Finite Group Databasebeta

← #33185810

Subgroups — #33185810

Order 18688.

978 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618329C211yesmaximal, characteristic, complemented
2#22611067C214yesmaximal, complemented
3#22610738C212yesmaximal
4#22610058C212yesmaximal, complemented
5#22609931C214yesmaximal, complemented
6C4×C4⋊#10596831C211yesmaximal, characteristic, complemented
7C4×#127×C73C211yesmaximal, characteristic
8C2×#14556591—21no—
9C2×#14556591—21no—
10C2×#14556591C2212yes—
11C2×C4⋊#10596831C2212yes—
12C2×#14556512C2218yes—
13C2×C4×#10596831C2212yes—
14#17988457—21no—
15#17988457—21no—
16#17988457C2212yes—
17#17988457C2212yes—
18#17988349C2218yes—
19#17988342C2212yes—
20C2×#127×C73C2211yescharacteristic
21C2×C4×D4×C73—21no—
22C2×C4×D4×C73—21no—
23C2×C4×D4×C73C2212yes—
24#390×C73—21no—
25#390×C73—21no—
26#390×C73C2212yes—
27#384×C73C2211yescharacteristic
28C43×C73C2211yescharacteristic
29C22×Dic146—22no—
30C22×#10596831C2314yes—
31#14556591—44no—
32#14556591—44no—
33#14556591—44no—
34C2×C4⋊Dic73—22no—
35C2×C4⋊Dic73—22no—
36C2×C4⋊Dic73—22no—
37C2×C4⋊Dic73—22no—
38C2×C4⋊Dic73—22no—
39C2×C4⋊Dic73C2314yes—
40C4⋊#10596831—24no—
41#14556512—48no—
42C4×#10596831—24no—
43C22×D4×C73—21no—
44C22×D4×C73—21no—
45C22×D4×C73C2312yes—
46C23×C4×C73—22no—
47#127×C73—22no—
48#127×C73Q814yescomplemented
49C4×D4×C73—28no—
50C4×D4×C73—44no—
51C4×D4×C73—44no—
52C2×C4⋊C4×C73—21no—
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73C2312yes—
55C2×C22⋊C4×C73—22no—
56C2×C22⋊C4×C73—22no—
57C2×C22⋊C4×C73—22no—
58C2×C42×C73D411yescharacteristic
59C2×C42×C73D411yescharacteristic
60C2×C42×C73D411yescharacteristic
61C2×C42×C73D411yescharacteristic
62C2×C42×C73C2311yescharacteristic
63C2×C42×C73C2311yescharacteristic
64C2×C42×C73C2311yescharacteristic
65C2×Dic146—44no—
66C2×Dic146—44no—
67C2×Dic146—44no—
68C22×Dic73—24no—
69C2×#10596831—24no—
70C2×#10596831—24no—
71C2×#10596831—24no—
72C4⋊Dic73—28no—
73C4⋊Dic73—44no—
74C4⋊Dic73—44no—
75C4⋊Dic73—44no—
76C4⋊Dic73—44no—
77C4⋊Dic73—44no—
78C24×C73—22no—
79C2×D4×C73—24no—
80C2×D4×C73—24no—
81C2×D4×C73—24no—
82C2×D4×C73—22no—
83C2×D4×C73—24no—
84C2×D4×C73—42no—
85C2×D4×C73—42no—
86C2×D4×C73—42no—
87C2×D4×C73—42no—
88C2×D4×C73C2×Q814yes—
89C22×C4×C73—22no—
90C22×C4×C73—44no—
91C22×C4×C73—44no—
92C22×C4×C73—44no—
93C22×C4×C73C2×D411yescharacteristic
94C22×C4×C73C2×D411yescharacteristic
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73C2×D411yescharacteristic
97C22×C4×C73C2×D411yescharacteristic
98C22×C4×C73C2×D411yescharacteristic
99C22×C4×C73C2411yescharacteristic
100C4⋊C4×C73—24no—
101C4⋊C4×C73—42no—
102C4⋊C4×C73—42no—
103C22⋊C4×C73—44no—
104C22⋊C4×C73—44no—
105C22⋊C4×C73—44no—
106C42×C73—22no—
107C42×C73—22no—
108C42×C73—22no—
109C42×C73D814yes—
110C42×C73SD1614yes—
111C42×C73C2×Q812yes—
112C42×C73#4114yes—
113C42×C73#4112yes—
114C23×C73—24no—
115C23×C73—42no—
116C23×C73—42no—
117C23×C73—42no—
118C23×C73—42no—
119C23×C73—42no—
120C23×C73—42no—
121C23×C73C22×D411yescharacteristic
