Finite Group Databasebeta

← #32982170

Subgroups — #32982170

Order 18688.

590 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988787C211yesmaximal, characteristic
2C2×Q8×#10596831C211yesmaximal, characteristic, complemented
3#22610736C211yesmaximal, characteristic
4C2×#17988330C211yesmaximal, characteristic
5#22607814C211yesmaximal, characteristic, complemented
6#22607765C211yesmaximal, characteristic, complemented
7C73×#1803C211yesmaximal, characteristic, complemented
8C22×#11924015—21no—
9C2×#14556588C2211yescharacteristic
10#17988787—22no—
11#17988787C414yescomplemented
12Q8×#10596831—22no—
13Q8×#10596831—22no—
14C2×C4⋊#10596831—21no—
15C2×C4⋊#10596831C2211yescharacteristic
16C2×C4⋊#10596831C2211yescharacteristic
17C2×#14556511—21no—
18C2×#14556511—21no—
19C2×#14556511—21no—
20C2×C4×#10596831—21no—
21C2×C4×#10596831C2211yescharacteristic
22#17988330—22no—
23C2×C4×Q8×C73C2211yescharacteristic
24C2×C4⋊C8×C73C2211yescharacteristic
25C2×#103×C73—21no—
26#384×C73C2211yescharacteristic
27#340×C73—21no—
28C22×D292—21no—
29C22×D292—21no—
30C22×D292—21no—
31C2×#11924015—24no—
32C2×#11924015—42no—
33C2×#11924015—42no—
34C22×#10596831—21no—
35C22×#10596831—21no—
36C22×#10596831D411yescharacteristic
37C22×#10596831D411yescharacteristic
38#14556588—21no—
39#14556588—21no—
40#14556588—21no—
41#14556588C2×C412yes—
42C4⋊#10596831—24no—
43C4⋊#10596831—21no—
44C4⋊#10596831—21no—
45C4⋊#10596831C2×C412yes—
46C4⋊#10596831D412yes—
47#14556511—24no—
48#14556511—42no—
49#14556511—42no—
50C4×#10596831—24no—
51C4×#10596831—21no—
52C4×#10596831—21no—
53C22×Q8×C73D411yescharacteristic
54C22×C8×C73—21no—
55C4×Q8×C73—21no—
56C4×Q8×C73—21no—
57C4×Q8×C73C2×C412yes—
58C4×Q8×C73D412yes—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—21no—
61C2×C4⋊C4×C73—21no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73D411yescharacteristic
64C2×C42×C73—21no—
65C2×C42×C73C2311yescharacteristic
66C4⋊C8×C73—21no—
67C4⋊C8×C73—21no—
68#103×C73—42no—
69C23×D73—41no—
70C2×D292—22no—
71C2×D292—22no—
72C2×D292—22no—
73C2×D292—41no—
74C2×D292—41no—
75C2×D292—41no—
76C2×D292—41no—
77C2×D292—41no—
78C2×D292—41no—
79C2×D292—41no—
80C2×D292—41no—
81C2×D292—41no—
82C2×D292—41no—
83C2×D292—41no—
84C2×D292—81no—
85C2×D292—81no—
86#11924015—48no—
87C2×#10596831—21no—
88C2×#10596831—21no—
89C2×#10596831—22no—
90C2×#10596831—21no—
91C2×#10596831—22no—
92C2×#10596831—21no—
93C2×#10596831—21no—
94C2×#10596831—41no—
95C2×#10596831—41no—
96C2×#10596831—41no—
97C2×#10596831—41no—
98C2×#10596831SD1612yescomplemented
99C2×Q8×C73—21no—
100C2×Q8×C73—21no—
101C2×Q8×C73—21no—
102C2×Q8×C73C22⋊C412yes—
103C2×Q8×C73D812yes—
104C2×Q8×C73SD1612yes—
105C22×C4×C73—21no—
106C22×C4×C73—21no—
107C22×C4×C73—41no—
108C22×C4×C73C2×D411yescharacteristic
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73#4111yescharacteristic
111C2×C8×C73—41no—
112C2×C8×C73—41no—
113C2×C8×C73—41no—
114C4⋊C4×C73—22no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—22no—
117C4⋊C4×C73—41no—
118C4⋊C4×C73—41no—
119C4⋊C4×C73—41no—
120C4⋊C4×C73—41no—
121C4⋊C4×C73—81no—
122C4⋊C4×C73C22⋊C412yes—
123C42×C73—22no—
124C42×C73—22no—
125C42×C73C22×C411yescharacteristic
126C42×C73C2×D411yescharacteristic
127C42×C73C2×D411yescharacteristic
128C42×C73#4111yescharacteristic
129C23×C73#12111yescharacteristic
130C22×D73—42no—
131C22×D73—81no—
132C22×D73—81no—
133C22×D73—81no—
134C22×D73—81no—
135C22×D73—81no—
136C22×D73—81no—
137Q8×C73—22no—
138Q8×C73—22no—
139Q8×C73—22no—
140Q8×C73#10214yescomplemented
141C2×C4×C73—22no—
