Finite Group Databasebeta

← #26127202

Subgroups — #26127202

Order 12032.

438 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 83 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#20248868C211yesmaximal, characteristic
2#20243468C211yesmaximal, characteristic, complemented
3#20241416C212yesmaximal, complemented
4#20241392C212yesmaximal, complemented
5C2×C47×#324C211yesmaximal, characteristic, complemented
6C2×#13093998C411yescharacteristic
7#15689087—21no—
8#15689062C411yescharacteristic
9#15689062C2211yescharacteristic
10#15689040C411yescharacteristic
11#15689040C2211yescharacteristic
12#15689013—21no—
13#15689013—21no—
14#15689002—21no—
15#15689002—21no—
16C4×C22⋊Dic47C411yescharacteristic
17#15688892—21no—
18#15688892C2212yes—
19C22×C4⋊C4×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C47×#324—21no—
22C47×#324C2212yes—
23C22×D188C2×C411yescharacteristic
24C22×C22⋊D47—21no—
25C2×#10625329—22no—
26#13093998—41no—
27#13093998—41no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C22⋊Dic47—21no—
30C2×(C2×D47)⋊C4—21no—
31C2×(C2×D47)⋊C4—21no—
32C2×(C2×D47)⋊C4—21no—
33C2×(C2×D47)⋊C4—21no—
34C2×(C2×D47)⋊C4—21no—
35C2×C4×Dic47—21no—
36C22⋊(C47⋊C8)—41no—
37C22⋊(C47⋊C8)—41no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C22×C8×C47—41no—
48C2×C4⋊C4×C47—41no—
49C2×C4⋊C4×C47—41no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C4⋊C4×C47C2311yescharacteristic
54C22⋊C8×C47—22no—
55C22⋊C8×C47—41no—
56C24×C47C22⋊C411yescharacteristic
57C23×D47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—22no—
65C22×C4×C47—41no—
66C22×C4×C47—81no—
67C22×C4×C47—81no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C2×C22⋊D47—42no—
75C2×C22⋊D47—42no—
76C2×C22⋊D47—41no—
77C2×C22⋊D47—41no—
78C22×Dic47—21no—
79C22×Dic47—21no—
80C22×Dic47—21no—
81C2×D188—41no—
82C2×D188—81no—
83C2×C8×C47—42no—
84C2×C8×C47—82no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C22⋊Dic47—42no—
90(C2×D47)⋊C4—42no—
91(C2×D47)⋊C4—42no—
92(C2×D47)⋊C4—42no—
93(C2×D47)⋊C4—81no—
94(C2×D47)⋊C4—81no—
95C4⋊Dic47—41no—
96C4×Dic47—81no—
97#10625329—42no—
98#10625329—42no—
99C2×C47⋊C8—42no—
100C23×C47—22no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—41no—
104C23×C47#9911yescharacteristic
105C23×C47#9911yescharacteristic
106C23×C47C2×C22⋊C411yescharacteristic
107C22×D47—41no—
108C22×D47—41no—
109C22×D47—41no—
110C22×D47—41no—
111C22×D47—41no—
112C22×D47—41no—
113C22×D47—41no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—81no—
131C2×C4×C47—81no—
132C2×C4×C47—81no—
133C2×C4×C47#13512yes—
134C22⋊D47—82no—
135C22⋊D47—82no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145D188—82no—
146C8×C47—82no—
147C47⋊C8—82no—
148#11852—471nomaximal, Sylow
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—42no—
157C22×C47—42no—
158C22×C47#35512yes—
159C22×C47C2×#9911yescharacteristic
160C22×C47#41611yescharacteristic
161C22×C47#42111yescharacteristic
162C2×D47—81no—
163C2×D47—81no—
164C2×D47—81no—
165C2×D47—81no—
166C2×D47—81no—
167C2×D47—81no—
168C2×D47—81no—
169C4×C47—22no—
170C4×C47—41no—
171C4×C47—41no—
172C4×C47—41no—
173C4×C47—41no—
174C4×C47—41no—
175Dic47—81no—
176Dic47—81no—
177Dic47—81no—
178#2591—471no—
179#1821—471no—
180C2×#324D4711yescharacteristic, complemented
181#1353—472no—
182#1345—472no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—22no—
186C2×C47—41no—
187C2×C47#151611yescharacteristic
188C2×C47#212111yescharacteristic
189C2×C47C2×#35511yescharacteristic
190D47—161no—
