Finite Group Databasebeta

← #26121248

Subgroups — #26121248

Order 12032.

356 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 93 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#20244186C211yesmaximal, characteristic
2#20243529C211yesmaximal, characteristic, complemented
3#20242163C211yesmaximal, characteristic, complemented
4#20242159C211yesmaximal, characteristic, complemented
5#20241814C211yesmaximal, characteristic
6#20241814C211yesmaximal, characteristic
7C47×#1275C211yesmaximal, characteristic, complemented
8C2×C4×D188C2211yescharacteristic
9#15689147C2211yescharacteristic
10#15689147C2211yescharacteristic
11C2×C4⋊(C47⋊C8)C2211yescharacteristic
12C2×(C2×D47)⋊C8—22no—
13#15688888—22no—
14#15688876—22no—
15#15688874—22no—
16#15688852—22no—
17#15688852—22no—
18C47×#431C2211yescharacteristic
19C47×#431C2211yescharacteristic
20C2×#97×C47C2211yescharacteristic
21C22×D188Q811yescharacteristic
22C22×C4×D47—21no—
23C2×#10625329—21no—
24C2×#10625329—21no—
25C22×C47⋊C8—21no—
26C4×D188—21no—
27C4×D188—21no—
28C4×D188—21no—
29C4×D188—21no—
30C2×C4⋊Dic47Q811yescharacteristic
31C2×(C2×D47)⋊C4—21no—
32C4⋊(C47⋊C8)—21no—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)—21no—
35C4⋊(C47⋊C8)—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38(C2×D47)⋊C8—44no—
39C22×C8×C47D412yes—
40C2×C42×C47C2311yescharacteristic
41C4⋊C8×C47—21no—
42C4⋊C8×C47—21no—
43C22⋊C8×C47—21no—
44C22⋊C8×C47—21no—
45#97×C47—21no—
46#97×C47—21no—
47C4×C8×C47D412yes—
48C4×C8×C47D412yes—
49C23×D47—21no—
50C22×C4×C47C2×D411yescharacteristic
51C22×C4×C47C2×Q811yescharacteristic
52C22×C4×C47#4111yescharacteristic
53C22×Dic47—21no—
54C2×D188—21no—
55C2×D188—21no—
56C2×D188—42no—
57C2×C4×D47—41no—
58C2×C4×D47—41no—
59C2×C4×D47—42no—
60C2×C4×D47—42no—
61C2×C8×C47—22no—
62C2×C8×C47—22no—
63C2×C8×C47—22no—
64C2×C8×C47—22no—
65C2×C8×C47—22no—
66C42×C47C2×D411yescharacteristic
67C42×C47C2×D411yescharacteristic
68C42×C47#4111yescharacteristic
69C42×C47#4111yescharacteristic
70(C2×D47)⋊C4—41no—
71(C2×D47)⋊C4—41no—
72C4⋊Dic47—21no—
73C4⋊Dic47—21no—
74#10625329—42no—
75#10625329—42no—
76C2×C47⋊C8—41no—
77C2×C47⋊C8—41no—
78C2×C47⋊C8—41no—
79C2×C47⋊C8—42no—
80C23×C47C22⋊Q811yescharacteristic
81C22×D47—22no—
82C22×D47—41no—
83C22×D47—42no—
84C22×D47—41no—
85C22×D47—42no—
86C2×C4×C47—41no—
87C2×C4×C47—41no—
88C2×C4×C47C22⋊Q811yescharacteristic
89C2×C4×C47C22⋊Q811yescharacteristic
90C2×C4×C47#12311yescharacteristic
91C2×C4×C47#12311yescharacteristic
92C2×C4×C47#12311yescharacteristic
93C2×C4×C47#12711yescharacteristic
94C2×C4×C47#13512yes—
95C2×C4×C47AGL(1,ℤ/8)12yes—
96C2×Dic47—41no—
97C2×Dic47—42no—
98D188—22no—
99D188—22no—
100D188—41no—
101D188—41no—
102C4×D47—82no—
103C4×D47—81no—
104C4×D47—81no—
105C8×C47—42no—
106C8×C47—42no—
107C8×C47—42no—
108C47⋊C8—81no—
109C47⋊C8—81no—
110C47⋊C8—81no—
111#10659—471nomaximal, Sylow
112C22×C47—21no—
113C22×C47—21no—
114C22×C47#39911yescharacteristic
115C22×C47#47812yes—
116C2×D47—44no—
117C2×D47—42no—
118C2×D47—82no—
119C2×D47—81no—
120C2×D47—81no—
121C4×C47—21no—
122C4×C47—21no—
123C4×C47—21no—
124C4×C47—21no—
125C4×C47#48212yes—
126C4×C47#49511yescharacteristic
127C4×C47#49611yescharacteristic
128Dic47—81no—
129#1955—471no—
130#1838—471no—
131#1571—471no—
132#1570—471no—
133#1486—471no—