122D4×C73—28no—
123D4×C73—44no—
124D4×C73—48no—
125D4×C73—48no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—42no—
133C2×C4×C73—42no—
134C2×C4×C73—42no—
135C2×C4×C73—88no—
136C2×C4×C73C22⋊Q812yes—
137C2×C4×C73C22⋊Q812yes—
138C2×C4×C73#12412yes—
139C2×C4×C73#12412yes—
140C2×C4×C73C2×D812yes—
141C2×C4×C73C2×D812yes—
142C2×C4×C73C2×D812yes—
143C2×C4×C73C2×SD1612yes—
144C2×C4×C73C2×SD1612yes—
145C2×C4×C73C2×SD1612yes—
146C2×C4×C73AGL(1,ℤ/8)12yes—
147C2×C4×C73#13712yes—
148C2×C4×C73C22×Q811yescharacteristic
149C2×C4×C73C2×#4112yes—
150C2×C4×C73C2×#4111yescharacteristic
151C2×C4×C73H(5,𝔽2)12yes—
152C2×Dic73—44no—
153C2×Dic73—44no—
154C2×Dic73—44no—
155Dic146—88no—
156#10596831—44no—
157C22×C73—44no—
158C22×C73—44no—
159C22×C73—44no—
160C22×C73—82no—
161C22×C73—82no—
162C22×C73—82no—
163C22×C73—82no—
164C22×C73C2×C22⋊Q811yescharacteristic
165C22×C73C2×#12411yescharacteristic
166C22×C73#53511yescharacteristic
167C22×C73C22×D811yescharacteristic
168C22×C73C22×SD1611yescharacteristic
169C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
170C22×C73C2×#13711yescharacteristic
171C4×C73—22no—
172C4×C73—22no—
173C4×C73—22no—
174C4×C73—82no—
175C4×C73#47414yes—
176C4×C73#47614yes—
177C4×C73#48614yes—
178C4×C73#55412yes—
179C4×C73#55512yes—
180Dic73—84no—
181#29139—731nomaximal, Sylow
182C2×C73—84no—
183C2×C73#256311yescharacteristic
184C2×C73C2×#47411yescharacteristic
185C2×C73C2×#47611yescharacteristic
186C2×C73C2×#48611yescharacteristic
187C2×C73#314111yescharacteristic
188C2×C73#320511yescharacteristic
189C2×C73#320611yescharacteristic
190#2868—731no—
191C4×#127D7311yescharacteristic
192#2023—734no—
193#1957—732no—
194#1834—732no—
195#1802—734no—
196C4×C4⋊C8—731no—
197C73#2913911yescharacteristic, complemented, Sylow
198C2×C4⋊Q8—732no—
199C2×C4⋊Q8—1461no—
200C2×C4⋊Q8—1461no—
201C2×#127C2×D7311yescharacteristic
202C2×C4×D4—21no—
203C2×C4×D4—21no—
204C2×C4×D4C2×D7312yes—
205C2×C4⋊C8—732no—
206C2×#102—738no—
207C2×C4×C8—732no—
208#390—21no—
209#390—21no—
210#390C2×D7312yes—
211#384—732no—
212#384—732no—
213#384—1461no—
214#384—1461no—
215#384C2×D7311yescharacteristic
216C43C2×D7311yescharacteristic
217#340—738no—
218#336—732no—
219C22×Q8—1462no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4C22×D7312yes—
223C23×C4—22no—
224C22×C8—734no—
225C4⋊Q8—2924no—
226C4⋊Q8—2924no—
227C4⋊Q8—2924no—
228#127—22no—
229#127Dic14614yescomplemented
230C4×D4—28no—
231C4×D4—44no—
232C4×D4—44no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—734no—
236C2×C4⋊C4—1462no—
237C2×C4⋊C4—1462no—
238C2×C4⋊C4—1462no—
239C2×C4⋊C4—1462no—
240C2×C4⋊C4—1462no—
241C2×C4⋊C4C22×D7312yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C42C22⋊D7311yescharacteristic
246C2×C42C22⋊D7311yescharacteristic
247C2×C42C22⋊D7311yescharacteristic
248C2×C42C22⋊D7311yescharacteristic
249C2×C42C22×D7311yescharacteristic
250C2×C42C22×D7311yescharacteristic
251C2×C42C22×D7311yescharacteristic
252C4⋊C8—1464no—
253#102—2928no—
254C4×C8—1464no—
255C24—22no—
