Finite Group Databasebeta

← #33071198

Subgroups — #33071198

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619724C211yesmaximal, characteristic
2C2×#17988789C211yesmaximal, characteristic
3#22618954C211yesmaximal, characteristic
4#22609351C214yesmaximal, complemented
5#22609303C214yesmaximal, complemented
6#22608702C211yesmaximal, characteristic, complemented
7#22608603C211yesmaximal, characteristic, complemented
8#22608588C211yesmaximal, characteristic, complemented
9C73×#3229C211yesmaximal, characteristic, complemented
10C22×#11924015—21no—
11C2×C4×D292C2211yescharacteristic
12#17988884—22no—
13#17988874C2212yes—
14#17988864C2212yes—
15#17988789—22no—
16#17988789C2214yes—
17C8×D292C2214yes—
18#17988652—22no—
19#17988640C2211yescharacteristic
20#17988630—21no—
21#17988611C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23#17988591—21no—
24C2×#14556515—21no—
25C22⋊C8×D73—22no—
26#17988338C2214yescomplemented
27#17988336C2212yescomplemented
28#17988333C2214yescomplemented
29#17988333C2212yescomplemented
30#533×C73C2211yescharacteristic
31C2×C4⋊Q8×C73C2211yescharacteristic
32#498×C73C2212yescomplemented
33#487×C73C2212yescomplemented
34#483×C73—22no—
35#475×C73—22no—
36#431×C73C2211yescharacteristic
37C22×D292D411yescharacteristic
38C22×C4×D73—21no—
39C2×#11924015—24no—
40C2×#11924015—44no—
41C2×#11924012—21no—
42C22×#10596831—21no—
43C2×C8×D73—24no—
44C4×D292—21no—
45C4×D292—21no—
46C4×D292D412yes—
47C4×D292C2312yes—
48C2×C4⋊Dic73D411yescharacteristic
49C2×#11924005—21no—
50#14556524—22no—
51#14556522—22no—
52C4⋊#10596831—21no—
53C4⋊#10596831—21no—
54C4⋊#10596831C2312yes—
55C4⋊#10596831C2312yes—
56C4⋊#10596831C2312yes—
57#14556515—24no—
58#14556511—22no—
59#14556511—22no—
60C22⋊#10596831—21no—
61#14556503—24no—
62C4×#10596831—21no—
63C22×Q8×C73—21no—
64C4⋊Q8×C73—21no—
65C4⋊Q8×C73—21no—
66C4⋊Q8×C73—21no—
67C4⋊Q8×C73C2311yescharacteristic
68C4⋊Q8×C73C2311yescharacteristic
69C4⋊Q8×C73C2312yes—
70#125×C73—21no—
71C22⋊Q8×C73—21no—
72C22⋊Q8×C73—21no—
73C4×Q8×C73—21no—
74C4×Q8×C73—21no—
75#117×C73—21no—
76C2×C4⋊C4×C73—21no—
77C2×C4⋊C4×C73—21no—
78C2×C42×C73C2311yescharacteristic
79#106×C73—24no—
80C4⋊C8×C73D412yes—
81#103×C73—24no—
82C22⋊C8×C73D412yes—
83C4×C8×C73C2312yes—
84C23×D73—21no—
85C2×D292—21no—
86C2×D292—22no—
87C2×D292SD1614yes—
88C2×D292C2×D412yes—
89C22×Dic73—21no—
90C2×C4×D73—22no—
91C2×C4×D73—22no—
92C2×C4×D73—82no—
93#11924015—48no—
94#11924012—42no—
95C2×#10596831—22no—
96C2×#10596831—22no—
97C2×#10596831—22no—
98C2×#10596831—42no—
99C8×D73—84no—
100C4⋊Dic73—21no—
101C4⋊Dic73C2×D412yes—
102#11924005—22no—
103#11924005—22no—
104C2×Q8×C73—21no—
105C2×Q8×C73—22no—
106C2×Q8×C73—21no—
107C2×Q8×C73—41no—
108C2×Q8×C73—42no—
109C2×Q8×C73—42no—
110C22×C4×C73—41no—
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73#4111yescharacteristic
114C2×C8×C73C2×D414yes—
115C4⋊C4×C73—22no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73—21no—
118C4⋊C4×C73—21no—
