Finite Group Databasebeta

← #33071155

Subgroups — #33071155

Order 18688.

708 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620853C211yesmaximal, characteristic, complemented
2C2×#17988867C211yesmaximal, characteristic, complemented
3#22618955C211yesmaximal, characteristic, complemented
4#22609356C214yesmaximal, complemented
5#22609270C214yesmaximal, complemented
6#22608703C211yesmaximal, characteristic, complemented
7#22608617C211yesmaximal, characteristic, complemented
8#22608580C211yesmaximal, characteristic, complemented
9C73×#3036C211yesmaximal, characteristic
10C2×#14556591C2211yescharacteristic
11#17988959C2212yescomplemented
12#17988939C2212yescomplemented
13#17988867—22no—
14#17988867C2214yes—
15#17988792—22no—
16#17988725—22no—
17D4×#10596831—22no—
18D4×#10596831C2214yes—
19#17988640C2211yescharacteristic
20#17988640C2211yescharacteristic
21C2×#14556524—21no—
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556512—21no—
24C2×C22⋊#10596831—22no—
25#17988337C2214yescomplemented
26#17988333C2212yescomplemented
27#17988332C2214yescomplemented
28#17988331C2212yescomplemented
29C2×AGL(1,ℤ/8)×C73—21no—
30#532×C73C2211yescharacteristic
31C2×C4×D4×C73C2211yescharacteristic
32#463×C73—22no—
33#461×C73C2212yes—
34#459×C73C2212yes—
35#423×C73C2211yescharacteristic
36#417×C73—21no—
37C22×Dic146—21no—
38C2×#11924016—24no—
39C22×#10596831—24no—
40C22×#10596831—21no—
41#14556591—21no—
42#14556591—21no—
43#14556591—21no—
44#14556591C2312yes—
45C2×C4⋊Dic73—21no—
46C2×C4⋊Dic73—21no—
47#14556524—24no—
48C4⋊#10596831—21no—
49C4⋊#10596831D412yes—
50C4⋊#10596831C2312yes—
51C4⋊#10596831C2312yes—
52#14556514—24no—
53#14556512—24no—
54C22⋊#10596831—24no—
55C22⋊#10596831—21no—
56C22⋊#10596831D412yes—
57C4×#10596831—21no—
58C4×#10596831C2312yes—
59C2×#41×C73—21no—
60C22×D4×C73D411yescharacteristic
61C23×C4×C73—21no—
62AGL(1,ℤ/8)×C73—44no—
63C2×SD16×C73—22no—
64C2×D8×C73—22no—
65C2×M16×C73—21no—
66C4⋊Q8×C73C2311yescharacteristic
67#127×C73C2311yescharacteristic
68#124×C73—21no—
69#121×C73—21no—
70#121×C73—21no—
71C4×D4×C73—21no—
72C4×D4×C73—21no—
73C4×D4×C73—21no—
74C4×D4×C73—21no—
75C4×D4×C73D412yes—
76C4×D4×C73C2312yes—
77C2×C4⋊C4×C73D411yescharacteristic
78C2×C22⋊C4×C73—21no—
79C2×C42×C73C2311yescharacteristic
80C4⋊C8×C73—21no—
81C4⋊C8×C73C2312yes—
82#103×C73—22no—
83#102×C73—22no—
84C2×Dic146—22no—
85C2×Dic146—41no—
86C2×Dic146—82no—
87C22×Dic73—41no—
88#11924016—84no—
89C2×#10596831—22no—
90C2×#10596831—22no—
91C2×#10596831—42no—
92C2×#10596831—84no—
93C2×#10596831C2×D414yes—
94C4⋊Dic73—22no—
95C4⋊Dic73—22no—
96C4⋊Dic73—41no—
97C4⋊Dic73—41no—
98C24×C73—21no—
99#41×C73—42no—
100#41×C73—42no—
101C2×Q8×C73—21no—
102C2×D4×C73—21no—
103C2×D4×C73—22no—
104C2×D4×C73—21no—
105C2×D4×C73—21no—
106C2×D4×C73—21no—
107C2×D4×C73—41no—
108C2×D4×C73SD1614yescomplemented
109C2×D4×C73C2×D412yes—
110C22×C4×C73—21no—
111C22×C4×C73—22no—
112C22×C4×C73—22no—
113C22×C4×C73—41no—
114C22×C4×C73—82no—
115C22×C4×C73C2×D411yescharacteristic
116C22×C4×C73C2×D411yescharacteristic
117C22×C4×C73#4111yescharacteristic
118SD16×C73—44no—
119D8×C73—44no—
120M16×C73—42no—
