Finite Group Databasebeta

← #26371186

Subgroups — #26371186

Order 12032.

534 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253039C211yesmaximal, characteristic
2#20251996C211yesmaximal, characteristic
3#20245606C218yesmaximal, complemented
4#20245162C211yesmaximal, characteristic, complemented
5#20245161C211yesmaximal, characteristic, complemented
6C2×#15688939C211yesmaximal, characteristic, complemented
7#20245008C211yesmaximal, characteristic
8C2×C47×#362C211yesmaximal, characteristic, complemented
9#15689889—21no—
10C2×#13094030C2211yescharacteristic
11#15689697—21no—
12C2×#13093964C2211yescharacteristic
13#15689378C2214yes—
14#15689361—22no—
15#15689171—21no—
16#15689170C2214yes—
17C2×#13093960C2211yescharacteristic
18#108×D47—22no—
19#15689161C2211yescharacteristic
20#15689132—21no—
21#15688947C2214yes—
22#15688946C2214yes—
23#15688939C2214yes—
24#15688904C2214yes—
25C22×Q16×C47C2211yescharacteristic
26C2×M32×C47C2211yescharacteristic
27C2×#108×C47C2211yescharacteristic
28C47×#362C2214yes—
29C2×#10625350D411yescharacteristic
30#13094032—42no—
31#13094030—21no—
32#13094030—21no—
33#13094030—21no—
34#13094030—21no—
35#13094030—42no—
36C2×#10625329—21no—
37C2×#10625329D411yescharacteristic
38C2×#10625324C2312yes—
39M16×D47—24no—
40C2×C8×D47D412yes—
41#13094022—41no—
42#13094013—41no—
43#13094007—42no—
44#13094000—24no—
45#13093983—21no—
46#13093964—21no—
47#13093964C2312yes—
48C2×C47⋊C16—21no—
49#13093960—21no—
50#13093960C2312yes—
51#13093959—24no—
52#13093958—21no—
53#13093956C2312yes—
54#13093948—24no—
55C22×Q8×C47—21no—
56C2×Q16×C47—21no—
57C2×Q16×C47—21no—
58C2×Q16×C47—21no—
59C2×Q16×C47—21no—
60C2×Q16×C47—21no—
61C2×Q16×C47C2312yes—
62C2×M16×C47—21no—
63C22×C8×C47C2311yescharacteristic
64M32×C47—21no—
65M32×C47C2312yes—
66C2×C16×C47—21no—
67#108×C47—21no—
68#108×C47C2312yes—
69C2×Q8×C47—22no—
70C2×Q8×C47—41no—
71C2×Q8×C47—41no—
72C2×Q8×C47—41no—
73C2×Q8×C47—41no—
74C2×Q8×C47—41no—
75C22×C4×C47—41no—
76C22×C4×C47C2×D411yescharacteristic
77C2×C22⋊D47SD1612yes—
78#10625350—21no—
79#10625350—21no—
80#10625350—21no—
81#10625350—21no—
82C2×D188SD1611yescharacteristic
83C2×C4×D47C2×D412yes—
84C2×Dic94SD1611yescharacteristic
85Q16×C47—22no—
86Q16×C47—22no—
87Q16×C47—41no—
88Q16×C47—41no—
89M16×C47—22no—
90M16×C47—41no—
91C2×C8×C47—41no—
92C2×C8×C47C2×D411yescharacteristic
93C2×C8×C47#4111yescharacteristic
94C2×C8×C47#4111yescharacteristic
95C2×C8×C47#4111yescharacteristic
96C2×C8×C47#4111yescharacteristic
97C2×C8×C47C2411yescharacteristic
98C22⋊Dic47—41no—
99(C2×D47)⋊C4—44no—
100C4⋊Dic47—22no—
101Dic47⋊C4—44no—
102C4×Dic47—41no—
103#10625329—21no—
104#10625329—21no—
105#10625329—22no—
106#10625329—41no—
107C2×C47⋊C8—41no—
108C2×C47⋊C8C2×D412yes—
109#10625324—22no—
110#10625324—22no—
111#10625324—44no—
112C8×D47—22no—
113C8×D47—22no—
114C8×D47—44no—
115C16×C47—22no—
116C47⋊C16—22no—
117C23×C47C2×SD1611yescharacteristic
118C22×D47C2×SD1612yes—
119Q8×C47—42no—
120Q8×C47—42no—
121Q8×C47—81no—
122Q8×C47—81no—
123C2×C4×C47—42no—
124C2×C4×C47—81no—
125C2×C4×C47—81no—
126C2×C4×C47#12311yescharacteristic
127C2×C4×C47#12311yescharacteristic
128C2×C4×C47C2×SD1611yescharacteristic
129C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C47C22×D411yescharacteristic
132C22⋊D47—22no—
133C22⋊D47—22no—
134C2×Dic47—42no—
135C2×Dic47C2×SD1612yes—
136D188—21no—
137D188—21no—
138C4×D47—22no—
139C4×D47—22no—
140Dic94—21no—
141Dic94—21no—
142C8×C47—42no—
143C8×C47#12311yescharacteristic
144C8×C47#12311yescharacteristic
145C8×C47C2×#4111yescharacteristic
146C8×C47C2×#4111yescharacteristic
147C47⋊C8—22no—
148C47⋊C8—42no—
149#33658—471nomaximal, Sylow
150C22×C47—41no—
151C22×C47#48111yescharacteristic
152C22×C47#48111yescharacteristic
153C22×C47C22×SD1611yescharacteristic
154C2×D47—22no—
155C2×D47—42no—
156C4×C47—82no—
157C4×C47#48111yescharacteristic
158C4×C47#48111yescharacteristic
159C4×C47C2×#12311yescharacteristic
160C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
161Dic47—22no—
162Dic47—82no—
