Finite Group Databasebeta

← #33056241

Subgroups — #33056241

Order 18688.

726 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#22619931C211yesmaximal, characteristic
2#22618642C211yesmaximal, characteristic, complemented
3C2×#17988665C211yesmaximal, characteristic, complemented
4#22608842C214yesmaximal, complemented
5#22608798C214yesmaximal, complemented
6#22608703C211yesmaximal, characteristic, complemented
7#22608331C211yesmaximal, characteristic, complemented
8#22608224C211yesmaximal, characteristic, complemented
9C73×#1486C211yesmaximal, characteristic, complemented
10C2×#14556599—21no—
11C22×#11924012—21no—
12#17989137C2211yescharacteristic
13#17988900C412yes—
14#17988867C412yes—
15#17988782C412yes—
16#17988721C412yes—
17C4×#11924013C2214yes—
18#17988665—21no—
19#17988665—21no—
20#17988665C2212yes—
21#17988665C2212yes—
22#17988664—21no—
23#17988664C2212yescomplemented
24#17988652—21no—
25#17988652C2212yescomplemented
26#17988640C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28#17988555—21no—
29C2×C22⋊#10596831—21no—
30C2×C4×#10596831C2211yescharacteristic
31#17988337C2214yescomplemented
32#17988316C2214yescomplemented
33#17988309C2212yescomplemented
34#17988308C2212yescomplemented
35C2×C4×D4×C73C2211yescharacteristic
36#431×C73C2211yescharacteristic
37#423×C73C2211yescharacteristic
38D4⋊C8×C73C2214yes—
39C2×#11924028—21no—
40C2×#11924013C2×C414yes—
41#14556599—44no—
42C2×#11924012—22no—
43C2×#11924012—22no—
44C2×#11924012—44no—
45C2×#11924011—22no—
46C22×#10596831—21no—
47C22×#10596831D411yescharacteristic
48C22×#10596831D411yescharacteristic
49C2×C8×D73—22no—
50#14556591C2×C411yescharacteristic
51#14556588C2×C411yescharacteristic
52#14556579—21no—
53#14556565—21no—
54#14556565C2×C412yes—
55C4×D292—21no—
56C4×D292C2312yes—
57#14556524C2×C414yes—
58C4⋊#10596831—21no—
59C4⋊#10596831C2×C412yes—
60C4⋊#10596831C2312yes—
61#14556512C2×C414yes—
62#14556511C2×C414yes—
63C22⋊#10596831—22no—
64C22⋊#10596831—22no—
65C22⋊#10596831C2×C412yes—
66#14556503—22no—
67#14556503—22no—
68C4×#10596831D411yescharacteristic
69C4×#10596831D411yescharacteristic
70C4×#10596831C2312yes—
71C4×#10596831C2311yescharacteristic
72C4×#10596831C2311yescharacteristic
73C22×D4×C73C2×C411yescharacteristic
74C23×C4×C73—21no—
75C2×M16×C73—21no—
76C4×D4×C73—21no—
77C4×D4×C73—21no—
78C4×D4×C73C2×C412yes—
79C4×D4×C73C2312yes—
80C2×C4⋊C4×C73C2×C411yescharacteristic
81C2×C22⋊C4×C73—21no—
82C2×C42×C73C2311yescharacteristic
83C4⋊C8×C73—21no—
84C4⋊C8×C73—21no—
85C4⋊C8×C73C2312yes—
86C22⋊C8×C73—21no—
87C4×C8×C73C2312yes—
88#11924028—42no—
89#11924028—42no—
90C2×Dic146—21no—
91C2×D292—21no—
92C2×D292—21no—
93C2×D292C22×C412yes—
94C2×C22⋊D73—22no—
95C2×C4×D73—22no—
96#11924013—44no—
97#11924012—44no—
98#11924012—44no—
99#11924011—44no—
100C2×#10596831—22no—
101C2×#10596831—22no—
102C2×#10596831—22no—
103C2×#10596831—42no—
104C2×#10596831C22×C414yes—
105C2×#10596831C2×D411yescharacteristic
106C2×#10596831C2×D411yescharacteristic
107C2×#10596831C2×D411yescharacteristic
108C2×#10596831C2×D411yescharacteristic
109C8×D73—44no—
110C4⋊Dic73—21no—
111C4⋊Dic73C22×C412yes—
112#11924005—22no—
113#11924005—22no—
114C24×C73—21no—
115C2×D4×C73—21no—
116C2×D4×C73—42no—
117C2×D4×C73C22×C412yes—
118C22×C4×C73—21no—
119C22×C4×C73—22no—
