Finite Group Databasebeta

← #33239430

Subgroups — #33239430

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
2C2×#17988709C211yesmaximal, characteristic, complemented
3#22618731C211yesmaximal, characteristic, complemented
4#22611826C214yesmaximal, complemented
5#22611819C214yesmaximal, complemented
6#22611703C211yesmaximal, characteristic, complemented
7#22611681C211yesmaximal, characteristic, complemented
8C2×#17988364C211yesmaximal, characteristic, complemented
9C73×#2145C211yesmaximal, characteristic, complemented
10C2×#14556613C2211yescharacteristic
11C2×#14556598—21no—
12#17989055—21no—
13#17989028C412yescomplemented
14#17989027C2212yes—
15#17989023C412yescomplemented
16#17989022C2212yes—
17#17989017C414yescomplemented
18C2×#14556531C2211yescharacteristic
19#17988709—22no—
20#17988709C2214yes—
21#17988698C2212yescomplemented
22#17988693C2212yescomplemented
23#17988686—22no—
24C2×#14556527C2211yescharacteristic
25C2×#14556518—21no—
26#17988581—21no—
27C2×#14556501C2211yescharacteristic
28#17988409C2214yescomplemented
29#17988404C2212yescomplemented
30#17988401C2212yescomplemented
31#17988364C2214yescomplemented
32C2×#135×C73C2211yescharacteristic
33C2×M32×C73C2211yescharacteristic
34#427×C73C2211yescharacteristic
35#360×C73C2214yes—
36C2×#11924028—21no—
37#14556613—21no—
38#14556613—21no—
39#14556613C2×C412yes—
40#14556613C2×C412yes—
41C2×Dic292C2311yescharacteristic
42C2×#11924017—21no—
43C2×D584C2311yescharacteristic
44#14556598—24no—
45#14556598—42no—
46C2×#11924012—21no—
47C2×#11924012—41no—
48C22×#10596831—41no—
49C22×#10596831D411yescharacteristic
50#14556556—22no—
51#14556553—22no—
52#14556551—42no—
53#14556548—41no—
54C2×C4×Dic73D411yescharacteristic
55C42×D73—41no—
56#14556537C2×C414yescomplemented
57#14556535C2×C414yescomplemented
58#14556534C2×C414yescomplemented
59#14556532C2×C414yescomplemented
60#14556531C2×C412yescomplemented
61#14556531C2312yes—
62C2×#11924003C2×C412yescomplemented
63#14556527—21no—
64#14556527C2312yes—
65#14556518—24no—
66#14556514—22no—
67#14556511—22no—
68#14556501D412yes—
69#14556501C2312yes—
70C2×#41×C73—21no—
71#135×C73—21no—
72#135×C73—21no—
73#135×C73C2×C412yescomplemented
74#135×C73C2312yes—
75C2×Q16×C73C2×C411yescharacteristic, complemented
76C2×SD16×C73—21no—
77C2×D8×C73C2×C411yescharacteristic, complemented
78C22×C8×C73C2311yescharacteristic
79#117×C73—21no—
80M32×C73—21no—
81M32×C73C2312yes—
82C2×C16×C73—21no—
83#107×C73—21no—
84#106×C73C2312yes—
85#11924028—22no—
86#11924028—22no—
87#11924028—41no—
88#11924028—41no—
89C2×Dic146—21no—
90C2×D292—21no—
91C2×C22⋊D73—41no—
92C22×Dic73—21no—
93C2×C4×D73—41no—
94C2×C4×D73—42no—
95Dic292—21no—
96Dic292C22×C412yes—
97#11924017—22no—
98#11924017—22no—
99D584—21no—
100D584C22×C412yes—
101#11924012—22no—
102#11924012—44no—
103#11924012—41no—
104C2×#10596831—21no—
105C2×#10596831—21no—
106C2×#10596831—44no—
107C2×#10596831—41no—
108C2×#10596831C2×D412yes—
109#11924007—42no—
110#11924005—42no—
111C4×Dic73—21no—
112C4×Dic73—42no—
113C4×Dic73C2×D412yes—
114#11924003C22×C414yes—
115#41×C73—22no—
116#41×C73—22no—
117#41×C73—41no—
118#41×C73—41no—
119C2×Q8×C73—21no—
120C2×D4×C73—21no—
121C2×D4×C73—41no—
122C22×C4×C73—41no—
123C22×C4×C73C2×D411yescharacteristic
124Q16×C73—21no—
125Q16×C73C22×C412yes—
126SD16×C73—22no—
127SD16×C73—22no—
128D8×C73—21no—
129D8×C73C22×C412yes—
130C2×C8×C73C22×C411yescharacteristic
131C2×C8×C73C22×C411yescharacteristic
132C2×C8×C73C2×D411yescharacteristic
133C2×C8×C73#4111yescharacteristic
134C2×C8×C73#4111yescharacteristic
135C2×C8×C73C2411yescharacteristic
136C4⋊C4×C73—22no—
137C22⋊C4×C73—41no—
138C42×C73—41no—
139C16×C73—22no—
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—41no—
150C2×C4×C73—42no—
151C2×C4×C73—42no—
152C2×C4×C73—42no—
153C2×C4×C73C4×D411yescharacteristic
154C2×C4×C73C4×D411yescharacteristic
155C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C73C22×D411yescharacteristic
159C22⋊D73—42no—
160C22⋊D73—42no—
161C2×Dic73—22no—
162C2×Dic73—41no—
163C2×Dic73—81no—
164D292—22no—
165D292—41no—
166C4×D73—42no—
167C4×D73—42no—
168C4×D73—84no—
169Dic146—22no—
170Dic146—41no—
171C8×C73C4×D411yescharacteristic
172C8×C73C4×D411yescharacteristic
173C8×C73C23×C411yescharacteristic
174C8×C73C2×#4111yescharacteristic
175#10596831—22no—
176#10596831—42no—
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—82no—
187C4×C73#44211yescharacteristic
188C4×C73#44211yescharacteristic
189C4×C73C2×C4×D411yescharacteristic