119C4⋊C4×C73—22no—
120C4⋊C4×C73—21no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73—41no—
123C4⋊C4×C73—41no—
124C4⋊C4×C73—41no—
125C22⋊C4×C73—41no—
126C42×C73—41no—
127C42×C73C2×D411yescharacteristic
128C42×C73C2×D411yescharacteristic
129C42×C73#4111yescharacteristic
130C42×C73C2411yescharacteristic
131C23×C73#12111yescharacteristic
132C22×D73—22no—
133C22×D73—24no—
134C22×D73—81no—
135C22×D73—81no—
136Q8×C73—42no—
137Q8×C73—42no—
138Q8×C73—42no—
139Q8×C73—81no—
140Q8×C73—82no—
141C2×C4×C73—22no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—42no—
146C2×C4×C73—82no—
147C2×C4×C73#12111yescharacteristic
148C2×C4×C73#12111yescharacteristic
149C2×C4×C73#12111yescharacteristic
150C2×C4×C73C2×SD1612yes—
151C2×C4×C73#13512yes—
152C2×C4×C73C22×D411yescharacteristic
153C2×C4×C73C22×D411yescharacteristic
154C2×C4×C73C2×#4111yescharacteristic
155C2×Dic73—22no—
156C2×Dic73—81no—
157D292—22no—
158D292—24no—
159D292C2×SD1614yes—
160C4×D73—44no—
161C4×D73—82no—
162C8×C73—24no—
163#10596831—42no—
164#10596831—42no—
165#10596831—42no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73#46512yes—
169C22×C73C2×#12111yescharacteristic
170C2×D73—24no—
171C2×D73—42no—
172C2×D73—42no—
173C2×D73—84no—
174C4×C73—21no—
175C4×C73—21no—
176C4×C73—21no—
177C4×C73—81no—
178C4×C73—82no—
179C4×C73—81no—
180C4×C73#46512yes—
181C4×C73C22×SD1611yescharacteristic
182C4×C73C2×#13511yescharacteristic
183C4×C73#57712yes—
184Dic73—42no—
185#19390—731nomaximal, Sylow
186C2×C73—22no—
187C2×C73#304211yescharacteristic
188C2×C73C2×#46511yescharacteristic
189C2×C73#306911yescharacteristic
190D73—44no—
191#3229D7311yescharacteristic, complemented
192#3077—731no—
193C2×#461—731no—
194#2972—731no—
195#1679—734no—
196#1671—734no—
197#1570—731no—
198#1546—731no—
199#1542—731no—
200C73#1939011yescharacteristic, complemented, Sylow
201C22×SD16—1461no—
202#533C2×D7311yescharacteristic
203C2×C4⋊Q8C2×D7311yescharacteristic
204C2×C4×D4—731no—
205#498C2×D7312yescomplemented
206#487C2×D7312yescomplemented
207#483—22no—
208#478—1462no—
209#476—732no—
210#475—22no—
211#474—732no—
212#461—734no—
213#461—1462no—
214C8×D4—734no—
215C8×D4—1462no—
216#431—731no—
217#431C2×D7311yescharacteristic
218#428—1461no—
219#423—731no—
220C2×C4⋊C8—731no—
221#418—1461no—
222C2×#103—1461no—
223C2×C22⋊C8—1462no—
224#334—734no—
225#333—732no—
226#332—732no—
227#332—734no—
228C22×Q8—21no—
229C22×D4—731no—
230C23×C4—1461no—
231C2×SD16—2924no—
232C2×SD16—1464no—
233C2×M16—1461no—
234C22×C8—1461no—
235C22×C8—1464no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8C22×D7312yes—
240C4⋊Q8C22×D7311yescharacteristic
241C4⋊Q8C22×D7311yescharacteristic
242#125—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×D4—732no—
248C4×D4—732no—
249C4×D4—1461no—
250C4×D4—1461no—
251#117—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—731no—
255C2×C22⋊C4—1461no—
256C2×C42C22×D7311yescharacteristic
257#107—1462no—
258#106—24no—
