Finite Group Databasebeta

← #33003041

Subgroups — #33003041

Order 18688.

590 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556532C211yesmaximal, characteristic
2C2×#17988388C211yesmaximal, characteristic, complemented
3C2×#17988387C211yesmaximal, characteristic, complemented
4#22610797C211yesmaximal, characteristic, complemented
5#22608022C211yesmaximal, characteristic, complemented
6#22608009C211yesmaximal, characteristic, complemented
7C2×C4×D8×C73C211yesmaximal, characteristic, complemented
8C22×D584C2211yescharacteristic
9C2×#14556532—22no—
10C2×#14556532—22no—
11C2×#14556532C414yescomplemented
12C22×#11924003C2211yescharacteristic
13C22×#11924003C2211yescharacteristic
14C2×#14556525C2211yescharacteristic
15C2×#14556513—21no—
16#17988489—21no—
17#17988388—22no—
18#17988387—22no—
19C22×D8×C73C2211yescharacteristic
20C2×C4×D4×C73—21no—
21C4×D8×C73—22no—
22C4×D8×C73—22no—
23C2×#107×C73C2211yescharacteristic
24C2×#102×C73—21no—
25C2×C4×C8×C73C2211yescharacteristic
26C22×D292—21no—
27C2×D584—21no—
28C2×D584—21no—
29C2×D584—21no—
30C2×D584—21no—
31C2×D584—21no—
32C2×D584C2×C412yescomplemented
33C2×C4⋊Dic73—21no—
34C2×#11924005—41no—
35C2×#11924005—41no—
36#14556532—28no—
37C2×#11924003—21no—
38C2×#11924003—21no—
39C2×#11924003—21no—
40C2×#11924003—21no—
41C2×#11924003—21no—
42C2×#11924003C2×C412yescomplemented
43#14556525—21no—
44#14556525—21no—
45#14556513—42no—
46C22×D4×C73—21no—
47C23×C4×C73—41no—
48C2×D8×C73—21no—
49C2×D8×C73—21no—
50C2×D8×C73—21no—
51C2×D8×C73C2×C412yes—
52C2×D8×C73D412yes—
53C2×D8×C73D412yes—
54C22×C8×C73D411yescharacteristic
55C22×C8×C73C2311yescharacteristic
56C4×D4×C73—24no—
57C4×D4×C73—42no—
58C2×C4⋊C4×C73—21no—
59C2×C22⋊C4×C73—41no—
60C2×C42×C73D411yescharacteristic
61#107×C73—21no—
62#107×C73D412yes—
63#102×C73—24no—
64C4×C8×C73—21no—
65C4×C8×C73—21no—
66C23×D73—41no—
67C2×D292—22no—
68C2×D292—41no—
69C2×D292—41no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—41no—
73C22×Dic73—41no—
74D584—22no—
75D584—22no—
76D584—41no—
77D584—41no—
78C4⋊Dic73—41no—
79C4⋊Dic73—41no—
80#11924005—82no—
81#11924005—82no—
82#11924003—22no—
83#11924003—41no—
84C24×C73—41no—
85C2×D4×C73—22no—
86C2×D4×C73—22no—
87C2×D4×C73—22no—
88C2×D4×C73—41no—
89C2×D4×C73—41no—
90C2×D4×C73—41no—
91C22×C4×C73—44no—
92C22×C4×C73—41no—
93C22×C4×C73—82no—
94C22×C4×C73—82no—
95C22×C4×C73D811yescharacteristic
96C22×C4×C73D811yescharacteristic
97C22×C4×C73C2×D411yescharacteristic
98D8×C73—22no—
99D8×C73—22no—
100D8×C73—22no—
101D8×C73C22⋊C414yescomplemented
102C2×C8×C73—21no—
103C2×C8×C73—21no—
104C2×C8×C73—21no—
105C2×C8×C73C22×C411yescharacteristic
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73C2×D411yescharacteristic
108C2×C8×C73C2×D411yescharacteristic
109C2×C8×C73#4111yescharacteristic
110C2×C8×C73#4111yescharacteristic
111C4⋊C4×C73—22no—
112C4⋊C4×C73—41no—
113C22⋊C4×C73—44no—
114C42×C73—21no—
115C42×C73—21no—
116C23×C73—42no—
117C23×C73—42no—
118C23×C73—42no—
119C23×C73—81no—
120C23×C73—81no—
121C23×C73—81no—
122C23×C73—81no—
123C23×C73C2×D811yescharacteristic
124C22×D73—42no—
125C22×D73—81no—
126C22×D73—81no—
127C22×D73—81no—
128C22×D73—81no—
129C22×D73—81no—
130C22×D73—81no—
131D4×C73—24no—
132D4×C73—42no—
133D4×C73—42no—
