Finite Group Databasebeta

← #26135392

Subgroups — #26135392

Order 12032.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#15689525C211yesmaximal, characteristic, complemented
2C2×#15688934C212yesmaximal
3#20244276C211yesmaximal, characteristic
4#20241487C212yesmaximal, complemented
5C47×#1376C211yesmaximal, characteristic, complemented
6C2×#13094046C2211yescharacteristic
7C2×#13094016—21no—
8#15689525—24no—
9C2×#13093960C2211yescharacteristic
10C2×#13093958—21no—
11C2×#13093956C2212yes—
12C2×#13093952—21no—
13#15689089—21no—
14#15688934—24no—
15C2×M32×C47C2212yes—
16C2×C4×C8×C47C2211yescharacteristic
17C2×#10625350—21no—
18#13094046—22no—
19#13094046—22no—
20C2×Dic188Q811yescharacteristic
21C2×#10625325—21no—
22C2×D376Q811yescharacteristic
23C2×#10625329—21no—
24#13094016—24no—
25#13094016—42no—
26C2×C4⋊Dic47—22no—
27C2×Dic47⋊C4—42no—
28#13093960—22no—
29#13093958—24no—
30#13093956—22no—
31#13093956D414yes—
32#13093952—42no—
33(C2×D47)⋊C8—42no—
34C22×C8×C47D411yescharacteristic
35C22×C8×C47C2311yescharacteristic
36C2×C42×C47D411yescharacteristic
37M32×C47—22no—
38M32×C47—22no—
39C2×C16×C47—22no—
40C4×C8×C47—22no—
41C22×C4×C47—21no—
42C22×C4×C47C2×D411yescharacteristic
43C2×C22⋊D47—41no—
44C22×Dic47—42no—
45#10625350—41no—
46#10625350—41no—
47#10625350—41no—
48#10625350—41no—
49C2×D188—21no—
50C2×C4×D47—41no—
51C2×Dic94—21no—
52C2×C8×C47—21no—
53C2×C8×C47—21no—
54C2×C8×C47SD1612yes—
55C2×C8×C47C2×D412yes—
56C2×C8×C47C2×Q811yescharacteristic
57C2×C8×C47#4112yes—
58C2×C8×C47#4111yescharacteristic
59C42×C47—21no—
60C42×C47D812yes—
61C4⋊Dic47—24no—
62C4⋊Dic47—42no—
63Dic47⋊C4—48no—
64#10625329—41no—
65#10625329—41no—
66C2×C47⋊C8—41no—
67Dic188—22no—
68D376—22no—
69#10625325—42no—
70C16×C47—42no—
71C23×C47AGL(1,ℤ/8)11yescharacteristic
72C22×D47—41no—
73C2×C4×C47—22no—
74C2×C4×C47—42no—
75C2×C4×C47#12011yescharacteristic
76C2×C4×C47C22⋊Q811yescharacteristic
77C2×C4×C47#12411yescharacteristic
78C2×C4×C47C2×D811yescharacteristic
79C2×C4×C47C2×SD1611yescharacteristic
80C2×C4×C47#13711yescharacteristic
81C22⋊D47—81no—
82C22⋊D47—81no—
83C2×Dic47—44no—
84C2×Dic47—41no—
85C2×Dic47—84no—
86D188—41no—
87D188—41no—
88C4×D47—81no—
89C4×D47—81no—
90Dic94—41no—
91Dic94—41no—
92C8×C47—22no—
93C8×C47C22⋊Q812yes—
94C8×C47#12412yes—
95C47⋊C8—81no—
96#13372—471nomaximal, Sylow
97C22×C47—22no—
98C22×C47#44711yescharacteristic
99C22×C47#47611yescharacteristic
100C22×C47#48911yescharacteristic
101C2×D47—82no—
102C2×D47—81no—
103C4×C47—42no—
104C4×C47#39311yescharacteristic
105C4×C47#44811yescharacteristic
106C4×C47#47411yescharacteristic
107C4×C47#48611yescharacteristic
108Dic47—84no—
109Dic47—81no—
110C2×#499—471no—
111C2×#360—472no—
112#1974—471no—
113#1383—472no—
114#1376D4711yescharacteristic, complemented
115C2×C47—21no—
116C2×C47—21no—
117C2×C47#202311yescharacteristic
118C2×C47#223112yes—
119D47—161no—
120C2×#135—471no—
