Finite Group Databasebeta

← #33240828

Subgroups — #33240828

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#22619579C211yesmaximal, characteristic
2#22612122C212yesmaximal, complemented
3#22612122C212yesmaximal, complemented
4#22612116C212yesmaximal, complemented
5#22612111C212yesmaximal, complemented
6#22611717C211yesmaximal, characteristic, complemented
7#22611713C211yesmaximal, characteristic, complemented
8#22611646C211yesmaximal, characteristic, complemented
9C2×#17988387C211yesmaximal, characteristic, complemented
10#22611545C211yesmaximal, characteristic, complemented
11C73×#2913C211yesmaximal, characteristic, complemented
12C22×D584C2211yescharacteristic
13#17989151—21no—
14C2×#14556531C2211yescharacteristic
15C22×#11924003C2211yescharacteristic
16#17988840—22no—
17#17988823C2212yes—
18#17988823C2212yes—
19#17988625C2211yescharacteristic
20#17988593—21no—
21#17988414C2211yescharacteristic, complemented
22#17988414C2211yescharacteristic, complemented
23#17988414C2212yescomplemented
24#17988412C2211yescharacteristic, complemented
25#17988412C2211yescharacteristic, complemented
26#17988412C2212yescomplemented
27#17988387C2212yes—
28#17988387C2212yes—
29#17988387C2212yes—
30#17988387C2212yes—
31C22⋊#11924003C2212yescomplemented
32C22⋊#11924003C2212yescomplemented
33C2×#131×C73C2211yescharacteristic
34#520×C73—21no—
35#428×C73—22no—
36#428×C73—21no—
37#427×C73C2211yescharacteristic
38#427×C73C2212yescomplemented
39C2×#107×C73C2211yescharacteristic
40C2×#107×C73C2212yescomplemented
41C22×D292—21no—
42C2×D584—21no—
43C2×D584—22no—
44C2×D584—22no—
45C2×D584C2312yes—
46#14556587—41no—
47#14556580—41no—
48#14556565—22no—
49#14556565—22no—
50#14556559—42no—
51#14556550—21no—
52#14556531—22no—
53C2×#11924003—22no—
54C2×#11924003C2311yescharacteristic
55C2×#11924003C2312yes—
56C2×#11924003C2311yescharacteristic
57#14556525C2311yescharacteristic
58#14556525C2311yescharacteristic
59#14556523—21no—
60#14556513—22no—
61#14556513—22no—
62C2×#41×C73D411yescharacteristic
63#131×C73—22no—
64#131×C73—22no—
65C2×M16×C73D412yes—
66C2×M16×C73D411yescharacteristic
67C22×C8×C73C2312yes—
68C22×C8×C73C2311yescharacteristic
69C4×Q8×C73—41no—
70C4×D4×C73—42no—
71C4×D4×C73—41no—
72#117×C73—21no—
73#117×C73—22no—
74C2×C4⋊C4×C73—21no—
75C2×C4⋊C4×C73—22no—
76#107×C73—21no—
77#107×C73—22no—
78#107×C73C2312yes—
79#107×C73C2311yescharacteristic
80#107×C73C2311yescharacteristic
81#106×C73—21no—
82#106×C73—22no—
83C23×D73—41no—
84C2×D292—22no—
85C2×D292—42no—
86C2×D292—42no—
87C2×D292—41no—
88C2×C22⋊D73—42no—
89C2×C22⋊D73—42no—
90D584—22no—
91D584—22no—
92D584—42no—
93C4⋊Dic73—21no—
94C4⋊Dic73—21no—
95C22⋊Dic73—41no—
96#11924005—42no—
97#11924005—42no—
98C4×Dic73—41no—
99#11924003—21no—
100#11924003—21no—
101#11924003—22no—
102#41×C73—22no—
103#41×C73—22no—
104C2×Q8×C73D811yescharacteristic
105C2×D4×C73D812yes—
106C2×D4×C73D811yescharacteristic
107C22×C4×C73—42no—
108C22×C4×C73—41no—