142C2×C4×C73—22no—
143C2×C4×C73—22no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—22no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—81no—
150C2×C4×C73—81no—
151C2×C4×C73—81no—
152C2×C4×C73C2×C22⋊C411yescharacteristic
153C2×C4×C73C4×D411yescharacteristic
154C2×C4×C73C4×D411yescharacteristic
155C2×C4×C73#12011yescharacteristic
156C2×C4×C73#12111yescharacteristic
157C2×C4×C73#12311yescharacteristic
158C2×C4×C73C2×D811yescharacteristic
159C2×C4×C73C2×SD1611yescharacteristic
160C2×C4×C73C2×SD1611yescharacteristic
161C2×C4×C73#13511yescharacteristic
162C2×C4×C73#13511yescharacteristic
163C2×C4×C73#13511yescharacteristic
164C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C73#13711yescharacteristic
166D292—42no—
167D292—42no—
168D292—42no—
169D292—42no—
170D292—82no—
171D292—82no—
172D292—81no—
173D292—81no—
174D292—81no—
175D292—81no—
176C8×C73—81no—
177#10596831—22no—
178#10596831—42no—
179#10596831—41no—
180#10596831—42no—
181C22×C73#38611yescharacteristic
182C22×C73#46011yescharacteristic
183C22×C73#46411yescharacteristic
184C22×C73#46511yescharacteristic
185C22×C73#46611yescharacteristic
186C22×C73#47111yescharacteristic
187C22×C73#47211yescharacteristic
188C2×D73—82no—
189C2×D73—82no—
190C2×D73—82no—
191C2×D73—161no—
192C2×D73—161no—
193C2×D73—161no—
194C2×D73—161no—
195C4×C73—24no—
196C4×C73—21no—
197C4×C73—21no—
198C4×C73—42no—
199C4×C73—161no—
200C4×C73C2×#10211yescharacteristic
201C4×C73#41611yescharacteristic
202C4×C73C4×SD1611yescharacteristic
203C4×C73#44011yescharacteristic
204C4×C73#44811yescharacteristic
205C4×C73#44911yescharacteristic
206C4×C73#46511yescharacteristic
207C4×C73#46611yescharacteristic
208#9222—731nomaximal, Sylow
209C2×C73#166611yescharacteristic
210C2×C73#166811yescharacteristic
211C2×C73#167811yescharacteristic
212C2×C73#168211yescharacteristic
213C2×C73#188011yescharacteristic
214C2×C73#189311yescharacteristic
215C2×C73#190711yescharacteristic
216D73—162no—
217C2×#460—731no—
218C2×C8×Q8—731no—
219#1957—731no—
220#1803D7311yescharacteristic, complemented
221C2×#331—731no—
222#1296—731no—
223#1277—731no—
224C73#922211yescharacteristic, complemented, Sylow
225C22×SD16—1461no—
226C2×#127—731no—
227C2×C4×Q8C2×D7311yescharacteristic
228#460—734no—
229#460—1462no—
230C8×Q8—1462no—
231C8×Q8—1462no—
232C2×C4⋊C8—731no—
233C2×C4⋊C8—731no—
234C2×C4⋊C8—1461no—
235C2×C4⋊C8C2×D7311yescharacteristic
236C2×#103—21no—
237C2×#102—1461no—
238C2×#102—1461no—
239C2×#102—1461no—
240C2×C4×C8—731no—
241C2×C4×C8—1461no—
242#384C2×D7311yescharacteristic
243#340—21no—
244#331—1462no—
245C22×Q8C22⋊D7311yescharacteristic
246C22×D4—1461no—
247C22×D4—1461no—
248C22×D4—1461no—
249C2×SD16—2922no—
250C2×SD16—2922no—
251C2×SD16—1464no—
252C22×C8—21no—
253C22×C8—731no—
254C22×C8—731no—
255C22×C8—1461no—
256C22×C8—1461no—
257#127—732no—
258#127—1461no—
259#127—1461no—
260#127—1461no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8C4×D7312yes—
264C4×Q8D29212yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C22⋊D7311yescharacteristic
270C2×C42—21no—
271C2×C42C22×D7311yescharacteristic
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—732no—
275C4⋊C8—732no—
276C4⋊C8—1464no—
277C4⋊C8—1461no—
278C4⋊C8—1461no—
279#103—42no—
280#102—2922no—
281#102—2922no—
282#102—1464no—
283C4×C8—1464no—
284C4×C8—1461no—
285C4×C8—1461no—
286C24—2921no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8#1192400512yes—
291C2×Q8#1192401312yes—
292C2×Q8#1192401512yes—
293C2×D4—2921no—
294C2×D4—2921no—
295C2×D4—2921no—
296C2×D4—2921no—
