Finite Group Databasebeta

← #33239221

Subgroups — #33239221

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989005C211yesmaximal, characteristic, complemented
2C2×#17989004C211yesmaximal, characteristic, complemented
3#22618832C211yesmaximal, characteristic
4#22611770C214yesmaximal, complemented
5#22611765C214yesmaximal, complemented
6#22611696C211yesmaximal, characteristic, complemented
7#22611558C211yesmaximal, characteristic, complemented
8#22611530C211yesmaximal, characteristic, complemented
9C73×#2117C211yesmaximal, characteristic, complemented
10#17989385C411yescharacteristic
11C2×#14556597C2211yescharacteristic
12#17989005—21no—
13#17989005—21no—
14#17989005C2212yescomplemented
15#17989005C2212yescomplemented
16#17989004—21no—
17#17989004—21no—
18#17989004C2212yescomplemented
19#17989004C2212yescomplemented
20C2×#14556531—21no—
21C2×#14556531C2211yescharacteristic
22C2×#14556530—21no—
23C22×#11924003—21no—
24C22×#11924003C2211yescharacteristic
25C2×C16×D73—21no—
26#17988707C414yes—
27#17988702C2214yes—
28#17988635C411yescharacteristic
29#17988604C411yescharacteristic
30#17988603C411yescharacteristic
31#17988542C2211yescharacteristic
32#17988407C2214yescomplemented
33#17988370C2214yescomplemented
34#17988362C2212yescomplemented
35C16×Dic73C2212yescomplemented
36C2×#131×C73C2211yescharacteristic
37C2×M32×C73C2211yescharacteristic
38C2×M32×C73C2211yescharacteristic
39#350×C73C2214yescomplemented
40C2×#11924028C2×C411yescharacteristic
41#14556622C812yes—
42#14556621C812yes—
43#14556617C812yes—
44#14556616C812yes—
45#14556611C2×C414yes—
46#14556597—21no—
47#14556597—21no—
48#14556597C2×C412yes—
49#14556597C2312yes—
50C2×#11924012C2×C411yescharacteristic
51C2×#11924012C2×C411yescharacteristic
52C2×#11924011—21no—
53C2×C8×D73—21no—
54#14556550C2×C411yescharacteristic
55#14556531—22no—
56#14556531—22no—
57#14556531—21no—
58#14556531C2312yes—
59#14556530—22no—
60#14556530—22no—
61C2×#11924003—22no—
62C2×#11924003—22no—
63C2×#11924003—21no—
64C2×#11924003D411yescharacteristic
65C2×#11924003D411yescharacteristic
66C2×#11924003D411yescharacteristic
67C2×#11924003D411yescharacteristic
68C2×#11924003C2311yescharacteristic
69C2×#11924003C2311yescharacteristic
70C16×D73—22no—
71C16×D73—22no—
72#14556526C2×C414yes—
73#14556518C2×C414yes—
74#14556517C2×C414yes—
75#14556501C2×C412yes—
76C8×Dic73C2312yes—
77C2×#41×C73C2×C411yescharacteristic
78#131×C73—21no—
79#131×C73—21no—
80#131×C73C2×C412yes—
81#131×C73C2312yes—
82C2×M16×C73—21no—
83C2×M16×C73C2×C411yescharacteristic
84C22×C8×C73—21no—
85C22×C8×C73C2311yescharacteristic
86M32×C73—21no—
87M32×C73—21no—
88M32×C73—21no—
89M32×C73C2312yes—
90C2×C16×C73—21no—
91C2×C16×C73C2312yes—
92#11924028—21no—
93#11924028—21no—
94#11924028C2×C812yes—
95#11924028C22×C412yes—
96C2×Dic146C2×C811yescharacteristic
97C2×D292C2×C811yescharacteristic
98C2×C22⋊D73—21no—
99C2×C4×D73—21no—
100#11924018C2×C814yes—
101#11924016C2×C814yes—
102#11924015C2×C814yes—
103#11924013C2×C814yes—
104#11924012—21no—
105#11924012C2×C812yes—
106#11924012C2×C812yes—
107#11924012C22×C412yes—
108#11924011—22no—
109#11924011—22no—
110C2×#10596831—21no—
111C2×#10596831C22×C412yes—
112C8×D73—22no—
113C8×D73—22no—
114C4⋊Dic73—21no—
115C22⋊Dic73—21no—
116C4×Dic73C22×C412yes—
117#11924003—22no—
118#11924003C2×D411yescharacteristic
119#11924003C2×D411yescharacteristic
120#11924003C2×D411yescharacteristic
121#11924003C2×D411yescharacteristic
