Finite Group Databasebeta

← #26130142

Subgroups — #26130142

Order 12032.

344 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 75 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#20245611C211yesmaximal, characteristic, complemented
2#20245525C211yesmaximal, characteristic, complemented
3#20242782C211yesmaximal, characteristic, complemented
4#20241434C211yesmaximal, characteristic, complemented
5Q8⋊(C47⋊C16)C211yesmaximal, characteristic, complemented
6#20241349C211yesmaximal, characteristic, complemented
7C47×#3391C211yesmaximal, characteristic
8#15689528C2211yescharacteristic
9#15689333—21no—
10#15688948—21no—
11C4⋊(C47⋊C16)C2211yescharacteristic
12#15688924—21no—
13#15688924—22no—
14C4×C47⋊C16C2211yescharacteristic
15#15688876C2211yescharacteristic
16#15688868—21no—
17C47×#550—21no—
18C47×#493—21no—
19C47×#493C2211yescharacteristic
20C47×#459—22no—
21C47×#459—21no—
22C8×Q8×C47C2211yescharacteristic
23C4×D8×C47—21no—
24C4×D8×C47C2211yescharacteristic
25C2×#10625337—41no—
26#13094024—21no—
27C2×C47⋊C16—21no—
28C2×C47⋊C16—21no—
29#13093958—21no—
30#13093958—21no—
31C4⋊(C47⋊C8)—21no—
32#13093947—41no—
33C2×#41×C47—41no—
34C2×D8×C47—22no—
35C2×D8×C47—21no—
36C2×D8×C47D411yescharacteristic
37C2×D8×C47D412yes—
38#127×C47—22no—
39#127×C47—21no—
40#121×C47—42no—
41#121×C47—41no—
42C4×Q8×C47D411yescharacteristic
43C4×D4×C47—22no—
44C4×D4×C47—21no—
45#107×C47D411yescharacteristic
46C4⋊C8×C47—21no—
47C4⋊C8×C47D411yescharacteristic
48#102×C47—22no—
49#102×C47—21no—
50C4×C8×C47—21no—
51C4×C8×C47C2311yescharacteristic
52#41×C47—82no—
53#41×C47—82no—
54C2×Q8×C47D811yescharacteristic
55C2×D4×C47—21no—
56C2×D4×C47—22no—
57C2×D4×C47—42no—
58C2×D4×C47—42no—
59C2×D4×C47—41no—
60C2×D4×C47—41no—
61C22×C4×C47—42no—
62C22×C4×C47—41no—
63C2×Dic94—41no—
64D8×C47—21no—
65D8×C47—21no—
66D8×C47—42no—
67D8×C47—41no—
68D8×C47—41no—
69D8×C47—82no—
70C2×C8×C47—21no—
71C2×C8×C47C2×D411yescharacteristic
72C2×C8×C47C2×D411yescharacteristic
73C4⋊C4×C47—21no—
74C4⋊C4×C47—21no—
75C4⋊C4×C47D811yescharacteristic
76C22⋊C4×C47—42no—
77C22⋊C4×C47—41no—
78C42×C47—21no—
79C42×C47C2×D411yescharacteristic
80#10625337—82no—
81C4⋊Dic47—41no—
82C4⋊Dic47—41no—
83C2×C47⋊C8—41no—
84C47⋊C16—41no—
85C47⋊C16—41no—
86C23×C47—42no—
87C23×C47—41no—
88Q8×C47—21no—
89Q8×C47D1612yes—
90D4×C47—41no—
91D4×C47—42no—
92D4×C47—42no—
93D4×C47—41no—
94D4×C47—84no—
95D4×C47—81no—
96D4×C47—81no—
97D4×C47—161no—
98D4×C47—162no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—41no—
102C2×C4×C47—84no—
103C2×C4×C47—81no—
104C2×C4×C47—81no—
105C2×C4×C47#12011yescharacteristic
106C2×C4×C47C2×D811yescharacteristic
107C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
108C2×Dic47—81no—
109Dic94—81no—
110Dic94—81no—
111C8×C47—21no—
112C8×C47—41no—
113C8×C47#13511yescharacteristic
114C8×C47#13711yescharacteristic
115C47⋊C8—81no—
116#12380—471nomaximal, Sylow
117C22×C47—81no—
118C22×C47—82no—
119C22×C47—82no—
120C22×C47—81no—
121C22×C47—82no—
122C22×C47—81no—
123C22×C47#44711yescharacteristic
124C4×C47—22no—
125C4×C47—21no—
126C4×C47—41no—
127C4×C47—81no—
128C4×C47#45211yescharacteristic
129C4×C47#45511yescharacteristic
130C4×C47C2×D1611yescharacteristic
131C4×C47#50911yescharacteristic
132Dic47—161no—
133#3391D4711yescharacteristic
134#2240—471no—
135#2224—471no—
136#1674—471no—
137#1361—471no—
138Q8⋊C16—471no—
