Finite Group Databasebeta

← #26119877

Subgroups — #26119877

Order 12032.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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×#15689325C211yesmaximal, characteristic
2#20243377C211yesmaximal, characteristic
3#20242160C211yesmaximal, characteristic, complemented
4#20242060C211yesmaximal, characteristic, complemented
5#20241780C211yesmaximal, characteristic
6#20241284C211yesmaximal, characteristic, complemented
7C47×#1546C211yesmaximal, characteristic, complemented
8C22×#10625336—21no—
9C2×C4×D188C2211yescharacteristic
10#15689325—22no—
11#15689325—22no—
12#15689176C2211yescharacteristic
13#15689147C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093948—21no—
16C2×#13093940—21no—
17#15688891—21no—
18#15688873—22no—
19#15688869—22no—
20#15688852—22no—
21C2×C4⋊Q8×C47C2211yescharacteristic
22C47×#431C2211yescharacteristic
23C47×#423C2211yescharacteristic
24C47×#332—22no—
25C22×D188D411yescharacteristic
26C22×C4×D47—21no—
27C2×#10625336—24no—
28C2×#10625336—42no—
29C2×#10625336—42no—
30C2×#10625329—21no—
31C22×C47⋊C8—21no—
32C4×D188—21no—
33C4×D188—21no—
34C4×D188—21no—
35C4×D188—21no—
36C2×C4⋊Dic47D411yescharacteristic
37C2×(C2×D47)⋊C4—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)D412yes—
42C4⋊(C47⋊C8)D412yes—
43#13093948—24no—
44#13093940—42no—
45C22⋊(C47⋊C8)—21no—
46C4×C47⋊C8—21no—
47C22×Q8×C47—21no—
48C2×M16×C47—21no—
49C4⋊Q8×C47—21no—
50C4⋊Q8×C47—21no—
51C4⋊Q8×C47—21no—
52C4⋊Q8×C47D412yes—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—21no—
57C4⋊C8×C47—21no—
58C4⋊C8×C47—21no—
59C22⋊C8×C47—21no—
60C4×C8×C47—21no—
61C23×D47—21no—
62C2×Q8×C47—22no—
63C2×Q8×C47—41no—
64C2×Q8×C47—41no—
65C2×Q8×C47—41no—
66C2×Q8×C47—41no—
67C2×Q8×C47—41no—
68C22×C4×C47—41no—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47#4111yescharacteristic
71C22×C4×C47#4111yescharacteristic
72C22×Dic47—21no—
73C2×D188—21no—
74C2×D188—21no—
75C2×D188—21no—
76C2×D188—21no—
77C2×D188—21no—
78C2×D188SD1612yes—
79C2×C4×D47—41no—
80C2×C4×D47—41no—
81C2×C4×D47—82no—
82M16×C47—41no—
83M16×C47—41no—
84C2×C8×C47—41no—
85C2×C8×C47—41no—
86C4⋊C4×C47—22no—
87C4⋊C4×C47—22no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C42×C47C2×D411yescharacteristic
91C42×C47C2×D411yescharacteristic
92C42×C47C2×D411yescharacteristic
93C42×C47#4111yescharacteristic
94#10625336—48no—
95(C2×D47)⋊C4—41no—
96(C2×D47)⋊C4—41no—
97C4⋊Dic47—21no—
98C4⋊Dic47—21no—
99#10625329—41no—
100#10625329—41no—
101C2×C47⋊C8—22no—
102C2×C47⋊C8—22no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C23×C47#12311yescharacteristic
107C22×D47—22no—
108C22×D47—41no—
109C22×D47—41no—
110C22×D47—81no—
111C22×D47—81no—
112Q8×C47—42no—
113Q8×C47—42no—
114Q8×C47—81no—
115Q8×C47—81no—
116C2×C4×C47—22no—
117C2×C4×C47—41no—
118C2×C4×C47—42no—
119C2×C4×C47—81no—
120C2×C4×C47—81no—
121C2×C4×C47#12011yescharacteristic
122C2×C4×C47#12111yescharacteristic
123C2×C4×C47#12111yescharacteristic
124C2×C4×C47#12111yescharacteristic
125C2×C4×C47#12311yescharacteristic
126C2×C4×C47#12411yescharacteristic
127C2×C4×C47C2×SD1611yescharacteristic
128C2×C4×C47#13511yescharacteristic
129C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
131C2×Dic47—41no—
132C2×Dic47—81no—
133D188—22no—
134D188—22no—
135D188—41no—
136D188—41no—
137C4×D47—82no—
138C4×D47—82no—
139C8×C47—81no—
140C8×C47—81no—
141C47⋊C8—42no—
142C47⋊C8—42no—
143C47⋊C8—81no—
144#10445—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—21no—
147C22×C47#39411yescharacteristic
148C22×C47#48111yescharacteristic
149C22×C47#48211yescharacteristic
150C2×D47—42no—
151C2×D47—82no—
152C2×D47—81no—
153C2×D47—81no—
154C2×D47—82no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—82no—
159C4×C47#44911yescharacteristic
160C4×C47#45011yescharacteristic
161C4×C47#46511yescharacteristic
162C4×C47#46911yescharacteristic
163C4×C47#47112yes—
164Dic47—81no—
165C2×#461—471no—
166#1800—471no—
167#1570—471no—
168#1546—471no—
