Finite Group Databasebeta

← #26114134

Subgroups — #26114134

Order 12032.

518 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 105 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#20252428C211yesmaximal, characteristic
2#20248746C211yesmaximal, characteristic
3#20243377C211yesmaximal, characteristic
4#20241817C211yesmaximal, characteristic, complemented
5#20241248C211yesmaximal, characteristic, complemented
6#20241247C211yesmaximal, characteristic, complemented
7C47×#1487C211yesmaximal, characteristic, complemented
8C2×C4×D188C2211yescharacteristic
9#15689202—22no—
10#15689200—22no—
11#15689176C2211yescharacteristic
12#15689147C2211yescharacteristic
13C2×#13093940—21no—
14#15689085—22no—
15#15689058C412yes—
16#15689034C412yes—
17#15689025—21no—
18#15689021—21no—
19#15688994—22no—
20#15688989C2211yescharacteristic
21#15688891—21no—
22Q8⋊(C47⋊C8)—22no—
23C2×C4×Q8×C47C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C47×#423C2211yescharacteristic
26C47×Q8⋊C8—22no—
27C22×D188D411yescharacteristic
28C22×C4×D47D412yes—
29C2×#10625329—21no—
30C2×#10625324—24no—
31C4×D188—21no—
32C4×D188—21no—
33C4×D188—21no—
34C4×D188—21no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47D411yescharacteristic
37C2×Dic47⋊C4—22no—
38C2×(C2×D47)⋊C4—22no—
39C2×(C2×D47)⋊C4—22no—
40C2×(C2×D47)⋊C4C2×C412yes—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44#13093940—42no—
45C22⋊(C47⋊C8)—21no—
46(C2×D47)⋊C8—24no—
47C4×C47⋊C8D412yes—
48C47⋊#95—21no—
49C47⋊#95—22no—
50C47⋊#95C2×C412yes—
51C22×Q8×C47C2×C411yescharacteristic
52C2×M16×C47—21no—
53C4×Q8×C47—21no—
54C4×Q8×C47—21no—
55C4×Q8×C47—21no—
56C4×Q8×C47—21no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47C2×C411yescharacteristic
59C2×C42×C47—21no—
60C2×C42×C47C2311yescharacteristic
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C22⋊C8×C47—21no—
65C4×C8×C47D412yes—
66C23×D47C22⋊C412yes—
67C2×Q8×C47—21no—
68C2×Q8×C47—21no—
69C2×Q8×C47—42no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×Dic47—22no—
76C22×Dic47C22⋊C412yes—
77C2×D188—21no—
78C2×D188—21no—
79C2×D188—21no—
80C2×D188—21no—
81C2×D188—21no—
82C2×D188—21no—
83C2×C4×D47—22no—
84C2×C4×D47—22no—
85C2×C4×D47—82no—
86M16×C47—41no—
87M16×C47—41no—
88C2×C8×C47—22no—
89C2×C8×C47—41no—
90C4⋊C4×C47—21no—
91C4⋊C4×C47—21no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—41no—
94C42×C47—41no—
95C42×C47—41no—
96C42×C47C2×D411yescharacteristic
97C42×C47C2×D411yescharacteristic
98C42×C47#4111yescharacteristic
99C42×C47#4111yescharacteristic
100(C2×D47)⋊C4—22no—
101(C2×D47)⋊C4—22no—
102(C2×D47)⋊C4—44no—
103(C2×D47)⋊C4—82no—
104C4⋊Dic47—21no—
105C4⋊Dic47—21no—
106C4⋊Dic47—41no—
107C4⋊Dic47—41no—
108Dic47⋊C4—82no—
109#10625329—41no—
110#10625329—41no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—41no—
113#10625324—84no—
114C23×C47C2×C22⋊C411yescharacteristic
115C22×D47—22no—
116C22×D47—22no—
117C22×D47—22no—
118C22×D47—42no—
119C22×D47—42no—
120Q8×C47—22no—
121Q8×C47—22no—
122Q8×C47—41no—
123Q8×C47—41no—
124C2×C4×C47—22no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47C4×D411yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×C4×C47#12011yescharacteristic
135C2×C4×C47#12111yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C2×C4×C47#12311yescharacteristic
138C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C47#13711yescharacteristic
140C2×C4×C47#13711yescharacteristic
141C2×C4×C47#13711yescharacteristic
142C2×Dic47—22no—
143C2×Dic47—42no—
144C2×Dic47—42no—
145C2×Dic47—42no—
146C2×Dic47—42no—
147D188—41no—
148D188—41no—
149D188—41no—
150D188—41no—
151C4×D47—44no—
152C4×D47—82no—
153C8×C47—42no—
154C8×C47—81no—
155C47⋊C8—42no—
156C47⋊C8—81no—
157#9487—471nomaximal, Sylow
158C22×C47—21no—
159C22×C47—21no—
160C22×C47#38611yescharacteristic
161C22×C47#41711yescharacteristic
162C22×C47#41911yescharacteristic
163C2×D47—44no—
164C2×D47—42no—
165C2×D47—42no—
166C2×D47—42no—
167C2×D47—82no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—42no—
173C4×C47—81no—
174C4×C47#46811yescharacteristic
175C4×C47#46811yescharacteristic
176C4×C47#47011yescharacteristic
177C4×C47#47011yescharacteristic
178Dic47—42no—
179Dic47—82no—
180#2975—471no—
181#2580—471no—