256C2×Q8—2924no—
257C2×Q8—2924no—
258C2×Q8—2924no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4C2×Dic14614yes—
269C22×C4—22no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—1464no—
274C22×C4C2×C22⋊D7311yescharacteristic
275C22×C4C2×C22⋊D7311yescharacteristic
276C22×C4C2×C22⋊D7311yescharacteristic
277C22×C4C2×C22⋊D7311yescharacteristic
278C22×C4C2×C22⋊D7311yescharacteristic
279C22×C4C2×C22⋊D7311yescharacteristic
280C22×C4C23×D7311yescharacteristic
281C2×C8—1464no—
282C2×C8—1464no—
283C2×C8—1464no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—2924no—
288C4⋊C4—2924no—
289C4⋊C4—2924no—
290C4⋊C4—2924no—
291C4⋊C4—2924no—
292C4⋊C4—1468no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42#1192401314yes—
300C42#1192401614yes—
301C42C2×Dic14612yes—
302C42#1192402714yes—
303C42#1192402812yes—
304C23—24no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23C22×C22⋊D7311yescharacteristic
312Q8—5848no—
313D4—28no—
314D4—48no—
315D4—48no—
316D4—44no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—88no—
327C2×C4—2924no—
328C2×C4—2924no—
329C2×C4—2924no—
330C2×C4#1455657012yes—
331C2×C4#1455657012yes—
332C2×C4#1455657812yes—
333C2×C4#1455657812yes—
334C2×C4C2×#1192401312yes—
335C2×C4C2×#1192401312yes—
336C2×C4C2×#1192401312yes—
337C2×C4C2×#1192401612yes—
338C2×C4C2×#1192401612yes—
339C2×C4C2×#1192401612yes—
340C2×C4#1455661612yes—
341C2×C4#1455662112yes—
342C2×C4C22×Dic14611yescharacteristic
343C2×C4C2×#1192402712yes—
344C2×C4C2×#1192402811yescharacteristic
345C2×C4#1455663712yes—
346C8—2924no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—82no—
351C22—82no—
352C22—82no—
353C22—82no—
354C22C2×#1455657011yescharacteristic
355C22C2×#1455657811yescharacteristic
356C22#1798915111yescharacteristic
357C22C22×#1192401311yescharacteristic
358C22C22×#1192401611yescharacteristic
359C22C2×#1455661611yescharacteristic
360C22C2×#1455662111yescharacteristic
361C4—22no—
362C4—22no—
363C4—22no—
364C4—82no—
365C4—5844no—
366C4#1798886714yes—
367C4#1798887714yes—
368C4#1798893314yes—
369C4#1798929412yes—
370C4#1798930212yes—
371C2—84no—
372C2#2261510411yescharacteristic
373C2C2×#1798886711yescharacteristic
374C2C2×#1798887711yescharacteristic
375C2C2×#1798893311yescharacteristic
376C2#2262032411yescharacteristic
377C2#2262084611yescharacteristic
378C2#2262085311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC4×#127×C73C2
socleC23×C73C22×D4
prime radicalC2×#127×C73C22
hypercenterC4×#127D73
hypocenterC73#29139
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C43 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. #22618329C2semidirect
2. #22611067C2semidirect
4. #22610058C2semidirect
5. #22609931C2semidirect
6. C4×C4⋊#10596831C2semidirect
48. #127×C73Q8semidirect
197. C73#29139semidirect
229. #127Dic146semidirect

Maximal subgroups

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

subgroupindex
#226183292
#226110672
#226107382
#226100582
#226099312
C4×C4⋊#105968312
C4×#127×C732
#2913973