297C2×D4—2921no—
298C2×D4—2921no—
299C2×D4—2921no—
300C2×D4—2921no—
301C2×D4—2921no—
302C2×D4—2921no—
303C2×D4—2921no—
304C2×D4—5841no—
305C2×D4—5841no—
306C2×D4—1462no—
307C2×D4—1462no—
308C2×D4—1462no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4D4×D7311yescharacteristic
313C22×C4C2×C22⋊D7311yescharacteristic
314C22×C4#1192402711yescharacteristic
315SD16—2928no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—2921no—
320C2×C8—2921no—
321C2×C8—2921no—
322C2×C8—2921no—
323C2×C8—732no—
324C2×C8—1462no—
325C2×C8—1462no—
326C2×C8—1461no—
327C2×C8—1461no—
328C2×C8—1461no—
329C2×C8—1461no—
330C2×C8—1461no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—81no—
339C4⋊C4#1192400512yes—
340C42—22no—
341C42—22no—
342C42C2×C4×D7311yescharacteristic
343C42D4×D7311yescharacteristic
344C42C2×D29211yescharacteristic
345C42#1192402711yescharacteristic
346C23—2922no—
347C23—5841no—
348C23—5841no—
349C23—5841no—
350C23—5841no—
351C23—5841no—
352C23—5841no—
353C23#1455656311yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—22no—
357Q8#1455651114yescomplemented
358D4—2922no—
359D4—2922no—
360D4—2922no—
361D4—2922no—
362D4—5842no—
363D4—5842no—
364D4—5841no—
365D4—5841no—
366D4—5841no—
367D4—5841no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4C2×#1192400511yescharacteristic
380C2×C4#1455655111yescharacteristic
381C2×C4#1455655111yescharacteristic
382C2×C4#1455656111yescharacteristic
383C2×C4#1455656311yescharacteristic
384C2×C4#1455657511yescharacteristic
385C2×C4C2×#1192401311yescharacteristic
386C2×C4SD16×D7311yescharacteristic
387C2×C4C2×#1192401511yescharacteristic
388C2×C4#1455661111yescharacteristic
389C2×C4#1455661111yescharacteristic
390C2×C4#1455661411yescharacteristic
391C2×C4#1455661811yescharacteristic
392C2×C4#1455662311yescharacteristic
393C8—81no—
394C8—2922no—
395C8—2921no—
396C8—2922no—
397C8—1462no—
398C22—5842no—
399C22—5842no—
400C22—5842no—
401C22—11681no—
402C22—11681no—
403C22—11681no—
404C22—11681no—
405C22#1798846411yescharacteristic
406C22#1798878511yescharacteristic
407C22#1798881111yescharacteristic
408C22#1798881811yescharacteristic
409C22#1798882111yescharacteristic
410C22#1798885211yescharacteristic
411C22#1798885511yescharacteristic
412C4—21no—
413C4—21no—
414C4—24no—
415C4—42no—
416C4—161no—
417C4C2×#1455651111yescharacteristic
418C4#1798858111yescharacteristic
419C4#1798867211yescharacteristic
420C4#1798869011yescharacteristic
421C4#1798872711yescharacteristic
422C4#1798873511yescharacteristic
423C4#1798881811yescharacteristic
424C4#1798882111yescharacteristic
425C2—11682no—
426C2#2260928111yescharacteristic
427C2#2260928911yescharacteristic
428C2#2260934911yescharacteristic
429C2#2260937711yescharacteristic
430C2#2261028411yescharacteristic
431C2#2261035711yescharacteristic
432C2#2261043711yescharacteristic
subgroupG/H
centerC23#14556563
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1803C2
socleC23×C73#121
prime radicalC2×#14556588C22
cosocleC2×C42×C73C23
hypercenter#1803D73
hypocenterC73#9222
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#10596831C2semidirect
5. #22607814C2semidirect
6. #22607765C2semidirect
7. C73×#1803C2semidirect
11. #17988787C4semidirect
98. C2×#10596831SD16semidirect
140. Q8×C73#102semidirect
220. #1803D73semidirect
224. C73#9222semidirect
357. Q8#14556511semidirect

Maximal subgroups

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

subgroupindex
C2×#179887872
C2×Q8×#105968312
#226107362
C2×#179883302
#226078142
#226077652
C73×#18032
#922273