182#1800—471no—
183#1487—471no—
184#1487D4711yescharacteristic, complemented
185#1286—471no—
186#1286—471no—
187C2×C47—21no—
188C2×C47—21no—
189C2×C47#189111yescharacteristic
190C2×C47#189411yescharacteristic
191C2×C47#191111yescharacteristic
192D47—82no—
193C2×C4×Q8C2×D4711yescharacteristic
194C2×C4×D4—471no—
195#436—942no—
196#436—942no—
197#431—471no—
198#431C2×D4711yescharacteristic
199#423—471no—
200#423C2×D4711yescharacteristic
201C2×#100—941no—
202#407—942no—
203#399—472no—
204#393—472no—
205#390—941no—
206#389—941no—
207#385—942no—
208#382—471no—
209#343—941no—
210Q8⋊C8—22no—
211Q8⋊C8—942no—
212C47#948711yescharacteristic, complemented, Sylow
213C22×Q8C4×D4711yescharacteristic
214C22×D4—471no—
215C23×C4—472no—
216C2×M16—21no—
217C2×M16—944no—
218C2×M16—941no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×D4—941no—
224C4×D4—941no—
225C4×D4—941no—
226C4×D4—941no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—471no—
229C2×C4⋊C4—942no—
230C2×C4⋊C4—941no—
231C2×C4⋊C4C4×D4711yescharacteristic
232C2×C22⋊C4—472no—
233C2×C22⋊C4—942no—
234C2×C22⋊C4—942no—
235C2×C42—21no—
236C2×C42C22×D4711yescharacteristic
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—941no—
241C4⋊C8—941no—
242C4⋊C8—941no—
243#100—1882no—
244C22⋊C8—21no—
245C22⋊C8—944no—
246C22⋊C8—941no—
247C4×C8—472no—
248C4×C8C22⋊D4712yes—
249#95—472no—
250#95—942no—
251#95—941no—
252C24—472no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—42no—
256C2×D4—941no—
257C2×D4—941no—
258C2×D4—941no—
259C2×D4—941no—
260C2×D4—941no—
261C2×D4—941no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—472no—
265C22×C4—942no—
266C22×C4—942no—
267C22×C4—942no—
268C22×C4—3762no—
269C22×C4C2×C4×D4711yescharacteristic
270C22×C4D4×D4711yescharacteristic
271C22×C4D4×D4711yescharacteristic
272M16—41no—
273M16—41no—
274M16—3764no—
275M16—1881no—
276M16—1881no—
277C2×C8—22no—
278C2×C8—41no—
279C2×C8—942no—
280C2×C8—1881no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—941no—
286C4⋊C4—941no—
287C4⋊C4—3762no—
288C4⋊C4—1881no—
289C4⋊C4—1881no—
290C22⋊C4—942no—
291C22⋊C4—942no—
292C22⋊C4—3762no—
293C22⋊C4—1884no—
294C42—41no—
295C42—41no—
296C42#1062535011yescharacteristic
297C42D4×D4711yescharacteristic
298C42#1062535211yescharacteristic
299C42C2×C22⋊D4711yescharacteristic
300C23—942no—
301C23—942no—
302C23—942no—
303C23—1882no—
304C23—1882no—
305C23C22⋊C4×D4711yescharacteristic
306Q8—22no—
307Q8—22no—
308Q8—41no—
309Q8—41no—
310D4—1881no—
311D4—1881no—
312D4—1881no—
313D4—1881no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—942no—
323C2×C4—3762no—
324C2×C4—1884no—
325C2×C4—1882no—
326C2×C4—1882no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4(C22⋊D47)⋊C411yescharacteristic
330C2×C4C4×C22⋊D4711yescharacteristic
331C2×C4#1309399211yescharacteristic
332C2×C4#1309399511yescharacteristic
333C2×C4#1309399711yescharacteristic
334C2×C4#1309400611yescharacteristic
335C2×C4#1309405111yescharacteristic
336C2×C4#1309405311yescharacteristic
337C2×C4#1309405611yescharacteristic
338C2×C4#1309405611yescharacteristic
339C8—42no—
340C8—81no—
341C8—3761no—
342C8—1882no—
343C22—21no—
344C22—21no—
345C22—3762no—
346C22—1884no—
347C22—1882no—
348C22—1882no—
349C22—1882no—
350C22#1568899911yescharacteristic
351C22#1568912311yescharacteristic
352C22#1568913011yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—81no—
359C4—3762no—
360C4—1882no—
361C4#1568936611yescharacteristic
362C4#1568937211yescharacteristic
363C4#1568937811yescharacteristic
364C4#1568938011yescharacteristic
365C2—21no—
366C2—21no—
367C2—3762no—
368C2#2024380311yescharacteristic
369C2#2024381811yescharacteristic
370C2#2024391711yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1487C2
socleC22×C47#386
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1487D47
hypocenterC47#9487
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
4. #20241817C2semidirect
5. #20241248C2semidirect
6. #20241247C2semidirect
7. C47×#1487C2semidirect
184. #1487D47semidirect
212. C47#9487semidirect

Maximal subgroups

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

subgroupindex
#202524282
#202487462
#202433772
#202418172
#202412482
#202412472
C47×#14872
#948747