Finite Group Databasebeta

← #33055920

Subgroups — #33055920

Order 18688.

766 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619932C211yesmaximal, characteristic
2#22618642C211yesmaximal, characteristic, complemented
3C2×#17988652C211yesmaximal, characteristic, complemented
4#22608794C214yesmaximal, complemented
5#22608754C214yesmaximal, complemented
6#22608355C211yesmaximal, characteristic, complemented
7#22608329C211yesmaximal, characteristic, complemented
8#22608224C211yesmaximal, characteristic, complemented
9C73×#1570C211yesmaximal, characteristic, complemented
10C2×#14556599—21no—
11C22×C8×D73—21no—
12#17989137C2211yescharacteristic
13C2×C4×D292C2211yescharacteristic
14#17988901C412yes—
15#17988868C412yes—
16#17988783C412yes—
17#17988723C412yes—
18C4×D584C2214yes—
19#17988664—21no—
20#17988664C2212yescomplemented
21#17988652—21no—
22#17988652—21no—
23#17988652—21no—
24#17988652C2212yes—
25#17988652C2212yes—
26#17988652C2212yescomplemented
27#17988640C2211yescharacteristic
28#17988555—21no—
29C2×#14556503—21no—
30C2×C4×#10596831C2211yescharacteristic
31#17988314C2214yes—
32#17988314C2214yescomplemented
33#17988309C2212yescomplemented
34#17988308C2212yescomplemented
35#431×C73C2211yescharacteristic
36#423×C73C2211yescharacteristic
37C2×C4⋊C8×C73C2211yescharacteristic
38#334×C73C2214yescomplemented
39C2×#11924028—21no—
40C22×D292C2×C411yescharacteristic
41C22×C4×D73—21no—
42C2×D584C2×C414yes—
43#14556599—44no—
44C2×#11924011—22no—
45C22×#10596831D411yescharacteristic
46C22×#10596831D411yescharacteristic
47C2×C8×D73—22no—
48C2×C8×D73—22no—
49C2×C8×D73—22no—
50C2×C8×D73—44no—
51#14556591C2×C411yescharacteristic
52#14556588C2×C411yescharacteristic
53#14556579—21no—
54#14556565—21no—
55#14556565C2×C412yes—
56C4×D292—21no—
57C4×D292—21no—
58C4×D292—21no—
59C4×D292C2×C412yes—
60C4×D292C2312yes—
61C4×D292C2312yes—
62C2×C4⋊Dic73C2×C411yescharacteristic
63C2×#11924005—21no—
64#14556525C2×C414yes—
65C4⋊#10596831—21no—
66#14556513C2×C414yes—
67#14556513C2×C414yes—
68C22⋊#10596831—21no—
69#14556503—22no—
70#14556503—22no—
71#14556503—22no—
72#14556503—22no—
73C4×#10596831D411yescharacteristic
74C4×#10596831D411yescharacteristic
75C4×#10596831C2312yes—
76C4×#10596831C2311yescharacteristic
77C4×#10596831C2311yescharacteristic
78C2×M16×C73—21no—
79C22×C8×C73—21no—
80C2×C42×C73C2311yescharacteristic
81C4⋊C8×C73—21no—
82C4⋊C8×C73—21no—
83C4⋊C8×C73C2×C412yes—
84C4⋊C8×C73C2312yes—
85C4⋊C8×C73C2312yes—
86C22⋊C8×C73C2×C412yes—
87C4×C8×C73C2312yes—
88C23×D73—21no—
89#11924028—42no—
90#11924028—42no—
91C2×Dic146—21no—
92C2×D292—21no—
93C2×D292—24no—
94C2×D292—21no—
95C2×D292—21no—
96C2×D292C22×C412yes—
97C2×D292C22×C412yes—
98C2×C22⋊D73—22no—
99C22×Dic73—21no—
100C2×C4×D73—22no—
101C2×C4×D73—22no—
102C2×C4×D73—22no—
103C2×C4×D73—42no—
104C2×C4×D73—42no—
105D584—28no—
106#11924011—44no—
107C2×#10596831—22no—
108C2×#10596831—22no—
109C2×#10596831—22no—
110C2×#10596831C2×D411yescharacteristic
111C2×#10596831C2×D411yescharacteristic
112C2×#10596831C2×D411yescharacteristic
113C2×#10596831C2×D411yescharacteristic
114C8×D73—44no—
115C8×D73—44no—
116C8×D73—44no—
117C4⋊Dic73—21no—
118C4⋊Dic73—21no—
119C4⋊Dic73C22×C412yes—
120C4⋊Dic73C22×C412yes—
121#11924005—22no—
122#11924005—22no—
123#11924005—22no—
124#11924005—22no—