163#3059—471no—
164#2916—471no—
165#2239—478no—
166#2155—471no—
167#2155—471no—
168C2×#362—471no—
169C2×#362D4711yescharacteristic, complemented
170#2115—471no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47C2×#48111yescharacteristic
175D47—42no—
176C22×Q16C2×D4711yescharacteristic
177#568—941no—
178C2×#131—471no—
179#536—941no—
180C2×M32—471no—
181C2×M32C2×D4711yescharacteristic
182#470—474no—
183#467—942no—
184#430—474no—
185#430—941no—
186C2×#108—471no—
187C2×#108—942no—
188C2×#108C2×D4711yescharacteristic
189#427—471no—
190#419—941no—
191#365—474no—
192#365—474no—
193#362—474no—
194#362C2×D4714yes—
195#349—474no—
196C47#3365811yescharacteristic, complemented, Sylow
197C2×#41—471no—
198C22×Q8—21no—
199C2×Q16—21no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—21no—
203C2×Q16—21no—
204C2×Q16C22×D4712yes—
205#131—941no—
206#131—941no—
207#131—941no—
208#131—941no—
209#131—1882no—
210#131—1882no—
211C2×M16—21no—
212C2×M16—471no—
213C2×M16—472no—
214C2×M16—944no—
215C2×M16—941no—
216C22×C8—472no—
217C22×C8C22×D4711yescharacteristic
218C4⋊Q8—1881no—
219#124—1881no—
220#123—1882no—
221C22⋊Q8—944no—
222#117—941no—
223M32—21no—
224M32—472no—
225M32—941no—
226M32C22×D4712yes—
227C2×C16—21no—
228C2×C16—941no—
229#108—21no—
230#108—472no—
231#108—944no—
232#108—941no—
233#108C22×D4712yes—
234#107—941no—
235#106—472no—
236#103—944no—
237#41—941no—
238#41—941no—
239#41—941no—
240#41—941no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—471no—
248C2×D4—472no—
249C2×D4—471no—
250C22×C4—41no—
251C22×C4—472no—
252C22×C4D4×D4711yescharacteristic
253Q16—22no—
254Q16—22no—
255Q16—41no—
256Q16—41no—
257M16—22no—
258M16—41no—
259M16—941no—
260M16—941no—
261M16—942no—
262M16—942no—
263M16—942no—
264M16—1884no—
265M16—1881no—
266C2×C8—41no—
267C2×C8—472no—
268C2×C8—942no—
269C2×C8—942no—
270C2×C8—1884no—
271C2×C8—1881no—
272C2×C8#1062535011yescharacteristic
273C2×C8#1062535011yescharacteristic
274C2×C8D4×D4711yescharacteristic
275C2×C8#1062535211yescharacteristic
276C2×C8#1062535211yescharacteristic
277C2×C8C23×D4711yescharacteristic
278C4⋊C4—942no—
279C4⋊C4—1884no—
280C22⋊C4—1884no—
281C22⋊C4—1881no—
282C42—1881no—
283C16—22no—
284C16—942no—
285C23—472no—
286C23SD16×D4711yescharacteristic
287Q8—42no—
288Q8—42no—
289Q8—81no—
290Q8—81no—
291Q8—941no—
292Q8—941no—
293D4—942no—
294D4—942no—
295D4—941no—
296D4—941no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—472no—
301C2×C4—942no—
302C2×C4—942no—
303C2×C4—1882no—
304C2×C4#1309400611yescharacteristic
305C2×C4#1309400611yescharacteristic
306C2×C4SD16×D4711yescharacteristic
307C2×C4#1309404811yescharacteristic
308C2×C4#1309404811yescharacteristic
309C2×C4C2×D4×D4711yescharacteristic
310C8—42no—
311C8—942no—
312C8—1882no—
313C8#1309400611yescharacteristic
314C8#1309400611yescharacteristic
315C8C2×#1062535211yescharacteristic
316C8C2×#1062535011yescharacteristic
317C22—41no—
318C22—942no—
319C22—1882no—
320C22#1568944011yescharacteristic
321C22#1568944011yescharacteristic
322C22C2×SD16×D4711yescharacteristic
323C4—82no—
324C4—942no—
325C4—3762no—
326C4#1568944011yescharacteristic
327C4#1568944011yescharacteristic
328C4C2×#1309400611yescharacteristic
329C4C2×#1309404811yescharacteristic
330C2—21no—
331C2—21no—
332C2—21no—
333C2—1882no—
334C2C2×#1568944011yescharacteristic
subgroupG/H
centerC2C2×#15689440
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC2×C47×#362C2
socleC2×C47C2×#481
prime radicalC2×#10625350D4
hypercenterC2×#362D47
hypocenterC47#33658
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#362

Complements

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

NH ≅ G/Ntype
3. #20245606C2semidirect
4. #20245162C2semidirect
5. #20245161C2semidirect
6. C2×#15688939C2semidirect
8. C2×C47×#362C2semidirect
169. C2×#362D47semidirect
196. C47#33658semidirect

Maximal subgroups

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

subgroupindex
#202530392
#202519962
#202456062
#202451622
#202451612
C2×#156889392
#202450082
C2×C47×#3622
#3365847