120C22×C4×C73—22no—
121C22×C4×C73—42no—
122C22×C4×C73—42no—
123C22×C4×C73C22×C411yescharacteristic
124C22×C4×C73C2×D411yescharacteristic
125C22×C4×C73#4111yescharacteristic
126M16×C73—42no—
127C2×C8×C73—22no—
128C2×C8×C73—22no—
129C4⋊C4×C73—21no—
130C4⋊C4×C73C22×C412yes—
131C22⋊C4×C73—22no—
132C22⋊C4×C73—22no—
133C42×C73C22×C411yescharacteristic
134C42×C73C2×D411yescharacteristic
135C42×C73#4111yescharacteristic
136C42×C73C2411yescharacteristic
137C23×C73—22no—
138C23×C73—42no—
139C23×C73—42no—
140C23×C73—42no—
141C23×C73C4×D411yescharacteristic
142C22×D73—22no—
143D4×C73—22no—
144D4×C73—22no—
145D4×C73—42no—
146C2×C4×C73—22no—
147C2×C4×C73—22no—
148C2×C4×C73—41no—
149C2×C4×C73—42no—
150C2×C4×C73—42no—
151C2×C4×C73—42no—
152C2×C4×C73—44no—
153C2×C4×C73C4×D411yescharacteristic
154C2×C4×C73C4×D411yescharacteristic
155C2×C4×C73C4×D411yescharacteristic
156C2×C4×C73AGL(1,ℤ/8)12yes—
157C2×C4×C73AGL(1,ℤ/8)12yes—
158C2×C4×C73C23×C411yescharacteristic
159C2×C4×C73C22×D411yescharacteristic
160C2×C4×C73C2×#4111yescharacteristic
161C22⋊D73—44no—
162C2×Dic73—22no—
163D292—24no—
164D292—42no—
165D292—42no—
166C4×D73—42no—
167C4×D73—42no—
168Dic146—42no—
169C8×C73—42no—
170C8×C73—42no—
171#10596831—24no—
172#10596831—21no—
173#10596831—21no—
174#10596831—21no—
175#10596831—21no—
176#10596831—42no—
177C22×C73—21no—
178C22×C73—21no—
179C22×C73—42no—
180C22×C73—42no—
181C22×C73—42no—
182C22×C73—84no—
183C22×C73#44212yes—
184C22×C73C2×C4×D411yescharacteristic
185C2×D73—42no—
186C2×D73—42no—
187C2×D73—42no—
188C4×C73—21no—
189C4×C73—21no—
190C4×C73—21no—
191C4×C73—42no—
192C4×C73#44212yes—
193C4×C73#56812yes—
194C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
195C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
196Dic73—42no—
197#17935—731nomaximal, Sylow
198C2×C73—22no—
199C2×C73—82no—
200C2×C73#293011yescharacteristic
201C2×C73C2×#44211yescharacteristic
202C2×C73#295511yescharacteristic
203D73—82no—
204#3100—731no—
205#2931—731no—
206C2×#436—731no—
207#1594—734no—
208#1586—734no—
209#1570—731no—
210#1486D7311yescharacteristic, complemented
211#1480—731no—
212#1452—731no—
213C73#1793511yescharacteristic, complemented, Sylow
214C2×#131—1461no—
215C22×M16—1461no—
216#532—731no—
217C2×C4×D4C2×D7311yescharacteristic
218#480—732no—
219#474—732no—
220#459—732no—
221#447—732no—
222C4×D8—734no—
223#436—732no—
224#436—732no—
225#436—732no—
226#436—1461no—
227#436—1461no—
228#436—1461no—
229C8×D4—732no—
230C8×D4—1461no—
231#431—731no—
232#431C2×D7311yescharacteristic
233#423C2×D7311yescharacteristic
234C2×C4⋊C8—731no—
235#408—1461no—
236C2×C22⋊C8—1461no—
237C2×C4×C8—731no—
238#334—734no—
239D4⋊C8—734no—
240D4⋊C8C2×D7314yes—
241#322—732no—
242#322—732no—
243C2×#41—1461no—
244C22×D4C4×D7311yescharacteristic
245C23×C4—21no—
246C2×D8—734no—
247#131—2924no—
248C2×M16—21no—
249C2×M16—2924no—
250C2×M16—1462no—
251C2×M16—1462no—
252C2×M16—1462no—
253C22×C8—731no—
254C22×C8—731no—
255C22×C8—1462no—
256C22×C8—1461no—
257C4⋊Q8—731no—
258#127—731no—
259#124—1461no—
260#121—732no—
261#121—1461no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—732no—
265C4×D4—1461no—
266C4×D4C4×D7312yes—
267C4×D4C22×D7312yes—
268C2×C4⋊C4C4×D7311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C42C22×D7311yescharacteristic