259#106—1462no—
260C4⋊C8—732no—
261C4⋊C8—732no—
262C4⋊C8—732no—
263C4⋊C8—1461no—
264C4⋊C8—1461no—
265C4⋊C8C22⋊D7312yes—
266#103—24no—
267#103—1464no—
268#102—1462no—
269#102—1462no—
270C22⋊C8—1461no—
271C22⋊C8—1464no—
272C22⋊C8C22⋊D7312yes—
273C4×C8—1461no—
274C4×C8C22×D7312yes—
275C24—1461no—
276C2×Q8—22no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×D4—734no—
283C2×D4—732no—
284C2×D4—1461no—
285C2×D4—1462no—
286C22×C4—41no—
287C22×C4—5842no—
288C22×C4—1462no—
289C22×C4—1461no—
290C22×C4—1462no—
291C22×C4D4×D7311yescharacteristic
292C22×C4C2×C22⋊D7311yescharacteristic
293C22×C4#1192402711yescharacteristic
294SD16—2928no—
295M16—2922no—
296C2×C8—2922no—
297C2×C8—5844no—
298C2×C8—1462no—
299C2×C8—1462no—
300C2×C8—1462no—
301C2×C8C2×C22⋊D7314yes—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—732no—
313C4⋊C4—1461no—
314C22⋊C4—41no—
315C22⋊C4—1462no—
316C22⋊C4—1462no—
317C42—41no—
318C42D4×D7311yescharacteristic
319C42C2×C22⋊D7311yescharacteristic
320C42#1192402711yescharacteristic
321C42C23×D7311yescharacteristic
322C23—5841no—
323C23—5841no—
324C23—1462no—
325C23—1464no—
326C23#1455656211yescharacteristic
327Q8—42no—
328Q8—42no—
329Q8—42no—
330Q8—81no—
331Q8—82no—
332D4—734no—
333D4—1464no—
334D4—1462no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—2924no—
342C2×C4—5842no—
343C2×C4—5841no—
344C2×C4—1462no—
345C2×C4#1455656211yescharacteristic
346C2×C4#1455656211yescharacteristic
347C2×C4#1455656211yescharacteristic
348C2×C4SD16×D7312yes—
349C2×C4#1455661212yes—
350C2×C4C2×D4×D7311yescharacteristic
351C2×C4C22×C22⋊D7311yescharacteristic
352C2×C4C2×#1192402711yescharacteristic
353C8—24no—
354C8—2922no—
355C8—2922no—
356C8—2922no—
357C22—21no—
358C22—21no—
359C22—2922no—
360C22—2922no—
361C22—5844no—
362C22—1464no—
363C22#1798881312yes—
364C22C2×#1455656211yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—81no—
370C4—82no—
371C4—2922no—
372C4#1798881312yes—
373C4C2×SD16×D7311yescharacteristic
374C4C2×#1455661211yescharacteristic
375C4#1798939512yes—
376C2—22no—
377C2—2924no—
378C2#2261943911yescharacteristic
379C2C2×#1798881311yescharacteristic
380C2#2261966411yescharacteristic
subgroupG/H
centerC22C2×#14556562
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3229C2
socleC22×C73C2×#121
prime radicalC22×D292D4
hypercenter#3229D73
hypocenterC73#19390
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
4. #22609351C2semidirect
5. #22609303C2semidirect
6. #22608702C2semidirect
7. #22608603C2semidirect
8. #22608588C2semidirect
9. C73×#3229C2semidirect
26. #17988338C22semidirect
27. #17988336C22semidirect
28. #17988333C22semidirect
29. #17988333C22semidirect
32. #498×C73C22semidirect
33. #487×C73C22semidirect
191. #3229D73semidirect
200. C73#19390semidirect
205. #498C2×D73semidirect
206. #487C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226197242
C2×#179887892
#226189542
#226093512
#226093032
#226087022
#226086032
#226085882
C73×#32292
#1939073