Finite Group Databasebeta

← #26111275

Subgroups — #26111275

Order 12032.

800 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 143 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×#15689043C211yesmaximal, characteristic
2#20246006C211yesmaximal, characteristic
3#20243324C212yesmaximal
4#20241268C212yesmaximal
5C2×C47×#324C211yesmaximal, characteristic
6C23×Dic94C2211yescharacteristic
7C2×#13093983—21no—
8C22×Dic47⋊C4—21no—
9C2×#13093949—24no—
10#15689043—22no—
11#15689043C414yescomplemented
12#15689022—24no—
13#15688983—21no—
14#15688983C2212yes—
15C2×C47⋊#95—21no—
16C2×C47⋊#95C2211yescharacteristic
17C22×C4⋊C4×C47C2211yescharacteristic
18C2×C22⋊C8×C47C2212yes—
19C47×#324—22no—
20C22×Dic94—22no—
21C22×Dic94—22no—
22C22×Dic94—21no—
23C22×Dic94—44no—
24C22×Dic94C2×C412yes—
25C23×Dic47—21no—
26C23×Dic47—21no—
27#13093983—42no—
28#13093983—42no—
29C2×C4⋊Dic47—22no—
30C2×C4⋊Dic47—22no—
31C2×C4⋊Dic47D414yes—
32C2×Dic47⋊C4—24no—
33C2×Dic47⋊C4—44no—
34C2×C22⋊Dic47—22no—
35C2×C4×Dic47—22no—
36#13093949—48no—
37C47⋊#95—22no—
38C47⋊#95—24no—
39C47⋊#95—21no—
40C47⋊#95—22no—
41C47⋊#95C2×C412yes—
42C23×C4×C47D411yescharacteristic
43C23×C4×C47D411yescharacteristic
44C23×C4×C47C2311yescharacteristic
45C22×C8×C47—22no—
46C2×C4⋊C4×C47—22no—
47C2×C4⋊C4×C47—22no—
48C2×C4⋊C4×C47C2×C412yes—
49C2×C4⋊C4×C47D412yes—
50C22⋊C8×C47—22no—
51C22⋊C8×C47—22no—
52C24×C47C2×D411yescharacteristic
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—42no—
60C22×C4×C47—42no—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22⋊C412yes—
63C22×C4×C47SD1612yes—
64C22×C4×C47Q1612yes—
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D412yes—
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47#4112yes—
69C22×Dic47—24no—
70C22×Dic47—24no—
71C22×Dic47—22no—
72C22×Dic47—22no—
73C22×Dic47—24no—
74C22×Dic47—42no—
75C22×Dic47—42no—
76C22×Dic47—41no—
77C22×Dic47—41no—
78C22×Dic47—41no—
79C22×Dic47—41no—
80C2×Dic94—28no—
81C2×Dic94—24no—
82C2×Dic94—44no—
83C2×Dic94—48no—
84C2×Dic94—44no—
85C2×Dic94—44no—
86C2×Dic94—42no—
87C2×C8×C47—44no—
88C2×C8×C47—42no—
89C4⋊C4×C47—24no—
90C4⋊C4×C47—24no—
91C22⋊Dic47—42no—
92C22⋊Dic47—42no—
93C4⋊Dic47—44no—
94C4⋊Dic47—44no—
95C4⋊Dic47—82no—
96Dic47⋊C4—48no—
97C4×Dic47—44no—
98C23×C47—22no—
99C23×C47—22no—
100C23×C47C2×C22⋊C411yescharacteristic
101C23×C47#12011yescharacteristic
102C23×C47#12311yescharacteristic
103C23×C47C2×SD1611yescharacteristic
104C23×C47C2×Q1611yescharacteristic
105C23×C47AGL(1,ℤ/8)11yescharacteristic
106C23×C47#13711yescharacteristic
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—24no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—81no—
121C2×C4×C47#10314yescomplemented
122C2×C4×C47C4×D412yes—
123C2×C4×C47#12112yes—
124C2×Dic47—24no—
125C2×Dic47—24no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—44no—
129C2×Dic47—44no—
130C2×Dic47—44no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—44no—
134C2×Dic47—44no—
135C2×Dic47—44no—
136C2×Dic47—44no—
137Dic94—28no—
138Dic94—48no—
139Dic94—44no—
140C8×C47—82no—
141#9015—471nomaximal, Sylow
142C22×C47—22no—
143C22×C47—22no—
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—21no—
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—21no—
150C22×C47#38611yescharacteristic
151C22×C47C2×#10311yescharacteristic
152C22×C47#41711yescharacteristic
153C22×C47#45011yescharacteristic
154C22×C47#45111yescharacteristic
155C22×C47#45612yes—
156C22×C47B(4,𝔽2)12yes—
157C22×C47#48311yescharacteristic
158C22×C47#48411yescharacteristic
159C4×C47—24no—
160C4×C47—22no—
161C4×C47—22no—
162C4×C47—42no—
163Dic47—44no—
164Dic47—44no—
165Dic47—84no—
166C2×#395—471no—
167#2316—471no—
168#1787—472no—
169C2×#324D4711yescharacteristic
170#1294—472no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47—21no—
