Finite Group Databasebeta

← #26131818

Subgroups — #26131818

Order 12032.

986 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 135 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#20245077C214yesmaximal, complemented
2#20243316C212yesmaximal, complemented
3C47×#3485C211yesmaximal, characteristic
4C2×#13093943C2214yescomplemented
5C2×#13093941—22no—
6#15689087—22no—
7#15688922—24no—
8#15688920—24no—
9C2×H(5,𝔽2)×C47C411yescharacteristic
10C22×#41×C47C2211yescharacteristic
11C47×#547—24no—
12C47×#546—24no—
13C47×#536C2212yes—
14C47×#534—22no—
15C47×#525C412yes—
16C2×#123×C47—22no—
17C47×#518C411yescharacteristic
18C2×#10625329—24no—
19#13093943—28no—
20#13093941—44no—
21C22⋊(C47⋊C8)—44no—
22H(5,𝔽2)×C47—44no—
23C2×#41×C47—22no—
24C2×#41×C47—24no—
25C2×#41×C47—44no—
26C2×#41×C47C2×C412yes—
27C22×Q8×C47C2311yescharacteristic
28C22×D4×C47—22no—
29C22×D4×C47—22no—
30C22×D4×C47C2×C411yescharacteristic
31C22×D4×C47D412yes—
32C23×C4×C47C2×C411yescharacteristic
33C23×C4×C47D412yes—
34C4⋊Q8×C47D414yes—
35#124×C47D414yes—
36#123×C47—28no—
37#123×C47—24no—
38#123×C47—22no—
39#123×C47—24no—
40C22⋊Q8×C47—22no—
41C22⋊Q8×C47—24no—
42#121×C47—28no—
43#121×C47—22no—
44#120×C47—24no—
45#120×C47—22no—
46C4×D4×C47—24no—
47C4×D4×C47—24no—
48#117×C47C2×C412yes—
49C2×C4⋊C4×C47—24no—
50C2×C22⋊C4×C47—24no—
51C2×C22⋊C4×C47—22no—
52C24×C47—22no—
53C24×C47C22⋊C411yescharacteristic
54C24×C47C22⋊C412yes—
55#41×C47—44no—
56#41×C47—416no—
57#41×C47—48no—
58C2×Q8×C47—42no—
59C2×Q8×C47C22⋊C412yes—
60C2×Q8×C47C2×D414yes—
61C2×D4×C47—24no—
62C2×D4×C47—24no—
63C2×D4×C47—24no—
64C2×D4×C47—22no—
65C2×D4×C47—24no—
66C2×D4×C47—22no—
67C2×D4×C47—24no—
68C2×D4×C47—42no—
69C2×D4×C47—48no—
70C2×D4×C47—44no—
71C2×D4×C47—44no—
72C2×D4×C47C22⋊C412yes—
73C22×C4×C47—22no—
74C22×C4×C47—24no—
75C22×C4×C47—24no—
76C22×C4×C47—22no—
77C22×C4×C47—22no—
78C22×C4×C47—24no—
79C22×C4×C47—42no—
80C22×C4×C47—42no—
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C2×D412yes—
84C4⋊C4×C47—24no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—28no—
87C22⋊C4×C47—24no—
88C22⋊C4×C47—24no—
89C22⋊C4×C47—24no—
90C22⋊C4×C47—22no—
91C22⋊C4×C47—28no—
92C42×C47C22⋊C414yes—
93#10625329—44no—
94#10625329—44no—
95C2×C47⋊C8—44no—
96C23×C47—22no—
97C23×C47—22no—
98C23×C47—22no—
99C23×C47—21no—
100C23×C47—22no—
101C23×C47—44no—
102C23×C47—82no—
103C23×C47—81no—
104C23×C47—84no—
105C23×C47#9914yes—
106C23×C47C2×C22⋊C411yescharacteristic
107Q8×C47—44no—
108Q8×C47—42no—
109D4×C47—28no—
110D4×C47—48no—
111D4×C47—42no—
112D4×C47—48no—
113D4×C47—44no—
114D4×C47—84no—
115C2×C4×C47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—24no—
119C2×C4×C47—21no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—48no—
123C2×C4×C47—44no—
124C2×C4×C47—44no—
125C2×C4×C47—44no—
126C2×C4×C47—84no—
127C2×C4×C47C2×C22⋊C412yes—
128C2×C4×C47#12014yes—
129C47⋊C8—84no—
130#12709—471nomaximal, Sylow
131C22×C47—24no—
132C22×C47—41no—
133C22×C47—44no—
134C22×C47—44no—
135C22×C47—42no—
136C22×C47—44no—
137C22×C47—41no—
138C22×C47—42no—
139C22×C47—82no—
140C22×C47—82no—
141C22×C47—164no—
142C22×C47#37911yescharacteristic
143C22×C47C2×#9912yes—
144C4×C47—24no—
145C4×C47—22no—
146C4×C47—44no—
