Finite Group Databasebeta

← #23774626

Subgroups — #23774626

Order 10112.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C4⋊Q8×D79C211yesmaximal, characteristic, complemented
2#19099092C211yesmaximal, characteristic
3#19098972C211yesmaximal, characteristic, complemented
4#19098969C211yesmaximal, characteristic, complemented
5#19098922C211yesmaximal, characteristic, complemented
6#19098897C212yesmaximal, complemented
7#19098880C212yesmaximal
8#19098846C212yesmaximal, complemented
9#19098841C212yesmaximal, complemented
10#19098618C211yesmaximal, characteristic, complemented
11#489×C79C211yesmaximal, characteristic, complemented
12C2×Q8×D79—21no—
13#14579161C2211yescharacteristic
14#14579160—21no—
15#14579160C2212yes—
16#14579159—21no—
17#14579154—21no—
18#14579153C2211yescharacteristic
19#14579143—21no—
20#14579137C2211yescharacteristic
21#14579132C2211yescharacteristic
22Q8×Dic79C2211yescharacteristic
23D4×Dic79C2211yescharacteristic
24(C22⋊D79)⋊C4—21no—
25#14579117—21no—
26C4⋊C4×D79—21no—
27C4⋊C4×D79C2212yes—
28C42×D79C2211yescharacteristic
29#14579095C2212yes—
30#14579094C2212yes—
31#14579093C2212yes—
32#14579092C2212yes—
33Dic79⋊C8C2212yes—
34#14579085C2212yes—
35#14579082C2212yes—
36(C2×D79)⋊C8C2212yes—
37#14579072C2211yescharacteristic
38C4⋊Q8×C79C2211yescharacteristic
39#124×C79C2211yescharacteristic
40#103×C79C2212yes—
41#102×C79C2212yes—
42#97×C79C2211yescharacteristic
43C2×Dic158—22no—
44C2×Dic158—21no—
45Q8×D79—44no—
46C2×C22⋊D79—21no—
47C22×Dic79—21no—
48C2×C4×D79—22no—
49C2×C4×D79D411yescharacteristic
50C2×C4×D79D411yescharacteristic
51C2×C4×D79C2311yescharacteristic
52C2×C79⋊C8C2312yes—
53C4⋊Dic79—21no—
54C4⋊Dic79—21no—
55C4⋊Dic79C2312yes—
56C4⋊Dic79C2311yescharacteristic
57C4⋊Dic79C2311yescharacteristic
58Dic79⋊C4—22no—
59Dic79⋊C4—22no—
60Dic79⋊C4—21no—
61Dic79⋊C4—21no—
62Dic79⋊C4—21no—
63C22⋊Dic79—21no—
64C22⋊Dic79—21no—
65(C2×D79)⋊C4—21no—
66C4×Dic79—21no—
67C4×Dic79D411yescharacteristic
68C4×Dic79D411yescharacteristic
69C4×Dic79C2311yescharacteristic
70C2×Q8×C79—21no—
71C2×Q8×C79C2311yescharacteristic
72C2×D4×C79C2311yescharacteristic
73C2×C8×C79C2312yes—
74C4⋊C4×C79—21no—
75C4⋊C4×C79C2312yes—
76C22⋊C4×C79—21no—
77C22⋊C4×C79—21no—
78C42×C79C2311yescharacteristic
79C23×C79—21no—
80C22×D79C2×D411yescharacteristic
81Q8×C79—22no—
82Q8×C79—42no—
83D4×C79—22no—
84C2×C4×C79—21no—
85C2×C4×C79—22no—
86C2×C4×C79C2×D411yescharacteristic
87C2×C4×C79C2×D411yescharacteristic
88C2×C4×C79C2411yescharacteristic
89C22⋊D79—42no—
90C2×Dic79—21no—
91C2×Dic79—21no—
92C2×Dic79—22no—
93C2×Dic79—42no—
94C2×Dic79C2×D411yescharacteristic
95C2×Dic79C2×D411yescharacteristic
96C2×Dic79C2×D411yescharacteristic
97C4×D79—22no—
98C4×D79—22no—
99C4×D79—22no—
100C4×D79—44no—
101Dic158—42no—
102Dic158—44no—
103C8×C79—22no—
104C79⋊C8—22no—
105C22×C79—41no—
106C22×C79—41no—
107C22×C79—41no—
108C22×C79C22×D411yescharacteristic
109C2×D79—21no—
110C2×D79—21no—
111C2×D79#13712yes—
112C4×C79—21no—
113C4×C79—21no—
114C4×C79—41no—
115C4×C79—42no—
116C4×C792−1+412yes—
117Dic79—21no—
118Dic79—21no—
119Dic79—42no—
120Dic79—41no—