109C22×C4×C73C2×D412yes—
110C22×C4×C73C2×D411yescharacteristic
111M16×C73—22no—
112M16×C73—24no—
113C2×C8×C73—22no—
114C2×C8×C73—22no—
115C2×C8×C73C2×D412yes—
116C2×C8×C73C2×D411yescharacteristic
117C2×C8×C73#4112yes—
118C2×C8×C73#4112yes—
119C2×C8×C73C2411yescharacteristic
120C4⋊C4×C73—21no—
121C4⋊C4×C73—21no—
122C4⋊C4×C73—22no—
123C4⋊C4×C73—42no—
124C4⋊C4×C73—41no—
125C22⋊C4×C73—42no—
126C22⋊C4×C73—41no—
127C42×C73—42no—
128C42×C73—41no—
129C23×C73C2×D812yes—
130C23×C73C2×D811yescharacteristic
131C22×D73—42no—
132C22×D73—82no—
133C22×D73—161no—
134C22×D73—161no—
135Q8×C73—22no—
136D4×C73—24no—
137D4×C73—22no—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73—41no—
141C2×C4×C73—42no—
142C2×C4×C73—41no—
143C2×C4×C73—161no—
144C2×C4×C73—162no—
145C2×C4×C73#12312yes—
146C2×C4×C73C2×D812yes—
147C2×C4×C73C2×D811yescharacteristic
148C2×C4×C73#13712yes—
149C2×C4×C73C22×D411yescharacteristic
150C22⋊D73—162no—
151C22⋊D73—162no—
152C2×Dic73—41no—
153C2×Dic73—41no—
154D292—42no—
155D292—42no—
156D292—82no—
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#48012yes—
167C22×C73C22×D811yescharacteristic
168C2×D73—82no—
169C2×D73—82no—
170C2×D73—82no—
171C2×D73—164no—
172C4×C73—22no—
173C4×C73—82no—
174C4×C73—81no—
175C4×C73—81no—
176C4×C73#48012yes—
177C4×C73C2×#12311yescharacteristic
178C4×C73C2×#13711yescharacteristic
179Dic73—81no—
180Dic73—81no—
181#33650—731nomaximal, Sylow
182C2×C73—22no—
183C2×C73—42no—
184C2×C73C2×#48011yescharacteristic
185C2×C73#341511yescharacteristic
186C2×C73#341611yescharacteristic
187D73—162no—
188#3058—731no—
189#2913D7311yescharacteristic, complemented
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×D8—731no—
201C2×#131C2×D7311yescharacteristic
202#535—1461no—
203#520—21no—
204C2×M32—731no—
205C22×C16—731no—
206#469—1462no—
207#466—732no—
208#466—732no—
209#428—22no—
210#428—21no—
211#427—731no—
212#427C2×D7311yescharacteristic
213#427C2×D7312yescomplemented
214C2×#107C2×D7311yescharacteristic
215C2×#107C2×D7312yescomplemented
216#418—1461no—
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—1461no—
231C2×D8—732no—
232C2×D8—1461no—
233C2×D8—1462no—
234C2×D8—1462no—
235#131—22no—
236#131—22no—
237C2×M16C22⋊D7311yescharacteristic
238C2×M16C22⋊D7312yes—
239C22×C8C22×D7312yes—
240C22×C8C22×D7311yescharacteristic
241#127—2921no—
242#124—2921no—
243#121—1462no—
244#121—1462no—
245#120—2922no—
246C4×Q8—41no—
247C4×D4—42no—
248C4×D4—41no—
249#117—21no—
250#117—22no—
251#117—1461no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254M32—1462no—
255C2×C16—731no—
256C2×C16—732no—
257C2×C16—731no—
258C2×C16—1462no—
259#107—21no—
260#107—22no—
261#107—731no—
262#107—731no—
263#107C22×D7312yes—
264#107C22×D7311yescharacteristic
265#107C22×D7311yescharacteristic
266#106—21no—
267#106—22no—
268#106—1461no—
269#102—1462no—
270#102—1462no—
271C24—2921no—
272#41—22no—
273#41—22no—
274C2×Q8#1192401311yescharacteristic
275C2×D4—2922no—
