Finite Group Databasebeta

← #26119876

Subgroups — #26119876

Order 12032.

440 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 101 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
2#20242158C211yesmaximal, characteristic, complemented
3#20242060C211yesmaximal, characteristic, complemented
4#20242060C211yesmaximal, characteristic, complemented
5#20241779C211yesmaximal, characteristic
6#20241285C211yesmaximal, characteristic, complemented
7C2×C47×#461C211yesmaximal, characteristic
8#15689176C2211yescharacteristic
9#15689176C2211yescharacteristic
10#15689147C2211yescharacteristic
11#15689147C2211yescharacteristic
12C2×#13093941—21no—
13#15688892—21no—
14#15688873—22no—
15#15688869—22no—
16#15688869—22no—
17D4⋊(C47⋊C8)—22no—
18C22×SD16×C47—21no—
19C2×C4⋊Q8×C47C2211yescharacteristic
20C2×C4×D4×C47C2211yescharacteristic
21C47×#461—22no—
22C47×#461—22no—
23C2×C4⋊C8×C47C2211yescharacteristic
24C2×#103×C47—21no—
25C2×#10625329—21no—
26C2×#10625329—21no—
27C4⋊(C47⋊C8)—21no—
28C4⋊(C47⋊C8)—21no—
29C4⋊(C47⋊C8)—21no—
30C4⋊(C47⋊C8)—21no—
31C4⋊(C47⋊C8)—21no—
32C4⋊(C47⋊C8)D412yes—
33#13093941—42no—
34C22⋊(C47⋊C8)—21no—
35C22⋊(C47⋊C8)—21no—
36C4×C47⋊C8—21no—
37C4×C47⋊C8—21no—
38C22×Q8×C47—21no—
39C22×D4×C47D411yescharacteristic
40C23×C4×C47—21no—
41C2×SD16×C47—24no—
42C2×SD16×C47—42no—
43C2×SD16×C47—42no—
44C22×C8×C47—21no—
45C4⋊Q8×C47—21no—
46C4⋊Q8×C47—21no—
47C4⋊Q8×C47—21no—
48C4⋊Q8×C47D412yes—
49C4×D4×C47—21no—
50C4×D4×C47—21no—
51C4×D4×C47—21no—
52C4×D4×C47—21no—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47D411yescharacteristic
56C2×C22⋊C4×C47—21no—
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—21no—
59C4⋊C8×C47D412yes—
60#103×C47—24no—
61C24×C47—21no—
62C2×Q8×C47—22no—
63C2×Q8×C47—41no—
64C2×Q8×C47—41no—
65C2×Q8×C47—41no—
66C2×Q8×C47—41no—
67C2×Q8×C47—41no—
68C2×D4×C47—21no—
69C2×D4×C47—21no—
70C2×D4×C47—21no—
71C2×D4×C47—21no—
72C2×D4×C47—21no—
73C2×D4×C47SD1612yes—
74C22×C4×C47—21no—
75C22×C4×C47—41no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47—82no—
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×C4×C47#4111yescharacteristic
82SD16×C47—48no—
83C2×C8×C47—22no—
84C2×C8×C47—41no—
85C2×C8×C47—41no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—22no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—22no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C22⋊C4×C47—41no—
93C22⋊C4×C47—41no—
94C42×C47C2×D411yescharacteristic
95C42×C47C2×D411yescharacteristic
96C42×C47C2×D411yescharacteristic
97C42×C47#4111yescharacteristic
98#10625329—41no—
99#10625329—41no—
100#10625329—41no—
101#10625329—41no—
102C2×C47⋊C8—22no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C23×C47—22no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47—81no—
110C23×C47—81no—
111C23×C47#12311yescharacteristic
112Q8×C47—42no—
113Q8×C47—42no—
114Q8×C47—81no—
115Q8×C47—81no—
116D4×C47—22no—
117D4×C47—22no—
118D4×C47—41no—
119D4×C47—41no—
120C2×C4×C47—22no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—82no—
125C2×C4×C47—81no—
126C2×C4×C47—81no—
127C2×C4×C47—82no—
128C2×C4×C47—81no—
129C2×C4×C47#12011yescharacteristic
130C2×C4×C47#12111yescharacteristic
131C2×C4×C47#12111yescharacteristic
132C2×C4×C47#12111yescharacteristic
133C2×C4×C47#12311yescharacteristic
134C2×C4×C47#12411yescharacteristic
135C2×C4×C47C2×SD1611yescharacteristic
136C2×C4×C47#13511yescharacteristic
137C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
139C8×C47—42no—
140C47⋊C8—42no—
141C47⋊C8—81no—
142C47⋊C8—81no—
143C47⋊C8—81no—
144#10445—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—21no—
147C22×C47—42no—
148C22×C47—82no—
149C22×C47—82no—
150C22×C47—81no—
151C22×C47—81no—
152C22×C47#39411yescharacteristic
153C22×C47#48111yescharacteristic
154C22×C47#48211yescharacteristic
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—82no—
159C4×C47—81no—