122#41×C73—21no—
123#41×C73—21no—
124#41×C73C2×C812yes—
125#41×C73C22×C412yes—
126C2×Q8×C73C2×C811yescharacteristic
127C2×D4×C73—21no—
128C2×D4×C73C2×C811yescharacteristic
129C22×C4×C73—21no—
130C22×C4×C73C22×C411yescharacteristic
131M16×C73—21no—
132M16×C73—22no—
133M16×C73—22no—
134M16×C73C22×C412yes—
135C2×C8×C73—21no—
136C2×C8×C73—22no—
137C2×C8×C73—22no—
138C2×C8×C73C22×C411yescharacteristic
139C2×C8×C73C2×D411yescharacteristic
140C2×C8×C73C2×D411yescharacteristic
141C2×C8×C73#4111yescharacteristic
142C2×C8×C73#4111yescharacteristic
143C2×C8×C73C2411yescharacteristic
144C16×C73—22no—
145C16×C73—22no—
146C23×C73—21no—
147C23×C73#13111yescharacteristic
148C22×D73—21no—
149Q8×C73—21no—
150Q8×C73C22×C812yes—
151D4×C73—22no—
152D4×C73—22no—
153D4×C73—21no—
154D4×C73C22×C812yes—
155C2×C4×C73—21no—
156C2×C4×C73—22no—
157C2×C4×C73—22no—
158C2×C4×C73C4×D411yescharacteristic
159C2×C4×C73C4×D411yescharacteristic
160C2×C4×C73C22×C811yescharacteristic
161C2×C4×C73C22×C811yescharacteristic
162C2×C4×C73#13111yescharacteristic
163C2×C4×C73C23×C411yescharacteristic
164C22⋊D73—22no—
165C22⋊D73—22no—
166C2×Dic73—21no—
167C2×Dic73—22no—
168D292—21no—
169D292C22×C812yes—
170C4×D73—22no—
171C4×D73—22no—
172Dic146—21no—
173Dic146C22×C812yes—
174C8×C73—22no—
175C8×C73C4×D411yescharacteristic
176C8×C73C4×D411yescharacteristic
177C8×C73C22×D411yescharacteristic
178C8×C73C2×#4111yescharacteristic
179#10596831—22no—
180#10596831C22×C814yes—
181C22×C73—22no—
182C22×C73—41no—
183C22×C73—42no—
184C22×C73C8×D411yescharacteristic
185C22×C73C8×D411yescharacteristic
186C22×C73C2×#13111yescharacteristic
187C2×D73—22no—
188C2×D73—42no—
189C4×C73—22no—
190C4×C73C8×D411yescharacteristic
191C4×C73C8×D411yescharacteristic
192C4×C73C2×C4×D411yescharacteristic
193C4×C73C23×C811yescharacteristic
194Dic73—22no—
195Dic73—42no—
196#33441—731nomaximal, Sylow
197C2×C73—21no—
198C2×C73—21no—
199C2×C73—21no—
200C2×C73—42no—
201C2×C73C2×C8×D411yescharacteristic
202D73—42no—
203C2×#504—731no—
204C2×#504—731no—
205#2955—731no—
206#2171—734no—
207#2170—734no—
208#2153—731no—
209#2117—731no—
210#2117D7311yescharacteristic, complemented
211#2108—731no—
212C73#3344111yescharacteristic, complemented, Sylow
213#575—731no—
214C2×#131—731no—
215C2×#131C2×D7311yescharacteristic
216#504—732no—
217#504—732no—
218#504—732no—
219#504—732no—
220#504—1461no—
221#504—1461no—
222#504—1461no—
223#504—1461no—
224C2×M32—731no—
225C2×M32—1461no—
226C2×M32—1461no—
227C2×M32C2×D7311yescharacteristic
228C2×M32C2×D7311yescharacteristic
229C22×C16—731no—
230C22×C16—1461no—
231C22×C16—1461no—
232#444—734no—
233#443—734no—
234#430—731no—
235#421—731no—
236#421—731no—
237#405—731no—
238#364—734no—
239#350—734no—
240#350C2×D7314yescomplemented
241#346—732no—
242C4×C16—732no—
243C2×#41—731no—
244C2×#41C4×D7311yescharacteristic
245#137—732no—
246#137—732no—
247AGL(1,ℤ/8)—732no—
248AGL(1,ℤ/8)—732no—
249#135—734no—
250#131—21no—
251#131—21no—
252#131—732no—
253#131—732no—
254#131—1461no—
255#131—1461no—
256#131C4×D7312yes—
257#131C22×D7312yes—
258C2×M16—21no—
259C2×M16—731no—
260C2×M16—731no—
261C2×M16—1461no—
262C2×M16C4×D7311yescharacteristic
263C22×C8—21no—
264C22×C8—1461no—
265C22×C8C22×D7311yescharacteristic
266#117—731no—
267M32—21no—
268M32—21no—
269M32—21no—
270M32—732no—
271M32—1462no—
272M32—1462no—
273M32—1462no—
274M32—1462no—
275M32—1461no—
276M32C22×D7312yes—
277C2×C16—21no—
278C2×C16—731no—
