Finite Group Databasebeta

← #33950320

Subgroups — #33950320

Order 18688.

2612 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22624705C211yesmaximal, characteristic
2#22624644C212yesmaximal, complemented
3#22618847C216yesmaximal, complemented
4#22618840C216yesmaximal, complemented
5#22618833C216yesmaximal, complemented
6C2×#17988703C216yesmaximal, complemented
7#22618476C212yesmaximal, complemented
8#131×Dic73C211yesmaximal, characteristic, complemented
9C73×#2916C211yesmaximal, characteristic, complemented
10#17989423C2212yes—
11#17989404C414yescomplemented
12#17989399C418yescomplemented
13#17989392C414yescomplemented
14#17989387C2216yes—
15C2×#14556613C2216yes—
16#17989353C22112yescomplemented
17#17989352—28no—
18C2×#14556599C22112yescomplemented
19C2×#14556598C2211yescharacteristic
20#131×D73—28no—
21#41×Dic73C2211yescharacteristic
22#17988708C22124yescomplemented
23#17988703C2214yescomplemented
24#17988703C22112yescomplemented
25#17988703C22112yescomplemented
26#17988703C22112yescomplemented
27#17988603C22112yescomplemented
28C2×#14556517C22112yescomplemented
29#17988545C2213yescomplemented
30#17988542C2213yescomplemented
31M16×Dic73C2213yescomplemented
32C2×C8×Dic73C2213yescomplemented
33#568×C73C2212yes—
34C2×#131×C73C2211yescharacteristic
35#430×C73C2216yescomplemented
36C2×#108×C73C2216yescomplemented
37#14556639—22no—
38#14556638C2×C418yes—
39#14556637C2×C418yes—
40#14556636—26no—
41C2×#11924028C23112yes—
42#41×D73—28no—
43#14556624C2×C4124yescomplemented
44#14556619C2×C4124yescomplemented
45#14556613C2×C4112yescomplemented
46#14556613C2×C4112yescomplemented
47#14556613C23112yes—
48#14556613C2314yes—
49C2×Dic292C2×C416yescomplemented
50C2×#11924017C2×C4112yescomplemented
51C2×D584C2×C416yescomplemented
52#14556599—212no—
53#14556599—212no—
54#14556599C2318yes—
55#14556599C23124yes—
56#14556598D414yes—
57#14556598D414yes—
58#14556597—224no—
59C2×#11924012C2313yes—
60C2×#11924011—212no—
61M16×D73—224no—
62C22×#10596831C2313yes—
63C2×C8×D73—212no—
64Q8×Dic73—21no—
65D4×Dic73—23no—
66#14556550C2313yes—
67C2×C4×Dic73C2313yes—
68#14556517C2318yes—
69#14556517C23124yes—
70#14556501C2316yes—
71C8×Dic73C2311yescharacteristic
72C8×Dic73C2313yes—
73C8×Dic73C2313yes—
74C8×Dic73C2313yes—
75C2×#41×C73C2311yescharacteristic
76#131×C73—28no—
77#131×C73C2×C414yes—
78#131×C73C2×C414yes—
79C2×M16×C73C23112yes—
80C2×M16×C73C2313yes—
81C22×C8×C73C2313yes—
82#108×C73C2314yes—
83#108×C73C23112yes—
84#11924029—28no—
85#11924028—212no—
86#11924028—212no—
87#11924028C22×C4124yes—
88#11924028C2418yes—
89#11924027—224no—
90C2×Dic146C22×C4112yes—
91Q8×D73—28no—
92C2×D292C22×C4112yes—
93C2×C22⋊D73—212no—
94D4×D73—224no—
95C22×Dic73—23no—
96C2×C4×D73—212no—
97Dic292C22×C414yes—
98Dic292C22×C4112yes—
99#11924017C22×C418yes—
100#11924017C22×C4124yes—
101D584C22×C414yes—
102D584C22×C4112yes—
103#11924012C2×D4112yes—
104#11924011—224no—
105#11924011—28no—
106C2×#10596831C2×D416yes—
107C2×#10596831C2×D416yes—
108C2×#10596831C2411yescharacteristic
109C2×#10596831C2413yes—
110C8×D73—28no—
111C8×D73—224no—
112C4⋊Dic73—23no—
113C22⋊Dic73—23no—
114C4×Dic73—23no—
115C4×Dic73C2411yescharacteristic
116C4×Dic73C2413yes—
117#41×C73C22×C414yes—
118#41×C73C2×D414yes—
119C2×Q8×C73#4111yescharacteristic
120C2×D4×C73#4113yes—
121C22×C4×C73C2413yes—
122M16×C73—212no—
123M16×C73C22×C4112yes—
124M16×C73C2418yes—
125C2×C8×C73—212no—
126C2×C8×C73C22×C416yes—
127C2×C8×C73C22×C416yes—
128C2×C8×C73C2411yescharacteristic
129C2×C8×C73C2413yes—
130C23×C73C2×#4113yes—
131C22×D73—212no—
132Q8×C73C4×D414yes—
133D4×C73C4×D4112yes—
134C2×C4×C73C23×C416yes—
135C2×C4×C73C22×D416yes—
136C2×C4×C73C2×#4113yes—
137C2×C4×C73C2511yescharacteristic
138C22⋊D73—224no—
139C22⋊D73—28no—
140C2×Dic73—212no—
141C2×Dic73—21no—
142C2×Dic73—23no—
143C2×Dic73—46no—
144D292—212no—
145D292C23×C418yes—
146C4×D73—224no—
147C4×D73—28no—
148Dic146—212no—
149Dic146C23×C418yes—
150C8×C73—28no—
151C8×C73C23×C411yescharacteristic
152C8×C73C23×C411yescharacteristic
153C8×C73C23×C416yes—
