Finite Group Databasebeta

← #33071217

Subgroups — #33071217

Order 18688.

728 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621052C211yesmaximal, characteristic, complemented
2#22619395C211yesmaximal, characteristic
3#22618955C211yesmaximal, characteristic, complemented
4#22609370C214yesmaximal, complemented
5#22609318C214yesmaximal, complemented
6#22608705C211yesmaximal, characteristic, complemented
7#22608603C211yesmaximal, characteristic, complemented
8#22608581C211yesmaximal, characteristic, complemented
9C2×#461×C73C211yesmaximal, characteristic, complemented
10C2×#14556617—21no—
11#17989137C2211yescharacteristic
12#17988990C2212yescomplemented
13#17988935C2212yescomplemented
14#17988900—22no—
15#17988864—22no—
16#17988802—22no—
17#17988792C2212yes—
18#17988782C2212yes—
19D4×#10596831—22no—
20D4×#10596831C2214yes—
21#17988640C2211yescharacteristic
22#17988640C2211yescharacteristic
23#17988611C2211yescharacteristic
24#17988586—21no—
25C2×C22⋊#10596831—22no—
26#17988339C2214yescomplemented
27#17988334C2212yescomplemented
28#17988333C2214yescomplemented
29#17988330C2212yescomplemented
30C22×SD16×C73—21no—
31C2×C4⋊Q8×C73C2211yescharacteristic
32C2×C4×D4×C73C2211yescharacteristic
33#461×C73—22no—
34#461×C73C2214yes—
35C2×C4⋊C8×C73C2211yescharacteristic
36C2×#103×C73—21no—
37C2×#11924028—21no—
38#14556617—44no—
39C2×#11924016—22no—
40C2×#11924013—22no—
41C2×#11924012—21no—
42C22×#10596831—24no—
43#14556591C2311yescharacteristic
44#14556588C2311yescharacteristic
45#14556579—21no—
46#14556565—21no—
47#14556565—21no—
48C4×D292—21no—
49C4×D292—21no—
50#14556524—24no—
51C4⋊#10596831—21no—
52C4⋊#10596831D412yes—
53C4⋊#10596831C2312yes—
54C4⋊#10596831C2312yes—
55#14556514—22no—
56#14556511—24no—
57#14556511—22no—
58C22⋊#10596831—24no—
59C22⋊#10596831—21no—
60C22⋊#10596831D412yes—
61C4×#10596831—21no—
62C4×#10596831C2312yes—
63C22×Q8×C73—21no—
64C22×D4×C73D411yescharacteristic
65C23×C4×C73—21no—
66C2×SD16×C73—24no—
67C2×SD16×C73—44no—
68C22×C8×C73—21no—
69C4⋊Q8×C73—21no—
70C4⋊Q8×C73—21no—
71C4⋊Q8×C73—21no—
72C4⋊Q8×C73C2312yes—
73C4×D4×C73—21no—
74C4×D4×C73—21no—
75C4×D4×C73D412yes—
76C4×D4×C73C2312yes—
77C2×C4⋊C4×C73—21no—
78C2×C4⋊C4×C73—21no—
79C2×C4⋊C4×C73D411yescharacteristic
80C2×C22⋊C4×C73—21no—
81C2×C42×C73C2311yescharacteristic
82C4⋊C8×C73—21no—
83C4⋊C8×C73C2312yes—
84#103×C73—24no—
85#11924028—42no—
86#11924028—42no—
87C2×Dic146—21no—
88C2×D292—21no—
89C2×D292—21no—
90C2×D292—21no—
91C2×C22⋊D73—41no—
92C2×C4×D73—41no—
93#11924016—44no—
94#11924013—44no—
95#11924012—42no—
96C2×#10596831—22no—
97C2×#10596831—22no—
98C2×#10596831—84no—
99C2×#10596831C2×D414yes—
100C4⋊Dic73—21no—
101C4⋊Dic73—21no—
102#11924005—41no—
103#11924005—41no—
104C24×C73—21no—
105C2×Q8×C73—22no—
106C2×Q8×C73—41no—
107C2×Q8×C73—42no—
108C2×Q8×C73—42no—
109C2×D4×C73—21no—
110C2×D4×C73—22no—
111C2×D4×C73D814yes—
112C2×D4×C73C2×D412yes—
113C22×C4×C73—21no—
114C22×C4×C73—22no—
115C22×C4×C73—22no—
116C22×C4×C73—41no—
117C22×C4×C73—82no—
118C22×C4×C73C2×D411yescharacteristic
119C22×C4×C73C2×D411yescharacteristic
120C22×C4×C73#4111yescharacteristic
121SD16×C73—48no—
122C2×C8×C73—22no—
123C2×C8×C73—42no—
