Finite Group Databasebeta

← #26369326

Subgroups — #26369326

Order 12032.

658 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689541C211yesmaximal, characteristic
2#20251985C211yesmaximal, characteristic
3#20251984C211yesmaximal, characteristic
4#20245176C211yesmaximal, characteristic, complemented
5#20245175C212yesmaximal, complemented
6#20245173C212yesmaximal, complemented
7#20245170C211yesmaximal, characteristic, complemented
8#20245169C212yesmaximal, complemented
9C2×#15688950C211yesmaximal, characteristic, complemented
10#367×D47C212yesmaximal, complemented
11C47×#2159C211yesmaximal, characteristic, complemented
12C2×#13094030C2211yescharacteristic
13#15689604—21no—
14#15689603—21no—
15#15689541C418yescomplemented
16C2×#13093963C2212yes—
17C22×C47⋊C16C2211yescharacteristic
18C2×C16×D47C2212yes—
19#15689166—21no—
20#15689164—22no—
21#15689163—21no—
22#15689162—22no—
23#15689161C2211yescharacteristic
24#15689160C2212yes—
25#15689160C2212yes—
26C2×#13093958C2211yescharacteristic
27#107×D47C2212yes—
28#107×D47C2212yes—
29#15688951C2212yes—
30#15688950C2211yescharacteristic
31#15688950C2212yes—
32#15688950C2212yes—
33#15688950C2211yescharacteristic
34#15688949C2212yes—
35#15688948C2211yescharacteristic
36#15688948C2211yescharacteristic
37C2×M32×C47C2211yescharacteristic
38C47×#427C2211yescharacteristic
39C2×#107×C47C2211yescharacteristic
40C47×#367C2212yes—
41C47×#366C2212yes—
42C2×#10625350D411yescharacteristic
43#13094030C2×C414yes—
44#13094030Q814yes—
45C2×#10625329D411yescharacteristic
46C2×#10625324D412yes—
47C2×C8×D47C2312yes—
48C4×Dic94—41no—
49#13093989—41no—
50C4×D188—41no—
51#13093986—41no—
52C4×C22⋊D47—42no—
53(C22⋊D47)⋊C4—42no—
54#13093983—21no—
55#13093982—22no—
56#13093982—22no—
57C2×C4⋊Dic47—21no—
58C4⋊C4×D47—22no—
59C4⋊C4×D47—22no—
60#13093963C2×C418yescomplemented
61C2×C47⋊C16C2×C414yescomplemented
62C2×C47⋊C16C2311yescharacteristic
63C2×C47⋊C16C2311yescharacteristic
64C16×D47C2×C418yescomplemented
65#13093958—21no—
66#13093958C2311yescharacteristic
67#13093958C2312yes—
68#13093958C2311yescharacteristic
69#13093957—22no—
70#13093957—22no—
71#13093956—21no—
72#13093955—22no—
73#13093955—22no—
74C22×C8×C47C2311yescharacteristic
75#117×C47—21no—
76C2×C4⋊C4×C47—21no—
77M32×C47C2×C414yescomplemented
78C2×C16×C47C2312yes—
79#107×C47—21no—
80#107×C47C2311yescharacteristic
81#107×C47C2312yes—
82#107×C47C2311yescharacteristic
83#106×C47—21no—
84C22×C4×C47—41no—
85C22×C4×C47C2×D411yescharacteristic
86C2×C22⋊D47D812yes—
87C22×Dic47—41no—
88#10625350C4⋊C414yes—
89#10625350Q1614yes—
90C2×D188D811yescharacteristic
91C2×C4×D47—42no—
92C2×C4×D47—42no—
93C2×C4×D47C2×D412yes—
94C2×Dic94D811yescharacteristic
95C2×C8×C47C22×C412yes—
96C2×C8×C47C2×D411yescharacteristic
97C2×C8×C47C2×Q812yes—
98C2×C8×C47C2411yescharacteristic
99C4⋊C4×C47—21no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—41no—
