Finite Group Databasebeta

← #23771414

Subgroups — #23771414

Order 10112.

478 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14579143C211yesmaximal, characteristic
2C2×#14579141C211yesmaximal, characteristic, complemented
3#19098495C212yesmaximal, complemented
4#19098494C211yesmaximal, characteristic, complemented
5#19098492C212yesmaximal, complemented
6#19098486C212yesmaximal, complemented
7#19098484C211yesmaximal, characteristic, complemented
8#19098479C212yesmaximal
9#19098475C211yesmaximal, characteristic, complemented
10#19098422C211yesmaximal, characteristic, complemented
11#397×C79C211yesmaximal, characteristic, complemented
12C22×Dic158C2211yescharacteristic
13C22×C22⋊D79C2211yescharacteristic
14C22×C4×D79C2211yescharacteristic
15#14579143—24no—
16#14579141—24no—
17C2×C4⋊Dic79C2212yes—
18C2×C4⋊Dic79C2211yescharacteristic
19C2×C4⋊Dic79C2212yes—
20C2×Dic79⋊C4C2211yescharacteristic
21C2×Dic79⋊C4C2212yes—
22C2×Dic79⋊C4C2212yes—
23C2×C22⋊Dic79C2211yescharacteristic
24C2×C22⋊Dic79C2212yes—
25C2×(C2×D79)⋊C4C2212yes—
26C2×(C2×D79)⋊C4C2211yescharacteristic
27C2×(C2×D79)⋊C4C2212yes—
28C79⋊#95C2211yescharacteristic
29C79⋊#95C2212yes—
30C79⋊#95C2212yes—
31C79⋊#95C2212yes—
32C2×C4⋊C4×C79C2211yescharacteristic
33C2×C22⋊C4×C79C2211yescharacteristic
34C2×C22⋊C4×C79C2212yes—
35#95×C79C2211yescharacteristic
36#95×C79C2212yes—
37C23×D79C2311yescharacteristic
38C2×Dic158—22no—
39C2×Dic158—42no—
40C2×C22⋊D79—22no—
41C2×C22⋊D79—42no—
42C22×Dic79C2312yes—
43C22×Dic79C2312yes—
44C22×Dic79C2311yescharacteristic
45C22×Dic79C2311yescharacteristic
46C22×Dic79C2311yescharacteristic
47C2×C4×D79—42no—
48C2×C4×D79D414yes—
49C4⋊Dic79—24no—
50C4⋊Dic79—22no—
51C4⋊Dic79—42no—
52Dic79⋊C4—22no—
53Dic79⋊C4—42no—
54Dic79⋊C4—42no—
55C22⋊Dic79—22no—
56C22⋊Dic79—42no—
57(C2×D79)⋊C4—24no—
58(C2×D79)⋊C4—22no—
59(C2×D79)⋊C4—42no—
60C24×C79C2311yescharacteristic
61C22×C4×C79C2311yescharacteristic
62C22×C4×C79C2312yes—
63C22×C4×C79C2312yes—
64C22×C4×C79C2311yescharacteristic
65C4⋊C4×C79—22no—
66C22⋊C4×C79—22no—
67C22⋊C4×C79—42no—
68C23×C79—21no—
69C23×C79—22no—
70C23×C79—21no—
71C23×C79—22no—
72C23×C79—21no—
73C23×C79C2411yescharacteristic
74C22×D79—21no—
75C22×D79—22no—
76C22×D79—22no—
77C22×D79—21no—
78C22×D79C2×D412yes—
79C2×C4×C79—22no—
80C2×C4×C79—21no—
81C2×C4×C79—22no—
82C2×C4×C79—22no—
83C2×C4×C79—22no—
84C2×C4×C79—22no—
85C2×C4×C79—22no—
86C2×C4×C79—22no—
87C2×C4×C79—22no—
88C2×C4×C79C2×D412yes—
89C22⋊D79—44no—
90C2×Dic79—22no—
91C2×Dic79—21no—
92C2×Dic79—22no—
93C2×Dic79—21no—
94C2×Dic79—22no—
95C2×Dic79—22no—
96C2×Dic79—22no—
97C2×Dic79—21no—
98C2×Dic79—22no—
99C2×Dic79—22no—
100C2×Dic79—21no—
101C2×Dic79—22no—
102C2×Dic79C2×D412yes—
103C4×D79—44no—
104Dic158—44no—
105C22×C79—41no—
106C22×C79—41no—
107C22×C79—41no—
108C22×C79—42no—
109C22×C79—42no—
110C22×C79C22×D411yescharacteristic
