Finite Group Databasebeta

← #33240822

Subgroups — #33240822

Order 18688.

588 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618531C211yesmaximal, characteristic, complemented
2#22612119C212yesmaximal, complemented
3#22612119C212yesmaximal, complemented
4#22612114C212yesmaximal, complemented
5#22612109C212yesmaximal, complemented
6#22611720C211yesmaximal, characteristic, complemented
7#22611713C211yesmaximal, characteristic, complemented
8#22611647C211yesmaximal, characteristic, complemented
9C2×#17988388C211yesmaximal, characteristic, complemented
10#22611545C211yesmaximal, characteristic, complemented
11C73×#3058C211yesmaximal, characteristic
12#17989070—21no—
13C2×#14556531C2211yescharacteristic
14C22×#11924003C2211yescharacteristic
15#17988629—21no—
16#17988629—22no—
17#17988625C2211yescharacteristic
18#17988625C2212yescomplemented
19C2×#14556525C2211yescharacteristic
20C2×#14556525C2212yescomplemented
21#17988414C2211yescharacteristic, complemented
22#17988414C2211yescharacteristic, complemented
23#17988414C2212yescomplemented
24#17988412C2211yescharacteristic, complemented
25#17988412C2211yescharacteristic, complemented
26#17988412C2212yescomplemented
27#17988388C2212yes—
28#17988388C2212yes—
29#17988388C2212yes—
30#17988388C2212yes—
31C22⋊#11924003C2212yescomplemented
32C22⋊#11924003C2212yescomplemented
33C22×D8×C73C2211yescharacteristic
34C2×#131×C73C2211yescharacteristic
35#535×C73—21no—
36#469×C73—22no—
37#466×C73C2212yes—
38#466×C73C2212yes—
39#427×C73C2211yescharacteristic
40#418×C73—21no—
41Q8×Dic73—41no—
42D4×Dic73—42no—
43D4×Dic73—41no—
44#14556550—21no—
45#14556550—22no—
46C2×C4⋊Dic73—21no—
47C2×C4⋊Dic73—22no—
48#14556531—22no—
49C2×#11924003—22no—
50C2×#11924003C2311yescharacteristic
51C2×#11924003C2312yes—
52C2×#11924003C2311yescharacteristic
53#14556525—21no—
54#14556525—22no—
55#14556525C2312yes—
56#14556525C2311yescharacteristic
57#14556525C2311yescharacteristic
58#14556523—21no—
59#14556523—22no—
60C2×#41×C73D411yescharacteristic
61C22×D4×C73—21no—
62C2×D8×C73—21no—
63C2×D8×C73—22no—
64C2×D8×C73—22no—
65C2×D8×C73C2312yes—
66#131×C73—22no—
67#131×C73—22no—
68C2×M16×C73D411yescharacteristic
69C2×M16×C73D412yes—
70C22×C8×C73C2312yes—
71C22×C8×C73C2311yescharacteristic
72#127×C73—41no—
73#124×C73—41no—
74#121×C73—22no—
75#121×C73—22no—
76#120×C73—42no—
77#117×C73—21no—
78#107×C73C2311yescharacteristic
79#107×C73C2311yescharacteristic
80#106×C73—21no—
81#102×C73—22no—
82#102×C73—22no—
83C22×Dic73—42no—
84C22×Dic73—41no—
85C4⋊Dic73—21no—
86C4⋊Dic73—21no—
87C4⋊Dic73—22no—
88C4⋊Dic73—42no—
89C4⋊Dic73—41no—
90C22⋊Dic73—42no—
91C22⋊Dic73—41no—
92C4×Dic73—42no—
93C4×Dic73—41no—
94#11924003—21no—
95#11924003—21no—
96#11924003—22no—
97C24×C73—41no—
98#41×C73—22no—
99#41×C73—22no—
100C2×Q8×C73D811yescharacteristic
101C2×D4×C73—22no—
102C2×D4×C73—42no—
103C2×D4×C73—42no—
104C2×D4×C73—42no—
105C2×D4×C73—41no—
106C2×D4×C73—42no—
107C2×D4×C73D812yes—
108C2×D4×C73D811yescharacteristic
109C22×C4×C73C2×D412yes—
110C22×C4×C73C2×D411yescharacteristic
111D8×C73—22no—
112D8×C73—22no—
113D8×C73—42no—
