Finite Group Databasebeta

← #33239434

Subgroups — #33239434

Order 18688.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623065C211yesmaximal, characteristic
2#22618733C211yesmaximal, characteristic, complemented
3#22611829C214yesmaximal, complemented
4#22611822C214yesmaximal, complemented
5#22611705C211yesmaximal, characteristic, complemented
6#22611678C211yesmaximal, characteristic, complemented
7#22611667C211yesmaximal, characteristic, complemented
8C2×#17988365C211yesmaximal, characteristic, complemented
9C2×#444×C73C211yesmaximal, characteristic, complemented
10C2×#14556613C2211yescharacteristic
11C4×#11924028—21no—
12#17989028C412yescomplemented
13#17989027C2212yes—
14#17989023C412yescomplemented
15#17989022C2212yes—
16#17989017C414yescomplemented
17C2×#14556531C2211yescharacteristic
18C2×#14556531C2211yescharacteristic
19#17988700C2212yescomplemented
20#17988695C2212yescomplemented
21#17988689—22no—
22#17988625C2211yescharacteristic
23#17988584—21no—
24#17988411C2214yescomplemented
25#17988402C2212yescomplemented
26#17988399C2212yescomplemented
27#17988397C2214yes—
28#17988365C2214yescomplemented
29C2×#135×C73C2211yescharacteristic
30C2×#131×C73—21no—
31#444×C73—22no—
32#444×C73C2214yes—
33C2×#108×C73C2211yescharacteristic
34C2×#104×C73—21no—
35C2×#97×C73C2211yescharacteristic
36C2×#11924028—21no—
37#14556613—21no—
38#14556613—21no—
39#14556613C2×C412yes—
40#14556613C2×C412yes—
41C2×Dic292C2311yescharacteristic
42C2×#11924017—21no—
43C2×D584C2311yescharacteristic
44C4×Dic146—22no—
45C4×D292—22no—
46C4×C22⋊D73—42no—
47#14556550—21no—
48#14556548—41no—
49C42×D73—41no—
50#14556537C2×C414yescomplemented
51#14556535C2×C414yescomplemented
52#14556534C2×C414yescomplemented
53#14556532C2×C414yescomplemented
54#14556531—21no—
55#14556531C2×C412yescomplemented
56#14556531C2312yes—
57#14556531C2312yes—
58C2×#11924003—21no—
59C2×#11924003C2×C412yescomplemented
60#14556525—21no—
61#14556523C2312yes—
62#14556516—22no—
63#14556513—22no—
64C2×#41×C73—21no—
65#135×C73—21no—
66#135×C73—21no—
67#135×C73C2×C412yescomplemented
68#135×C73C2312yes—
69C2×Q16×C73C2×C411yescharacteristic, complemented
70C2×SD16×C73—21no—
71C2×D8×C73C2×C411yescharacteristic, complemented
72#131×C73—24no—
73#131×C73—42no—
74C2×M16×C73—21no—
75C2×M16×C73—41no—
76C22×C8×C73—41no—
77C22×C8×C73D411yescharacteristic
78C22×C8×C73C2311yescharacteristic
79C2×C42×C73D411yescharacteristic
80#108×C73—21no—
81#108×C73C2312yes—
82#104×C73—24no—
83#97×C73D412yes—
84#97×C73C2312yes—
85#11924028—22no—
86#11924028—22no—
87#11924028—41no—
88#11924028—41no—
89C2×Dic146—21no—
90C2×D292—21no—
91C2×C22⋊D73—41no—
92C2×C4×D73—41no—
93C2×C4×D73—42no—
94Dic292—21no—
95Dic292C22×C412yes—
96#11924017—22no—
97#11924017—22no—
98D584—21no—
99D584C22×C412yes—
100C4⋊Dic73—22no—
101#11924007—42no—
102C22⋊Dic73—41no—
103#11924005—42no—
104C4×Dic73—42no—
105C4×Dic73—41no—
106#11924003—22no—
107#11924003C22×C414yes—
108#41×C73—22no—
109#41×C73—22no—
110#41×C73—41no—
111#41×C73—41no—
112C2×Q8×C73—21no—
113C2×D4×C73—21no—
114C2×D4×C73—41no—
115C22×C4×C73—21no—
116C22×C4×C73—41no—
117C22×C4×C73C2×D411yescharacteristic
118Q16×C73—21no—
119Q16×C73C22×C412yes—
120SD16×C73—22no—
121SD16×C73—22no—
122D8×C73—21no—
123D8×C73C22×C412yes—
124M16×C73—22no—
125M16×C73—44no—
126M16×C73—41no—
127C2×C8×C73—21no—
128C2×C8×C73—21no—
129C2×C8×C73—44no—
130C2×C8×C73—41no—
131C2×C8×C73C22×C411yescharacteristic
