Finite Group Databasebeta

← #26113672

Subgroups — #26113672

Order 12032.

514 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 109 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#20248557C211yesmaximal, characteristic
3#20243377C211yesmaximal, characteristic
4#20242070C211yesmaximal, characteristic, complemented
5#20242068C211yesmaximal, characteristic, complemented
6#20241291C211yesmaximal, characteristic, complemented
7C47×#1303C211yesmaximal, characteristic, complemented
8C2×#13094024C2211yescharacteristic
9C2×C4×D188C2211yescharacteristic
10#15689202—22no—
11#15689200—22no—
12#15689176C2211yescharacteristic
13#15689147C2211yescharacteristic
14C2×#13093940—21no—
15#15689085—22no—
16#15689036C414yes—
17#15689025—21no—
18#15688989—21no—
19#15688980—22no—
20#15688891—21no—
21#15688871—22no—
22#15688871—22no—
23C47×#431C2211yescharacteristic
24C47×#423C2211yescharacteristic
25C4×C4⋊C4×C47C2211yescharacteristic
26C22×Dic94C2×C412yes—
27C22×D188D411yescharacteristic
28C22×C4×D47D412yes—
29C2×#10625329—21no—
30C2×#10625324—24no—
31#13094024—21no—
32#13094024—21no—
33#13094024—21no—
34#13094024—21no—
35C4×D188—21no—
36C4×D188—21no—
37C4×D188—21no—
38C4×D188—21no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47D411yescharacteristic
42C2×(C2×D47)⋊C4—24no—
43C2×(C2×D47)⋊C4C2×C412yes—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)D412yes—
47#13093940—42no—
48C22⋊(C47⋊C8)—21no—
49(C2×D47)⋊C8—24no—
50C4×C47⋊C8—21no—
51C47⋊#95—24no—
52C2×M16×C47—21no—
53C2×C4⋊C4×C47—21no—
54C2×C42×C47—21no—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—21no—
57C4⋊C8×C47—21no—
58C4⋊C8×C47D412yes—
59C22⋊C8×C47—21no—
60C4×C8×C47—21no—
61#95×C47C2×C412yes—
62C23×D47C22⋊C412yes—
63C22×C4×C47—22no—
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×Dic47—22no—
68C22×Dic47C22⋊C412yes—
69C2×D188—21no—
70C2×D188—21no—
71C2×D188—21no—
72C2×D188—21no—
73C2×D188—21no—
74C2×D188—21no—
75C2×C4×D47—22no—
76C2×C4×D47—22no—
77C2×C4×D47—44no—
78C2×Dic94—22no—
79C2×Dic94—22no—
80C2×Dic94—82no—
81M16×C47—41no—
82M16×C47—41no—
83C2×C8×C47—22no—
84C2×C8×C47—41no—
85C4⋊C4×C47—42no—
86C42×C47—41no—
87C42×C47—41no—
88C42×C47C2×D411yescharacteristic
89C42×C47C2×D411yescharacteristic
90C42×C47#4111yescharacteristic
91C42×C47#4111yescharacteristic
92(C2×D47)⋊C4—22no—
93(C2×D47)⋊C4—22no—
94(C2×D47)⋊C4—84no—
95C4⋊Dic47—21no—
96C4⋊Dic47—21no—
97C4⋊Dic47—41no—
98C4⋊Dic47—41no—
99C4⋊Dic47—41no—
100C4⋊Dic47—41no—
101#10625329—41no—
102#10625329—41no—
103C2×C47⋊C8—22no—
104C2×C47⋊C8—41no—
105#10625324—84no—
106C23×C47C2×C22⋊C411yescharacteristic
107C22×D47—22no—
108C22×D47—22no—
109C22×D47—22no—
110C22×D47—42no—
111C22×D47—42no—
112C2×C4×C47—22no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47C4×D411yescharacteristic
118C2×C4×C47C4×D411yescharacteristic
119C2×C4×C47#12011yescharacteristic
120C2×C4×C47#12111yescharacteristic
121C2×C4×C47#12111yescharacteristic
122C2×C4×C47#12311yescharacteristic
123C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C47#13711yescharacteristic
125C2×C4×C47#13711yescharacteristic
126C2×C4×C47#13711yescharacteristic
127C2×Dic47—22no—
128C2×Dic47—42no—
129C2×Dic47—44no—
130C2×Dic47—42no—
131D188—41no—
132D188—41no—
133D188—41no—
134D188—41no—
135C4×D47—44no—
136C4×D47—82no—
137Dic94—82no—
138Dic94—82no—
139C8×C47—42no—
140C8×C47—81no—
141C47⋊C8—42no—
142C47⋊C8—81no—
143#9399—471nomaximal, Sylow
144C22×C47—21no—
145C22×C47—21no—
146C22×C47#38611yescharacteristic
147C22×C47#41711yescharacteristic
148C22×C47#41911yescharacteristic
149C2×D47—44no—
150C2×D47—42no—
151C2×D47—42no—
152C2×D47—42no—
153C2×D47—82no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—82no—
158C4×C47#45512yes—
159C4×C47#46811yescharacteristic
160C4×C47#46811yescharacteristic
161C4×C47#47011yescharacteristic
162C4×C47#47011yescharacteristic
163Dic47—42no—
164Dic47—82no—
165#2975—471no—
166#2563—471no—
167#1800—471no—
168#1548—471no—
169#1548—471no—
170#1303—471no—