114M16×C73—22no—
115M16×C73—24no—
116C2×C8×C73—22no—
117C2×C8×C73—22no—
118C2×C8×C73C2×D412yes—
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73#4112yes—
121C2×C8×C73#4112yes—
122C2×C8×C73C2411yescharacteristic
123C4⋊C4×C73—21no—
124C4⋊C4×C73—21no—
125C22⋊C4×C73—41no—
126C22⋊C4×C73—42no—
127C22⋊C4×C73—42no—
128C42×C73—41no—
129C23×C73—42no—
130C23×C73—82no—
131C23×C73—161no—
132C23×C73—161no—
133C23×C73C2×D812yes—
134C23×C73C2×D811yescharacteristic
135Q8×C73—22no—
136D4×C73—24no—
137D4×C73—22no—
138D4×C73—42no—
139D4×C73—42no—
140D4×C73—82no—
141D4×C73—162no—
142D4×C73—162no—
143C2×C4×C73—22no—
144C2×C4×C73—22no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73#12312yes—
148C2×C4×C73C2×D812yes—
149C2×C4×C73C2×D811yescharacteristic
150C2×C4×C73#13712yes—
151C2×C4×C73C22×D411yescharacteristic
152C2×Dic73—41no—
153C2×Dic73—42no—
154C2×Dic73—41no—
155C2×Dic73—161no—
156C2×Dic73—162no—
157C8×C73—22no—
158C8×C73#12312yes—
159C8×C73C2×#4111yescharacteristic
160C8×C73C2×#4111yescharacteristic
161C22×C73—21no—
162C22×C73—21no—
163C22×C73—22no—
164C22×C73—22no—
165C22×C73—22no—
166C22×C73—82no—
167C22×C73—82no—
168C22×C73—82no—
169C22×C73—164no—
170C22×C73#48012yes—
171C22×C73C22×D811yescharacteristic
172C4×C73—22no—
173C4×C73—81no—
174C4×C73—81no—
175C4×C73#48012yes—
176C4×C73C2×#12311yescharacteristic
177C4×C73C2×#13711yescharacteristic
178Dic73—82no—
179Dic73—81no—
180Dic73—81no—
181#33650—731nomaximal, Sylow
182C2×C73—22no—
183C2×C73—42no—
184C2×C73—162no—
185C2×C73C2×#48011yescharacteristic
186C2×C73#341511yescharacteristic
187C2×C73#341611yescharacteristic
188#3058D7311yescharacteristic
189#2913—731no—
190#2234—732no—
191#2234—732no—
192#2233—732no—
193#2232—732no—
194#2159—731no—
195#2158—731no—
196#2140—731no—
197C2×#357—731no—
198#2113—731no—
199C73#3365011yescharacteristic, complemented, Sylow
200C22×D8C2×D7311yescharacteristic
201C2×#131C2×D7311yescharacteristic
202#535—21no—
203#520—1461no—
204C2×M32—731no—
205C22×C16—731no—
206#469—22no—
207#466C2×D7312yes—
208#466C2×D7312yes—
209#428—1461no—
210#428—1462no—
211#427—731no—
212#427—732no—
213#427C2×D7311yescharacteristic
214C2×#107—731no—
215C2×#107—732no—
216#418—21no—
217#367—731no—
218#367—731no—
219#367—732no—
220#366—731no—
221#366—731no—
222#366—732no—
223#357—732no—
224#357—732no—
225#357—732no—
226#357—732no—
227C22⋊C16—732no—
228C22⋊C16—732no—
229C2×#41C22⋊D7311yescharacteristic
230C22×D4—21no—
231C2×D8—21no—
232C2×D8—22no—
233C2×D8—22no—
234C2×D8C22×D7312yes—
235#131—22no—
236#131—22no—
237C2×M16C22⋊D7311yescharacteristic
238C2×M16C22⋊D7312yes—
239C22×C8C22×D7312yes—
240C22×C8C22×D7311yescharacteristic
241#127—41no—
242#124—41no—
243#121—22no—
244#121—22no—
245#120—42no—
246C4×Q8—2921no—
247C4×D4—2922no—
248C4×D4—2921no—
249#117—21no—
250#117—1462no—
251#117—1461no—
252C2×C4⋊C4—1462no—
253C2×C4⋊C4—1461no—
254M32—1462no—
255C2×C16—731no—
256C2×C16—732no—
257C2×C16—731no—
258C2×C16—1462no—
259#107—732no—
260#107—731no—
261#107—731no—
262#107—1462no—
263#107—1461no—
264#107C22×D7311yescharacteristic
265#107C22×D7311yescharacteristic