124C4⋊C4×C73—21no—
125C4⋊C4×C73—22no—
126C4⋊C4×C73—22no—
127C4⋊C4×C73—41no—
128C4⋊C4×C73—41no—
129C4⋊C4×C73C2×D412yes—
130C22⋊C4×C73—22no—
131C22⋊C4×C73—22no—
132C42×C73C2×D411yescharacteristic
133C42×C73C2×D411yescharacteristic
134C42×C73#4111yescharacteristic
135C42×C73C2411yescharacteristic
136C23×C73—22no—
137C23×C73—24no—
138C23×C73—81no—
139C23×C73—81no—
140C23×C73#12111yescharacteristic
141C22×D73—41no—
142Q8×C73—42no—
143Q8×C73—42no—
144Q8×C73—82no—
145D4×C73—22no—
146D4×C73—24no—
147D4×C73C2×D814yes—
148C2×C4×C73—22no—
149C2×C4×C73—22no—
150C2×C4×C73—41no—
151C2×C4×C73—44no—
152C2×C4×C73—42no—
153C2×C4×C73—82no—
154C2×C4×C73—81no—
155C2×C4×C73—82no—
156C2×C4×C73#12111yescharacteristic
157C2×C4×C73#12111yescharacteristic
158C2×C4×C73#12111yescharacteristic
159C2×C4×C73C2×D812yes—
160C2×C4×C73#13512yes—
161C2×C4×C73C22×D411yescharacteristic
162C2×C4×C73C22×D411yescharacteristic
163C2×C4×C73C2×#4111yescharacteristic
164C22⋊D73—82no—
165C2×Dic73—41no—
166D292—41no—
167D292—41no—
168D292—42no—
169D292—81no—
170C4×D73—82no—
171Dic146—41no—
172Dic146—41no—
173C8×C73—42no—
174#10596831—24no—
175#10596831—42no—
176#10596831—42no—
177C22×C73—21no—
178C22×C73—21no—
179C22×C73—24no—
180C22×C73—42no—
181C22×C73—42no—
182C22×C73—84no—
183C22×C73#46612yes—
184C22×C73C2×#12111yescharacteristic
185C2×D73—81no—
186C2×D73—81no—
187C2×D73—81no—
188C4×C73—21no—
189C4×C73—21no—
190C4×C73—21no—
191C4×C73—42no—
192C4×C73—82no—
193C4×C73#46612yes—
194C4×C73C22×D811yescharacteristic
195C4×C73C2×#13511yescharacteristic
196C4×C73#57712yes—
197Dic73—81no—
198#19392—731nomaximal, Sylow
199C2×C73—22no—
200C2×C73—44no—
201C2×C73#304211yescharacteristic
202C2×C73C2×#46611yescharacteristic
203C2×C73#306911yescharacteristic
204D73—161no—
205#3226—731no—
206#3036—731no—
207C2×#461D7311yescharacteristic, complemented
208#2972—731no—
209#1681—734no—
210#1673—734no—
211#1571—731no—
212#1546—731no—
213#1540—731no—
214C73#1939211yescharacteristic, complemented, Sylow
215C2×AGL(1,ℤ/8)—1461no—
216C22×SD16—21no—
217#532—731no—
218C2×C4⋊Q8C2×D7311yescharacteristic
219C2×C4×D4C2×D7311yescharacteristic
220#500—732no—
221#486—732no—
222#480—1462no—
223#474—1462no—
224#463—1462no—
225#461—22no—
226#461—732no—
227#461C2×D7314yes—
228#459—732no—
229C8×D4—734no—
230C8×D4—1462no—
231#431—731no—
232#431—731no—
233#423—731no—
234C2×C4⋊C8C2×D7311yescharacteristic
235#417—1461no—
236C2×#103—21no—
237C2×C22⋊C8—1462no—
238#335—734no—
239#332—734no—
240#332—732no—
241#331—732no—
242C2×#41—1461no—
243C22×Q8—21no—
244C22×D4C22⋊D7311yescharacteristic
245C23×C4—21no—
246AGL(1,ℤ/8)—2924no—
247C2×SD16—24no—
248C2×SD16—44no—
249C2×SD16—1462no—
250C2×D8—1462no—
251C2×M16—1461no—
252C22×C8—21no—
253C22×C8—1464no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—731no—
258C4⋊Q8C22×D7312yes—
259#127—731no—
260#124—1461no—
261#121—1461no—
262#121—1461no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—1461no—
266C4×D4—1461no—
267C4×D4C22⋊D7312yes—
268C4×D4C22×D7312yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C22⋊D7311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C42C22×D7311yescharacteristic
274#107—1464no—