171#1303D4711yescharacteristic, complemented
172C2×C47—21no—
173C2×C47—21no—
174C2×C47#188511yescharacteristic
175C2×C47#189411yescharacteristic
176C2×C47#191411yescharacteristic
177D47—82no—
178C2×C4⋊Q8—471no—
179C2×C4×D4—471no—
180#436—942no—
181#436—942no—
182#431—471no—
183#431C2×D4711yescharacteristic
184#423—471no—
185#423C2×D4711yescharacteristic
186C2×#100—941no—
187#407—942no—
188#393—474no—
189#390—941no—
190#382—941no—
191#380—942no—
192C4×C4⋊C4C2×D4711yescharacteristic
193#343—941no—
194#333—942no—
195#333—942no—
196C47#939911yescharacteristic, complemented, Sylow
197C22×Q8—472no—
198C22×D4—471no—
199C23×C4—472no—
200C2×M16—21no—
201C2×M16—944no—
202C2×M16—941no—
203C4⋊Q8—941no—
204C4⋊Q8—941no—
205C4⋊Q8—941no—
206C4⋊Q8—941no—
207C4×D4—941no—
208C4×D4—941no—
209C4×D4—941no—
210C4×D4—941no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—471no—
213C2×C4⋊C4—941no—
214C2×C4⋊C4—941no—
215C2×C22⋊C4—472no—
216C2×C22⋊C4—944no—
217C2×C42—21no—
218C2×C42C22×D4711yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—472no—
222C4⋊C8—941no—
223C4⋊C8—941no—
224C4⋊C8D18812yes—
225#100—1882no—
226C22⋊C8—21no—
227C22⋊C8—944no—
228C22⋊C8—941no—
229C4×C8—21no—
230C4×C8—941no—
231#95—944no—
232#95C4×D4712yes—
233C24—472no—
234C2×Q8—942no—
235C2×Q8—942no—
236C2×Q8—3762no—
237C2×D4—941no—
238C2×D4—941no—
239C2×D4—941no—
240C2×D4—941no—
241C2×D4—941no—
242C2×D4—941no—
243C22×C4—22no—
244C22×C4—472no—
245C22×C4—942no—
246C22×C4—942no—
247C22×C4—942no—
248C22×C4—1884no—
249C22×C4C2×C4×D4711yescharacteristic
250C22×C4D4×D4711yescharacteristic
251C22×C4D4×D4711yescharacteristic
252M16—41no—
253M16—41no—
254M16—3764no—
255M16—1881no—
256M16—1881no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—942no—
260C2×C8—1881no—
261C4⋊C4—42no—
262C4⋊C4—941no—
263C4⋊C4—941no—
264C4⋊C4—1881no—
265C4⋊C4—1881no—
266C4⋊C4—1881no—
267C4⋊C4—1881no—
268C22⋊C4—942no—
269C22⋊C4—942no—
270C22⋊C4—3764no—
271C42—41no—
272C42—41no—
273C42C2×D18811yescharacteristic
274C42#1062535011yescharacteristic
275C42D4×D4711yescharacteristic
276C42#1062535411yescharacteristic
277C23—942no—
278C23—942no—
279C23—942no—
280C23—1882no—
281C23—1882no—
282C23C22⋊C4×D4711yescharacteristic
283Q8—3762no—
284Q8—3762no—
285D4—1881no—
286D4—1881no—
287D4—1881no—
288D4—1881no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—942no—
295C2×C4—3762no—
296C2×C4—1884no—
297C2×C4—1882no—
298C2×C4—1884no—
299C2×C4—1882no—
300C2×C4#1309398611yescharacteristic
301C2×C4C4×D18811yescharacteristic
302C2×C4#1309399411yescharacteristic
303C2×C4#1309399711yescharacteristic
304C2×C4#1309399911yescharacteristic
305C2×C4#1309400911yescharacteristic
306C2×C4#1309404811yescharacteristic
307C2×C4#1309405311yescharacteristic
308C2×C4#1309405311yescharacteristic
309C2×C4#1309405611yescharacteristic
310C8—42no—
311C8—81no—
312C8—3761no—
313C8—1882no—
314C22—21no—
315C22—21no—
316C22—3762no—
317C22—1884no—
318C22—1882no—
319C22—1882no—
320C22—1882no—
321C22#1568900111yescharacteristic
322C22#1568912111yescharacteristic
323C22#1568913011yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—82no—
328C4—3762no—
329C4—1882no—
330C4#1568930212yes—
331C4#1568936811yescharacteristic
332C4#1568937011yescharacteristic
333C4#1568938011yescharacteristic
334C4#1568938211yescharacteristic
335C2—21no—
336C2—21no—
337C2—3762no—
338C2#2024377511yescharacteristic
339C2#2024382011yescharacteristic
340C2#2024393411yescharacteristic
subgroupG/H
centerC22#15689001
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1303C2
socleC22×C47#386
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1303D47
hypocenterC47#9399
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
4. #20242070C2semidirect
5. #20242068C2semidirect
6. #20241291C2semidirect
7. C47×#1303C2semidirect
171. #1303D47semidirect
196. C47#9399semidirect

Maximal subgroups

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

subgroupindex
#202524282
#202485572
#202433772
#202420702
#202420682
#202412912
C47×#13032
#939947