266#106—21no—
267#106—1462no—
268#106—1461no—
269#102—22no—
270#102—22no—
271C24—41no—
272#41—22no—
273#41—22no—
274C2×Q8#1192401311yescharacteristic
275C2×D4—22no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4#1192401312yes—
282C2×D4#1192401311yescharacteristic
283C22×C4—2922no—
284C22×C4—2921no—
285C22×C4C2×C22⋊D7312yes—
286C22×C4C2×C22⋊D7311yescharacteristic
287D8—22no—
288D8—22no—
289D8—42no—
290M16—22no—
291M16—24no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8C2×C22⋊D7312yes—
295C2×C8C2×C22⋊D7311yescharacteristic
296C2×C8#1192402712yes—
297C2×C8#1192402712yes—
298C2×C8C23×D7311yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—2922no—
302C4⋊C4—2921no—
303C4⋊C4—1461no—
304C4⋊C4—1461no—
305C4⋊C4—1462no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—2922no—
310C22⋊C4—2921no—
311C42—41no—
312C42—2922no—
313C42—2921no—
314C16—1461no—
315C16—1461no—
316C16—1462no—
317C23—42no—
318C23—82no—
319C23—161no—
320C23—161no—
321C23C2×#1192401312yes—
322C23C2×#1192401311yescharacteristic
323Q8—22no—
324D4—24no—
325D4—22no—
326D4—42no—
327D4—42no—
328D4—82no—
329D4—162no—
330D4—162no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—2921no—
336C2×C4—2922no—
337C2×C4—2921no—
338C2×C4—11681no—
339C2×C4—11682no—
340C2×C4#1455657212yes—
341C2×C4C2×#1192401312yes—
342C2×C4C2×#1192401311yescharacteristic
343C2×C4#1455662112yes—
344C2×C4C22×C22⋊D7311yescharacteristic
345C8—22no—
346C8#1455657212yes—
347C8C2×#1192402711yescharacteristic
348C8C2×#1192402711yescharacteristic
349C22—21no—
350C22—21no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—82no—
355C22—82no—
356C22—82no—
357C22—164no—
358C22#1798889812yes—
359C22C22×#1192401311yescharacteristic
360C4—22no—
361C4—81no—
362C4—81no—
363C4—5842no—
364C4—5841no—
365C4—5841no—
366C4#1798889812yes—
367C4C2×#1455657211yescharacteristic
368C4C2×#1455662111yescharacteristic
369C2—22no—
370C2—42no—
371C2—162no—
372C2C2×#1798889811yescharacteristic
373C2#2262307011yescharacteristic
374C2#2262307811yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
Fitting, prime radicalC73×#3058C2
socleC22×C73C22×D8
hypercenter#3058D73
hypocenterC73#33650
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3058

Complements

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

NH ≅ G/Ntype
1. #22618531C2semidirect
2. #22612119C2semidirect
3. #22612119C2semidirect
4. #22612114C2semidirect
5. #22612109C2semidirect
6. #22611720C2semidirect
7. #22611713C2semidirect
8. #22611647C2semidirect
9. C2×#17988388C2semidirect
10. #22611545C2semidirect
18. #17988625C22semidirect
20. C2×#14556525C22semidirect
21. #17988414C22semidirect
22. #17988414C22semidirect
23. #17988414C22semidirect
24. #17988412C22semidirect
25. #17988412C22semidirect
26. #17988412C22semidirect
31. C22⋊#11924003C22semidirect
32. C22⋊#11924003C22semidirect
199. C73#33650semidirect

Maximal subgroups

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

subgroupindex
#226185312
#226121192
#226121142
#226121092
#226117202
#226117132
#226116472
C2×#179883882
#226115452
C73×#30582
#3365073