275C4⋊C8—21no—
276C4⋊C8—732no—
277C4⋊C8—732no—
278C4⋊C8—732no—
279C4⋊C8—1461no—
280C4⋊C8C22×D7312yes—
281#103—24no—
282#103—1462no—
283#102—1464no—
284#102—1462no—
285C22⋊C8—732no—
286C22⋊C8—1464no—
287C22⋊C8—1461no—
288C4×C8—732no—
289C4×C8—1461no—
290C24—21no—
291#41—2922no—
292#41—2922no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—1461no—
298C2×D4—21no—
299C2×D4—22no—
300C2×D4—2921no—
301C2×D4—1461no—
302C2×D4—1461no—
303C2×D4—1461no—
304C2×D4#1192401314yes—
305C2×D4C2×C22⋊D7312yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—82no—
311C22×C4—2921no—
312C22×C4D4×D7311yescharacteristic
313C22×C4C2×C22⋊D7311yescharacteristic
314C22×C4#1192402711yescharacteristic
315SD16—48no—
316SD16—2924no—
317D8—2924no—
318M16—2922no—
319C2×C8—22no—
320C2×C8—42no—
321C2×C8—5844no—
322C2×C8—734no—
323C2×C8—1462no—
324C2×C8—1462no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—1461no—
331C4⋊C4—1461no—
332C4⋊C4C2×C22⋊D7312yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—2921no—
336C22⋊C4—2921no—
337C42D4×D7311yescharacteristic
338C42C2×C22⋊D7311yescharacteristic
339C42#1192402711yescharacteristic
340C42C23×D7311yescharacteristic
341C23—22no—
342C23—24no—
343C23—81no—
344C23—81no—
345C23—2921no—
346C23#1455656311yescharacteristic
347Q8—42no—
348Q8—42no—
349Q8—82no—
350Q8—2921no—
351Q8—2921no—
352D4—24no—
353D4—22no—
354D4—2922no—
355D4—2921no—
356D4—2921no—
357D4—5841no—
358D4—5842no—
359D4C2×#1192401314yes—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—2921no—
369C2×C4—5842no—
370C2×C4#1455656311yescharacteristic
371C2×C4#1455656311yescharacteristic
372C2×C4#1455656311yescharacteristic
373C2×C4C2×#1192401312yes—
374C2×C4#1455661112yes—
375C2×C4C2×D4×D7311yescharacteristic
376C2×C4C22×C22⋊D7311yescharacteristic
377C2×C4C2×#1192402711yescharacteristic
378C8—42no—
379C8—2922no—
380C8—2922no—
381C8—1464no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—84no—
388C22—5841no—
389C22—5841no—
390C22—5841no—
391C22#1798882112yes—
392C22C2×#1455656311yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—82no—
398C4—5841no—
399C4#1798882112yes—
400C4C22×#1192401311yescharacteristic
401C4C2×#1455661111yescharacteristic
402C4#1798939312yes—
403C2—22no—
404C2—44no—
405C2—11681no—
406C2#2261944011yescharacteristic
407C2C2×#1798882111yescharacteristic
408C2#2261966511yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#461×C73C2
socleC22×C73C2×#121
prime radical#22619395C2
hypercenterC2×#461D73
hypocenterC73#19392
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. #22621052C2semidirect
3. #22618955C2semidirect
4. #22609370C2semidirect
5. #22609318C2semidirect
6. #22608705C2semidirect
7. #22608603C2semidirect
8. #22608581C2semidirect
9. C2×#461×C73C2semidirect
12. #17988990C22semidirect
13. #17988935C22semidirect
26. #17988339C22semidirect
27. #17988334C22semidirect
28. #17988333C22semidirect
29. #17988330C22semidirect
207. C2×#461D73semidirect
214. C73#19392semidirect

Maximal subgroups

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

subgroupindex
#226210522
#226193952
#226189552
#226093702
#226093182
#226087052
#226086032
#226085812
C2×#461×C732
#1939273