132C2×C8×C73C22×C411yescharacteristic
133C2×C8×C73C2×D412yes—
134C2×C8×C73C2×D411yescharacteristic
135C2×C8×C73#4111yescharacteristic
136C2×C8×C73#4111yescharacteristic
137C2×C8×C73C2411yescharacteristic
138C42×C73—21no—
139C42×C73C2×D412yes—
140C23×C73—41no—
141C23×C73AGL(1,ℤ/8)11yescharacteristic
142C22×D73—41no—
143Q8×C73—22no—
144Q8×C73—41no—
145D4×C73—22no—
146D4×C73—42no—
147D4×C73—42no—
148D4×C73—41no—
149C2×C4×C73—22no—
150C2×C4×C73—41no—
151C2×C4×C73—42no—
152C2×C4×C73—42no—
153C2×C4×C73—81no—
154C2×C4×C73C4×D411yescharacteristic
155C2×C4×C73C4×D411yescharacteristic
156C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C73C22×D411yescharacteristic
160C22⋊D73—42no—
161C22⋊D73—42no—
162C2×Dic73—41no—
163C2×Dic73—42no—
164D292—22no—
165D292—41no—
166C4×D73—42no—
167C4×D73—42no—
168C4×D73—84no—
169Dic146—22no—
170Dic146—41no—
171C8×C73—22no—
172C8×C73—42no—
173C8×C73C4×D411yescharacteristic
174C8×C73C4×D411yescharacteristic
175C8×C73C23×C411yescharacteristic
176C8×C73C2×#4111yescharacteristic
177C22×C73—41no—
178C22×C73—42no—
179C22×C73—82no—
180C22×C73#44211yescharacteristic
181C22×C73#44211yescharacteristic
182C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
183C2×D73—42no—
184C2×D73—82no—
185C4×C73—42no—
186C4×C73—42no—
187C4×C73#44211yescharacteristic
188C4×C73#44211yescharacteristic
189C4×C73C2×C4×D411yescharacteristic
190C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
191Dic73—42no—
192Dic73—82no—
193#33468—731nomaximal, Sylow
194C2×C73—21no—
195C2×C73—21no—
196C2×C73—21no—
197C2×C73—82no—
198C2×C73C2×#44211yescharacteristic
199D73—82no—
200#3414—731no—
201C2×#444D7311yescharacteristic, complemented
202#2943—731no—
203AGL(1,ℤ/16)—734no—
204#2180—734no—
205#2155—731no—
206#2148—731no—
207#2145—731no—
208C2×#347—731no—
209C73#3346811yescharacteristic, complemented, Sylow
210C2×#135—731no—
211C2×#135C2×D7311yescharacteristic
212C2×#131—21no—
213C4×#41—1461no—
214#510—732no—
215#510—732no—
216#509—732no—
217#509—732no—
218#508—734no—
219C2×M32—731no—
220C2×M32—731no—
221#444—22no—
222#444C2×D7314yes—
223#442—732no—
224#441—732no—
225#440—1462no—
226C2×#108C2×D7311yescharacteristic
227#427—731no—
228C2×#104—21no—
229#416—1461no—
230C2×#97C2×D7311yescharacteristic
231#365—734no—
232#362—732no—
233#361—732no—
234#360—734no—
235#347—734no—
236C2×#41—21no—
237C2×#41—1461no—
238#135—21no—
239#135—21no—
240#135—732no—
241#135—732no—
242#135—1461no—
243#135—1461no—
244#135C4×D7312yescomplemented
245#135C22×D7312yes—
246C2×Q16—731no—
247C2×Q16C4×D7311yescharacteristic, complemented
248C2×SD16—21no—
249C2×SD16—1461no—
250C2×D8—731no—
251C2×D8C4×D7311yescharacteristic, complemented
252#131—24no—
253#131—42no—
254C2×M16—21no—
255C2×M16—41no—
256C22×C8—41no—
257C22×C8C22⋊D7311yescharacteristic
258C22×C8C22×D7311yescharacteristic
259C4×Q8—1462no—
260C4×D4—2922no—
261C4×D4—1462no—
262#117—2921no—
263#117—1461no—
264C2×C42—2921no—
265C2×C42C22⋊D7311yescharacteristic
266Q32—734no—
267SD32—734no—
268SD32—734no—
269D16—734no—
270M32—732no—
271M32—732no—
272M32—732no—
273M32—1461no—
274C2×C16—732no—
275C2×C16—1461no—
276#108—21no—
277#108C22×D7312yes—
278#107—1461no—
279#106—732no—
280#104—24no—
281#103—1462no—
282#102—1462no—
283#97C22⋊D7312yes—
284#97C22×D7312yes—
285#41—22no—
286#41—22no—
287#41—41no—
288#41—41no—
289#41—2921no—
290#41—2921no—
291#41—1462no—
292#41—1462no—
293C2×Q8—21no—