276C2×D4—2922no—
277C2×D4—2922no—
278C2×D4—2921no—
279C2×D4—2922no—
280C2×D4—1462no—
281C2×D4#1192401312yes—
282C2×D4#1192401311yescharacteristic
283C22×C4—42no—
284C22×C4—41no—
285C22×C4C2×C22⋊D7312yes—
286C22×C4C2×C22⋊D7311yescharacteristic
287D8—2922no—
288D8—1462no—
289D8—1462no—
290M16—22no—
291M16—24no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8C2×C22⋊D7312yes—
295C2×C8C2×C22⋊D7311yescharacteristic
296C2×C8#1192402812yes—
297C2×C8#1192402812yes—
298C2×C8C23×D7311yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—1461no—
305C4⋊C4—1461no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—2921no—
309C22⋊C4—2922no—
310C22⋊C4—2922no—
311C42—42no—
312C42—41no—
313C42—2921no—
314C16—1461no—
315C16—1461no—
316C16—1462no—
317C23—2922no—
318C23—5842no—
319C23—11681no—
320C23—11681no—
321C23C2×#1192401312yes—
322C23C2×#1192401311yescharacteristic
323Q8—22no—
324D4—24no—
325D4—22no—
326D4—2922no—
327D4—2922no—
328D4—5842no—
329D4—11682no—
330D4—11682no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—161no—
337C2×C4—162no—
338C2×C4—2921no—
339C2×C4—2921no—
340C2×C4#1455657412yes—
341C2×C4C2×#1192401312yes—
342C2×C4C2×#1192401311yescharacteristic
343C2×C4#1455662212yes—
344C2×C4C22×C22⋊D7311yescharacteristic
345C8—22no—
346C8#1455657412yes—
347C8C2×#1192402811yescharacteristic
348C8C2×#1192402811yescharacteristic
349C22—21no—
350C22—21no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—5842no—
355C22—5842no—
356C22—5842no—
357C22—11684no—
358C22#1798890012yes—
359C22C22×#1192401311yescharacteristic
360C4—22no—
361C4—82no—
362C4—81no—
363C4—81no—
364C4—5841no—
365C4—5841no—
366C4#1798890012yes—
367C4C2×#1455657411yescharacteristic
368C4C2×#1455662211yescharacteristic
369C2—22no—
370C2—42no—
371C2—11682no—
372C2C2×#1798890011yescharacteristic
373C2#2262307011yescharacteristic
374C2#2262307511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2913C2
socleC22×C73C22×D8
prime radical#22619579C2
hypercenter#2913D73
hypocenterC73#33650
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #2913

Complements

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

NH ≅ G/Ntype
2. #22612122C2semidirect
3. #22612122C2semidirect
4. #22612116C2semidirect
5. #22612111C2semidirect
6. #22611717C2semidirect
7. #22611713C2semidirect
8. #22611646C2semidirect
9. C2×#17988387C2semidirect
10. #22611545C2semidirect
11. C73×#2913C2semidirect
21. #17988414C22semidirect
22. #17988414C22semidirect
23. #17988414C22semidirect
24. #17988412C22semidirect
25. #17988412C22semidirect
26. #17988412C22semidirect
31. C22⋊#11924003C22semidirect
32. C22⋊#11924003C22semidirect
38. #427×C73C22semidirect
40. C2×#107×C73C22semidirect
189. #2913D73semidirect
199. C73#33650semidirect
213. #427C2×D73semidirect
215. C2×#107C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226195792
#226121222
#226121162
#226121112
#226117172
#226117132
#226116462
C2×#179883872
#226115452
C73×#29132
#3365073