Finite Group Databasebeta

← #26104503

Subgroups — #26104503

Order 12032.

594 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 127 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×#15689021C211yesmaximal, characteristic
2#20243150C211yesmaximal, characteristic
3#20241210C214yesmaximal, complemented
4C47×#1749C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic
6C22×C4⋊Dic47C411yescharacteristic
7C22×Dic47⋊C4—21no—
8C2×C22⋊(C47⋊C8)—21no—
9C2×#13093935—21no—
10#15689021—21no—
11#15689021—21no—
12#15689021C412yes—
13#15689021C2212yes—
14C2×C47⋊#95—21no—
15#15688878C2212yes—
16C22×C42×C47C2211yescharacteristic
17C2×C22⋊C8×C47—21no—
18C2×#97×C47—21no—
19C47×#336C2212yes—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—24no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—42no—
27C2×C4⋊Dic47C2×C412yes—
28C2×C4⋊Dic47C2×C412yes—
29C2×Dic47⋊C4—22no—
30C2×Dic47⋊C4—22no—
31C2×Dic47⋊C4—44no—
32C22⋊(C47⋊C8)—24no—
33#13093935—42no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C23×C4×C47C2×C411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C22×C8×C47—22no—
40C2×C42×C47—42no—
41C2×C42×C47C2×C411yescharacteristic
42C2×C42×C47D411yescharacteristic
43C2×C42×C47D411yescharacteristic
44C2×C42×C47C2311yescharacteristic
45C22⋊C8×C47—24no—
46#97×C47—42no—
47C24×C47C22⋊C411yescharacteristic
48C22×C4×C47—22no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×Dic47—24no—
64C22×Dic47—22no—
65C22×Dic47—22no—
66C22×Dic47—42no—
67C22×Dic47—42no—
68C22×Dic47—42no—
69C22×Dic47—42no—
70C22×Dic47—42no—
71C2×C8×C47—24no—
72C2×C8×C47—42no—
73C2×C8×C47—42no—
74C42×C47—22no—
75C42×C47—22no—
76C42×C47—41no—
77C42×C47—41no—
78C4⋊Dic47—24no—
79C4⋊Dic47—22no—
80C4⋊Dic47—42no—
81C4⋊Dic47—42no—
82C4⋊Dic47M1614yes—
83Dic47⋊C4—44no—
84Dic47⋊C4—44no—
85C2×C47⋊C8—24no—
86C2×C47⋊C8—42no—
87C2×C47⋊C8—42no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—21no—
92C23×C47#10112yes—
93C23×C47#10412yes—
94C23×C47#10412yes—
95C23×C47C2×C22⋊C411yescharacteristic
96C2×C4×C47—28no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—44no—
112C2×C4×C47—44no—
113C2×C4×C47#9912yes—
114C2×C4×C47C2×M1612yes—
115C2×C4×C47AGL(1,ℤ/8)12yes—
116C2×C4×C47#13712yes—
117C2×Dic47—24no—
118C2×Dic47—48no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—84no—
125C8×C47—44no—
126C47⋊C8—44no—
127#7627—471nomaximal, Sylow
128C22×C47—22no—
129C22×C47—22no—
130C22×C47—22no—
131C22×C47—22no—
132C22×C47—22no—
133C22×C47—22no—
134C22×C47#33014yes—
135C22×C47#40711yescharacteristic
136C22×C47C2×#9911yescharacteristic
137C22×C47C2×#10111yescharacteristic
138C22×C47#41811yescharacteristic
139C22×C47#41911yescharacteristic
140C22×C47C2×#10411yescharacteristic
141C22×C47C2×#10411yescharacteristic
142C4×C47—22no—
143C4×C47—22no—
144C4×C47—22no—
145C4×C47—22no—
146C4×C47—41no—
147C4×C47—41no—
148Dic47—44no—
149Dic47—82no—
150C2×#389—471no—
151#1749—471no—
152#1749D4711yescharacteristic
153#1271—474no—
154C2×C47—24no—
155C2×C47#146311yescharacteristic
156C2×C47#149011yescharacteristic
157C2×C47#149111yescharacteristic
158C2×C47#150711yescharacteristic
159C2×C47#150811yescharacteristic
160C2×C47C2×#33011yescharacteristic