154#10596831C22×D411yescharacteristic
155#10596831C22×D416yes—
156#10596831C22×D411yescharacteristic
157C22×C73—43no—
158C22×C73C2×C4×D416yes—
159C22×C73C22×#4111yescharacteristic
160C2×D73—28no—
161C2×D73—812no—
162C4×C73C2×C4×D416yes—
163C4×C73C24×C411yescharacteristic
164C4×C73C23×D411yescharacteristic
165Dic73—22no—
166Dic73—28no—
167Dic73—43no—
168#60183—731nomaximal, Sylow
169C2×C73—21no—
170C2×C73—26no—
171C2×C73C22×C4×D411yescharacteristic
172D73—48no—
173#3584—731no—
174#3573—732no—
175#2957—736no—
176#2956—736no—
177#2955—736no—
178C2×#443—736no—
179#2916D7311yescharacteristic, complemented
180#2901—732no—
181C4×#131—731no—
182C73#6018311yescharacteristic, complemented, Sylow
183#585—732no—
184#578—734no—
185#577—738no—
186#576—734no—
187#575—736no—
188C2×#135—736no—
189#568—7312no—
190#568—1468no—
191#568C2×D7312yes—
192C2×#131—731no—
193C2×#131—7312no—
194C2×#131—1468no—
195C2×#131C2×D7311yescharacteristic
196C4×#41—731no—
197#444—7324no—
198#443—734no—
199#443—7312no—
200#443—7312no—
201#443—7312no—
202#430C2×D7316yescomplemented
203C2×#108C2×D7316yescomplemented
204#421—7312no—
205C2×#104—7312no—
206#405—733no—
207#405—733no—
208C4×M16—733no—
209C2×C4×C8—733no—
2102−1+4—738no—
2112−1+4—1462no—
212H(5,𝔽2)—738no—
213H(5,𝔽2)—1466no—
214C2×#41—7312no—
215C2×#41—1468no—
216C2×#41C22×D7311yescharacteristic
217#137—7324no—
218AGL(1,ℤ/8)—7324no—
219#135—7312no—
220#135—7312no—
221#135—734no—
222#135—7312no—
223C2×Q16—736no—
224C2×SD16—7312no—
225C2×D8—736no—
226#131—28no—
227#131—734no—
228#131—734no—
229#131—738no—
230#131—7324no—
231#131—14624no—
232#131—14612no—
233#131—14612no—
234#131C4×D7314yes—
235#131C4×D7314yes—
236C2×M16—733no—
237C2×M16—14612no—
238C2×M16—14624no—
239C2×M16C22×D73112yes—
240C2×M16C22×D7313yes—
241C22×C8—733no—
242C22×C8—14612no—
243C22×C8C22×D7313yes—
244C4×Q8—1461no—
245C4×D4—1463no—
246#117—733no—
247C2×C42—733no—
248#108C22×D7314yes—
249#108C22×D73112yes—
250#104—738no—
251#104—7324no—
252#97—736no—
253C4×C8—731no—
254C4×C8—733no—
255C4×C8—733no—
256C4×C8—733no—
257#41—7324no—
258#41—738no—
259#41—1468no—
260#41—14624no—
261#41—14612no—
262#41—14612no—
263#41C2×C4×D7314yes—
264#41D4×D7314yes—
265C2×Q8—7312no—
266C2×Q8—1468no—
267C2×Q8#1192402911yescharacteristic
268C2×D4—7312no—
269C2×D4—14612no—
270C2×D4—14624no—
271C2×D4#1192402913yes—
272C22×C4—1463no—
273C22×C4—14612no—
274C22×C4C23×D7313yes—
275Q16—734no—
276Q16—7312no—
277SD16—738no—
278SD16—7324no—
279D8—734no—
280D8—7312no—
281M16—212no—
282M16—7312no—
283M16—1468no—
284M16—14624no—
285M16C2×C4×D73112yes—
286M16C23×D7318yes—
287C2×C8—212no—
288C2×C8—736no—
289C2×C8—736no—
290C2×C8—731no—
291C2×C8—733no—
292C2×C8—1468no—
293C2×C8—14624no—
294C2×C8C2×C4×D7316yes—
295C2×C8C2×C4×D7316yes—
296C2×C8C23×D7311yescharacteristic
297C2×C8C23×D7313yes—
298C4⋊C4—1463no—
299C22⋊C4—1463no—
300C42—731no—
301C42—733no—
302C42—1463no—
303C23—14612no—
304C23C2×#1192402913yes—
305Q8—738no—
306Q8—14612no—
307Q8#1455655314yes—
308D4—738no—
309D4—1468no—
310D4—14624no—
311D4—14612no—
312D4#14556553112yes—
313C2×C4—2926no—
314C2×C4—1463no—
315C2×C4—1461no—
316C2×C4—14612no—
317C2×C4—1468no—
318C2×C4—14624no—
319C2×C4C22×C4×D7316yes—
320C2×C4C2×D4×D7316yes—
321C2×C4C2×#1192402913yes—
322C2×C4C24×D7311yescharacteristic
323C8—28no—
324C8—731no—
325C8—731no—
326C8—736no—
327C8C22×C4×D7311yescharacteristic
328C8C22×C4×D7311yescharacteristic
329C8C22×C4×D7316yes—
330C22—43no—
331C22—58412no—
332C22—1468no—
333C22C2×#1455655316yes—
334C22C22×#1192402911yescharacteristic
335C4—2923no—
336C4—1462no—
337C4—1468no—
338C4C2×#1455655316yes—
339C4C23×C4×D7311yescharacteristic
340C4C22×D4×D7311yescharacteristic
341C2—21no—
342C2—26no—
343C2—2928no—
344C2C22×#1455655311yescharacteristic
subgroupG/H
centerC4C22×D4×D73
commutatorC4×C73C24×C4
FrattiniC2×C4C24×D73
FittingC73×#2916C2
socleC2×C73C22×C4×D4
prime radical#22624705C2
cosocleC2×C4×C73C25
hypercenter#2916D73
hypocenterC73#60183
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×#41 ◁ #2916