121C2×C8×C73—22no—
122C4⋊C4×C73—21no—
123C4⋊C4×C73—21no—
124C4⋊C4×C73—21no—
125C4⋊C4×C73C2×D412yes—
126C22⋊C4×C73—22no—
127C22⋊C4×C73—22no—
128C22⋊C4×C73—41no—
129C22⋊C4×C73—41no—
130C42×C73C2×D411yescharacteristic
131C42×C73C2×D411yescharacteristic
132C42×C73#4111yescharacteristic
133C42×C73C2411yescharacteristic
134C23×C73—22no—
135C23×C73—24no—
136C23×C73—41no—
137C23×C73—81no—
138C23×C73—81no—
139C23×C73#12111yescharacteristic
140Q8×C73—41no—
141Q8×C73—41no—
142D4×C73—22no—
143D4×C73—24no—
144D4×C73—41no—
145D4×C73—41no—
146D4×C73—42no—
147D4×C73—81no—
148D4×C73—82no—
149D4×C73C2×SD1614yes—
150C2×C4×C73—22no—
151C2×C4×C73—22no—
152C2×C4×C73—41no—
153C2×C4×C73—44no—
154C2×C4×C73—41no—
155C2×C4×C73—82no—
156C2×C4×C73—81no—
157C2×C4×C73—82no—
158C2×C4×C73#12111yescharacteristic
159C2×C4×C73#12111yescharacteristic
160C2×C4×C73#12111yescharacteristic
161C2×C4×C73C2×SD1612yes—
162C2×C4×C73#13512yes—
163C2×C4×C73C22×D411yescharacteristic
164C2×C4×C73C22×D411yescharacteristic
165C2×C4×C73C2×#4111yescharacteristic
166C2×Dic73—42no—
167C2×Dic73—82no—
168Dic146—44no—
169Dic146—82no—
170C8×C73—42no—
171#10596831—24no—
172#10596831—42no—
173#10596831—42no—
174C22×C73—21no—
175C22×C73—21no—
176C22×C73—24no—
177C22×C73—42no—
178C22×C73—42no—
179C22×C73—84no—
180C22×C73—81no—
181C22×C73—81no—
182C22×C73—81no—
183C22×C73#46512yes—
184C22×C73C2×#12111yescharacteristic
185C4×C73—21no—
186C4×C73—21no—
187C4×C73—21no—
188C4×C73—42no—
189C4×C73—81no—
190C4×C73#46512yes—
191C4×C73C22×SD1611yescharacteristic
192C4×C73C2×#13511yescharacteristic
193C4×C73#57612yes—
194Dic73—82no—
195#19385—731nomaximal, Sylow
196C2×C73—22no—
197C2×C73—44no—
198C2×C73—161no—
199C2×C73#304111yescharacteristic
200C2×C73C2×#46511yescharacteristic
201C2×C73#306911yescharacteristic
202#3206—731no—
203C2×#474—731no—
204#3036D7311yescharacteristic
205#2972—731no—
206#1679—734no—
207#1665—734no—
208#1570—731no—
209#1549—731no—
210#1540—731no—
211C73#1938511yescharacteristic, complemented, Sylow
212C2×AGL(1,ℤ/8)—21no—
213#532C2×D7311yescharacteristic
214C2×C4⋊Q8—731no—
215C2×C4×D4C2×D7311yescharacteristic
216#492—732no—
217#487—732no—
218#474—734no—
219#474—1462no—
220#463—22no—
221#461—1462no—
222#461C2×D7312yes—
223#459C2×D7312yes—
224#448—1462no—
225C8×D4—734no—
226C8×D4—1462no—
227#431—731no—
228#431—731no—
229C2×#107—1461no—
230#423C2×D7311yescharacteristic
231C2×C4⋊C8—731no—
232#417—21no—
233C2×#102—1461no—
234C2×C22⋊C8—1462no—
235#334—734no—
236#332—732no—
237#332—734no—
238#331—732no—
239C2×#41—21no—
240C22×Q8—1461no—
241C22×D4C22⋊D7311yescharacteristic
242C23×C4—21no—
243AGL(1,ℤ/8)—44no—
244C2×SD16—22no—
245C2×SD16—1464no—
246C2×D8—22no—
247C2×M16—21no—
248C22×C8—1461no—
249C22×C8—1464no—
250C4⋊Q8—732no—
251C4⋊Q8—1461no—
252C4⋊Q8—1461no—
253C4⋊Q8—1461no—
254C4⋊Q8C22×D7311yescharacteristic
255#127C22×D7311yescharacteristic
256#124—21no—
257#121—21no—
258#121—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4C22⋊D7312yes—
264C4×D4C22×D7312yes—
265C2×C4⋊C4—1461no—
266C2×C4⋊C4—1461no—
267C2×C4⋊C4C22⋊D7311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C42C22×D7311yescharacteristic
