Finite Group Databasebeta

← #26134792

Subgroups — #26134792

Order 12032.

462 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689501C211yesmaximal, characteristic
2C2×#15688943C211yesmaximal, characteristic
3C2×#15688938C211yesmaximal, characteristic
4#20245109C211yesmaximal, characteristic
5#20245108C211yesmaximal, characteristic
6#20243270C211yesmaximal, characteristic
7C2×C47×#362C211yesmaximal, characteristic
8C22×Dic188C2211yescharacteristic
9C22×#10625337—21no—
10C2×#13094022—21no—
11C2×#13093964C2211yescharacteristic
12#15689501—22no—
13#15689501C414yescomplemented
14#15689161C2211yescharacteristic
15#15689081—21no—
16C2×C8×Dic47C2211yescharacteristic
17#15688943—22no—
18#15688938—22no—
19#15688928—22no—
20#15688927—22no—
21#15688884—21no—
22C22×Q16×C47C2211yescharacteristic
23C2×M32×C47C2211yescharacteristic
24C2×#108×C47C2211yescharacteristic
25C47×#362—22no—
26C22×Dic94—21no—
27C2×Dic188—21no—
28C2×Dic188—21no—
29C2×Dic188—21no—
30C2×Dic188—21no—
31C2×Dic188—21no—
32C2×Dic188C2×C412yes—
33C2×#10625337—24no—
34C2×#10625337—41no—
35C2×#10625337—41no—
36C2×#10625337—42no—
37C22×C47⋊C8D411yescharacteristic
38#13094022—24no—
39#13094022—42no—
40#13093983—21no—
41C2×Dic47⋊C4—42no—
42C2×C4×Dic47D411yescharacteristic
43#13093964—21no—
44#13093964—21no—
45C2×C47⋊C16—21no—
46#13093958D412yes—
47#13093956—21no—
48#13093934—41no—
49C8×Dic47—41no—
50C8×Dic47C2×C412yes—
51C8×Dic47D412yes—
52C22×Q8×C47—21no—
53C2×Q16×C47—21no—
54C2×Q16×C47—21no—
55C2×Q16×C47—21no—
56C2×Q16×C47—21no—
57C2×Q16×C47—21no—
58C2×Q16×C47C2×C412yes—
59C2×M16×C47—21no—
60C22×C8×C47C2311yescharacteristic
61M32×C47—21no—
62M32×C47—21no—
63C2×C16×C47—21no—
64#108×C47—21no—
65#108×C47D412yes—
66C2×Q8×C47—22no—
67C2×Q8×C47—41no—
68C2×Q8×C47—41no—
69C2×Q8×C47—41no—
70C2×Q8×C47—41no—
71C2×Q8×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47C2×D411yescharacteristic
74C22×Dic47—21no—
75C22×Dic47—41no—
76C2×Dic94—22no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×Dic94—41no—
80C2×Dic94—41no—
81C2×Dic94—41no—
82Q16×C47—22no—
83Q16×C47—22no—
84Q16×C47—41no—
85Q16×C47—41no—
86M16×C47—22no—
87M16×C47—41no—
88C2×C8×C47—41no—
89C2×C8×C47C22×C411yescharacteristic
90C2×C8×C47C2×D411yescharacteristic
91C2×C8×C47C2×D411yescharacteristic
92C2×C8×C47C2×D411yescharacteristic
93C2×C8×C47#4111yescharacteristic
94C2×C8×C47#4111yescharacteristic
95C22⋊Dic47—41no—
96#10625337—44no—
97#10625337—44no—
98C4⋊Dic47—22no—
99Dic47⋊C4—48no—
100C4×Dic47—21no—
101C4×Dic47—41no—
102C4×Dic47C22⋊C412yes—
103C2×C47⋊C8C22⋊C412yes—
104C2×C47⋊C8D811yescharacteristic
105C2×C47⋊C8D811yescharacteristic
106C2×C47⋊C8SD1612yes—
107Dic188—22no—
108Dic188—22no—
109Dic188—41no—
110Dic188—41no—
111C16×C47—41no—
112C47⋊C16—41no—
113C23×C47AGL(1,ℤ/8)11yescharacteristic
114Q8×C47—42no—
115Q8×C47—42no—
116Q8×C47—81no—
117Q8×C47—81no—
118C2×C4×C47—42no—
119C2×C4×C47—81no—
120C2×C4×C47—81no—
121C2×C4×C47C2×C22⋊C411yescharacteristic
122C2×C4×C47#12111yescharacteristic
123C2×C4×C47#12111yescharacteristic
124C2×C4×C47C2×D811yescharacteristic
125C2×C4×C47C2×SD1611yescharacteristic
126C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
127C2×Dic47—24no—
128C2×Dic47—22no—
129C2×Dic47—42no—
130C2×Dic47—42no—
131C2×Dic47—81no—
132C2×Dic47—81no—
133Dic94—42no—
134Dic94—42no—
135Dic94—81no—
136Dic94—81no—
137C8×C47—42no—
138C8×C47C4×D411yescharacteristic
139C8×C47C4×D411yescharacteristic
140C8×C47#12411yescharacteristic
141C8×C47#12711yescharacteristic
142C47⋊C8#10212yes—
143C47⋊C8#10212yes—
144#13262—471nomaximal, Sylow
145C22×C47—22no—
146C22×C47#41811yescharacteristic
147C22×C47#45911yescharacteristic
148C22×C47#46011yescharacteristic
149C4×C47—82no—
150C4×C47#39011yescharacteristic
151C4×C47C2×#10211yescharacteristic
152C4×C47#45911yescharacteristic
153C4×C47#46011yescharacteristic
154Dic47—24no—