279C2×C16—731no—
280C2×C16—731no—
281C2×C16—731no—
282C2×C16—731no—
283C2×C16—731no—
284C2×C16—1462no—
285C2×C16—1462no—
286C2×C16—1462no—
287C2×C16—1462no—
288C2×C16—1461no—
289C2×C16C22×D7312yes—
290#108—734no—
291#104—734no—
292#104—734no—
293#97—732no—
294C4×C8—732no—
295#41—21no—
296#41—21no—
297#41—732no—
298#41—732no—
299#41—1461no—
300#41—1461no—
301#41C8×D7312yes—
302#41C2×C4×D7312yes—
303C2×Q8—731no—
304C2×Q8C8×D7311yescharacteristic
305C2×D4—21no—
306C2×D4—731no—
307C2×D4—1461no—
308C2×D4C8×D7311yescharacteristic
309C22×C4—21no—
310C22×C4—1461no—
311C22×C4C2×C4×D7311yescharacteristic
312Q16—734no—
313SD16—734no—
314SD16—734no—
315D8—734no—
316M16—21no—
317M16—22no—
318M16—22no—
319M16—732no—
320M16—732no—
321M16—732no—
322M16—1461no—
323M16—1462no—
324M16—1462no—
325M16C2×C4×D7312yes—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—732no—
330C2×C8—1461no—
331C2×C8—1462no—
332C2×C8—1462no—
333C2×C8C2×C4×D7311yescharacteristic
334C2×C8D4×D7311yescharacteristic
335C2×C8D4×D7311yescharacteristic
336C2×C8#1192402711yescharacteristic
337C2×C8#1192402711yescharacteristic
338C2×C8C23×D7311yescharacteristic
339C4⋊C4—1461no—
340C22⋊C4—1461no—
341C42—732no—
342C16—22no—
343C16—22no—
344C16—731no—
345C16—731no—
346C16—731no—
347C16—731no—
348C16—1462no—
349C23—21no—
350C23—1461no—
351C23#1455659911yescharacteristic
352Q8—21no—
353Q8—732no—
354Q8—1461no—
355Q8C2×C8×D7312yes—
356D4—22no—
357D4—22no—
358D4—21no—
359D4—732no—
360D4—1462no—
361D4—1462no—
362D4—1461no—
363D4C2×C8×D7312yes—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—1462no—
368C2×C4—1461no—
369C2×C4—1462no—
370C2×C4—1462no—
371C2×C4#1455655111yescharacteristic
372C2×C4#1455655111yescharacteristic
373C2×C4C2×C8×D7311yescharacteristic
374C2×C4C2×C8×D7311yescharacteristic
375C2×C4#1455659911yescharacteristic
376C2×C4C22×C4×D7311yescharacteristic
377C8—22no—
378C8—734no—
379C8—1462no—
380C8#1455655111yescharacteristic
381C8#1455655111yescharacteristic
382C8C2×D4×D7311yescharacteristic
383C8C2×#1192402711yescharacteristic
384C22—22no—
385C22—41no—
386C22—42no—
387C22—2922no—
388C22—1462no—
389C22#1798865011yescharacteristic
390C22#1798865011yescharacteristic
391C22C2×#1455659911yescharacteristic
392C4—22no—
393C4—2922no—
394C4—1462no—
395C4#1798865011yescharacteristic
396C4#1798865011yescharacteristic
397C4C2×#1455655111yescharacteristic
398C4C22×C8×D7311yescharacteristic
399C2—21no—
400C2—21no—
401C2—21no—
402C2—42no—
403C2—2922no—
404C2C2×#1798865011yescharacteristic
subgroupG/H
centerC8C2×D4×D73
commutatorC4×C73C23×C8
FrattiniC2×C8C23×D73
FittingC73×#2117C2
socleC2×C73C2×C8×D4
prime radical#17989385C4
cosocleC2×C8×C73C24
hypercenter#2117D73
hypocenterC73#33441
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
1. C2×#17989005C2semidirect
2. C2×#17989004C2semidirect
4. #22611770C2semidirect
5. #22611765C2semidirect
6. #22611696C2semidirect
7. #22611558C2semidirect
8. #22611530C2semidirect
9. C73×#2117C2semidirect
14. #17989005C22semidirect
15. #17989005C22semidirect
18. #17989004C22semidirect
19. #17989004C22semidirect
32. #17988407C22semidirect
33. #17988370C22semidirect
34. #17988362C22semidirect
35. C16×Dic73C22semidirect
39. #350×C73C22semidirect
210. #2117D73semidirect
212. C73#33441semidirect
240. #350C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890052
C2×#179890042
#226188322
#226117702
#226117652
#226116962
#226115582
#226115302
C73×#21172
#3344173