Finite Group Databasebeta

← #26108051

Subgroups — #26108051

Order 12032.

802 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 151 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#20243380C211yesmaximal, characteristic, complemented
2#20242073C212yesmaximal, complemented
3#20241813C212yesmaximal, complemented
4#20241709C211yesmaximal, characteristic, complemented
5C2×C47×#555C211yesmaximal, characteristic
6#15689176C2212yes—
7#15689147C2212yes—
8C2×#13093941—21no—
9#15688892—21no—
10#15688869—24no—
11D4⋊(C47⋊C8)—24no—
12#15688847—22no—
13C47×#555—24no—
14C47×#555—24no—
15C2×C4⋊Q8×C47C2211yescharacteristic
16C2×#124×C47—22no—
17C2×#121×C47—21no—
18C2×#121×C47C412yes—
19C2×C4×D4×C47C2211yescharacteristic
20C2×C4×D4×C47C2211yescharacteristic
21C2×#117×C47—21no—
22C2×#117×C47C412yes—
23C2×#10625329—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)—22no—
27#13093941—42no—
28C22⋊(C47⋊C8)—22no—
29C4×C47⋊C8—22no—
30C22×Q8×C47—21no—
31C22×D4×C47—22no—
32C22×D4×C47C2×C411yescharacteristic
33C22×D4×C47D411yescharacteristic
34C23×C4×C47—21no—
35C23×C4×C47C2×C412yes—
36C4⋊Q8×C47—22no—
37C4⋊Q8×C47D414yes—
38#124×C47—216no—
39#121×C47—28no—
40#121×C47—24no—
41#121×C47—24no—
42C4×D4×C47—22no—
43C4×D4×C47—22no—
44C4×D4×C47—22no—
45C4×D4×C47D414yes—
46#117×C47—28no—
47#117×C47—24no—
48#117×C47—24no—
49C2×C4⋊C4×C47—21no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47C2×C412yes—
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C22⋊C4×C47—22no—
54C2×C22⋊C4×C47—22no—
55C2×C22⋊C4×C47—21no—
56C2×C22⋊C4×C47D412yes—
57C2×C42×C47—22no—
58C2×C42×C47C2311yescharacteristic
59C24×C47—21no—
60C24×C47C22⋊C412yes—
61C2×Q8×C47—24no—
62C2×Q8×C47—82no—
63C2×D4×C47—28no—
64C2×D4×C47—21no—
65C2×D4×C47—22no—
66C2×D4×C47—22no—
67C2×D4×C47—21no—
68C2×D4×C47—21no—
69C2×D4×C47—42no—
70C2×D4×C47—84no—
71C2×D4×C47C22⋊C412yes—
72C22×C4×C47—21no—
73C22×C4×C47—22no—
74C22×C4×C47—22no—
75C22×C4×C47—22no—
76C22×C4×C47—24no—
77C22×C4×C47—44no—
78C22×C4×C47—42no—
79C22×C4×C47—41no—
80C22×C4×C47—41no—
81C22×C4×C47C22⋊C412yes—
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C4⋊C4×C47—21no—
86C4⋊C4×C47—22no—
87C4⋊C4×C47—24no—
88C4⋊C4×C47—22no—
89C4⋊C4×C47C22⋊C412yes—
90C4⋊C4×C47C22⋊C414yes—
91C22⋊C4×C47—28no—
92C22⋊C4×C47—28no—
93C22⋊C4×C47—24no—
94C22⋊C4×C47—24no—
95C42×C47—28no—
96C42×C47C2×D412yes—
97C42×C47C2×D412yes—
98#10625329—42no—
99#10625329—42no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—42no—
102C23×C47—22no—
103C23×C47—22no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—42no—
107C23×C47—42no—
108C23×C47—42no—
109C23×C47—42no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47C2×C22⋊C411yescharacteristic
113Q8×C47—84no—
114D4×C47—24no—
115D4×C47—42no—
116D4×C47—42no—
117D4×C47—42no—
118D4×C47—88no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—24no—
123C2×C4×C47—41no—
124C2×C4×C47—44no—
125C2×C4×C47—44no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—44no—
129C2×C4×C47—44no—
130C2×C4×C47—44no—
131C2×C4×C47—42no—
132C2×C4×C47C2×C22⋊C411yescharacteristic
133C2×C4×C47C2×C22⋊C411yescharacteristic
134C2×C4×C47#12012yes—
135C2×C4×C47#12012yes—
136C2×C4×C47#13512yes—
137C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C47#13711yescharacteristic
139C47⋊C8—82no—
140C47⋊C8—82no—
141#8397—471nomaximal, Sylow
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—42no—
145C22×C47—42no—
146C22×C47—42no—
147C22×C47—42no—
148C22×C47—42no—
149C22×C47—42no—
150C22×C47—82no—
151C22×C47—82no—
152C22×C47—82no—
153C22×C47—82no—
154C22×C47#37911yescharacteristic
155C22×C47#41611yescharacteristic