160C4×C47#44911yescharacteristic
161C4×C47#45011yescharacteristic
162C4×C47#46511yescharacteristic
163C4×C47#46911yescharacteristic
164C4×C47#47112yes—
165C2×#461D4711yescharacteristic
166#1800—471no—
167#1570—471no—
168#1546—471no—
169#1546—471no—
170#1478—471no—
171#1301—471no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47—82no—
175C2×C47#203411yescharacteristic
176C2×C47#204611yescharacteristic
177C2×C47#204911yescharacteristic
178C22×SD16—21no—
179C2×C4⋊Q8C2×D4711yescharacteristic
180C2×C4×D4C2×D4711yescharacteristic
181#461—22no—
182#461—22no—
183#431—471no—
184#431—471no—
185#423—471no—
186#423—471no—
187C2×C4⋊C8C2×D4711yescharacteristic
188C2×#103—21no—
189C2×#100—941no—
190#343—941no—
191#334—942no—
192#332—942no—
193#332—942no—
194D4⋊C8—942no—
195C47#1044511yescharacteristic, complemented, Sylow
196C22×Q8—21no—
197C22×D4C22⋊D4711yescharacteristic
198C23×C4—21no—
199C2×SD16—24no—
200C2×SD16—42no—
201C2×SD16—42no—
202C2×M16—941no—
203C2×M16—941no—
204C22×C8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8C22⋊D4712yes—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C22⋊D4711yescharacteristic
216C2×C22⋊C4—21no—
217C2×C42C22×D4711yescharacteristic
218C4⋊C8—21no—
219C4⋊C8—472no—
220C4⋊C8—941no—
221C4⋊C8—941no—
222C4⋊C8—941no—
223C4⋊C8—941no—
224C4⋊C8—941no—
225C4⋊C8C22⋊D4712yes—
226#103—24no—
227#100—1882no—
228C22⋊C8—941no—
229C22⋊C8—941no—
230C4×C8—941no—
231C4×C8—941no—
232C24—21no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4#1062533512yes—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—82no—
250C22×C4#1062535211yescharacteristic
251C22×C4#1062535211yescharacteristic
252C22×C4C2×C22⋊D4711yescharacteristic
253SD16—48no—
254M16—1881no—
255M16—1881no—
256M16—1881no—
257M16—1881no—
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—942no—
262C2×C8—1881no—
263C2×C8—1881no—
264C2×C8—1881no—
265C4⋊C4—21no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C42D4×D4711yescharacteristic
274C42#1062535211yescharacteristic
275C42C2×C22⋊D4711yescharacteristic
276C42C2×C22⋊D4711yescharacteristic
277C23—22no—
278C23—41no—
279C23—41no—
280C23—81no—
281C23—81no—
282C23#1309400511yescharacteristic
283Q8—42no—
284Q8—42no—
285Q8—81no—
286Q8—81no—
287D4—22no—
288D4—22no—
289D4—41no—
290D4—41no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4#1309399311yescharacteristic
301C2×C4#1309399511yescharacteristic
302C2×C4#1309399611yescharacteristic
303C2×C4#1309399611yescharacteristic
304C2×C4#1309400511yescharacteristic
305C2×C4#1309401111yescharacteristic
306C2×C4C2×#1062533511yescharacteristic
307C2×C4#1309404411yescharacteristic
308C2×C4#1309404911yescharacteristic
309C2×C4#1309405011yescharacteristic
310C8—42no—
311C8—3761no—
312C8—3761no—
313C8—3761no—
314C8—1882no—
315C22—21no—
316C22—21no—
317C22—42no—
318C22—82no—
319C22—82no—
320C22—81no—
321C22—81no—
322C22#1568903711yescharacteristic
323C22#1568943911yescharacteristic
324C22#1568944411yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—82no—
329C4—81no—
330C4#1568926811yescharacteristic
331C4#1568927411yescharacteristic
332C4#1568935011yescharacteristic
333C4#1568937411yescharacteristic
334C4#1568938312yes—
335C2—21no—
336C2—21no—
337C2—82no—
338C2#2024457411yescharacteristic
339C2#2024464611yescharacteristic
340C2#2024466511yescharacteristic
subgroupG/H
centerC22#15689037
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC2×C47×#461C2
socleC22×C47#394
prime radicalC22×D4×C47D4
hypercenterC2×#461D47
hypocenterC47#10445
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
2. #20242158C2semidirect
3. #20242060C2semidirect
4. #20242060C2semidirect
6. #20241285C2semidirect
195. C47#10445semidirect

Maximal subgroups

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

subgroupindex
#202433802
#202421582
#202420602
#202417792
#202412852
C2×C47×#4612
#1044547