Finite Group Databasebeta

← #35314945

Subgroups — #35314945

Order 21248.

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

normal subgroups only

This group has 316 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#24918113C211yesmaximal, characteristic
2#24909279C211yesmaximal, characteristic, complemented
3#24905453C212yesmaximal, complemented
4#24905447C212yesmaximal, complemented
5C83×#1747C211yesmaximal, characteristic, complemented
6#19128306—21no—
7#19128103C411yescharacteristic, complemented
8#19128078C411yescharacteristic, complemented
9#19128066—21no—
10#19127954C2212yes—
11#19127937—21no—
12#19127814C412yescomplemented
13C4×D664—21no—
14#19127522—21no—
15#19127483C2212yes—
16#19127479—21no—
17#405×C83—21no—
18#405×C83C2211yescharacteristic
19#338×C83C2211yescharacteristic
20H(3,ℤ/4)×C83—21no—
21H(3,ℤ/4)×C83C2211yescharacteristic
22C2×D664—21no—
23C2×D664C2×C412yes—
24#14581789—21no—
25#14581784—21no—
26#14581777—42no—
27#14581766—22no—
28#14581765—42no—
29C4×D332—22no—
30#14581732—22no—
31#14581726—21no—
32#14581724C2×C412yes—
33#14581714—22no—
34#14581714—22no—
35C2×M16×C83—21no—
36C2×M16×C83C2311yescharacteristic
37C22×C8×C83D411yescharacteristic
38#117×C83—21no—
39#117×C83D411yescharacteristic
40#97×C83—22no—
41#97×C83C2×C411yescharacteristic
42#97×C83D411yescharacteristic
43C4×C8×C83—22no—
44C4×C8×C83C2×C411yescharacteristic
45C4×C8×C83D411yescharacteristic
46C2×D332—22no—
47C2×D332—42no—
48C2×C22⋊D83—42no—
49C2×C4×D83—42no—
50D664—42no—
51D664—41no—
52D664—41no—
53C4⋊Dic83—22no—
54Dic83⋊C4—42no—
55(C2×D83)⋊C4—42no—
56(C2×D83)⋊C4—42no—
57C83⋊C16—42no—
58C22×C4×C83C2×D411yescharacteristic
59M16×C83—41no—
60M16×C83—41no—
61M16×C83—41no—
62C2×C8×C83—21no—
63C2×C8×C83—21no—
64C2×C8×C83—22no—
65C2×C8×C83C22⋊C411yescharacteristic
66C2×C8×C83C22⋊C411yescharacteristic
67C2×C8×C83C22×C411yescharacteristic
68C2×C8×C83C2×D411yescharacteristic
69C4⋊C4×C83—21no—
70C4⋊C4×C83—41no—
71C22⋊C4×C83—21no—
72C22⋊C4×C83—41no—
73C42×C83—22no—
74C42×C83C22⋊C411yescharacteristic
75C42×C83C22⋊C411yescharacteristic
76C23×C83AGL(1,ℤ/8)11yescharacteristic
77C22×D83—42no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—42no—
83C2×C4×C83C2×C22⋊C411yescharacteristic
84C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
85C22⋊D83—162no—
86C2×Dic83—42no—
87D332—42no—
88D332—42no—
89D332—162no—
90C4×D83—162no—
91C8×C83—21no—
92C8×C83—21no—
93C8×C83—21no—
94C8×C83—42no—
95C8×C83#10411yescharacteristic
96C8×C83#10411yescharacteristic
97C22×C83—21no—
98C22×C83—41no—
99C22×C83#41811yescharacteristic
100C2×D83—82no—
101C2×D83—82no—
102C2×D83—82no—
103C4×C83—21no—
104C4×C83—41no—
105C4×C83—41no—
106C4×C83—82no—
107C4×C83#41011yescharacteristic
108C4×C83C2×#10411yescharacteristic
109Dic83—82no—
110#11770—831nomaximal, Sylow
111C2×C83—21no—
112C2×C83—41no—
113C2×C83#150511yescharacteristic
114D83—162no—
115#3176—831no—
116#2166—831no—
117#1747D8311yescharacteristic, complemented