191C2×#121—471no—
192C22×C4⋊C4C2×D4711yescharacteristic
193#407—941no—
194C2×C22⋊C8—21no—
195#400—471no—
196#400—471no—
197#394—471no—
198#394—471no—
199#388—941no—
200#388—941no—
201#386—941no—
202#386—941no—
203C4×C22⋊C4—471no—
204#343—472no—
205#343—941no—
206#324—21no—
207#324C2×D4712yes—
208C47#1185211yescharacteristic, complemented, Sylow
209C22×D4—471no—
210C22×D4—941no—
211C23×C4C4×D4711yescharacteristic
212C23×C4D18811yescharacteristic
213C23×C4C22⋊D4711yescharacteristic
214C2×M16—942no—
215C22×C8—41no—
216#121—1881no—
217#121—1881no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—471no—
221C2×C4⋊C4C4×D4711yescharacteristic
222C2×C4⋊C4D18811yescharacteristic
223C2×C4⋊C4C22⋊D4711yescharacteristic
224C2×C4⋊C4C22×D4711yescharacteristic
225C2×C22⋊C4—941no—
226C2×C22⋊C4—941no—
227C2×C22⋊C4—941no—
228C2×C22⋊C4—941no—
229C2×C22⋊C4—941no—
230C2×C22⋊C4—941no—
231C2×C42—941no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234C22⋊C8—1881no—
235C22⋊C8—1881no—
236#95—471no—
237#95—471no—
238#95—941no—
239#95—941no—
240#95—941no—
241#95—941no—
242C24—941no—
243C24(C2×D47)⋊C411yescharacteristic
244C2×D4—3761no—
245C2×D4—1881no—
246C2×D4—1882no—
247C2×D4—1882no—
248C2×D4—1881no—
249C2×D4—1881no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—941no—
261C22×C4—941no—
262C22×C4—941no—
263C22×C4(C2×D47)⋊C411yescharacteristic
264C22×C4(C2×D47)⋊C411yescharacteristic
265C22×C4(C2×D47)⋊C411yescharacteristic
266C22×C4C2×C4×D4711yescharacteristic
267C22×C4C2×D18811yescharacteristic
268C22×C4C2×C22⋊D4711yescharacteristic
269M16—1882no—
270M16—1882no—
271C2×C8—42no—
272C2×C8—82no—
273C2×C8—1882no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—1881no—
279C22⋊C4—3761no—
280C22⋊C4—3761no—
281C22⋊C4—1882no—
282C22⋊C4—1882no—
283C22⋊C4—1882no—
284C22⋊C4—1882no—
285C42—3761no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—41no—
290C23—1881no—
291C23—1881no—
292C23—1881no—
293C23—1881no—
294C23—1881no—
295C23—1881no—
296C23—1881no—
297C23#1309393811yescharacteristic
298C23#1309393811yescharacteristic
299C23C2×(C2×D47)⋊C411yescharacteristic
300D4—3762no—
301D4—3762no—
302D4—3762no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—1881no—
323C2×C4—1881no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4—1881no—
327C2×C4—1881no—
328C2×C4—1881no—
329C2×C4—1881no—
330C2×C4—1881no—
331C2×C4#1309404612yes—
332C8—82no—
333C8—3762no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—3761no—
344C22—3761no—
345C22—3761no—
346C22—3761no—
347C22—3761no—
348C22—3761no—
349C22—3761no—
350C22#1568891912yes—
351C22C2×#1309393811yescharacteristic
352C22#1568912011yescharacteristic
353C22#1568913911yescharacteristic
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3761no—
361C4—3761no—
362C4—3761no—
363C2—21no—
364C2—21no—
365C2—22no—
366C2—41no—
367C2—7521no—
368C2#2024194511yescharacteristic
369C2#2024503211yescharacteristic
370C2C2×#1568891911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC2×C47×#324C2
socleC22×C47C2×#99
prime radicalC2×#13093998C4
cosocleC2×C4⋊C4×C47C23
hypercenterC2×#324D47
hypocenterC47#11852
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #20243468C2semidirect
3. #20241416C2semidirect
4. #20241392C2semidirect
5. C2×C47×#324C2semidirect
180. C2×#324D47semidirect
208. C47#11852semidirect

Maximal subgroups

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

subgroupindex
#202488682
#202434682
#202414162
#202413922
C2×C47×#3242
#1185247