175C2×C47#160412yes—
176C2×C47#160512yes—
177C2×C47#187211yescharacteristic
178C2×C47#188611yescharacteristic
179C2×C47#188911yescharacteristic
180C23×Q8—471no—
181C2×#117—941no—
182C22×C4⋊C4—941no—
183C22×C4⋊C4C2×D4711yescharacteristic
184C2×#103—944no—
185C2×C22⋊C8C2×D4712yes—
186#395—474no—
187#395—942no—
188#389—944no—
189#380—472no—
190#380—941no—
191C2×#95—471no—
192C2×#95—941no—
193#324—22no—
194C47#901511yescharacteristic, complemented, Sylow
195C22×Q8—472no—
196C22×Q8—941no—
197C22×Q8—942no—
198C22×Q8—942no—
199C22×Q8—1884no—
200C23×C4—941no—
201C23×C4—941no—
202C23×C4D18811yescharacteristic
203C23×C4D18811yescharacteristic
204C23×C4C22×D4711yescharacteristic
205C22×C8—22no—
206#117—1882no—
207#117—1882no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—474no—
211C2×C4⋊C4—942no—
212C2×C4⋊C4—942no—
213C2×C4⋊C4—944no—
214C2×C4⋊C4—1884no—
215C2×C4⋊C4C4×D4712yes—
216C2×C4⋊C4C22⋊D4712yes—
217C2×C22⋊C4—942no—
218C2×C42—942no—
219#103—1888no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222#95—472no—
223#95—942no—
224#95—944no—
225#95—942no—
226#95—941no—
227C24C2×D18811yescharacteristic
228C2×Q8—948no—
229C2×Q8—944no—
230C2×Q8—1884no—
231C2×Q8—1884no—
232C2×Q8—1884no—
233C2×Q8—1888no—
234C2×Q8—1882no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—944no—
244C22×C4—944no—
245C22×C4—942no—
246C22×C4—942no—
247C22×C4—944no—
248C22×C4—1882no—
249C22×C4—1882no—
250C22×C4—1881no—
251C22×C4—1881no—
252C22×C4—1881no—
253C22×C4—1881no—
254C22×C4#1062532512yes—
255C22×C4Dic18812yes—
256C22×C4(C2×D47)⋊C412yes—
257C22×C4(C2×D47)⋊C412yes—
258C22×C4C2×C4×D4711yescharacteristic
259C22×C4D4×D4712yes—
260C22×C4#1062535212yes—
261C22×C4C2×C22⋊D4711yescharacteristic
262C2×C8—44no—
263C2×C8—42no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—3762no—
267C4⋊C4—1884no—
268C4⋊C4—1884no—
269C4⋊C4—1888no—
270C22⋊C4—1882no—
271C22⋊C4—1882no—
272C42—1884no—
273C23—22no—
274C23—22no—
275C23C2×(C2×D47)⋊C411yescharacteristic
276C23#1309399411yescharacteristic
277C23#1309400811yescharacteristic
278C23C2×#1062532511yescharacteristic
279C23C2×Dic18811yescharacteristic
280C23#1309405211yescharacteristic
281C23#1309405711yescharacteristic
282Q8—948no—
283Q8—1888no—
284Q8—1884no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—944no—
300C2×C4—944no—
301C2×C4—1884no—
302C2×C4—1884no—
303C2×C4—1884no—
304C2×C4—1882no—
305C2×C4—1884no—
306C2×C4—1884no—
307C2×C4—1882no—
308C2×C4—1884no—
309C2×C4—1882no—
310C2×C4—1884no—
311C2×C4—1882no—
312C2×C4#1309394914yescomplemented
313C2×C4(C22⋊D47)⋊C412yes—
314C2×C4#1309399612yes—
315C8—82no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22#1568900011yescharacteristic
325C22C2×#1309394911yescharacteristic
326C22#1568912511yescharacteristic
327C22#1568927611yescharacteristic
328C22#1568928111yescharacteristic
329C22#1568930812yes—
330C22#1568931112yes—
331C22#1568945411yescharacteristic
332C22#1568945911yescharacteristic
333C4—22no—
334C4—22no—
335C4—24no—
336C4—42no—
337C4—3764no—
338C4—1884no—
339C4—1884no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#2024235912yes—
345C2#2024236412yes—
346C2#2024369311yescharacteristic
347C2#2024377911yescharacteristic
348C2#2024379411yescharacteristic
subgroupG/H
centerC23#13093994
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#324C2
socleC23×C47#120
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenterC2×#324D47
hypocenterC47#9015
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
11. #15689043C4semidirect
121. C2×C4×C47#103semidirect
194. C47#9015semidirect
312. C2×C4#13093949semidirect

Maximal subgroups

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

subgroupindex
C2×#156890432
#202460062
#202433242
#202412682
C2×C47×#3242
#901547