155Dic47—82no—
156Dic47—82no—
157C2×#494—471no—
158C2×#364—471no—
159C2×#362—471no—
160C2×#362D4711yescharacteristic
161#2141—471no—
162#2141—471no—
163#1772—471no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47#198211yescharacteristic
167C2×C47#221412yes—
168C22×Q16—471no—
169C22×Q16—941no—
170C22×Q16C2×D4711yescharacteristic
171C2×C4⋊Q8—941no—
172C2×M32—471no—
173C2×M32C2×D4711yescharacteristic
174#494—474no—
175#494—942no—
176C2×#108C2×D4711yescharacteristic
177#427—471no—
178#405—941no—
179C2×C4×C8—471no—
180#364—942no—
181#362—22no—
182#362—942no—
183#358—942no—
184#358—942no—
185#339—941no—
186C47#1326211yescharacteristic, complemented, Sylow
187C22×Q8—21no—
188C22×Q8—941no—
189C2×Q16—21no—
190C2×Q16—21no—
191C2×Q16—21no—
192C2×Q16—21no—
193C2×Q16—21no—
194C2×Q16—472no—
195C2×Q16—944no—
196C2×Q16—941no—
197C2×Q16—941no—
198C2×Q16—941no—
199C2×Q16—941no—
200C2×Q16—941no—
201C2×Q16—1881no—
202C2×Q16—1881no—
203C2×Q16—1882no—
204C2×Q16C4×D4712yes—
205C2×M16—21no—
206C22×C8—471no—
207C22×C8C22×D4711yescharacteristic
208C4⋊Q8—944no—
209C4⋊Q8—1882no—
210#117—941no—
211C2×C4⋊C4—1882no—
212C2×C42—471no—
213M32—21no—
214M32—21no—
215M32—941no—
216M32—941no—
217C2×C16—21no—
218C2×C16—941no—
219#108—21no—
220#108C22⋊D4712yes—
221#107—472no—
222#106—941no—
223#97—1881no—
224C4×C8—472no—
225C4×C8—472no—
226C4×C8—1881no—
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—942no—
234C2×Q8—1881no—
235C2×Q8—1881no—
236C2×Q8—1881no—
237C2×Q8—1881no—
238C2×Q8—1881no—
239C22×C4—41no—
240C22×C4—941no—
241C22×C4—1881no—
242C22×C4D4×D4711yescharacteristic
243Q16—22no—
244Q16—22no—
245Q16—41no—
246Q16—41no—
247Q16—942no—
248Q16—942no—
249Q16—1884no—
250Q16—1884no—
251Q16—1881no—
252Q16—1881no—
253M16—22no—
254M16—41no—
255C2×C8—41no—
256C2×C8—472no—
257C2×C8—471no—
258C2×C8—471no—
259C2×C8—472no—
260C2×C8C2×C4×D4711yescharacteristic
261C2×C8#1062535011yescharacteristic
262C2×C8D4×D4711yescharacteristic
263C2×C8D4×D4711yescharacteristic
264C2×C8#1062535211yescharacteristic
265C2×C8C2×C22⋊D4711yescharacteristic
266C4⋊C4—942no—
267C4⋊C4—1888no—
268C22⋊C4—1881no—
269C42—472no—
270C42—941no—
271C42—1881no—
272C16—41no—
273C16—1881no—
274C23#1309405111yescharacteristic
275Q8—42no—
276Q8—42no—
277Q8—81no—
278Q8—81no—
279Q8—3761no—
280Q8—3761no—
281Q8—1882no—
282Q8—1882no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—944no—
287C2×C4—942no—
288C2×C4—3761no—
289C2×C4—3761no—
290C2×C4—1882no—
291C2×C4—1882no—
292C2×C4C22⋊C4×D4711yescharacteristic
293C2×C4#1309399611yescharacteristic
294C2×C4#1309399711yescharacteristic
295C2×C4D8×D4711yescharacteristic
296C2×C4SD16×D4711yescharacteristic
297C2×C4#1309404811yescharacteristic
298C8—42no—
299C8—472no—
300C8—472no—
301C8(C22⋊D47)⋊C411yescharacteristic
302C8C4×C22⋊D4711yescharacteristic
303C8#1309401011yescharacteristic
304C8#1309402011yescharacteristic
305C22—22no—
306C22#1568912611yescharacteristic
307C22#1568931711yescharacteristic
308C22#1568932111yescharacteristic
309C4—82no—
310C4—944no—
311C4—3762no—
312C4—3762no—
313C4#1568902311yescharacteristic
314C4#102×D4711yescharacteristic
315C4#1568931611yescharacteristic
316C4#1568932211yescharacteristic
317C2—21no—
318C2—21no—
319C2#2024431311yescharacteristic
320C2#2024546812yes—
subgroupG/H
centerC22#15689317
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#362C2
socleC22×C47#459
prime radicalC23×C47AGL(1,ℤ/8)
cosocleC22×C8×C47C23
hypercenterC2×#362D47
hypocenterC47#13262
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#362

Complements

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

NH ≅ G/Ntype
13. #15689501C4semidirect
186. C47#13262semidirect

Maximal subgroups

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

subgroupindex
C2×#156895012
C2×#156889432
C2×#156889382
#202451092
#202451082
#202432702
C2×C47×#3622
#1326247