Finite Group Databasebeta

← #32996217

Subgroups — #32996217

Order 18688.

504 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22611542C211yesmaximal, characteristic, complemented
2C2×#17988350C211yesmaximal, characteristic, complemented
3#22607943C212yesmaximal, complemented
4#22607940C212yesmaximal, complemented
5C73×#3050C211yesmaximal, characteristic
6C22×C4⋊Dic73—21no—
7C2×#14556531—21no—
8C22⋊#11924003—21no—
9C22⋊#11924003C2212yes—
10#17988350—21no—
11#17988350C2212yes—
12C2×#135×C73C411yescharacteristic, complemented
13C23×C8×C73C2211yescharacteristic
14#525×C73—21no—
15#467×C73C411yescharacteristic, complemented
16#467×C73C2211yescharacteristic
17#466×C73C411yescharacteristic, complemented
18#466×C73C2211yescharacteristic
19#465×C73—21no—
20#465×C73—21no—
21#427×C73C411yescharacteristic, complemented
22#416×C73—21no—
23C23×Dic73—41no—
24C2×C4⋊Dic73—24no—
25C2×C4⋊Dic73—22no—
26C2×C4⋊Dic73—42no—
27C2×C4⋊Dic73—41no—
28#14556531—42no—
29C2×#11924003—22no—
30C2×#41×C73—21no—
31C23×C4×C73D411yescharacteristic
32#135×C73—42no—
33C2×Q16×C73C2×C411yescharacteristic, complemented
34C2×SD16×C73—21no—
35C2×D8×C73C2×C411yescharacteristic, complemented
36C22×C8×C73—42no—
37C22×C8×C73C2×C411yescharacteristic
38C22×C8×C73C2×C411yescharacteristic
39C22×C8×C73D411yescharacteristic
40C22×C8×C73D411yescharacteristic
41C22×C8×C73D411yescharacteristic
42C22×C8×C73C2311yescharacteristic
43#123×C73—41no—
44C22⋊Q8×C73—21no—
45C22⋊Q8×C73—21no—
46#121×C73—21no—
47#121×C73—21no—
48#120×C73—41no—
49C4×D4×C73—41no—
50C4×D4×C73—41no—
51#117×C73—21no—
52#107×C73C2×C411yescharacteristic, complemented
53#107×C73C2×C411yescharacteristic, complemented
54#106×C73—21no—
55#103×C73—21no—
56#103×C73—21no—
57#102×C73—21no—
58#102×C73—21no—
59C22×Dic73—44no—
60C22×Dic73—42no—
61C22×Dic73—82no—
62C22×Dic73—82no—
63C4⋊Dic73—24no—
64C4⋊Dic73—44no—
65C4⋊Dic73—42no—
66#11924003—42no—
67C24×C73D811yescharacteristic
68#41×C73—42no—
69#41×C73—42no—
70C2×Q8×C73—21no—
71C2×D4×C73—21no—
72C2×D4×C73—41no—
73C2×D4×C73—41no—
74C2×D4×C73—41no—
75C22×C4×C73—22no—
76C22×C4×C73—22no—
77C22×C4×C73—41no—
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73D811yescharacteristic
81C22×C4×C73SD1611yescharacteristic
82C22×C4×C73SD1611yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84Q16×C73—41no—
85SD16×C73—42no—
86D8×C73—41no—
87C2×C8×C73—22no—
88C2×C8×C73—22no—
89C2×C8×C73—41no—
90C2×C8×C73—41no—
91C2×C8×C73—41no—
92C2×C8×C73—41no—
93C2×C8×C73C22⋊C411yescharacteristic
94C2×C8×C73C22⋊C411yescharacteristic
95C2×C8×C73C22×C411yescharacteristic
96C2×C8×C73C2×D411yescharacteristic
97C4⋊C4×C73—21no—
98C4⋊C4×C73—21no—
99C4⋊C4×C73—41no—
100C4⋊C4×C73—41no—
101C22⋊C4×C73—41no—
102C22⋊C4×C73—41no—
103C22⋊C4×C73—41no—
104C22⋊C4×C73—41no—
105C22⋊C4×C73—41no—
106C42×C73—41no—
107C23×C73—22no—
108C23×C73—22no—
109C23×C73—21no—
110C23×C73—21no—
111C23×C73—41no—
112C23×C73#10211yescharacteristic
113C23×C73#10211yescharacteristic
