Finite Group Databasebeta

← #33239446

Subgroups — #33239446

Order 18688.

646 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#22618847C211yesmaximal, characteristic, complemented
3#22618815C211yesmaximal, characteristic, complemented
4#22611832C214yesmaximal, complemented
5#22611825C214yesmaximal, complemented
6C2×#17988411C211yesmaximal, characteristic, complemented
7C2×#17988398C211yesmaximal, characteristic, complemented
8#22611535C211yesmaximal, characteristic, complemented
9C73×#2148C211yesmaximal, characteristic, complemented
10C2×#14556613C2211yescharacteristic
11#17989353—21no—
12#17989070—21no—
13#17989028C2212yes—
14#17989027C412yescomplemented
15#17989023C2212yes—
16#17989022C412yescomplemented
17#17989017C414yescomplemented
18C2×#14556531C2211yescharacteristic
19#17988708C2214yes—
20#17988703—22no—
21#17988699C2212yescomplemented
22#17988694C2212yescomplemented
23SD16×Dic73—22no—
24C2×#14556523C2211yescharacteristic
25#17988603—21no—
26#17988588—21no—
27#17988542C2211yescharacteristic
28#17988411C2214yescomplemented
29#17988398C2214yes—
30#17988364C2214yescomplemented
31C2×#135×C73C2211yescharacteristic
32C2×M32×C73C2211yescharacteristic
33C2×#108×C73C2211yescharacteristic
34#362×C73C2212yescomplemented
35#361×C73C2212yescomplemented
36C2×#11924028—21no—
37#14556613—21no—
38#14556613—21no—
39#14556613C2×C412yescomplemented
40#14556613C2312yes—
41C2×Dic292C2×C411yescharacteristic, complemented
42C2×#11924017—21no—
43C2×D584C2×C411yescharacteristic, complemented
44#14556599—24no—
45#14556597—42no—
46C2×#11924012D411yescharacteristic
47M16×D73—42no—
48Q8×Dic73—22no—
49D4×Dic73—22no—
50D4×Dic73—42no—
51#14556550—41no—
52#14556550D411yescharacteristic
53C2×C4⋊Dic73—21no—
54C2×C4⋊Dic73—41no—
55#14556537C2×C414yescomplemented
56#14556535C2×C414yescomplemented
57#14556534C2×C414yescomplemented
58#14556532C2×C414yescomplemented
59#14556531C2×C412yescomplemented
60#14556531C2312yes—
61C2×#11924003C2×C412yescomplemented
62#14556523—21no—
63#14556523C2312yes—
64#14556517—24no—
65#14556515—22no—
66#14556512—22no—
67#14556501C2312yes—
68C8×Dic73D412yes—
69C2×#41×C73—21no—
70#135×C73—21no—
71#135×C73—21no—
72#135×C73C2×C412yes—
73#135×C73C2×C412yes—
74C2×Q16×C73C2311yescharacteristic
75C2×SD16×C73—21no—
76C2×D8×C73C2311yescharacteristic
77C2×M16×C73—21no—
78C22×C8×C73C2311yescharacteristic
79M32×C73—21no—
80M32×C73C2312yes—
81C2×C16×C73—21no—
82#108×C73—21no—
83#108×C73C2312yes—
84#11924028—22no—
85#11924028—22no—
86#11924028—41no—
87#11924028—41no—
88C2×Dic146—21no—
89C2×D292—21no—
90C2×C22⋊D73—41no—
91C22×Dic73—42no—
92C22×Dic73—41no—
93C2×C4×D73—41no—
94Dic292—21no—
95Dic292C22×C412yes—
96#11924017—22no—
97#11924017—22no—
98D584—21no—
99D584C22×C412yes—
100#11924012—21no—
101#11924012—21no—
102#11924011—44no—
103C2×#10596831C2×D412yes—
104C8×D73—44no—
105C4⋊Dic73—21no—
106C4⋊Dic73—22no—
107C4⋊Dic73—42no—
108C4⋊Dic73—41no—
109C22⋊Dic73—21no—
110C22⋊Dic73—42no—
111C4×Dic73—42no—
112C4×Dic73C2×D412yes—
113#11924003C22×C414yes—
114#41×C73—22no—
115#41×C73—22no—
116#41×C73—41no—
117#41×C73—41no—
118C2×Q8×C73—21no—
119C2×D4×C73—21no—
120C2×D4×C73—41no—
121C22×C4×C73—41no—
122C22×C4×C73C2×D411yescharacteristic
123Q16×C73—21no—
124Q16×C73C22×C412yes—
125SD16×C73—22no—
126SD16×C73—22no—
127D8×C73—21no—
