D4—2922no—
253C4×D4—1462no—
254C4×D4—1462no—
255C4×D4—1462no—
256#117—21no—
257#117—2921no—
258#117—731no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—2921no—
261C2×C22⋊C4—2922no—
262SD32—738no—
263D16—738no—
264M32C4×D7314yescomplemented
265C2×C16—731no—
266C2×C16—731no—
267C2×C16—1462no—
268C2×C16C22×D7312yes—
269#107—21no—
270#107C22×D7312yes—
271#107C22×D7311yescharacteristic
272#107C22×D7311yescharacteristic
273#106—21no—
274#103—1462no—
275#102—1462no—
276#102—1462no—
277#102—1462no—
278#97—732no—
279C4×C8—732no—
280C24—2921no—
281#41—2922no—
282#41—1464no—
283C2×Q8—1461no—
284C2×D4—2921no—
285C2×D4—2921no—
286C2×D4—2922no—
287C2×D4—1464no—
288C2×D4—1462no—
289C2×D4—1461no—
290C22×C4—41no—
291C22×C4—2922no—
292C22×C4—2922no—
293C22×C4—2922no—
294C22×C4—2921no—
295C22×C4D4×D7311yescharacteristic
296Q16—732no—
297Q16—1461no—
298SD16—1464no—
299D8—732no—
300D8—734no—
301D8—1464no—
302D8—1462no—
303D8—1461no—
304M16—1462no—
305C2×C8—732no—
306C2×C8C2×C4×D7312yes—
307C2×C8D4×D7311yescharacteristic
308C2×C8#1192402912yes—
309C2×C8C23×D7311yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—2922no—
315C4⋊C4—731no—
316C4⋊C4—731no—
317C22⋊C4—41no—
318C22⋊C4—2922no—
319C22⋊C4—2922no—
320C22⋊C4—2922no—
321C22⋊C4—732no—
322C42—41no—
323C42—2922no—
324C42—732no—
325C16—1461no—
326C16—1461no—
327C16C2×C4×D7314yes—
328C23—2921no—
329C23—2922no—
330C23—2924no—
331C23—11681no—
332C23—11681no—
333C23D8×D7311yescharacteristic
334Q8—2921no—
335Q8—1462no—
336D4—2924no—
337D4—2924no—
338D4—2922no—
339D4—2921no—
340D4—1464no—
341D4—1462no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—161no—
346C2×C4—2921no—
347C2×C4—2924no—
348C2×C4—732no—
349C2×C4—732no—
350C2×C4—11682no—
351C2×C4—11682no—
352C2×C4—11682no—
353C2×C4#1455655312yes—
354C2×C4D8×D7311yescharacteristic
355C2×C4#1455661412yes—
356C2×C4C2×D4×D7311yescharacteristic
357C8—1462no—
358C8#1455655312yes—
359C8C22×C4×D7311yescharacteristic
360C8C2×#1192402911yescharacteristic
361C22—21no—
362C22—21no—
363C22—2922no—
364C22—2924no—
365C22—5841no—
366C22—5842no—
367C22—5842no—
368C22—5841no—
369C22—11684no—
370C22#1798866912yes—
371C22C2×D8×D7311yescharacteristic
372C4—82no—
373C4—81no—
374C4—81no—
375C4—2922no—
376C4—1462no—
377C4—1462no—
378C4#1798866912yes—
379C4C2×#1455655311yescharacteristic
380C4C2×#1455661411yescharacteristic
381C2—22no—
382C2—5844no—
383C2—5842no—
384C2C2×#1798866911yescharacteristic
385C2#2262307111yescharacteristic
386C2#2262307611yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2159C2
socleC22×C73C22×D8
prime radicalC2×#17989025C2
cosocleC2×C8×C73C24
hypercenter#2159D73
hypocenterC73#33454
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
2. #22618805C2semidirect
3. #22618782C2semidirect
4. #22611806C2semidirect
5. #22611794C2semidirect
6. #22611782C2semidirect
7. #22611777C2semidirect
8. #22611636C2semidirect
9. C2×#17988387C2semidirect
10. #22611530C2semidirect
11. C73×#2159C2semidirect
16. #17989025C4semidirect
22. #17988681C22semidirect
25. #17988669C22semidirect
29. #17988390C22semidirect
30. #17988390C22semidirect
33. #17988387C22semidirect
34. #17988387C22semidirect
35. #17988362C22semidirect
36. C16×Dic73C22semidirect
40. #367×C73C22semidirect
41. #366×C73C22semidirect
45. #14556613C2×C4semidirect
50. C2×D584C2×C4semidirect
65. #14556536C2×C4semidirect
66. D1168C2×C4semidirect
79. M32×C73C2×C4semidirect
201. #2159D73semidirect
205. C73#33454semidirect
226. #367C2×D73semidirect
227. #366C2×D73semidirect
264. M32C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890252
#226188052
#226187822
#226118062
#226117942
#226117822
#226117772
#226116362
C2×#179883872
#226115302
C73×#21592
#3345473