156C22×C47#41711yescharacteristic
157C4×C47—22no—
158C4×C47—21no—
159C4×C47—44no—
160C4×C47—42no—
161C4×C47—42no—
162C4×C47#41312yes—
163C4×C47#44912yes—
164C4×C47#45212yes—
165C2×#555D4711yescharacteristic
166#1800—471no—
167#1549—472no—
168#1486—472no—
169#1460—471no—
170C2×C47—21no—
171C2×C47—21no—
172C2×C47—82no—
173C2×C47—82no—
174C2×C47#178511yescharacteristic
175C2×C47#178811yescharacteristic
176C2×C47#179511yescharacteristic
177#555—24no—
178#555—24no—
179C2×C4⋊Q8C2×D4711yescharacteristic
180C2×#124—22no—
181C2×#121—21no—
182C2×#121Dic4712yes—
183C2×C4×D4C2×D4711yescharacteristic
184C2×C4×D4C2×D4711yescharacteristic
185C2×#117—21no—
186C2×#117Dic4712yes—
187#431—472no—
188#423—472no—
189C2×#100—941no—
190#343—941no—
191#332—944no—
192D4⋊C8—944no—
193#323—942no—
194C47#839711yescharacteristic, complemented, Sylow
195C22×Q8—21no—
196C22×D4—22no—
197C22×D4C2×Dic4711yescharacteristic
198C22×D4C22⋊D4711yescharacteristic
199C23×C4—21no—
200C23×C4C2×Dic4712yes—
201C2×M16—942no—
202C4⋊Q8—22no—
203C4⋊Q8C22⋊D4714yes—
204#124—216no—
205#121—28no—
206#121—24no—
207#121—24no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4C22⋊D4714yes—
212#117—28no—
213#117—24no—
214#117—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C2×Dic4711yescharacteristic
217C2×C4⋊C4C2×Dic4712yes—
218C2×C4⋊C4C22⋊D4711yescharacteristic
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4C22⋊D4712yes—
223C2×C42—22no—
224C2×C42C22×D4711yescharacteristic
225C4⋊C8—942no—
226C4⋊C8—942no—
227C4⋊C8—942no—
228#100—1882no—
229C22⋊C8—942no—
230C4×C8—942no—
231C24—21no—
232C24C22⋊Dic4712yes—
233C2×Q8—24no—
234C2×Q8—82no—
235C2×D4—28no—
236C2×D4—21no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—42no—
242C2×D4—84no—
243C2×D4C22⋊Dic4712yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C22⋊Dic4712yes—
254C22×C4C22×Dic4711yescharacteristic
255C22×C4C2×C22⋊D4711yescharacteristic
256C22×C4C2×C22⋊D4711yescharacteristic
257M16—1882no—
258M16—1882no—
259C2×C8—1882no—
260C2×C8—1882no—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4C22⋊Dic4712yes—
266C4⋊C4C22⋊Dic4714yes—
267C22⋊C4—28no—
268C22⋊C4—28no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C42—28no—
272C42C2×C22⋊D4712yes—
273C42C2×C22⋊D4712yes—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—41no—
283C23—41no—
284C23C2×C22⋊Dic4711yescharacteristic
285Q8—84no—
286D4—24no—
287D4—42no—
288D4—42no—
289D4—42no—
290D4—88no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4C2×C22⋊Dic4711yescharacteristic
305C2×C4C2×C22⋊Dic4711yescharacteristic
306C2×C4#1309399312yes—
307C2×C4#1309399312yes—
308C2×C4#1309404412yes—
309C2×C4#1309404911yescharacteristic
310C2×C4#1309405411yescharacteristic
311C8—3762no—
312C8—3762no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22#1568897911yescharacteristic
326C22#1568911911yescharacteristic
327C22#1568912411yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—42no—
332C4—44no—
333C4#1568910712yes—
334C4#1568926812yes—
335C4#1568928512yes—
336C2—21no—
337C2—21no—
338C2—82no—
339C2—82no—
340C2#2024331611yescharacteristic
341C2#2024332811yescharacteristic
342C2#2024335811yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC2×C47×#555C2
socleC22×C47#379
cosocleC2×C42×C47C23
hypercenterC2×#555D47
hypocenterC47#8397
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #20243380C2semidirect
2. #20242073C2semidirect
3. #20241813C2semidirect
4. #20241709C2semidirect
194. C47#8397semidirect

Maximal subgroups

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

subgroupindex
#202433802
#202420732
#202418132
#202417092
C2×C47×#5552
#839747