128D8×C73C22×C412yes—
129M16×C73—22no—
130M16×C73—41no—
131C2×C8×C73—41no—
132C2×C8×C73C22×C411yescharacteristic
133C2×C8×C73C22×C411yescharacteristic
134C2×C8×C73C2×D411yescharacteristic
135C2×C8×C73#4111yescharacteristic
136C2×C8×C73#4111yescharacteristic
137C2×C8×C73C2411yescharacteristic
138C16×C73—22no—
139C23×C73—41no—
140C23×C73AGL(1,ℤ/8)11yescharacteristic
141C22×D73—41no—
142Q8×C73—22no—
143Q8×C73—41no—
144D4×C73—22no—
145D4×C73—42no—
146D4×C73—42no—
147D4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—42no—
150C2×C4×C73—42no—
151C2×C4×C73C4×D411yescharacteristic
152C2×C4×C73C4×D411yescharacteristic
153C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C73C22×D411yescharacteristic
157C22⋊D73—42no—
158C22⋊D73—42no—
159C2×Dic73—22no—
160C2×Dic73—41no—
161C2×Dic73—42no—
162C2×Dic73—162no—
163C2×Dic73—161no—
164D292—22no—
165D292—41no—
166C4×D73—42no—
167C4×D73—42no—
168Dic146—22no—
169Dic146—41no—
170C8×C73—42no—
171C8×C73C4×D411yescharacteristic
172C8×C73C4×D411yescharacteristic
173C8×C73C23×C411yescharacteristic
174C8×C73C2×#4111yescharacteristic
175#10596831—22no—
176C22×C73—41no—
177C22×C73—42no—
178C22×C73—81no—
179C22×C73—81no—
180C22×C73#44211yescharacteristic
181C22×C73#44211yescharacteristic
182C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
183C2×D73—42no—
184C2×D73—161no—
185C4×C73—42no—
186C4×C73#44211yescharacteristic
187C4×C73#44211yescharacteristic
188C4×C73C2×C4×D411yescharacteristic
189C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
190Dic73—42no—
191Dic73—42no—
192Dic73—82no—
193#33469—731nomaximal, Sylow
194C2×C73—21no—
195C2×C73—21no—
196C2×C73—21no—
197C2×C73—82no—
198C2×C73C2×#44211yescharacteristic
199D73—82no—
200#3414—731no—
201#2957—731no—
202#2952—731no—
203AGL(1,ℤ/16)—734no—
204#2180—734no—
205C2×#365—731no—
206#2148D7311yescharacteristic, complemented
207C2×#360—731no—
208#2110—731no—
209C73#3346911yescharacteristic, complemented, Sylow
210C2×#135—731no—
211C2×#135C2×D7311yescharacteristic
212#568—1461no—
213#520—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#443—1462no—
223#442—732no—
224#441—732no—
225C4×SD16—1462no—
226C2×#108C2×D7311yescharacteristic
227C2×#106—731no—
228#421—1461no—
229#417—1461no—
230#405—731no—
231#365—734no—
232#362C2×D7312yescomplemented
233#361C2×D7312yescomplemented
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×D7312yes—
245#135C4×D7312yes—
246C2×Q16—731no—
247C2×Q16C22×D7311yescharacteristic
248C2×SD16—21no—
249C2×SD16—1461no—
250C2×D8—731no—
251C2×D8C22×D7311yescharacteristic
252#131—2922no—
253#131—1464no—
254C2×M16—21no—
255C2×M16—2922no—
256C2×M16—731no—
257C22×C8C22×D7311yescharacteristic
258C4×Q8—1462no—
259C4×D4—2922no—
260C4×D4—1462no—
261#117—2921no—
262#117—731no—
263C2×C4⋊C4—2921no—
264C2×C4⋊C4—1461no—
265Q32—734no—
266SD32—734no—
267SD32—734no—
268D16—734no—
269M32—21no—
270M32—732no—
271M32—732no—
272M32C22×D7312yes—
273C2×C16—21no—
274C2×C16—732no—
275#108—21no—
276#108C22×D7312yes—
277#106—732no—
278#106—1461no—
279#104—1464no—
280#103—1462no—
281#102—1462no—
282#97—732no—
283C4×C8—732no—
284#41—22no—
285#41—22no—
286#41—41no—
287#41—41no—
288#41—2921no—
289#41—2921no—
290#41—1462no—
291#41—1462no—
292C2×Q8—21no—
293C2×Q8—1461no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—2921no—