169#1546D4711yescharacteristic, complemented
170#1478—471no—
171#1301—471no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47#203411yescharacteristic
175C2×C47#204611yescharacteristic
176C2×C47#204911yescharacteristic
177D47—82no—
178C22×SD16—941no—
179C2×C4⋊Q8C2×D4711yescharacteristic
180C2×C4×D4—471no—
181#461—942no—
182#461—942no—
183#431—471no—
184#431C2×D4711yescharacteristic
185#423—471no—
186#423C2×D4711yescharacteristic
187C2×C4⋊C8—471no—
188C2×#103—941no—
189C2×#100—941no—
190#343—941no—
191#334—942no—
192#332—22no—
193#332—942no—
194D4⋊C8—942no—
195C47#1044511yescharacteristic, complemented, Sylow
196C22×Q8—21no—
197C22×D4—471no—
198C23×C4—941no—
199C2×SD16—944no—
200C2×SD16—1882no—
201C2×SD16—1882no—
202C2×M16—21no—
203C2×M16—941no—
204C22×C8—941no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8C22⋊D4712yes—
209C4×D4—941no—
210C4×D4—941no—
211C4×D4—941no—
212C4×D4—941no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—471no—
216C2×C22⋊C4—941no—
217C2×C42C22×D4711yescharacteristic
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—472no—
222C4⋊C8—472no—
223C4⋊C8—941no—
224C4⋊C8—941no—
225C4⋊C8—941no—
226#103—944no—
227#100—1882no—
228C22⋊C8—21no—
229C22⋊C8—941no—
230C4×C8—21no—
231C4×C8—941no—
232C24—941no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×D4—472no—
240C2×D4—941no—
241C2×D4—941no—
242C2×D4—941no—
243C2×D4—941no—
244C2×D4—941no—
245C22×C4—41no—
246C22×C4—941no—
247C22×C4—3762no—
248C22×C4—1881no—
249C22×C4—1881no—
250C22×C4#1062535011yescharacteristic
251C22×C4D4×D4711yescharacteristic
252C22×C4#1062535211yescharacteristic
253SD16—1888no—
254M16—41no—
255M16—41no—
256M16—1881no—
257M16—1881no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—942no—
261C2×C8—942no—
262C2×C8—1881no—
263C2×C8—1881no—
264C2×C8—1881no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—941no—
270C4⋊C4—941no—
271C22⋊C4—1881no—
272C22⋊C4—1881no—
273C42#1062535011yescharacteristic
274C42D4×D4711yescharacteristic
275C42D4×D4711yescharacteristic
276C42C2×C22⋊D4711yescharacteristic
277C23—942no—
278C23—3761no—
279C23—3761no—
280C23—1881no—
281C23—1881no—
282C23#1309400611yescharacteristic
283Q8—42no—
284Q8—42no—
285Q8—81no—
286Q8—81no—
287D4—942no—
288D4—942no—
289D4—1881no—
290D4—1881no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—3762no—
297C2×C4—3762no—
298C2×C4—3761no—
299C2×C4—1881no—
300C2×C4#1309399211yescharacteristic
301C2×C4#1309399611yescharacteristic
302C2×C4#1309399711yescharacteristic
303C2×C4#1309399711yescharacteristic
304C2×C4#1309400711yescharacteristic
305C2×C4#1309401011yescharacteristic
306C2×C4SD16×D4711yescharacteristic
307C2×C4#1309404311yescharacteristic
308C2×C4#1309404811yescharacteristic
309C2×C4#1309405111yescharacteristic
310C8—81no—
311C8—81no—
312C8—3761no—
313C8—1882no—
314C8—1882no—
315C22—21no—
316C22—21no—
317C22—3762no—
318C22—3762no—
319C22—3761no—
320C22—3761no—
321C22—1882no—
322C22#1568903811yescharacteristic
323C22#1568944011yescharacteristic
324C22#1568944511yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—82no—
329C4—3761no—
330C4#1568926511yescharacteristic
331C4#1568927211yescharacteristic
332C4#1568935111yescharacteristic
333C4#1568937511yescharacteristic
334C4#1568938912yes—
335C2—21no—
336C2—21no—
337C2—3762no—
338C2#2024457511yescharacteristic
339C2#2024464711yescharacteristic
340C2#2024466611yescharacteristic
subgroupG/H
centerC22#15689038
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1546C2
socleC22×C47#394
prime radicalC22×D188D4
hypercenter#1546D47
hypocenterC47#10445
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
3. #20242160C2semidirect
4. #20242060C2semidirect
6. #20241284C2semidirect
7. C47×#1546C2semidirect
169. #1546D47semidirect
195. C47#10445semidirect

Maximal subgroups

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

subgroupindex
C2×#156893252
#202433772
#202421602
#202420602
#202417802
#202412842
C47×#15462
#1044547