294C2×Q8—1461no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—2921no—
298C2×D4—1461no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—2922no—
302C22×C4—2921no—
303C22×C4C2×C22⋊D7311yescharacteristic
304Q16—21no—
305Q16—732no—
306Q16—1461no—
307Q16C2×C4×D7312yes—
308SD16—22no—
309SD16—22no—
310SD16—1462no—
311SD16—1462no—
312D8—21no—
313D8—732no—
314D8—1461no—
315D8C2×C4×D7312yes—
316M16—22no—
317M16—44no—
318M16—41no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—44no—
322C2×C8—41no—
323C2×C8C2×C4×D7311yescharacteristic
324C2×C8C2×C4×D7311yescharacteristic
325C2×C8C2×C22⋊D7312yes—
326C2×C8C2×C22⋊D7311yescharacteristic
327C2×C8#1192402811yescharacteristic
328C2×C8#1192402811yescharacteristic
329C2×C8C23×D7311yescharacteristic
330C4⋊C4—2922no—
331C4⋊C4—1462no—
332C22⋊C4—2922no—
333C22⋊C4—2921no—
334C42—21no—
335C42—2922no—
336C42—2921no—
337C42C2×C22⋊D7312yes—
338C16—734no—
339C16—1462no—
340C23—41no—
341C23—2921no—
342C23#1455661611yescharacteristic
343Q8—22no—
344Q8—41no—
345Q8—2921no—
346Q8—1462no—
347D4—22no—
348D4—42no—
349D4—42no—
350D4—41no—
351D4—2922no—
352D4—2922no—
353D4—2921no—
354D4—1462no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—2921no—
361C2×C4—2922no—
362C2×C4—2922no—
363C2×C4—2922no—
364C2×C4—5844no—
365C2×C4C4×C22⋊D7311yescharacteristic
366C2×C4C4×C22⋊D7311yescharacteristic
367C2×C4#1455661611yescharacteristic
368C2×C4#1455661711yescharacteristic
369C2×C4#1455661711yescharacteristic
370C2×C4C22×C22⋊D7311yescharacteristic
371C8—22no—
372C8—42no—
373C8C4×C22⋊D7311yescharacteristic
374C8C4×C22⋊D7311yescharacteristic
375C8C22×C4×D7311yescharacteristic
376C8C2×#1192402811yescharacteristic
377C22—41no—
378C22—42no—
379C22—82no—
380C22—2922no—
381C22—5842no—
382C22#1798869711yescharacteristic
383C22#1798869711yescharacteristic
384C22C2×#1455661611yescharacteristic
385C4—42no—
386C4—42no—
387C4—2922no—
388C4—5842no—
389C4#1798869711yescharacteristic
390C4#1798869711yescharacteristic
391C4C2×C4×C22⋊D7311yescharacteristic
392C4C2×#1455661711yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—5842no—
398C2C2×#1798869711yescharacteristic
subgroupG/H
centerC4#17988697
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC2×#444×C73C2
socleC2×C73C2×#442
prime radical#22623065C2
cosocleC2×C8×C73C24
hypercenterC2×#444D73
hypocenterC73#33468
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. #22618733C2semidirect
3. #22611829C2semidirect
4. #22611822C2semidirect
5. #22611705C2semidirect
6. #22611678C2semidirect
7. #22611667C2semidirect
8. C2×#17988365C2semidirect
9. C2×#444×C73C2semidirect
12. #17989028C4semidirect
14. #17989023C4semidirect
16. #17989017C4semidirect
19. #17988700C22semidirect
20. #17988695C22semidirect
24. #17988411C22semidirect
25. #17988402C22semidirect
26. #17988399C22semidirect
28. #17988365C22semidirect
50. #14556537C2×C4semidirect
51. #14556535C2×C4semidirect
52. #14556534C2×C4semidirect
53. #14556532C2×C4semidirect
55. #14556531C2×C4semidirect
59. C2×#11924003C2×C4semidirect
67. #135×C73C2×C4semidirect
69. C2×Q16×C73C2×C4semidirect
71. C2×D8×C73C2×C4semidirect
201. C2×#444D73semidirect
209. C73#33468semidirect
244. #135C4×D73semidirect
247. C2×Q16C4×D73semidirect
251. C2×D8C4×D73semidirect

Maximal subgroups

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

subgroupindex
#226230652
#226187332
#226118292
#226118222
#226117052
#226116782
#226116672
C2×#179883652
C2×#444×C732
#3346873