Finite Group Databasebeta

← #26136497

Subgroups — #26136497

Order 12032.

486 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#20245053C211yesmaximal, characteristic, complemented
2#20242050C211yesmaximal, characteristic, complemented
3#20241533C214yesmaximal, complemented
4C47×#3018C211yesmaximal, characteristic
5#15689176—21no—
6#15689097—21no—
7#15688918—21no—
8#15688916C2212yescomplemented
9#15688872—21no—
10#15688867C2212yescomplemented
11C47×#585C411yescharacteristic, complemented
12C47×#575—21no—
13C47×#532C2211yescharacteristic
14C47×#456—21no—
15C47×#455C412yescomplemented
16C47×#454—21no—
17C47×#453C2212yes—
18C2×#104×C47—21no—
19C47×#413C411yescharacteristic, complemented
20#13093983—41no—
21C4⋊(C47⋊C8)—22no—
22#13093939—22no—
23#13093939—41no—
24C22⋊(C47⋊C8)—41no—
25C4×C47⋊C8—41no—
262−1+4×C47—21no—
272−1+4×C47—41no—
28H(5,𝔽2)×C47—41no—
29H(5,𝔽2)×C47—41no—
30H(5,𝔽2)×C47C2×C412yes—
31C2×#41×C47—21no—
32C2×#41×C47—41no—
33C2×#41×C47—42no—
34C2×#41×C47C2×C411yescharacteristic
35C2×#41×C47D411yescharacteristic
36#137×C47—22no—
37#137×C47—41no—
38AGL(1,ℤ/8)×C47—22no—
39AGL(1,ℤ/8)×C47—41no—
40#135×C47—42no—
41C2×M16×C47—21no—
42C4⋊Q8×C47D411yescharacteristic
43#127×C47C2311yescharacteristic
44#124×C47C2×C411yescharacteristic
45#124×C47D411yescharacteristic
46#121×C47—41no—
47C4×D4×C47—41no—
48C2×C42×C47D411yescharacteristic
49#104×C47—22no—
50#104×C47—22no—
51#101×C47—21no—
52#100×C47C2×C412yes—
53#41×C47—22no—
54#41×C47—22no—
55#41×C47—41no—
56#41×C47—42no—
57#41×C47—41no—
58#41×C47—41no—
59#41×C47—41no—
60#41×C47—41no—
61#41×C47—42no—
62#41×C47—42no—
63#41×C47—44no—
64#41×C47—42no—
65#41×C47—41no—
66#41×C47—42no—
67C2×Q8×C47—21no—
68C2×Q8×C47—42no—
69C2×Q8×C47—41no—
70C2×Q8×C47C22⋊C411yescharacteristic
71C2×Q8×C47C22⋊C411yescharacteristic
72C2×D4×C47—21no—
73C2×D4×C47—21no—
74C2×D4×C47—21no—
75C2×D4×C47—41no—
76C2×D4×C47—42no—
77C2×D4×C47—44no—
78C2×D4×C47—41no—
79C2×D4×C47—42no—
80C2×D4×C47—41no—
81C2×D4×C47C22×C411yescharacteristic
82C2×D4×C47C2×D411yescharacteristic
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—41no—
87C22×C4×C47—42no—
88C22×C4×C47C22⋊C411yescharacteristic
89Q16×C47—42no—
90SD16×C47—42no—
91SD16×C47—42no—
92D8×C47—42no—
93M16×C47—22no—
94M16×C47—41no—
95C2×C8×C47—41no—
96C4⋊C4×C47—41no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—41no—
99C42×C47—21no—
100C42×C47C22⋊C411yescharacteristic
101C42×C47C2×D411yescharacteristic
102C22⋊Dic47—42no—
103C4⋊Dic47—81no—
104C4×Dic47—81no—
105C2×C47⋊C8—42no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—41no—
109C23×C47—42no—
110C23×C47#9911yescharacteristic
111Q8×C47—22no—
112Q8×C47—41no—
113Q8×C47—41no—
114Q8×C47—42no—
115Q8×C47—41no—
116D4×C47—22no—
117D4×C47—41no—
118D4×C47—41no—
119D4×C47—42no—
120D4×C47—42no—
121D4×C47—42no—
122D4×C47—44no—
123D4×C47—42no—
124D4×C47—41no—
125D4×C47—82no—
126D4×C47—81no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—42no—
134C2×C4×C47—42no—
135C2×C4×C47—41no—
136C2×C4×C47—44no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47#9911yescharacteristic
144C2×C4×C47C2×C22⋊C411yescharacteristic
145C2×Dic47—82no—
146C8×C47—42no—
147C47⋊C8—82no—
148#13560—471nomaximal, Sylow
149C22×C47—21no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47—42no—
153C22×C47—41no—
154C22×C47—81no—
155C22×C47—81no—
156C22×C47—161no—
157C22×C47—162no—
158C22×C47C2×#9911yescharacteristic
159C4×C47—22no—
160C4×C47—21no—
161C4×C47—41no—
162C4×C47—41no—
163C4×C47—41no—
164C4×C47—42no—
165C4×C47AΣL(1,𝔽16)11yescharacteristic
166C4×C47AΣL(1,𝔽16)11yescharacteristic
167Dic47—162no—
168#3018D4711yescharacteristic
169#2126—471no—
170#1543—471no—