125C22×C4×C73C22×C411yescharacteristic
126C22×C4×C73C2×D411yescharacteristic
127C22×C4×C73#4111yescharacteristic
128M16×C73—42no—
129C2×C8×C73—22no—
130C2×C8×C73—22no—
131C2×C8×C73—42no—
132C2×C8×C73C22×C414yes—
133C42×C73C22×C411yescharacteristic
134C42×C73C2×D411yescharacteristic
135C42×C73#4111yescharacteristic
136C42×C73C2411yescharacteristic
137C23×C73C4×D411yescharacteristic
138C22×D73—24no—
139C22×D73—22no—
140C22×D73—22no—
141C22×D73—42no—
142C22×D73—42no—
143C2×C4×C73—22no—
144C2×C4×C73—41no—
145C2×C4×C73C4×D411yescharacteristic
146C2×C4×C73C4×D411yescharacteristic
147C2×C4×C73C4×D411yescharacteristic
148C2×C4×C73AGL(1,ℤ/8)12yes—
149C2×C4×C73AGL(1,ℤ/8)12yes—
150C2×C4×C73C23×C411yescharacteristic
151C2×C4×C73C22×D411yescharacteristic
152C2×C4×C73C2×#4111yescharacteristic
153C22⋊D73—44no—
154C2×Dic73—22no—
155C2×Dic73—22no—
156C2×Dic73—42no—
157D292—24no—
158D292—22no—
159D292—24no—
160D292—42no—
161D292—42no—
162D292#13114yescomplemented
163C4×D73—42no—
164C4×D73—42no—
165C4×D73—42no—
166C4×D73—42no—
167C4×D73—44no—
168Dic146—42no—
169C8×C73—24no—
170C8×C73—42no—
171C8×C73—42no—
172#10596831—21no—
173#10596831—21no—
174#10596831—21no—
175#10596831—21no—
176#10596831—42no—
177C22×C73—21no—
178C22×C73—21no—
179C22×C73#44212yes—
180C22×C73C2×C4×D411yescharacteristic
181C2×D73—24no—
182C2×D73—48no—
183C2×D73—42no—
184C2×D73—42no—
185C2×D73—42no—
186C2×D73—42no—
187C2×D73—42no—
188C4×C73—21no—
189C4×C73—21no—
190C4×C73—21no—
191C4×C73#44212yes—
192C4×C73C2×#13112yes—
193C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
194C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
195Dic73—42no—
196Dic73—42no—
197#17907—731nomaximal, Sylow
198C2×C73—22no—
199C2×C73#292911yescharacteristic
200C2×C73C2×#44211yescharacteristic
201C2×C73#295511yescharacteristic
202D73—44no—
203D73—82no—
204#3100—731no—
205#2931—731no—
206C2×C8×D4—731no—
207#1586—734no—
208#1580—734no—
209#1570D7311yescharacteristic, complemented
210#1486—731no—
211#1480—731no—
212#1452—731no—
213C73#1790711yescharacteristic, complemented, Sylow
214C2×#131—1461no—
215C23×C8—1461no—
216#532—731no—
217C2×C4×D4—731no—
218#480—732no—
219#474—732no—
220#459—732no—
221#447—732no—
222C4×D8—734no—
223#436—732no—
224#436—1461no—
225C8×D4—732no—
226C8×D4—732no—
227C8×D4—732no—
228C8×D4—1461no—
229C8×D4—1461no—
230C8×D4—1461no—
231#431—731no—
232#431C2×D7311yescharacteristic
233#423C2×D7311yescharacteristic
234C2×C4⋊C8C2×D7311yescharacteristic
235#408—1461no—
236C2×C22⋊C8—1461no—
237C2×C4×C8—731no—
238#334C2×D7314yescomplemented
239D4⋊C8—734no—
240D4⋊C8—734no—
241#322—732no—
242#322—732no—
243C2×#41—1461no—
244C22×D4—731no—
245C23×C4—1461no—
246C2×D8—734no—
247#131—2924no—
248C2×M16—21no—
249C2×M16—1462no—
250C22×C8—21no—
251C22×C8—2924no—
252C22×C8—731no—
253C22×C8—731no—
254C22×C8—1462no—
255C22×C8—1462no—
256C22×C8—1462no—
257C4⋊Q8—731no—
258#127—731no—
259#124—1461no—
260#121—732no—
261#121—1461no—
262C4×D4—732no—
263C4×D4—732no—
264C4×D4—732no—
265C4×D4—1461no—
266C4×D4—1461no—
267C4×D4—1461no—
268C2×C4⋊C4—731no—
269C2×C22⋊C4—1461no—
270C2×C42C22×D7311yescharacteristic
271#107—734no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—1461no—
275C4⋊C8C4×D7312yes—
276C4⋊C8C22×D7312yes—
277C4⋊C8C22×D7312yes—
278#102—734no—
279#102—734no—
280C22⋊C8—1462no—