134D4×C73—42no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—42no—
141C2×C4×C73—88no—
142C2×C4×C73—81no—
143C2×C4×C73—81no—
144C2×C4×C73D1612yescomplemented
145C2×C4×C73C4×D411yescharacteristic
146C2×C4×C73#12011yescharacteristic
147C2×C4×C73#12111yescharacteristic
148C2×C4×C73#13511yescharacteristic
149C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
151C2×Dic73—81no—
152C2×Dic73—81no—
153C2×Dic73—81no—
154D292—42no—
155D292—42no—
156D292—81no—
157D292—81no—
158C8×C73—41no—
159C8×C73C2×C22⋊C411yescharacteristic
160C8×C73C4×D411yescharacteristic
161C8×C73#12111yescharacteristic
162C8×C73#12311yescharacteristic
163C22×C73—44no—
164C22×C73—82no—
165C22×C73—82no—
166C22×C73—82no—
167C22×C73—82no—
168C22×C73—82no—
169C22×C73—82no—
170C22×C73C4×D811yescharacteristic
171C22×C73#44711yescharacteristic
172C22×C73#45911yescharacteristic
173C22×C73C2×D1611yescharacteristic
174C22×C73#50811yescharacteristic
175C22×C73#50911yescharacteristic
176C22×C73#50911yescharacteristic
177C2×D73—82no—
178C2×D73—82no—
179C2×D73—82no—
180C2×D73—161no—
181C2×D73—161no—
182C2×D73—161no—
183C2×D73—161no—
184C4×C73—22no—
185C4×C73—41no—
186C4×C73—82no—
187C4×C73#38611yescharacteristic
188C4×C73#44211yescharacteristic
189C4×C73#44911yescharacteristic
190C4×C73#46311yescharacteristic
191Dic73—161no—
192#13012—731nomaximal, Sylow
193C2×C73—84no—
194C2×C73#187911yescharacteristic
195C2×C73C4×D1611yescharacteristic
196C2×C73#217711yescharacteristic
197C2×C73#218411yescharacteristic
198C2×C73#218611yescharacteristic
199C2×C73#220611yescharacteristic
200C2×C73#221011yescharacteristic
201D73—162no—
202C22×D16—731no—
203C2×C4×D8D7311yescharacteristic, complemented
204C2×#357—731no—
205C2×#357—731no—
206#1970—731no—
207#1380—731no—
208#1374—731no—
209C73#1301211yescharacteristic, complemented, Sylow
210C22×D8—731no—
211C22×D8C2×D7311yescharacteristic
212C2×C4×D4—21no—
213C2×D16—734no—
214C2×D16—1462no—
215C2×D16—1462no—
216C22×C16—731no—
217C22×C16—731no—
218C4×D8—22no—
219C4×D8—22no—
220C2×#107—731no—
221C2×#107C2×D7311yescharacteristic
222C2×#102—21no—
223C2×#102—1461no—
224C2×C4×C8C2×D7311yescharacteristic
225#390—1461no—
226#357—1462no—
227#357—1462no—
228C22×D4—21no—
229C22×D4—1461no—
230C23×C4—41no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—732no—
235C2×D8—1461no—
236C2×D8—1461no—
237C2×D8—1461no—
238C2×D8—1461no—
239C2×D8—1461no—
240C2×D8C4×D7312yes—
241C2×D8D29212yes—
242C2×D8C22⋊D7312yes—
243C22×C8C22⋊D7311yescharacteristic
244C22×C8C22×D7311yescharacteristic
245C4×D4—24no—
246C4×D4—42no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—1461no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—2921no—
251C2×C22⋊C4—2921no—
252C2×C42C22⋊D7311yescharacteristic
253D16—1468no—
254C2×C16—732no—
255C2×C16—1461no—
256C2×C16—1461no—
257C2×C16—1461no—
258C2×C16—1461no—
259C2×C16—1461no—
260#107—21no—
261#107—1461no—
262#107—1461no—
263#107C22⋊D7312yes—
264#102—24no—
265#102—2922no—
266C4×C8—21no—
267C4×C8—21no—
268C24—41no—
269C24—2921no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—2921no—
277C2×D4—2921no—
278C2×D4—2921no—
279C2×D4—2921no—
280C2×D4—2921no—
281C2×D4—1462no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—2921no—
287C22×C4#1192401311yescharacteristic
288C22×C4#1192401311yescharacteristic