121C2×#125—941no—
122C2×M32C2×D4712yes—
123#499—944no—
124C2×#108—471no—
125C2×#107—941no—
126C2×#106—472no—
127C2×#104—941no—
128#408—941no—
129C2×C4×C8C2×D4711yescharacteristic
130#360—944no—
131C47#1337211yescharacteristic, complemented, Sylow
132C2×#41—941no—
133#135—942no—
134#135—942no—
135C2×Q16—471no—
136C2×SD16—941no—
137C2×D8—471no—
138C2×M16—941no—
139C22×C8D18811yescharacteristic
140C22×C8C22×D4711yescharacteristic
141#125—944no—
142#125—1882no—
143C2×C4⋊C4—942no—
144C2×C4⋊C4—1882no—
145C2×C42D18811yescharacteristic
146M32—22no—
147M32—22no—
148C2×C16—22no—
149#108—942no—
150#107—944no—
151#106—474no—
152#106—942no—
153#104—1882no—
154C22⋊C8—1882no—
155C4×C8—22no—
156#41—1881no—
157#41—1881no—
158#41—1881no—
159#41—1881no—
160C2×Q8—941no—
161C2×D4—941no—
162C2×D4—1881no—
163C22×C4—21no—
164C22×C4—1882no—
165C22×C4—1881no—
166C22×C4C2×D18811yescharacteristic
167Q16—942no—
168SD16—1882no—
169D8—942no—
170M16—1881no—
171M16—1881no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—1881no—
175C2×C8#1062532512yes—
176C2×C8#1062535012yes—
177C2×C8D4×D4712yes—
178C2×C8#1062535211yescharacteristic
179C2×C8Q8×D4711yescharacteristic
180C4⋊C4—944no—
181C4⋊C4—1888no—
182C4⋊C4—1882no—
183C42—21no—
184C42D37612yes—
185C16—42no—
186C23—1881no—
187C23#1309405211yescharacteristic
188Q8—1881no—
189Q8—1881no—
190D4—3761no—
191D4—3761no—
192D4—1881no—
193D4—1881no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—3764no—
197C2×C4—3761no—
198C2×C4—3761no—
199C2×C4—1884no—
200C2×C4—1881no—
201C2×C4#1309399411yescharacteristic
202C2×C4#1309400411yescharacteristic
203C2×C4#1309401211yescharacteristic
204C2×C4C2×D37611yescharacteristic
205C2×C4C2×#1062532511yescharacteristic
206C2×C4#1309405711yescharacteristic
207C8—22no—
208C8—3761no—
209C8#1309400212yes—
210C8#1309401012yes—
211C22—22no—
212C22—3761no—
213C22—3762no—
214C22#1568925911yescharacteristic
215C22#1568941411yescharacteristic
216C22#1568948311yescharacteristic
217C4—42no—
218C4—3764no—
219C4—3761no—
220C4#1568903611yescharacteristic
221C4#1568926411yescharacteristic
222C4#1568940411yescharacteristic
223C4#1568947011yescharacteristic
224C2—21no—
225C2—21no—
226C2—7521no—
227C2#2024452511yescharacteristic
228C2#2024556512yes—
subgroupG/H
centerC22#15689414
commutator, cosocleC22×C8×C47C23
FrattiniC22×C8C22×D47
FittingC47×#1376C2
socleC22×C47#476
prime radicalC2×#13094046C22
hypercenter#1376D47
hypocenterC47#13372
LCSG ▷ C22×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1376

Complements

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

NH ≅ G/Ntype
1. C2×#15689525C2semidirect
4. #20241487C2semidirect
5. C47×#1376C2semidirect
114. #1376D47semidirect
131. C47#13372semidirect

Maximal subgroups

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

subgroupindex
C2×#156895252
C2×#156889342
#202442762
#202414872
C47×#13762
#1337247