281C22⋊C8—1462no—
282C22⋊C8—1462no—
283C22⋊C8—1462no—
284C22⋊C8—1461no—
285C22⋊C8C4×D7312yes—
286C4×C8—731no—
287C4×C8—731no—
288C4×C8—731no—
289C4×C8—731no—
290C4×C8—732no—
291C4×C8C22×D7312yes—
292C24—1461no—
293#41—2922no—
294#41—2922no—
295C2×Q8—1461no—
296C2×D4—732no—
297C2×D4—732no—
298C2×D4—1464no—
299C2×D4—1462no—
300C2×D4—1461no—
301C2×D4—1461no—
302C2×D4—1461no—
303C22×C4—2922no—
304C22×C4—2922no—
305C22×C4—1461no—
306C22×C4—1462no—
307C22×C4—1462no—
308C22×C4—1462no—
309C22×C4C2×C4×D7311yescharacteristic
310C22×C4D4×D7311yescharacteristic
311C22×C4#1192402911yescharacteristic
312D8—1468no—
313M16—42no—
314M16—2924no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—42no—
318C2×C8—2924no—
319C2×C8—2924no—
320C2×C8—2924no—
321C2×C8—731no—
322C2×C8—731no—
323C2×C8—731no—
324C2×C8—731no—
325C2×C8—1462no—
326C2×C8—1462no—
327C2×C8—1462no—
328C2×C8C2×C4×D7314yes—
329C4⋊C4—732no—
330C4⋊C4—732no—
331C4⋊C4—1461no—
332C4⋊C4—1461no—
333C22⋊C4—1462no—
334C22⋊C4—1462no—
335C22⋊C4—1462no—
336C22⋊C4—1462no—
337C42C2×C4×D7311yescharacteristic
338C42D4×D7311yescharacteristic
339C42#1192402911yescharacteristic
340C42C23×D7311yescharacteristic
341C23—2922no—
342C23—2922no—
343C23—1464no—
344C23—1462no—
345C23—1462no—
346C23#1455655311yescharacteristic
347Q8—2922no—
348D4—2922no—
349D4—2924no—
350D4—2922no—
351D4—734no—
352D4—1464no—
353D4—1464no—
354D4—1462no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—2922no—
358C2×C4—2924no—
359C2×C4—2922no—
360C2×C4—2922no—
361C2×C4—2922no—
362C2×C4—2922no—
363C2×C4—1462no—
364C2×C4—1462no—
365C2×C4#1455655311yescharacteristic
366C2×C4#1455655311yescharacteristic
367C2×C4#1455655311yescharacteristic
368C2×C4#1455661812yes—
369C2×C4#1455661812yes—
370C2×C4C22×C4×D7311yescharacteristic
371C2×C4C2×D4×D7311yescharacteristic
372C2×C4C2×#1192402911yescharacteristic
373C8—24no—
374C8—42no—
375C8—42no—
376C8—2922no—
377C8—1461no—
378C8—1461no—
379C8—1461no—
380C8—1461no—
381C22—21no—
382C22—21no—
383C22—2928no—
384C22—2922no—
385C22—2922no—
386C22—2922no—
387C22—2922no—
388C22—2922no—
389C22—1464no—
390C22#1798869812yes—
391C22C2×#1455655311yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—2922no—
396C4—2922no—
397C4#1798869812yes—
398C4#131×D7312yes—
399C4C2×#1455661811yescharacteristic
400C4C2×#1455661811yescharacteristic
401C2—22no—
402C2—2924no—
403C2—5842no—
404C2#2261862411yescharacteristic
405C2C2×#1798869811yescharacteristic
406C2#2261883311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1570C2
socleC22×C73C2×C4×D4
prime radical#22619932C2
cosocleC42×C73C24
hypercenter#1570D73
hypocenterC73#17907
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
2. #22618642C2semidirect
3. C2×#17988652C2semidirect
4. #22608794C2semidirect
5. #22608754C2semidirect
6. #22608355C2semidirect
7. #22608329C2semidirect
8. #22608224C2semidirect
9. C73×#1570C2semidirect
20. #17988664C22semidirect
26. #17988652C22semidirect
32. #17988314C22semidirect
33. #17988309C22semidirect
34. #17988308C22semidirect
38. #334×C73C22semidirect
162. D292#131semidirect
209. #1570D73semidirect
213. C73#17907semidirect
238. #334C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226199322
#226186422
C2×#179886522
#226087942
#226087542
#226083552
#226083292
#226082242
C73×#15702
#1790773