171#1402—474no—
172C2×C47—21no—
173C2×C47—41no—
174C2×C47—81no—
175C2×C47—81no—
176C2×C47—82no—
177C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
178#585Dic4711yescharacteristic, complemented
179#575—21no—
180#532C2×D4711yescharacteristic
181#456—21no—
182#455Dic4712yescomplemented
183#454—21no—
184#453C2×D4712yes—
185#431—941no—
186C2×#104—21no—
187#413Dic4711yescharacteristic, complemented
188#410—941no—
189#354—941no—
190#353—472no—
191#333—941no—
192#331—472no—
193C47#1356011yescharacteristic, complemented, Sylow
1942−1+4—21no—
1952−1+4—41no—
196H(5,𝔽2)—41no—
197H(5,𝔽2)—41no—
198H(5,𝔽2)C2×Dic4712yes—
199C2×#41—21no—
200C2×#41—41no—
201C2×#41—42no—
202C2×#41C2×Dic4711yescharacteristic
203C2×#41C22⋊D4711yescharacteristic
204#137—22no—
205#137—41no—
206AGL(1,ℤ/8)—22no—
207AGL(1,ℤ/8)—41no—
208#135—42no—
209C2×M16—21no—
210C4⋊Q8C22⋊D4711yescharacteristic
211#127C22×D4711yescharacteristic
212#124C2×Dic4711yescharacteristic
213#124C22⋊D4711yescharacteristic
214#121—41no—
215C4×D4—41no—
216#117—1881no—
217C2×C42C22⋊D4711yescharacteristic
218C4⋊C8—942no—
219#104—22no—
220#104—22no—
221#101—21no—
222#100C2×Dic4712yes—
223#99—942no—
224#99—1881no—
225C22⋊C8—1881no—
226C4×C8—1881no—
227#41—22no—
228#41—22no—
229#41—41no—
230#41—42no—
231#41—41no—
232#41—41no—
233#41—41no—
234#41—41no—
235#41—42no—
236#41—42no—
237#41—44no—
238#41—42no—
239#41—41no—
240#41—42no—
241C2×Q8—21no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×Q8C22⋊Dic4711yescharacteristic
245C2×Q8C22⋊Dic4711yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4C22×Dic4711yescharacteristic
256C2×D4C2×C22⋊D4711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4C22⋊Dic4711yescharacteristic
263Q16—42no—
264SD16—42no—
265SD16—42no—
266D8—42no—
267M16—22no—
268M16—41no—
269C2×C8—41no—
270C2×C8—1882no—
271C4⋊C4—41no—
272C4⋊C4—3761no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—1882no—
276C42—21no—
277C42—3761no—
278C42C22⋊Dic4711yescharacteristic
279C42C2×C22⋊D4711yescharacteristic
280C23—21no—
281C23—21no—
282C23—41no—
283C23—42no—
284C23#1309393911yescharacteristic
285Q8—22no—
286Q8—41no—
287Q8—41no—
288Q8—42no—
289Q8—41no—
290D4—22no—
291D4—41no—
292D4—41no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—44no—
297D4—42no—
298D4—41no—
299D4—82no—
300D4—81no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—3762no—
318C2×C4#1309393911yescharacteristic
319C2×C4C2×C22⋊Dic4711yescharacteristic
320C8—42no—
321C8—3762no—
322C22—21no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—161no—
330C22—162no—
331C22C2×#1309393911yescharacteristic
332C4—22no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—42no—
338C4—7522no—
339C4#1568891011yescharacteristic
340C4#1568891011yescharacteristic
341C2—21no—
342C2—41no—
343C2—81no—
344C2—81no—
345C2—82no—
346C2C2×#1568891011yescharacteristic
subgroupG/H
centerC4#15688910
commutatorC2×D4×C47C22×C4
Frattini#127C22×D47
Fitting, prime radicalC47×#3018C2
socleC2×C47C2×AΣL(1,𝔽16)
cosocle#127×C47C23
hypercenter#3018D47
hypocenterC47#13560
LCSG ▷ C2×D4×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3018

Complements

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

NH ≅ G/Ntype
1. #20245053C2semidirect
2. #20242050C2semidirect
3. #20241533C2semidirect
8. #15688916C22semidirect
10. #15688867C22semidirect
11. C47×#585C4semidirect
15. C47×#455C4semidirect
19. C47×#413C4semidirect
178. #585Dic47semidirect
182. #455Dic47semidirect
187. #413Dic47semidirect
193. C47#13560semidirect

Maximal subgroups

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

subgroupindex
#202450532
#202420502
#202415332
C47×#30182
#1356047