114C23×C73C2×D811yescharacteristic
115Q8×C73—41no—
116Q8×C73—41no—
117D4×C73—41no—
118D4×C73—41no—
119D4×C73—82no—
120D4×C73—82no—
121D4×C73—81no—
122D4×C73—81no—
123C2×C4×C73—22no—
124C2×C4×C73—22no—
125C2×C4×C73—22no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—82no—
134C2×C4×C73#9912yes—
135C2×C4×C73#10211yescharacteristic
136C2×C4×C73#10211yescharacteristic
137C2×C4×C73#10412yes—
138C2×C4×C73C2×C22⋊C411yescharacteristic
139C2×C4×C73C2×SD1611yescharacteristic
140C2×Dic73—44no—
141C2×Dic73—88no—
142C2×Dic73—82no—
143C2×Dic73—82no—
144C8×C73—21no—
145C8×C73—21no—
146C8×C73—21no—
147C8×C73—21no—
148C22×C73—24no—
149C22×C73—24no—
150C22×C73—21no—
151C22×C73—21no—
152C22×C73—21no—
153C22×C73—21no—
154C22×C73—21no—
155C22×C73—21no—
156C22×C73—81no—
157C22×C73—81no—
158C22×C73—81no—
159C22×C73#32712yes—
160C22×C73#36012yes—
161C22×C73C2×#10211yescharacteristic
162C22×C73#50911yescharacteristic
163C22×C73#51011yescharacteristic
164C4×C73—21no—
165C4×C73—21no—
166C4×C73—81no—
167C4×C73—81no—
168C4×C73—81no—
169C4×C73#32712yes—
170C4×C73C2×#9911yescharacteristic
171C4×C73C2×#10411yescharacteristic
172Dic73—84no—
173#11704—731nomaximal, Sylow
174C2×C73—21no—
175C2×C73—22no—
176C2×C73—22no—
177C2×C73—21no—
178C2×C73—161no—
179C2×C73C2×#32711yescharacteristic
180C2×C73#213911yescharacteristic
181C2×C73C2×#36011yescharacteristic
182#3050D7311yescharacteristic
183#2112—731no—
184C2×#340—731no—
185#1348—732no—
186#1347—732no—
187C73#1170411yescharacteristic, complemented, Sylow
188C2×#135Dic7311yescharacteristic, complemented
189C23×C8C2×D7311yescharacteristic
190#525—21no—
191C22×C4⋊C4—1461no—
192C2×M32—1461no—
193#467Dic7311yescharacteristic, complemented
194#467C2×D7311yescharacteristic
195#466Dic7311yescharacteristic, complemented
196#466C2×D7311yescharacteristic
197#465—21no—
198#465—21no—
199#427Dic7311yescharacteristic, complemented
200#416—21no—
201C22⋊C16—732no—
202C22⋊C16—1461no—
203#340—732no—
204#340—1461no—
205C2×#41—21no—
206C23×C4—2921no—
207C23×C4C22⋊D7311yescharacteristic
208#135—42no—
209C2×Q16C2×Dic7311yescharacteristic, complemented
210C2×SD16—21no—
211C2×D8C2×Dic7311yescharacteristic, complemented
212C22×C8—42no—
213C22×C8C2×Dic7311yescharacteristic
214C22×C8C2×Dic7311yescharacteristic
215C22×C8C22⋊D7311yescharacteristic
216C22×C8C22⋊D7311yescharacteristic
217C22×C8C22⋊D7311yescharacteristic
218C22×C8C22×D7311yescharacteristic
219#123—41no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222#121—21no—
223#121—21no—
224#120—41no—
225C4×D4—41no—
226C4×D4—41no—
227#117—21no—
228C2×C4⋊C4—2922no—
229C2×C4⋊C4—2921no—
230C2×C4⋊C4—1464no—
231C2×C4⋊C4—1462no—
232M32—2922no—
233C2×C16—1462no—
234#107C2×Dic7311yescharacteristic, complemented
235#107C2×Dic7311yescharacteristic, complemented
236#106—21no—
237#103—21no—
238#103—21no—
239#102—21no—
240#102—21no—
241C24#1192401311yescharacteristic
242#41—42no—
243#41—42no—
244C2×Q8—21no—