Complements

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

NH ≅ G/Ntype
2. #22624644C2semidirect
3. #22618847C2semidirect
4. #22618840C2semidirect
5. #22618833C2semidirect
6. C2×#17988703C2semidirect
7. #22618476C2semidirect
8. #131×Dic73C2semidirect
9. C73×#2916C2semidirect
11. #17989404C4semidirect
12. #17989399C4semidirect
13. #17989392C4semidirect
16. #17989353C22semidirect
18. C2×#14556599C22semidirect
22. #17988708C22semidirect
23. #17988703C22semidirect
24. #17988703C22semidirect
25. #17988703C22semidirect
26. #17988703C22semidirect
27. #17988603C22semidirect
28. C2×#14556517C22semidirect
29. #17988545C22semidirect
30. #17988542C22semidirect
31. M16×Dic73C22semidirect
32. C2×C8×Dic73C22semidirect
35. #430×C73C22semidirect
36. C2×#108×C73C22semidirect
43. #14556624C2×C4semidirect
44. #14556619C2×C4semidirect
45. #14556613C2×C4semidirect
46. #14556613C2×C4semidirect
49. C2×Dic292C2×C4semidirect
50. C2×#11924017C2×C4semidirect
51. C2×D584C2×C4semidirect
179. #2916D73semidirect
182. C73#60183semidirect
202. #430C2×D73semidirect
203. C2×#108C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226247052
#226246442
#226188472
#226188402
#226188332
C2×#179887032
#226184762
#131×Dic732
C73×#29162
#6018373