190C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
191Dic73—42no—
192Dic73—42no—
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×#444—731no—
202#2943—731no—
203AGL(1,ℤ/16)—734no—
204#2180—734no—
205#2155—731no—
206#2148—731no—
207#2145D7311yescharacteristic, complemented
208C2×#347—731no—
209C73#3346811yescharacteristic, complemented, Sylow
210C2×#135—731no—
211C2×#135C2×D7311yescharacteristic
212C2×#131—1461no—
213C4×#41—1461no—
214#510—732no—
215#510—732no—
216#509—732no—
217#509—732no—
218#508—734no—
219C2×M32—731no—
220C2×M32C2×D7311yescharacteristic
221#444—734no—
222#444—1462no—
223#442—732no—
224#441—732no—
225#440—1462no—
226C2×#108—731no—
227#427C2×D7311yescharacteristic
228C2×#104—1461no—
229#416—1461no—
230C2×#97—731no—
231#365—734no—
232#362—732no—
233#361—732no—
234#360C2×D7314yes—
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—2922no—
253#131—1464no—
254C2×M16—2921no—
255C2×M16—1461no—
256C22×C8—2921no—
257C22×C8—731no—
258C22×C8C22×D7311yescharacteristic
259C4×Q8—1462no—
260C4×D4—2922no—
261C4×D4—1462no—
262#117—21no—
263#117—2921no—
264C2×C42—2921no—
265C2×C42—731no—
266Q32—734no—
267SD32—734no—
268SD32—734no—
269D16—734no—
270M32—21no—
271M32—732no—
272M32—732no—
273M32C22×D7312yes—
274C2×C16—21no—
275C2×C16—732no—
276#108—732no—
277#108—1461no—
278#107—21no—
279#106C22×D7312yes—
280#104—1464no—
281#103—1462no—
282#102—1462no—
283#97—732no—
284#97—732no—
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—41no—
300C22×C4—2922no—
301C22×C4—2921no—
302C22×C4—1461no—
303C22×C4D4×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—2921no—
317M16—2924no—
318M16—1462no—
319C2×C8—2921no—
320C2×C8—2924no—
321C2×C8—732no—
322C2×C8—1461no—
323C2×C8—1461no—
324C2×C8C2×C4×D7311yescharacteristic
325C2×C8C2×C4×D7311yescharacteristic
326C2×C8D4×D7311yescharacteristic
327C2×C8#1192402711yescharacteristic
328C2×C8#1192402711yescharacteristic
329C2×C8C23×D7311yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—2922no—
332C22⋊C4—41no—
333C22⋊C4—2922no—
334C42—41no—
335C42—2922no—
336C42—732no—
337C42—1461no—
338C16—22no—
339C16—734no—
340C23—41no—
341C23—2921no—
342C23#1455661511yescharacteristic
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—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—2921no—
360C2×C4—2922no—
361C2×C4—2922no—
362C2×C4—5844no—
363C2×C4—5841no—
364C2×C4—1462no—
365C2×C4#1455655111yescharacteristic
366C2×C4#1455655111yescharacteristic
367C2×C4#1455661511yescharacteristic
368C2×C4#1455661811yescharacteristic
369C2×C4#1455661811yescharacteristic
370C2×C4C2×D4×D7311yescharacteristic
371C8—2922no—
372C8—1462no—
373C8#1455655111yescharacteristic
374C8#1455655111yescharacteristic
375C8C22×C4×D7311yescharacteristic
376C8C2×#1192402711yescharacteristic
377C22—41no—
378C22—42no—
379C22—82no—
380C22—2922no—
381C22—5842no—
382C22#1798869611yescharacteristic
383C22#1798869611yescharacteristic
384C22C2×#1455661511yescharacteristic
385C4—42no—
386C4—82no—
387C4—2922no—
388C4—2922no—
389C4#1798869611yescharacteristic
390C4#1798869611yescharacteristic
391C4C2×#1455655111yescharacteristic
392C4C2×#1455661811yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—5842no—
398C2C2×#1798869611yescharacteristic
subgroupG/H
centerC4#17988696
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2145C2
socleC2×C73C2×#442
prime radical#22623065C2
cosocleC2×C8×C73C24
hypercenter#2145D73
hypocenterC73#33468
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2145

Complements

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

NH ≅ G/Ntype
2. C2×#17988709C2semidirect
3. #22618731C2semidirect
4. #22611826C2semidirect
5. #22611819C2semidirect
6. #22611703C2semidirect
7. #22611681C2semidirect
8. C2×#17988364C2semidirect
9. C73×#2145C2semidirect
13. #17989028C4semidirect
15. #17989023C4semidirect
17. #17989017C4semidirect
21. #17988698C22semidirect
22. #17988693C22semidirect
28. #17988409C22semidirect
29. #17988404C22semidirect
30. #17988401C22semidirect
31. #17988364C22semidirect
56. #14556537C2×C4semidirect
57. #14556535C2×C4semidirect
58. #14556534C2×C4semidirect
59. #14556532C2×C4semidirect
60. #14556531C2×C4semidirect
62. C2×#11924003C2×C4semidirect
73. #135×C73C2×C4semidirect
75. C2×Q16×C73C2×C4semidirect
77. C2×D8×C73C2×C4semidirect
207. #2145D73semidirect
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
C2×#179887092
#226187312
#226118262
#226118192
#226117032
#226116812
C2×#179883642
C73×#21452
#3346873