Finite Group Databasebeta

← #33003125

Subgroups — #33003125

Order 18688.

466 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#22623065C211yesmaximal, characteristic
2#22611680C211yesmaximal, characteristic, complemented
3#22611658C211yesmaximal, characteristic, complemented
4#22610824C211yesmaximal, characteristic, complemented
5#22608042C211yesmaximal, characteristic, complemented
6#22608015C211yesmaximal, characteristic, complemented
7C2×#444×C73C211yesmaximal, characteristic, complemented
8C2×#14556613C2211yescharacteristic
9#17989028—21no—
10#17989027C412yescomplemented
11#17989023—21no—
12#17989022C412yescomplemented
13#17989017—22no—
14C2×#14556531C2211yescharacteristic
15C2×#14556531C2211yescharacteristic
16C2×#14556527C2211yescharacteristic
17C2×#14556519—21no—
18#17988553—21no—
19#17988403—21no—
20#17988400—21no—
21#17988393—22no—
22C2×#135×C73C2211yescharacteristic
23C2×#131×C73—21no—
24#444×C73—22no—
25#444×C73—22no—
26C2×#108×C73C2211yescharacteristic
27C2×#104×C73—21no—
28C2×#97×C73C2211yescharacteristic
29C2×#11924028—21no—
30#14556613—21no—
31#14556613—21no—
32#14556613—21no—
33#14556613—21no—
34C2×Dic292C2×C411yescharacteristic
35C2×#11924017—21no—
36C2×D584C2×C411yescharacteristic
37C2×#11924012—21no—
38#14556537—22no—
39#14556535—22no—
40#14556534—22no—
41#14556532—22no—
42#14556531—21no—
43#14556531—21no—
44#14556531—21no—
45#14556531C2×C412yes—
46C2×#11924003—21no—
47C2×#11924003—21no—
48#14556527—21no—
49#14556527—21no—
50#14556519—42no—
51#14556503—42no—
52C2×#41×C73—21no—
53#135×C73—21no—
54#135×C73—21no—
55#135×C73C2×C412yes—
56#135×C73D412yes—
57C2×Q16×C73D411yescharacteristic
58C2×SD16×C73—21no—
59C2×D8×C73D411yescharacteristic
60#131×C73—24no—
61#131×C73—42no—
62C2×M16×C73—21no—
63C2×M16×C73—41no—
64C22×C8×C73—41no—
65C22×C8×C73D411yescharacteristic
66C22×C8×C73C2311yescharacteristic
67C2×C42×C73D411yescharacteristic
68#108×C73—21no—
69#108×C73D412yes—
70#104×C73—24no—
71#97×C73—21no—
72#97×C73—21no—
73#11924028—41no—
74#11924028—41no—
75#11924028—41no—
76#11924028—41no—
77C2×Dic146—21no—
78C2×D292—21no—
79C2×C22⋊D73—41no—
80C2×C4×D73—41no—
81Dic292—21no—
82Dic292—21no—
83#11924017—41no—
84#11924017—41no—
85D584—21no—
86D584—21no—
87#11924012—41no—
88#11924012—41no—
89C2×#10596831—41no—
90#11924003—22no—
91#11924003—41no—
92#41×C73—22no—
93#41×C73—22no—
94#41×C73—41no—
95#41×C73—41no—
96C2×Q8×C73—21no—
97C2×D4×C73—21no—
98C2×D4×C73—41no—
99C22×C4×C73—21no—
100C22×C4×C73—41no—
101C22×C4×C73C2×D411yescharacteristic
102Q16×C73—21no—
103Q16×C73C22⋊C412yes—
104SD16×C73—22no—
105SD16×C73—22no—
106D8×C73—21no—
107D8×C73C22⋊C412yes—
108M16×C73—22no—
109M16×C73—44no—
110M16×C73—41no—
111C2×C8×C73—21no—
112C2×C8×C73—21no—
113C2×C8×C73—44no—
114C2×C8×C73—41no—
115C2×C8×C73D812yes—
116C2×C8×C73C22×C411yescharacteristic
117C2×C8×C73C2×D411yescharacteristic