118#1341—832no—
119#1339—832no—
120C83#1177011yescharacteristic, complemented, Sylow
121#540—1661no—
122#499—831no—
123#493—831no—
124#490—1661no—
125#469—832no—
126#466—1661no—
127#442—832no—
128C4×D8—1661no—
129#405—21no—
130#405C2×D8311yescharacteristic
131#364—1661no—
132#349—832no—
133#347—1661no—
134#338C2×D8311yescharacteristic
135H(3,ℤ/4)—21no—
136H(3,ℤ/4)C2×D8311yescharacteristic
137C2×D8—832no—
138C2×D8—1661no—
139C2×M16—21no—
140C2×M16C22×D8311yescharacteristic
141C22×C8C22⋊D8311yescharacteristic
142#127—1661no—
143#125—1661no—
144#123—3322no—
145#121—3322no—
146#121—1662no—
147C4×D4—1662no—
148#117—21no—
149#117C22⋊D8311yescharacteristic
150M32—1662no—
151#107—1661no—
152#106—832no—
153#102—1662no—
154#102—1662no—
155#97—22no—
156#97C4×D8311yescharacteristic
157#97D33211yescharacteristic
158C4×C8—22no—
159C4×C8C4×D8311yescharacteristic
160C4×C8D33211yescharacteristic
161C2×D4—3322no—
162C2×D4—3322no—
163C2×D4—1662no—
164C22×C4—3322no—
165C22×C4C2×C22⋊D8311yescharacteristic
166D8—3322no—
167D8—3321no—
168D8—3321no—
169M16—41no—
170M16—41no—
171M16—41no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—22no—
175C2×C8(C2×D83)⋊C411yescharacteristic
176C2×C8(C2×D83)⋊C411yescharacteristic
177C2×C8C2×C4×D8311yescharacteristic
178C2×C8C2×D33211yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—41no—
181C4⋊C4—3322no—
182C4⋊C4—1662no—
183C22⋊C4—21no—
184C22⋊C4—41no—
185C22⋊C4—3322no—
186C22⋊C4—3322no—
187C42—22no—
188C42(C2×D83)⋊C411yescharacteristic
189C42(C2×D83)⋊C411yescharacteristic
190C16—3322no—
191C23—3322no—
192C23#1458181711yescharacteristic
193D4—13282no—
194D4—13282no—
195D4—3322no—
196D4—3322no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—13282no—
203C2×C4—3322no—
204C2×C4C2×(C2×D83)⋊C411yescharacteristic
205C2×C4#1458181711yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—42no—
210C8#1458172011yescharacteristic
211C8#1458172011yescharacteristic
212C22—21no—
213C22—41no—
214C22—6642no—
215C22—6642no—
216C22—6642no—
217C22#1912770511yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—6642no—
223C4#1912767411yescharacteristic
224C4C2×#1458172011yescharacteristic
225C2—21no—
226C2—41no—
227C2—13282no—
228C2#2490597011yescharacteristic
subgroupG/H
centerC2#24905970
commutatorC2×C8×C83C22×C4
FrattiniC2×M16C22×D83
FittingC83×#1747C2
socleC2×C83#1505
prime radical#24918113C2
cosocleC2×M16×C83C23
hypercenter#1747D83
hypocenterC83#11770
LCSG ▷ C2×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #1747

Complements

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

NH ≅ G/Ntype
2. #24909279C2semidirect
3. #24905453C2semidirect
4. #24905447C2semidirect
5. C83×#1747C2semidirect
7. #19128103C4semidirect
8. #19128078C4semidirect
12. #19127814C4semidirect
117. #1747D83semidirect
120. C83#11770semidirect

Maximal subgroups

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

subgroupindex
#249181132
#249092792
#249054532
#249054472
C83×#17472
#1177083