297C2×D4—1461no—
298C22×C4—41no—
299C22×C4—2921no—
300C22×C4—2921no—
301C22×C4—2922no—
302C22×C4D4×D7311yescharacteristic
303Q16—21no—
304Q16—732no—
305Q16—1461no—
306Q16C2×C4×D7312yes—
307SD16—22no—
308SD16—22no—
309SD16—1462no—
310SD16—1462no—
311D8—21no—
312D8—732no—
313D8—1461no—
314D8C2×C4×D7312yes—
315M16—22no—
316M16—41no—
317M16—2924no—
318M16—1461no—
319M16—1461no—
320C2×C8—41no—
321C2×C8—2924no—
322C2×C8—732no—
323C2×C8C2×C4×D7311yescharacteristic
324C2×C8C2×C4×D7311yescharacteristic
325C2×C8D4×D7311yescharacteristic
326C2×C8#1192402711yescharacteristic
327C2×C8#1192402711yescharacteristic
328C2×C8C23×D7311yescharacteristic
329C4⋊C4—2922no—
330C4⋊C4—2921no—
331C4⋊C4—1462no—
332C4⋊C4—1461no—
333C22⋊C4—2922no—
334C22⋊C4—1461no—
335C42—2922no—
336C42—732no—
337C16—22no—
338C16—734no—
339C23—41no—
340C23—2921no—
341C23#1455661811yescharacteristic
342Q8—22no—
343Q8—41no—
344Q8—2921no—
345Q8—1462no—
346D4—22no—
347D4—42no—
348D4—42no—
349D4—41no—
350D4—2922no—
351D4—2922no—
352D4—2921no—
353D4—1462no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—2921no—
358C2×C4—2922no—
359C2×C4—2922no—
360C2×C4—2922no—
361C2×C4—11682no—
362C2×C4—11681no—
363C2×C4—1462no—
364C2×C4#1455655111yescharacteristic
365C2×C4#1455655111yescharacteristic
366C2×C4#1455661511yescharacteristic
367C2×C4#1455661511yescharacteristic
368C2×C4#1455661811yescharacteristic
369C2×C4C2×D4×D7311yescharacteristic
370C8—42no—
371C8—1462no—
372C8#1455655111yescharacteristic
373C8#1455655111yescharacteristic
374C8C22×C4×D7311yescharacteristic
375C8C2×#1192402711yescharacteristic
376C22—41no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22—2922no—
381C22—11681no—
382C22#1798869611yescharacteristic
383C22#1798869611yescharacteristic
384C22C2×#1455661811yescharacteristic
385C4—42no—
386C4—2922no—
387C4—2922no—
388C4—5842no—
389C4#1798869611yescharacteristic
390C4#1798869611yescharacteristic
391C4C2×#1455655111yescharacteristic
392C4C2×#1455661511yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—5842no—
398C2C2×#1798869611yescharacteristic
subgroupG/H
centerC2C2×#17988696
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2148C2
socleC2×C73C2×#442
prime radical#22623065C2
cosocleC2×C8×C73C24
hypercenter#2148D73
hypocenterC73#33469
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. #22618847C2semidirect
3. #22618815C2semidirect
4. #22611832C2semidirect
5. #22611825C2semidirect
6. C2×#17988411C2semidirect
7. C2×#17988398C2semidirect
8. #22611535C2semidirect
9. C73×#2148C2semidirect
14. #17989027C4semidirect
16. #17989022C4semidirect
17. #17989017C4semidirect
21. #17988699C22semidirect
22. #17988694C22semidirect
28. #17988411C22semidirect
30. #17988364C22semidirect
34. #362×C73C22semidirect
35. #361×C73C22semidirect
39. #14556613C2×C4semidirect
41. C2×Dic292C2×C4semidirect
43. C2×D584C2×C4semidirect
55. #14556537C2×C4semidirect
56. #14556535C2×C4semidirect
57. #14556534C2×C4semidirect
58. #14556532C2×C4semidirect
59. #14556531C2×C4semidirect
61. C2×#11924003C2×C4semidirect
206. #2148D73semidirect
209. C73#33469semidirect
232. #362C2×D73semidirect
233. #361C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226230652
#226188472
#226188152
#226118322
#226118252
C2×#179884112
C2×#179883982
#226115352
C73×#21482
#3346973