121Dic79—41no—
122Dic79AGL(1,ℤ/8)12yes—
123C2×C79—81no—
124C2×C79#53611yescharacteristic
125C2×C79C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C79C2×#13711yescharacteristic
127D79—22no—
128#3239—791nomaximal, Sylow
129C79#323911yescharacteristic, complemented, Sylow
130C2×C4⋊Q8—791no—
131#529—791no—
132#501—791no—
133#501—791no—
134#489—791no—
135#489D7911yescharacteristic, complemented
136#484—792no—
137#481—792no—
138#475—792no—
139#474—792no—
140#432—791no—
141C22×Q8—1581no—
142C4⋊Q8—792no—
143C4⋊Q8—791no—
144C4⋊Q8—1581no—
145C4⋊Q8—1581no—
146C4⋊Q8C2×D7911yescharacteristic
147#126—1581no—
148#125—791no—
149#124C2×D7911yescharacteristic
150#123—1581no—
151C22⋊Q8—791no—
152#121—791no—
153C4×Q8—791no—
154C4×D4—791no—
155C4×D4—1581no—
156#117—1581no—
157C2×C4⋊C4—792no—
158C2×C4⋊C4—1581no—
159C2×C42—791no—
160#107—792no—
161#107—792no—
162#106—792no—
163#106—792no—
164C4⋊C8—792no—
165#103—792no—
166#103C2×D7912yes—
167#102—792no—
168#102C2×D7912yes—
169C22⋊C8—792no—
170#97—791no—
171#97C2×D7911yescharacteristic
172C2×Q8—21no—
173C2×Q8—3164no—
174C2×Q8—1582no—
175C2×Q8—1581no—
176C2×Q8C22×D7911yescharacteristic
177C2×D4—1581no—
178C2×D4C22×D7911yescharacteristic
179C22×C4—791no—
180C22×C4—791no—
181C22×C4—791no—
182C22×C4—1582no—
183C22×C4—1581no—
184C2×C8—792no—
185C2×C8C22×D7912yes—
186C4⋊C4—21no—
187C4⋊C4—792no—
188C4⋊C4—791no—
189C4⋊C4—791no—
190C4⋊C4—1582no—
191C4⋊C4—1582no—
192C4⋊C4—1581no—
193C4⋊C4—1581no—
194C4⋊C4—1581no—
195C4⋊C4—1581no—
196C4⋊C4—1581no—
197C4⋊C4C22×D7912yes—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—1581no—
201C22⋊C4—1581no—
202C22⋊C4—1581no—
203C42—791no—
204C42—791no—
205C42—791no—
206C42—1581no—
207C42C22×D7911yescharacteristic
208C23—21no—
209C23—791no—
210Q8—22no—
211Q8—42no—
212Q8—3162no—
213Q8—3164no—
214D4—22no—
215D4—3162no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—3164no—
219C2×C4—3162no—
220C2×C4—791no—
221C2×C4—791no—
222C2×C4—791no—
223C2×C4—1582no—
224C2×C4—1582no—
225C2×C4—1581no—
226C2×C4—1581no—
227C2×C4—1582no—
228C2×C4—1582no—
229C2×C4D4×D7911yescharacteristic
230C2×C4D4×D7911yescharacteristic
231C2×C4C23×D7911yescharacteristic
232C8—22no—
233C8—1582no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—792no—
238C22—1581no—
239C22—1581no—
240C22C2×D4×D7911yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—3162no—
246C4—3161no—
247C4—3161no—
248C4—792no—
249C4—1581no—
250C4—1581no—
251C4#1457920812yes—
252C2—81no—
253C2—1582no—
254C2#1909913311yescharacteristic
255C2AGL(1,ℤ/8)×D7911yescharacteristic
256C2#137×D7911yescharacteristic
subgroupG/H
centerC22C2×D4×D79
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
Fitting#489×C79C2
socleC22×C79C22×D4
prime radical#14579132C22
hypercenter#489D79
hypocenterC79#3239
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D79C2semidirect
3. #19098972C2semidirect
4. #19098969C2semidirect
5. #19098922C2semidirect
6. #19098897C2semidirect
8. #19098846C2semidirect
9. #19098841C2semidirect
10. #19098618C2semidirect
11. #489×C79C2semidirect
129. C79#3239semidirect
135. #489D79semidirect