103C22⋊C4×C47—41no—
104C42×C47—41no—
105C22⋊Dic47—41no—
106(C2×D47)⋊C4—42no—
107(C2×D47)⋊C4—42no—
108C4⋊Dic47—21no—
109C4⋊Dic47—22no—
110C4⋊Dic47—21no—
111C4⋊Dic47—41no—
112Dic47⋊C4—42no—
113Dic47⋊C4—42no—
114C4×Dic47—41no—
115C4×Dic47—42no—
116C4×Dic47—42no—
117#10625329C4⋊C414yes—
118C2×C47⋊C8C2×D412yes—
119#10625324C4⋊C418yes—
120C8×D47C22×C414yes—
121C8×D47C2×Q814yes—
122C16×C47C22×C414yes—
123C47⋊C16C22×C412yes—
124C47⋊C16C22×C412yes—
125C23×C47C2×D811yescharacteristic
126C22×D47C2×D812yes—
127C2×C4×C47—41no—
128C2×C4×C47—42no—
129C2×C4×C47—41no—
130C2×C4×C47—161no—
131C2×C4×C47C2×C4⋊C412yes—
132C2×C4×C47C2×D811yescharacteristic
133C2×C4×C47C2×Q1612yes—
134C2×C4×C47C22×D411yescharacteristic
135C22⋊D47#10718yes—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—42no—
139C2×Dic47—161no—
140C2×Dic47C2×D812yes—
141D188#10714yes—
142C4×D47—162no—
143C4×D47—162no—
144C4×D47C2×C4⋊C414yes—
145C4×D47C2×Q1614yes—
146Dic94#10714yes—
147C8×C47C2×C4⋊C412yes—
148C8×C47C23×C411yescharacteristic
149C8×C47C22×Q811yescharacteristic
150C47⋊C8C2×C4⋊C414yes—
151#33411—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47C2×#10712yes—
155C22×C47C22×D811yescharacteristic
156C2×D47—22no—
157C2×D47—22no—
158C2×D47C2×#10714yes—
159C4×C47—81no—
160C4×C47—82no—
161C4×C47—81no—
162C4×C47C2×#10712yes—
163C4×C47C22×C4⋊C411yescharacteristic
164C4×C47C22×Q1611yescharacteristic
165Dic47—81no—
166Dic47—82no—
167Dic47—81no—
168Dic47C2×#10714yes—
169C2×#504—471no—
170#2913—471no—
171#2913—471no—
172#2159—471no—
173#2159—472no—
174#2159—472no—
175#2159D4711yescharacteristic, complemented
176#2158—471no—
177#2158—472no—
178C2×#367—471no—
179C2×#367—472no—
180C2×C47—22no—
181C2×C47C22×#10711yescharacteristic
182C2×C47#341611yescharacteristic
183C2×C47#341611yescharacteristic
184D47—24no—
185C2×#131—471no—
186#520—941no—
187#520—941no—
188#504—478no—
189C2×M32—472no—
190C2×M32C2×D4711yescharacteristic
191C22×C16—471no—
192C22×C16—472no—
193#428—942no—
194#428—942no—
195#428—941no—
196#428—941no—
197#427—471no—
198#427—472no—
199#427—472no—
200#427C2×D4711yescharacteristic
201C2×#107—471no—
202C2×#107—472no—
203C2×#107—472no—
204C2×#107C2×D4711yescharacteristic
205#367—472no—
206#367—471no—
207#367—472no—
208#367—472no—
209#367—471no—
210#367C2×D4712yes—
211#366—472no—
212#366—471no—
213#366—471no—
214#366C2×D4712yes—
215C47#3341111yescharacteristic, complemented, Sylow
216C2×#41—471no—
217#131—474no—
218#131—474no—
219C2×M16—471no—
220C2×M16—472no—
221C22×C8—472no—
222C22×C8C22×D4711yescharacteristic
223C4×Q8—1881no—
224C4×Q8—1881no—
225C4×D4—1881no—
226C4×D4—1882no—
227C4×D4—1881no—
228C4×D4—1882no—
229#117—21no—
230#117—942no—
231#117—942no—
232#117—941no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—942no—
235C2×C4⋊C4—942no—
236C2×C4⋊C4—941no—
237M32—478no—
238M32C4×D4714yescomplemented