139#1331—471no—
140C2×C47—161no—
141C2×C47—162no—
142C2×C47#164111yescharacteristic
143C2×C47#218411yescharacteristic
144C2×C47#219411yescharacteristic
145#550—21no—
146#500—471no—
147#493—21no—
148#493C2×D4711yescharacteristic
149#462—941no—
150#459—22no—
151#459—21no—
152C8×Q8C2×D4711yescharacteristic
153C4×D8—21no—
154C4×D8C2×D4711yescharacteristic
155#366—941no—
156C4⋊C16—471no—
157#357—941no—
158#357—942no—
159C4×C16—471no—
160#335—471no—
161#332—941no—
162C47#1238011yescharacteristic, complemented, Sylow
163C2×#41—41no—
164C2×Q16—1881no—
165C2×D8—22no—
166C2×D8—21no—
167C2×D8C22⋊D4711yescharacteristic
168C2×D8C22⋊D4712yes—
169C4⋊Q8—941no—
170#127—22no—
171#127—21no—
172#121—42no—
173#121—41no—
174C4×Q8C22⋊D4711yescharacteristic
175C4×D4—22no—
176C4×D4—21no—
177C2×C16—941no—
178C2×C16—941no—
179#107—941no—
180#107—941no—
181#107C22⋊D4711yescharacteristic
182C4⋊C8—21no—
183C4⋊C8—941no—
184C4⋊C8C22⋊D4711yescharacteristic
185#103—1881no—
186#102—22no—
187#102—21no—
188C4×C8—21no—
189C4×C8C22×D4711yescharacteristic
190#41—82no—
191#41—82no—
192C2×Q8—1881no—
193C2×Q8#1062533411yescharacteristic
194C2×D4—21no—
195C2×D4—22no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—41no—
199C2×D4—41no—
200C22×C4—42no—
201C22×C4—41no—
202Q16—3762no—
203D8—21no—
204D8—21no—
205D8—42no—
206D8—41no—
207D8—41no—
208D8—82no—
209C2×C8—21no—
210C2×C8—1881no—
211C2×C8C2×C22⋊D4711yescharacteristic
212C2×C8C2×C22⋊D4711yescharacteristic
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—1881no—
216C4⋊C4—1881no—
217C4⋊C4#1062533411yescharacteristic
218C22⋊C4—42no—
219C22⋊C4—41no—
220C42—21no—
221C42C2×C22⋊D4711yescharacteristic
222C16—1881no—
223C16—1881no—
224C23—42no—
225C23—41no—
226Q8—21no—
227Q8—3761no—
228Q8—3761no—
229Q8#1309396512yes—
230D4—41no—
231D4—42no—
232D4—42no—
233D4—41no—
234D4—84no—
235D4—81no—
236D4—81no—
237D4—161no—
238D4—162no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—84no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—3761no—
246C2×C4#1309399311yescharacteristic
247C2×C4C2×#1062533411yescharacteristic
248C2×C4#1309405011yescharacteristic
249C8—21no—
250C8—41no—
251C8—3761no—
252C8#1309404411yescharacteristic
253C8#1309405411yescharacteristic
254C22—82no—
255C22—81no—
256C22—82no—
257C22—81no—
258C22—81no—
259C22—82no—
260C22#1568925711yescharacteristic
261C4—22no—
262C4—21no—
263C4—41no—
264C4—81no—
265C4—7521no—
266C4#1568928511yescharacteristic
267C4#1568930111yescharacteristic
268C4C2×#1309396511yescharacteristic
269C4#1568955911yescharacteristic
270C2—162no—
271C2—161no—
272C2#2024258111yescharacteristic
273C2#2024530011yescharacteristic
274C2#2024535911yescharacteristic
subgroupG/H
centerC22#15689257
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
Fitting, prime radicalC47×#3391C2
socleC22×C47#447
hypercenter#3391D47
hypocenterC47#12380
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3391

Complements

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

NH ≅ G/Ntype
1. #20245611C2semidirect
2. #20245525C2semidirect
3. #20242782C2semidirect
4. #20241434C2semidirect
5. Q8⋊(C47⋊C16)C2semidirect
6. #20241349C2semidirect
162. C47#12380semidirect

Maximal subgroups

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

subgroupindex
#202456112
#202455252
#202427822
#202414342
Q8⋊(C47⋊C16)2
#202413492
C47×#33912
#1238047