Finite Group Databasebeta

← #23776255

Subgroups — #23776255

Order 10112.

370 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 133 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#19099293C211yesmaximal, characteristic, complemented
2D4×Dic158C211yesmaximal, characteristic
3#19098960C212yesmaximal, complemented
4#19098959C211yesmaximal, characteristic, complemented
5#19098851C212yesmaximal
6#19098849C211yesmaximal, characteristic, complemented
7#19098847C212yesmaximal, complemented
8#19098687C211yesmaximal, characteristic, complemented
9#19098654C212yesmaximal
10C4×#11929612C211yesmaximal, characteristic
11#476×C79C211yesmaximal, characteristic
12C22×Dic158—21no—
13C2×#11929612C2211yescharacteristic
14#14579161C2211yescharacteristic
15#14579160—21no—
16#14579160C2212yes—
17#14579160C2212yes—
18#14579159—22no—
19#14579140—21no—
20#14579138—22no—
21C4×Dic158C2211yescharacteristic
22C4×Dic158C2211yescharacteristic
23#14579126—21no—
24#14579126C2212yes—
25D4×Dic79C2212yes—
26#14579093C2212yes—
27#14579092C2211yescharacteristic
28#14579092C2212yes—
29#14579092C2212yes—
30Dic79⋊C8C2212yes—
31#14579086C2212yes—
32#14579084C2211yescharacteristic
33#14579082C2211yescharacteristic
34C4×C79⋊C8C2211yescharacteristic
35C8×Dic79C2212yes—
36C4⋊Q8×C79C2211yescharacteristic
37C4×D4×C79C2211yescharacteristic
38#106×C79C2212yes—
39C4⋊C8×C79C2211yescharacteristic
40#102×C79C2212yes—
41C2×Dic158—22no—
42C2×Dic158—22no—
43C2×Dic158—21no—
44C2×Dic158—44no—
45C2×Dic158D411yescharacteristic
46C2×Dic158C2311yescharacteristic
47C22×Dic79—22no—
48#11929612Q814yescomplemented
49C2×C79⋊C8C2311yescharacteristic
50C2×C79⋊C8C2311yescharacteristic
51C4⋊Dic79—21no—
52C4⋊Dic79D411yescharacteristic
53C4⋊Dic79C2311yescharacteristic
54C4⋊Dic79C2312yes—
55Dic79⋊C4—22no—
56Dic79⋊C4—22no—
57Dic79⋊C4—22no—
58Dic79⋊C4—21no—
59Dic79⋊C4D412yes—
60C22⋊Dic79—22no—
61C4×Dic79—22no—
62C4×Dic79—21no—
63C4×Dic79C2312yes—
64C2×Q8×C79—21no—
65C2×D4×C79C2311yescharacteristic
66C22×C4×C79—21no—
67C2×C8×C79C2312yes—
68C4⋊C4×C79—21no—
69C4⋊C4×C79C2312yes—
70C4⋊C4×C79C2311yescharacteristic
71C22⋊C4×C79—21no—
72C42×C79C2311yescharacteristic
73C23×C79—21no—
74Q8×C79—81no—
75D4×C79—21no—
76D4×C79C2×Q812yes—
77C2×C4×C79—22no—
78C2×C4×C79—21no—
79C2×C4×C79—42no—
80C2×C4×C79C2×D411yescharacteristic
81C2×C4×C79C2×D411yescharacteristic
82C2×C4×C79C2411yescharacteristic
83C2×Dic79—21no—
84C2×Dic79—21no—
85C2×Dic79—22no—
86C2×Dic79—44no—
87C2×Dic79C2×D412yes—
88C2×Dic79C2×D412yes—
89Dic158—21no—
90Dic158—42no—
91Dic158—44no—
92Dic158—82no—
93Dic158SD1614yescomplemented
94Dic158C2×Q812yescomplemented
95C8×C79—22no—
96C79⋊C8—21no—
97C79⋊C8C2×Q812yescomplemented
98C22×C79—22no—
99C22×C79—41no—
100C22×C79—41no—
101C22×C79C22×D411yescharacteristic
102C4×C79—21no—
103C4×C79—41no—
104C4×C79—42no—
105C4×C79C2×SD1612yes—
106C4×C79C22×Q811yescharacteristic
107C4×C792−1+411yescharacteristic
108Dic79—22no—