270#107—1464no—
271C4⋊C8—21no—
272C4⋊C8—732no—
273C4⋊C8—732no—
274C4⋊C8—732no—
275C4⋊C8—1461no—
276C4⋊C8C22×D7312yes—
277#103—22no—
278#103—1464no—
279#102—22no—
280#102—1464no—
281C22⋊C8—732no—
282C22⋊C8—1464no—
283C22⋊C8—1461no—
284C4×C8—732no—
285C4×C8—1461no—
286C24—21no—
287#41—42no—
288#41—42no—
289C2×Q8—21no—
290C2×Q8—2921no—
291C2×Q8—5842no—
292C2×Q8—1462no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4#1192401614yescomplemented
300C2×D4C2×C22⋊D7312yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—2921no—
307C22×C4D4×D7311yescharacteristic
308C22×C4C2×C22⋊D7311yescharacteristic
309C22×C4#1192402711yescharacteristic
310SD16—44no—
311SD16—5844no—
312D8—44no—
313M16—42no—
314C2×C8—22no—
315C2×C8—2922no—
316C2×C8—5844no—
317C2×C8—734no—
318C2×C8—1462no—
319C2×C8—1462no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—2921no—
324C4⋊C4—2921no—
325C4⋊C4—1462no—
326C4⋊C4—1462no—
327C4⋊C4C2×C22⋊D7312yes—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C42D4×D7311yescharacteristic
333C42C2×C22⋊D7311yescharacteristic
334C42#1192402711yescharacteristic
335C42C23×D7311yescharacteristic
336C23—22no—
337C23—24no—
338C23—41no—
339C23—81no—
340C23—81no—
341C23#1455656311yescharacteristic
342Q8—41no—
343Q8—41no—
344Q8—2924no—
345Q8—5842no—
346D4—24no—
347D4—22no—
348D4—42no—
349D4—41no—
350D4—41no—
351D4—81no—
352D4—82no—
353D4C2×#1192401614yes—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—41no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—2922no—
363C2×C4—5842no—
364C2×C4#1455656311yescharacteristic
365C2×C4#1455656311yescharacteristic
366C2×C4#1455656311yescharacteristic
367C2×C4C2×#1192401612yes—
368C2×C4#1455661112yes—
369C2×C4C2×D4×D7311yescharacteristic
370C2×C4C22×C22⋊D7311yescharacteristic
371C2×C4C2×#1192402711yescharacteristic
372C8—42no—
373C8—2922no—
374C8—2922no—
375C8—1464no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#1798881412yes—
386C22C2×#1455656311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—81no—
392C4—5842no—
393C4#1798881412yes—
394C4C22×#1192401611yescharacteristic
395C4C2×#1455661111yescharacteristic
396C4#1798939112yes—
397C2—22no—
398C2—44no—
399C2—161no—
400C2#2261943611yescharacteristic
401C2C2×#1798881411yescharacteristic
402C2#2261967011yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC73×#3036C2
socleC22×C73C2×#121
hypercenter#3036D73
hypocenterC73#19385
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #22620853C2semidirect
2. C2×#17988867C2semidirect
3. #22618955C2semidirect
4. #22609356C2semidirect
5. #22609270C2semidirect
6. #22608703C2semidirect
7. #22608617C2semidirect
8. #22608580C2semidirect
11. #17988959C22semidirect
12. #17988939C22semidirect
25. #17988337C22semidirect
26. #17988333C22semidirect
27. #17988332C22semidirect
28. #17988331C22semidirect
108. C2×D4×C73SD16semidirect
211. C73#19385semidirect
299. C2×D4#11924016semidirect

Maximal subgroups

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

subgroupindex
#226208532
C2×#179888672
#226189552
#226093562
#226092702
#226087032
#226086172
#226085802
C73×#30362
#1938573