147C4×C47—41no—
148C4×C47—42no—
149#3485D4711yescharacteristic
150#2133—474no—
151#1785—472no—
152C2×C47—21no—
153C2×C47—22no—
154C2×C47—44no—
155C2×C47—81no—
156C2×C47—82no—
157C2×C47#178111yescharacteristic
158C2×H(5,𝔽2)Dic4711yescharacteristic
159C22×#41C2×D4711yescharacteristic
160#547—24no—
161#546—24no—
162#536C2×D4712yes—
163#534—22no—
164#525Dic4712yes—
165C2×#123—22no—
166#518Dic4711yescharacteristic
167C2×#101—474no—
168C2×#100—942no—
169#407—942no—
170#356—944no—
171#355—944no—
172C47#1270911yescharacteristic, complemented, Sylow
173H(5,𝔽2)—44no—
174C2×#41—22no—
175C2×#41—24no—
176C2×#41—44no—
177C2×#41C2×Dic4712yes—
178C22×Q8C22×D4711yescharacteristic
179C22×D4—22no—
180C22×D4—22no—
181C22×D4C2×Dic4711yescharacteristic
182C22×D4C22⋊D4712yes—
183C23×C4C2×Dic4711yescharacteristic
184C23×C4C22⋊D4712yes—
185C2×M16—944no—
186C4⋊Q8C22⋊D4714yes—
187#124C22⋊D4714yes—
188#123—28no—
189#123—24no—
190#123—22no—
191#123—24no—
192C22⋊Q8—22no—
193C22⋊Q8—24no—
194#121—28no—
195#121—22no—
196#120—24no—
197#120—22no—
198C4×D4—24no—
199C4×D4—24no—
200#117C2×Dic4712yes—
201C2×C4⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204#101—948no—
205#100—1884no—
206C22⋊C8—1884no—
207C24—22no—
208C24C22⋊Dic4711yescharacteristic
209C24C22⋊Dic4712yes—
210#41—44no—
211#41—416no—
212#41—48no—
213C2×Q8—42no—
214C2×Q8C22⋊Dic4712yes—
215C2×Q8C2×C22⋊D4714yes—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—48no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4C22⋊Dic4712yes—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C22⋊Dic4711yescharacteristic
237C22×C4C22×Dic4711yescharacteristic
238C22×C4C2×C22⋊D4712yes—
239M16—1884no—
240M16—1884no—
241C2×C8—1884no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—28no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—22no—
249C22⋊C4—28no—
250C42C22⋊Dic4714yes—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—44no—
257C23—82no—
258C23—81no—
259C23—84no—
260C23#1309393914yes—
261C23C2×C22⋊Dic4711yescharacteristic
262Q8—44no—
263Q8—42no—
264D4—28no—
265D4—48no—
266D4—42no—
267D4—48no—
268D4—44no—
269D4—84no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—84no—
282C2×C4C2×C22⋊Dic4712yes—
283C2×C4#1309399314yes—
284C8—3764no—
285C22—24no—
286C22—41no—
287C22—44no—
288C22—44no—
289C22—42no—
290C22—44no—
291C22—41no—
292C22—42no—
293C22—82no—
294C22—82no—
295C22—164no—
296C22#1568897911yescharacteristic
297C22C2×#1309393912yes—
298C4—24no—
299C4—22no—
300C4—44no—
301C4—41no—
302C4—42no—
303C2—21no—
304C2—22no—
305C2—44no—
306C2—81no—
307C2—82no—
308C2#2024330311yescharacteristic
subgroupG/H
centerC2#20243303
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
Fitting, prime radicalC47×#3485C2
socleC2×C47#1781
cosocleC22×Q8×C47C23
hypercenter#3485D47
hypocenterC47#12709
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ #3485

Complements

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

NH ≅ G/Ntype
1. #20245077C2semidirect
2. #20243316C2semidirect
4. C2×#13093943C22semidirect
172. C47#12709semidirect

Maximal subgroups

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

subgroupindex
#202450772
#202433162
C47×#34852
#1270947