109Dic79—22no—
110Dic79—42no—
111Dic79—41no—
112Dic79C2×SD1614yes—
113C2×C79—42no—
114C2×C79D4×Q811yescharacteristic
115C2×C79C22×SD1611yescharacteristic
116C2×C79#57711yescharacteristic
117Q8×SD16—791nomaximal, Sylow
118C79Q8×SD1611yescharacteristic, complemented, Sylow
119Q82—791no—
120D4×Q8—791no—
121#498—791no—
122#498—792no—
123#476—792no—
124#476D7911yescharacteristic
125#475—792no—
126#475—791no—
127C8×Q8—791no—
128C4×SD16—791no—
129C4×SD16—792no—
130C22×Q8—1581no—
131C2×SD16—791no—
132C4⋊Q8—792no—
133C4⋊Q8—792no—
134C4⋊Q8—791no—
135C4⋊Q8—1581no—
136C4⋊Q8—1582no—
137C4⋊Q8C2×D7911yescharacteristic
138C22⋊Q8—1582no—
139C22⋊Q8—1581no—
140C4×Q8—791no—
141C4×Q8—792no—
142C4×Q8—791no—
143C4×Q8—1581no—
144C4×D4—792no—
145C4×D4C2×D7911yescharacteristic
146#106—792no—
147#106—791no—
148#106—792no—
149#106—792no—
150#106C2×D7912yes—
151C4⋊C8—792no—
152C4⋊C8C2×D7911yescharacteristic
153#103—792no—
154#103—791no—
155#102—791no—
156#102C2×D7912yes—
157C4×C8—792no—
158C4×C8—791no—
159C2×Q8—21no—
160C2×Q8—3164no—
161C2×Q8—791no—
162C2×Q8—791no—
163C2×Q8—1582no—
164C2×Q8—1582no—
165C2×Q8—1581no—
166C2×D4C22×D7911yescharacteristic
167C22×C4—21no—
168C22×C4—1582no—
169SD16—794no—
170C2×C8—791no—
171C2×C8—791no—
172C2×C8C22×D7912yes—
173C4⋊C4—21no—
174C4⋊C4—792no—
175C4⋊C4—791no—
176C4⋊C4—791no—
177C4⋊C4—792no—
178C4⋊C4—1582no—
179C4⋊C4—1582no—
180C4⋊C4—1582no—
181C4⋊C4—1581no—
182C4⋊C4—1581no—
183C4⋊C4C22×D7912yes—
184C4⋊C4C22×D7911yescharacteristic
185C22⋊C4—21no—
186C22⋊C4—1582no—
187C42—792no—
188C42—1582no—
189C42—1581no—
190C42C22×D7911yescharacteristic
191C23—21no—
192Q8—81no—
193Q8—3164no—
194Q8—3162no—
195Q8—794no—
196Q8—792no—
197Q8—6322no—
198Q8—1581no—
199D4—21no—
200D4Q8×D7912yes—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—3164no—
205C2×C4—792no—
206C2×C4—792no—
207C2×C4—1581no—
208C2×C4—1582no—
209C2×C4—1581no—
210C2×C4D4×D7911yescharacteristic
211C2×C4D4×D7911yescharacteristic
212C2×C4C23×D7911yescharacteristic
213C8—22no—
214C8—792no—
215C8—1581no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22C2×D4×D7911yescharacteristic
220C4—21no—
221C4—41no—
222C4—42no—
223C4—3162no—
224C4—3161no—
225C4—794no—
226C4—1582no—
227C4—1582no—
228C4SD16×D7912yes—
229C4C2×Q8×D7911yescharacteristic
230C4#1457920811yescharacteristic
231C2—42no—
232C2#1909922611yescharacteristic
233C2C2×SD16×D7911yescharacteristic
234C2#1909937311yescharacteristic
subgroupG/H
centerC22C2×D4×D79
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
Fitting#476×C79C2
socleC22×C79C22×D4
prime radicalC2×D4×C79C23
hypercenter#476D79
hypocenterC79Q8×SD16
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #19099293C2semidirect
3. #19098960C2semidirect
4. #19098959C2semidirect
6. #19098849C2semidirect
7. #19098847C2semidirect
8. #19098687C2semidirect
48. #11929612Q8semidirect
93. Dic158SD16semidirect
94. Dic158C2×Q8semidirect
97. C79⋊C8C2×Q8semidirect
118. C79Q8×SD16semidirect