134#1486—471no—
135#1275D4711yescharacteristic, complemented
136C2×C47—22no—
137C2×C47#209111yescharacteristic
138C2×C47#209211yescharacteristic
139C2×C47#209811yescharacteristic
140D47—82no—
141C2×C4×D4—471no—
142#431C2×D4711yescharacteristic
143#431C2×D4711yescharacteristic
144#423—471no—
145#423—471no—
146C2×C4⋊C8—471no—
147C2×C22⋊C8—942no—
148C2×#97C2×D4711yescharacteristic
149#341—942no—
150#335—942no—
151#334—942no—
152D4⋊C8—942no—
153D4⋊C8—942no—
154C47#1065911yescharacteristic, complemented, Sylow
155C22×D4—471no—
156C23×C4—941no—
157C2×M16—941no—
158C2×M16—941no—
159C22×C8—941no—
160C22×C8D18812yes—
161C4×D4—941no—
162C4×D4—941no—
163C4×D4—941no—
164C4×D4—941no—
165C2×C4⋊C4—471no—
166C2×C22⋊C4—941no—
167C2×C42C22×D4711yescharacteristic
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—941no—
171C4⋊C8—941no—
172C4⋊C8—941no—
173C4⋊C8—941no—
174C4⋊C8—941no—
175C4⋊C8—941no—
176C22⋊C8—21no—
177C22⋊C8—21no—
178C22⋊C8—1884no—
179#97—21no—
180#97—21no—
181C4×C8D18812yes—
182C4×C8D18812yes—
183C24—941no—
184C2×D4—941no—
185C2×D4—941no—
186C2×D4—1882no—
187C22×C4—941no—
188C22×C4—1881no—
189C22×C4—1881no—
190C22×C4—1882no—
191C22×C4—1882no—
192C22×C4C2×D18811yescharacteristic
193C22×C4#1062535211yescharacteristic
194C22×C4Q8×D4711yescharacteristic
195M16—1882no—
196M16—1882no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—1881no—
203C2×C8—1881no—
204C2×C8—1881no—
205C2×C8—1882no—
206C4⋊C4—941no—
207C4⋊C4—941no—
208C22⋊C4—1881no—
209C22⋊C4—1881no—
210C42C2×D18811yescharacteristic
211C42C2×D18811yescharacteristic
212C42#1062535211yescharacteristic
213C42#1062535211yescharacteristic
214C23—942no—
215C23—1881no—
216C23—1882no—
217C23—1881no—
218C23—1882no—
219C23#1309400411yescharacteristic
220D4—942no—
221D4—942no—
222D4—1881no—
223D4—1881no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—3761no—
227C2×C4—3761no—
228C2×C4—3762no—
229C2×C4—1881no—
230C2×C4—1882no—
231C2×C4#1309400411yescharacteristic
232C2×C4#1309400411yescharacteristic
233C2×C4#1309400811yescharacteristic
234C2×C4#1309400811yescharacteristic
235C2×C4#1309400811yescharacteristic
236C2×C4#1309402111yescharacteristic
237C2×C4#1309404612yes—
238C2×C4#1309405212yes—
239C8—42no—
240C8—42no—
241C8—42no—
242C8—3761no—
243C8—3761no—
244C8—3761no—
245C22—21no—
246C22—21no—
247C22—3762no—
248C22—3761no—
249C22—3761no—
250C22—1884no—
251C22—1882no—
252C22#1568906011yescharacteristic
253C22#1568942412yes—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—21no—
258C4—3761no—
259C4#1568944812yes—
260C4#1568950711yescharacteristic
261C4#1568951211yescharacteristic
262C2—22no—
263C2—3762no—
264C2#2024490511yescharacteristic
265C2#2024490811yescharacteristic
266C2#2024494211yescharacteristic
subgroupG/H
centerC2×C4#13094021
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1275C2
socleC22×C47#399
prime radicalC22×D188Q8
hypercenter#1275D47
hypocenterC47#10659
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
2. #20243529C2semidirect
3. #20242163C2semidirect
4. #20242159C2semidirect
7. C47×#1275C2semidirect
135. #1275D47semidirect
154. C47#10659semidirect

Maximal subgroups

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

subgroupindex
#202441862
#202435292
#202421632
#202421592
#202418142
C47×#12752
#1065947