118C2×C8×C73C2×D411yescharacteristic
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73#4111yescharacteristic
121C2×C8×C73#4111yescharacteristic
122C42×C73—21no—
123C42×C73—21no—
124C23×C73—41no—
125C23×C73#13511yescharacteristic
126C22×D73—41no—
127Q8×C73—22no—
128Q8×C73—41no—
129D4×C73—22no—
130D4×C73—42no—
131D4×C73—42no—
132D4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—42no—
135C2×C4×C73—42no—
136C2×C4×C73—41no—
137C2×C4×C73—81no—
138C2×C4×C73C4×D411yescharacteristic
139C2×C4×C73#12011yescharacteristic
140C2×C4×C73#12111yescharacteristic
141C2×C4×C73C2×D811yescharacteristic
142C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
144C22⋊D73—81no—
145C22⋊D73—81no—
146C2×Dic73—41no—
147D292—41no—
148D292—41no—
149C4×D73—81no—
150C4×D73—81no—
151Dic146—41no—
152Dic146—41no—
153C8×C73—22no—
154C8×C73—42no—
155C8×C73C2×C22⋊C411yescharacteristic
156C8×C73C4×D411yescharacteristic
157C8×C73#12111yescharacteristic
158C8×C73#12311yescharacteristic
159#10596831—81no—
160C22×C73—41no—
161C22×C73—42no—
162C22×C73—82no—
163C22×C73C4×D811yescharacteristic
164C22×C73#44911yescharacteristic
165C22×C73#46311yescharacteristic
166C2×D73—82no—
167C2×D73—81no—
168C4×C73—42no—
169C4×C73—81no—
170C4×C73#38611yescharacteristic
171C4×C73#44211yescharacteristic
172C4×C73#44711yescharacteristic
173C4×C73#45911yescharacteristic
174Dic73—81no—
175#13024—731nomaximal, Sylow
176C2×C73—21no—
177C2×C73—21no—
178C2×C73—21no—
179C2×C73—82no—
180C2×C73#187911yescharacteristic
181D73—161no—
182#3414—731no—
183C2×#444D7311yescharacteristic, complemented
184#2148—731no—
185#2143—731no—
186#1975—731no—
187#1388—731no—
188#1377—731no—
189C73#1302411yescharacteristic, complemented, Sylow
190C2×#135—731no—
191C2×#135C2×D7311yescharacteristic
192C2×#131—21no—
193#510—732no—
194#510—1461no—
195#509—732no—
196#509—1461no—
197#508—1462no—
198C2×M32—731no—
199C2×M32—731no—
200#444—22no—
201#444—22no—
202C2×#108—731no—
203C2×#108C2×D7311yescharacteristic
204C2×#104—21no—
205C2×#104—1461no—
206#408—1461no—
207C2×#97C2×D7311yescharacteristic
208#362—1461no—
209#361—1461no—
210#359—1462no—
211C2×#41—21no—
212C2×#41—1461no—
213#135—21no—
214#135—21no—
215#135—1461no—
216#135—1461no—
217#135—1461no—
218#135—1461no—
219#135C4×D7312yes—
220#135C22⋊D7312yes—
221C2×Q16—731no—
222C2×Q16D29211yescharacteristic
223C2×SD16—21no—
224C2×SD16—1461no—
225C2×D8—731no—
226C2×D8D29211yescharacteristic
227#131—24no—
228#131—42no—
229C2×M16—21no—
230C2×M16—41no—
231C2×M16—1461no—
232C22×C8—41no—
233C22×C8C22⋊D7311yescharacteristic
234C22×C8C22×D7311yescharacteristic
235C2×C42C22⋊D7311yescharacteristic
236Q32—1462no—
237SD32—1462no—
238SD32—1462no—
239D16—1462no—
240M32—732no—
241M32—1461no—
242M32—1461no—
243M32—1461no—
244C2×C16—1461no—
245C2×C16—1461no—
246#108—21no—
247#108—1461no—
248#108—1461no—
249#108C22⋊D7312yes—
250#104—24no—
251#104—2922no—
252C22⋊C8—2922no—