289C22×C4C2×C22⋊D7311yescharacteristic
290D8—22no—
291D8—22no—
292D8—22no—
293D8—2921no—
294D8—2921no—
295D8—1462no—
296D8—1462no—
297D8#1192400514yescomplemented
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8C2×C4×D7311yescharacteristic
302C2×C8C2×C22⋊D7311yescharacteristic
303C2×C8C2×C22⋊D7311yescharacteristic
304C2×C8C2×D29211yescharacteristic
305C2×C8#1192402811yescharacteristic
306C2×C8#1192402811yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4—2921no—
310C4⋊C4—2921no—
311C22⋊C4—44no—
312C22⋊C4—5842no—
313C22⋊C4—5842no—
314C42—21no—
315C42—21no—
316C16—2921no—
317C16—1462no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—2922no—
326C23—5841no—
327C23—5841no—
328C23—5841no—
329C23—5841no—
330C23—5841no—
331C23—5841no—
332C23C2×#1192401311yescharacteristic
333D4—24no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—2922no—
338D4—2922no—
339D4—5841no—
340D4—5841no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—88no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—5841no—
351C2×C4—5841no—
352C2×C4—5841no—
353C2×C4#1455653212yescomplemented
354C2×C4C4×C22⋊D7311yescharacteristic
355C2×C4#1455656011yescharacteristic
356C2×C4#1455656511yescharacteristic
357C2×C4#1455661111yescharacteristic
358C2×C4#1455661611yescharacteristic
359C2×C4#1455661711yescharacteristic
360C8—41no—
361C8C2×#1192400511yescharacteristic
362C8C4×C22⋊D7311yescharacteristic
363C8#1455656511yescharacteristic
364C8#1455657411yescharacteristic
365C22—44no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—5842no—
373C22—5842no—
374C22—5842no—
375C22—11681no—
376C22—11681no—
377C22—11681no—
378C22—11681no—
379C22C4×#1192401311yescharacteristic
380C22#1798872111yescharacteristic
381C22#1798878211yescharacteristic
382C22C2×#1455653211yescharacteristic
383C22#1798901711yescharacteristic
384C22#1798902211yescharacteristic
385C22#1798902311yescharacteristic
386C4—22no—
387C4—41no—
388C4—82no—
389C4—11681no—
390C4#1798846611yescharacteristic
391C4#1798869711yescharacteristic
392C4#1798873211yescharacteristic
393C4#1798880211yescharacteristic
394C2—84no—
395C2—11682no—
396C2#2261027911yescharacteristic
397C2C4×#1455653211yescharacteristic
398C2#2261180511yescharacteristic
399C2#2261184311yescharacteristic
400C2#2261185411yescharacteristic
401C2#2261196611yescharacteristic
402C2#2261198611yescharacteristic
subgroupG/H
centerC23C2×#11924013
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×C4×D8×C73C2
socleC23×C73C2×D8
prime radicalC22×#14556532C2
cosocleC22×C8×C73C23
hypercenterC2×C4×D8D73
hypocenterC73#13012
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
2. C2×#17988388C2semidirect
3. C2×#17988387C2semidirect
4. #22610797C2semidirect
5. #22608022C2semidirect
6. #22608009C2semidirect
7. C2×C4×D8×C73C2semidirect
11. C2×#14556532C4semidirect
32. C2×D584C2×C4semidirect
42. C2×#11924003C2×C4semidirect
101. D8×C73C22⋊C4semidirect
144. C2×C4×C73D16semidirect
203. C2×C4×D8D73semidirect
209. C73#13012semidirect
297. D8#11924005semidirect
353. C2×C4#14556532semidirect

Maximal subgroups

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

subgroupindex
C22×#145565322
C2×#179883882
C2×#179883872
#226107972
#226080222
#226080092
C2×C4×D8×C732
#1301273