245C2×D4—21no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—2924no—
253C22×C4—2922no—
254C22×C4—5842no—
255C22×C4—5842no—
256C22×C4C22⋊Dic7311yescharacteristic
257C22×C4C22⋊Dic7311yescharacteristic
258C22×C4#1192401311yescharacteristic
259C22×C4#1192401611yescharacteristic
260C22×C4#1192401611yescharacteristic
261C22×C4C2×C22⋊D7311yescharacteristic
262Q16—41no—
263SD16—42no—
264D8—41no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8C22⋊Dic7311yescharacteristic
272C2×C8C22⋊Dic7311yescharacteristic
273C2×C8C22×Dic7311yescharacteristic
274C2×C8C2×C22⋊D7311yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—2922no—
280C4⋊C4—2924no—
281C4⋊C4—1464no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42—41no—
288C16—2922no—
289C23—22no—
290C23—21no—
291C23—21no—
292C23—22no—
293C23—41no—
294C23#1455651211yescharacteristic
295C23#1455651211yescharacteristic
296C23C2×#1192401311yescharacteristic
297Q8—41no—
298Q8—41no—
299D4—41no—
300D4—41no—
301D4—82no—
302D4—81no—
303D4—81no—
304D4—82no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—82no—
316C2×C4—2924no—
317C2×C4—5848no—
318C2×C4—5842no—
319C2×C4—5842no—
320C2×C4#1455650612yes—
321C2×C4#1455651211yescharacteristic
322C2×C4#1455651211yescharacteristic
323C2×C4#1455651812yes—
324C2×C4C2×C22⋊Dic7311yescharacteristic
325C2×C4C2×#1192401611yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C22—24no—
331C22—24no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#1798832112yes—
342C22#1798839712yes—
343C22C2×#1455651211yescharacteristic
344C22#1798902211yescharacteristic
345C22#1798902711yescharacteristic
346C4—21no—
347C4—21no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—5844no—
352C4#1798832112yes—
353C4C2×#1455650611yescharacteristic
354C4C2×#1455651811yescharacteristic
355C2—21no—
356C2—22no—
357C2—21no—
358C2—22no—
359C2—161no—
360C2C2×#1798832111yescharacteristic
361C2#2261164311yescharacteristic
362C2C2×#1798839711yescharacteristic
subgroupG/H
centerC22C2×#14556512
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
Fitting, prime radicalC73×#3050C2
socleC22×C73C2×#102
cosocleC22×C8×C73C23
hypercenter#3050D73
hypocenterC73#11704
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. #22611542C2semidirect
2. C2×#17988350C2semidirect
3. #22607943C2semidirect
4. #22607940C2semidirect
12. C2×#135×C73C4semidirect
15. #467×C73C4semidirect
17. #466×C73C4semidirect
21. #427×C73C4semidirect
33. C2×Q16×C73C2×C4semidirect
35. C2×D8×C73C2×C4semidirect
52. #107×C73C2×C4semidirect
53. #107×C73C2×C4semidirect
187. C73#11704semidirect
188. C2×#135Dic73semidirect
193. #467Dic73semidirect
195. #466Dic73semidirect
199. #427Dic73semidirect
209. C2×Q16C2×Dic73semidirect
211. C2×D8C2×Dic73semidirect
234. #107C2×Dic73semidirect
235. #107C2×Dic73semidirect

Maximal subgroups

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

subgroupindex
#226115422
C2×#179883502
#226079432
#226079402
C73×#30502
#1170473