271#107—734no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—732no—
275C4⋊C8—732no—
276C4⋊C8—1461no—
277C4⋊C8C22×D7312yes—
278#102—734no—
279#102—734no—
280C22⋊C8—21no—
281C22⋊C8—732no—
282C22⋊C8—1462no—
283C22⋊C8—1462no—
284C22⋊C8—1462no—
285C22⋊C8—1462no—
286C4×C8—731no—
287C4×C8—731no—
288C4×C8—732no—
289C4×C8—731no—
290C4×C8—731no—
291C4×C8C22×D7312yes—
292C24—21no—
293#41—2922no—
294#41—2922no—
295C2×Q8—1461no—
296C2×D4—21no—
297C2×D4—42no—
298C2×D4—732no—
299C2×D4—1462no—
300C2×D4—1461no—
301C2×D4—1461no—
302C2×D4C2×C4×D7312yes—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—1462no—
309C22×C4C2×C4×D7311yescharacteristic
310C22×C4D4×D7311yescharacteristic
311C22×C4#1192402711yescharacteristic
312D8—2924no—
313M16—42no—
314M16—2924no—
315M16—2924no—
316M16—2924no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—2924no—
320C2×C8—2922no—
321C2×C8—734no—
322C2×C8—731no—
323C2×C8—731no—
324C2×C8—731no—
325C2×C8—731no—
326C2×C8—1462no—
327C2×C8—1462no—
328C2×C8—1462no—
329C4⋊C4—21no—
330C4⋊C4—732no—
331C4⋊C4—1461no—
332C4⋊C4C2×C4×D7312yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—1462no—
336C22⋊C4—1462no—
337C42C2×C4×D7311yescharacteristic
338C42D4×D7311yescharacteristic
339C42#1192402711yescharacteristic
340C42C23×D7311yescharacteristic
341C23—22no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—1462no—
346C23#1455655111yescharacteristic
347Q8—2922no—
348D4—22no—
349D4—22no—
350D4—42no—
351D4—2922no—
352D4—2924no—
353D4—2922no—
354D4—1464no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—2922no—
363C2×C4—2922no—
364C2×C4—1462no—
365C2×C4#1455655111yescharacteristic
366C2×C4#1455655111yescharacteristic
367C2×C4#1455655111yescharacteristic
368C2×C4#1455661512yes—
369C2×C4#1455661812yes—
370C2×C4C22×C4×D7311yescharacteristic
371C2×C4C2×D4×D7311yescharacteristic
372C2×C4C2×#1192402711yescharacteristic
373C8—42no—
374C8—42no—
375C8—2922no—
376C8—1464no—
377C8—1461no—
378C8—1461no—
379C8—1461no—
380C8—1461no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—84no—
387C22—2922no—
388C22—2922no—
389C22—2922no—
390C22#1798869612yes—
391C22C2×#1455655111yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—2922no—
397C4#1798869612yes—
398C4#1798935312yes—
399C4C2×#1455661511yescharacteristic
400C4C2×#1455661811yescharacteristic
401C2—22no—
402C2—82no—
403C2—5842no—
404C2#2261863011yescharacteristic
405C2C2×#1798869611yescharacteristic
406C2#2261883011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1486C2
socleC22×C73C2×C4×D4
prime radical#22619931C2
cosocleC42×C73C24
hypercenter#1486D73
hypocenterC73#17935
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #22618642C2semidirect
3. C2×#17988665C2semidirect
4. #22608842C2semidirect
5. #22608798C2semidirect
6. #22608703C2semidirect
7. #22608331C2semidirect
8. #22608224C2semidirect
9. C73×#1486C2semidirect
23. #17988664C22semidirect
25. #17988652C22semidirect
31. #17988337C22semidirect
32. #17988316C22semidirect
33. #17988309C22semidirect
34. #17988308C22semidirect
210. #1486D73semidirect
213. C73#17935semidirect

Maximal subgroups

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

subgroupindex
#226199312
#226186422
C2×#179886652
#226088422
#226087982
#226087032
#226083312
#226082242
C73×#14862
#1793573