253#97—21no—
254#97—21no—
255#41—22no—
256#41—22no—
257#41—41no—
258#41—41no—
259#41—2921no—
260#41—2921no—
261#41—2921no—
262#41—2921no—
263C2×Q8—21no—
264C2×Q8—1461no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—2921no—
268C2×D4—1461no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—2921no—
272C22×C4C2×C22⋊D7311yescharacteristic
273Q16—21no—
274Q16—1461no—
275Q16—1461no—
276Q16#1192400512yes—
277SD16—22no—
278SD16—22no—
279SD16—2921no—
280SD16—2921no—
281D8—21no—
282D8—1461no—
283D8—1461no—
284D8#1192400512yes—
285M16—22no—
286M16—44no—
287M16—41no—
288M16—2921no—
289M16—2921no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—44no—
293C2×C8—41no—
294C2×C8—2921no—
295C2×C8#1192401312yes—
296C2×C8C2×C4×D7311yescharacteristic
297C2×C8C2×C22⋊D7311yescharacteristic
298C2×C8C2×C22⋊D7311yescharacteristic
299C2×C8C2×D29211yescharacteristic
300C2×C8#1192402811yescharacteristic
301C2×C8#1192402811yescharacteristic
302C42—21no—
303C42—21no—
304C16—2921no—
305C16—1462no—
306C23—41no—
307C23—2921no—
308C23#1455661111yescharacteristic
309Q8—22no—
310Q8—41no—
311Q8—2921no—
312Q8—2921no—
313D4—22no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—2921no—
318D4—2921no—
319D4—5841no—
320D4—5841no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—2921no—
327C2×C4—5841no—
328C2×C4—5841no—
329C2×C4C4×C22⋊D7311yescharacteristic
330C2×C4#1455656011yescharacteristic
331C2×C4#1455656511yescharacteristic
332C2×C4C2×#1192401311yescharacteristic
333C2×C4#1455661611yescharacteristic
334C2×C4#1455661711yescharacteristic
335C8—22no—
336C8—42no—
337C8—5841no—
338C8C2×#1192400511yescharacteristic
339C8C4×C22⋊D7311yescharacteristic
340C8#1455656511yescharacteristic
341C8#1455657411yescharacteristic
342C22—41no—
343C22—42no—
344C22—82no—
345C22—5842no—
346C22—5841no—
347C22C4×#1192401311yescharacteristic
348C22#1798873211yescharacteristic
349C22#1798880211yescharacteristic
350C4—42no—
351C4—81no—
352C4—5841no—
353C4#1798846611yescharacteristic
354C4#1798869711yescharacteristic
355C4#1798872111yescharacteristic
356C4#1798878211yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—82no—
361C2—11681no—
362C2#2261027911yescharacteristic
subgroupG/H
centerC4#17988721
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#444×C73C2
socleC2×C73#1879
prime radical#22623065C2
cosocleC22×C8×C73C23
hypercenterC2×#444D73
hypocenterC73#13024
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. #22611680C2semidirect
3. #22611658C2semidirect
4. #22610824C2semidirect
5. #22608042C2semidirect
6. #22608015C2semidirect
7. C2×#444×C73C2semidirect
10. #17989027C4semidirect
12. #17989022C4semidirect
183. C2×#444D73semidirect
189. C73#13024semidirect

Maximal subgroups

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

subgroupindex
#226230652
#226116802
#226116582
#226108242
#226080422
#226080152
C2×#444×C732
#1302473