239C2×C16—474no—
240C2×C16—471no—
241C2×C16—471no—
242C2×C16—478no—
243C2×C16C22×D4712yes—
244#107—21no—
245#107—471no—
246#107—471no—
247#107—472no—
248#107—942no—
249#107—942no—
250#107—941no—
251#107C22×D4711yescharacteristic
252#107C22×D4711yescharacteristic
253#107C22×D4712yes—
254#106—21no—
255#106—942no—
256#106—942no—
257#106—941no—
258#41—474no—
259#41—474no—
260C2×Q8—471no—
261C2×D4—472no—
262C2×D4—471no—
263C22×C4—41no—
264C22×C4—472no—
265C22×C4—1882no—
266C22×C4—1882no—
267C22×C4—1881no—
268C22×C4D4×D4711yescharacteristic
269M16—474no—
270M16—478no—
271C2×C8—472no—
272C2×C8—474no—
273C2×C8—474no—
274C2×C8C2×C4×D4712yes—
275C2×C8D4×D4711yescharacteristic
276C2×C8Q8×D4712yes—
277C2×C8C23×D4711yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—941no—
283C4⋊C4—942no—
284C4⋊C4—941no—
285C4⋊C4—1882no—
286C4⋊C4—1882no—
287C4⋊C4—1881no—
288C22⋊C4—41no—
289C22⋊C4—1882no—
290C22⋊C4—1882no—
291C22⋊C4—1881no—
292C42—41no—
293C42—1882no—
294C42—1882no—
295C42—1881no—
296C16—472no—
297C16—472no—
298C16C2×C4×D4714yes—
299C23—472no—
300C23D8×D4711yescharacteristic
301Q8—474no—
302D4—478no—
303D4—474no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—161no—
308C2×C4—472no—
309C2×C4—474no—
310C2×C4—474no—
311C2×C4—1881no—
312C2×C4—1882no—
313C2×C4—1881no—
314C2×C4—7522no—
315C2×C4—7522no—
316C2×C4—7521no—
317C2×C4C4⋊C4×D4712yes—
318C2×C4D8×D4711yescharacteristic
319C2×C4Q16×D4712yes—
320C2×C4C2×D4×D4711yescharacteristic
321C8—474no—
322C8C4⋊C4×D4712yes—
323C8C22×C4×D4711yescharacteristic
324C8C2×Q8×D4711yescharacteristic
325C22—21no—
326C22—21no—
327C22—474no—
328C22—942no—
329C22—942no—
330C22#107×D4712yes—
331C22C2×D8×D4711yescharacteristic
332C4—81no—
333C4—82no—
334C4—81no—
335C4—474no—
336C4—3762no—
337C4—3761no—
338C4—3761no—
339C4#107×D4712yes—
340C4C2×C4⋊C4×D4711yescharacteristic
341C4C2×Q16×D4711yescharacteristic
342C2—22no—
343C2—944no—
344C2C2×#107×D4711yescharacteristic
345C2#2025653311yescharacteristic
346C2#2025653411yescharacteristic
subgroupG/H
centerC22C2×D8×D47
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC47×#2159C2
socleC22×C47C22×D8
prime radicalC2×#10625350D4
cosocleC2×C8×C47C24
hypercenter#2159D47
hypocenterC47#33411
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. #20245176C2semidirect
5. #20245175C2semidirect
6. #20245173C2semidirect
7. #20245170C2semidirect
8. #20245169C2semidirect
9. C2×#15688950C2semidirect
10. #367×D47C2semidirect
11. C47×#2159C2semidirect
15. #15689541C4semidirect
60. #13093963C2×C4semidirect
61. C2×C47⋊C16C2×C4semidirect
64. C16×D47C2×C4semidirect
77. M32×C47C2×C4semidirect
175. #2159D47semidirect
215. C47#33411semidirect
238. M32C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156895412
#202519852
#202519842
#202451762
#202451752
#202451732
#202451702
#202451692
C2×#156889502
#367×D472
C47×#21592
#3341147