161C2×C47#155611yescharacteristic
162C22×C4⋊C4—471no—
163C22×C4⋊C4—471no—
164C22×C4⋊C4—941no—
165C22×C42C2×D4711yescharacteristic
166C2×C22⋊C8—21no—
167C2×C22⋊C8—941no—
168C2×#97—21no—
169C2×#97—941no—
170#389—472no—
171#389—472no—
172#389—941no—
173#389—941no—
174C2×#95—941no—
175#336—472no—
176#336C2×D4712yes—
177C47#762711yescharacteristic, complemented, Sylow
178C23×C4—941no—
179C23×C4—941no—
180C23×C4C4×D4711yescharacteristic
181C23×C4D18811yescharacteristic
182C23×C4C22⋊D4711yescharacteristic
183C22×C8—22no—
184C22×C8—942no—
185C2×C4⋊C4—472no—
186C2×C4⋊C4—472no—
187C2×C4⋊C4—944no—
188C2×C4⋊C4—941no—
189C2×C4⋊C4—941no—
190C2×C4⋊C4—942no—
191C2×C4⋊C4—942no—
192C2×C4⋊C4—1884no—
193C2×C4⋊C4—1882no—
194C2×C42—42no—
195C2×C42C4×D4711yescharacteristic
196C2×C42D18811yescharacteristic
197C2×C42C22⋊D4711yescharacteristic
198C2×C42C22×D4711yescharacteristic
199C22⋊C8—24no—
200C22⋊C8—944no—
201#97—42no—
202#97—1882no—
203#95—942no—
204#95—942no—
205C24(C2×D47)⋊C411yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—944no—
216C22×C4—942no—
217C22×C4—942no—
218C22×C4—1882no—
219C22×C4—1882no—
220C22×C4—1882no—
221C22×C4—1882no—
222C22×C4—1882no—
223C22×C4(C2×D47)⋊C411yescharacteristic
224C22×C4(C2×D47)⋊C411yescharacteristic
225C22×C4(C2×D47)⋊C411yescharacteristic
226C22×C4C2×C4×D4711yescharacteristic
227C22×C4C2×D18811yescharacteristic
228C22×C4C2×C22⋊D4711yescharacteristic
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—944no—
233C2×C8—1882no—
234C2×C8—1882no—
235C4⋊C4—474no—
236C4⋊C4—944no—
237C4⋊C4—942no—
238C4⋊C4—1882no—
239C4⋊C4—1884no—
240C4⋊C4—1882no—
241C4⋊C4—1884no—
242C42—22no—
243C42—22no—
244C42—41no—
245C42—41no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#1309394212yes—
251C23#1309395012yes—
252C23#1309395212yes—
253C23C2×(C2×D47)⋊C411yescharacteristic
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—944no—
272C2×C4—3764no—
273C2×C4—1882no—
274C2×C4—1882no—
275C2×C4—1882no—
276C2×C4—1888no—
277C2×C4—1882no—
278C2×C4—1882no—
279C2×C4#1309393812yes—
280C2×C4M16×D4712yes—
281C2×C4#1309405212yes—
282C2×C4#1309405412yes—
283C8—44no—
284C8—1884no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22#1568886414yes—
292C22#1568908611yescharacteristic
293C22C2×#1309393811yescharacteristic
294C22C2×#1309394211yescharacteristic
295C22#1568912911yescharacteristic
296C22#1568913111yescharacteristic
297C22C2×#1309395011yescharacteristic
298C22C2×#1309395211yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—3762no—
306C4—1884no—
307C2—24no—
308C2#2024171711yescharacteristic
309C2#2024183211yescharacteristic
310C2#2024183411yescharacteristic
311C2#2024190511yescharacteristic
312C2#2024190711yescharacteristic
313C2C2×#1568886411yescharacteristic
314C2#2024210411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1749C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1749D47
hypocenterC47#7627
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #20241210C2semidirect
177. C47#7627semidirect

Maximal subgroups

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

subgroupindex
C2×#156890212
#202431502
#202412102
C47×#17492
#762747