111C22×C79H(5,𝔽2)11yescharacteristic
112C22×C79H(5,𝔽2)12yes—
113C22×C792−1+412yes—
114C22×C792−1+411yescharacteristic
115C2×D79—22no—
116C2×D79—22no—
117C2×D79—22no—
118C2×D79—42no—
119C2×D79—42no—
120C4×C79—22no—
121C4×C79—41no—
122C4×C79—42no—
123C4×C79—42no—
124Dic79—22no—
125Dic79—41no—
126Dic79—42no—
127Dic79—41no—
128Dic79—42no—
129C2×C79—81no—
130C2×C79#53411yescharacteristic
131C2×C79#53611yescharacteristic
132C2×C79#53711yescharacteristic
133C2×C79#56212yes—
134C2×C79#56312yes—
135D79—42no—
136#2814—791nomaximal, Sylow
137C79#281411yescharacteristic, complemented, Sylow
138C2×#123—791no—
139C2×C22⋊Q8—791no—
140#398—791no—
141#398—792no—
142#397—792no—
143#397D7911yescharacteristic, complemented
144#396—792no—
145#395—791no—
146#394—792no—
147#393—791no—
148#380—791no—
149C22×Q8—791no—
150C22×D4—791no—
151C23×C4—791no—
152#123—1584no—
153C22⋊Q8—1584no—
154C2×C4⋊C4—792no—
155C2×C4⋊C4—792no—
156C2×C4⋊C4—792no—
157C2×C4⋊C4—792no—
158C2×C4⋊C4—791no—
159C2×C4⋊C4—791no—
160C2×C4⋊C4C2×D7911yescharacteristic
161C2×C22⋊C4—792no—
162C2×C22⋊C4—792no—
163C2×C22⋊C4—791no—
164C2×C22⋊C4—792no—
165C2×C22⋊C4—791no—
166C2×C22⋊C4C2×D7911yescharacteristic
167C2×C22⋊C4C2×D7912yes—
168#95—791no—
169#95—792no—
170#95—792no—
171#95—792no—
172#95C2×D7911yescharacteristic
173#95C2×D7912yes—
174C24—791no—
175C24C22×D7911yescharacteristic
176C2×Q8—3162no—
177C2×Q8—1582no—
178C2×D4—3162no—
179C2×D4—1582no—
180C22×C4—3162no—
181C22×C4—794no—
182C22×C4—791no—
183C22×C4—792no—
184C22×C4—791no—
185C22×C4—792no—
186C22×C4—791no—
187C22×C4C22×D7911yescharacteristic
188C22×C4C22×D7912yes—
189C22×C4C22×D7912yes—
190C22×C4C22×D7911yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—3162no—
193C4⋊C4—3162no—
194C4⋊C4—3162no—
195C4⋊C4—1584no—
196C4⋊C4—1582no—
197C4⋊C4—1582no—
198C22⋊C4—22no—
199C22⋊C4—42no—
200C22⋊C4—3162no—
201C22⋊C4—3162no—
202C22⋊C4—1584no—
203C22⋊C4—1582no—
204C22⋊C4—1582no—
205C23—21no—
206C23—22no—
207C23—21no—
208C23—22no—
209C23—21no—
210C23—792no—
211C23—1581no—
212C23—1582no—
213C23—1582no—
214C23—1581no—
215C23C23×D7911yescharacteristic
216Q8—3164no—
217D4—3164no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—3164no—
228C2×C4—792no—
229C2×C4—1582no—
230C2×C4—1581no—
231C2×C4—1582no—
232C2×C4—1582no—
233C2×C4—1582no—
234C2×C4—1582no—
235C2×C4—1581no—
236C2×C4—1582no—
237C2×C4—1581no—
238C2×C4—1582no—
239C2×C4—1581no—
240C2×C4—1582no—
241C2×C4D4×D7912yes—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22—3162no—
248C22—3162no—
249C22—1582no—
250C22—1582no—
251C22—1582no—
252C22C2×D4×D7911yescharacteristic
253C22#1457920611yescharacteristic
254C22#1457920712yes—
255C22#1457920811yescharacteristic
256C22#1457920812yes—
257C4—22no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—3161no—
262C4—3162no—
263C4—3161no—
264C4—3162no—
265C4—1582no—
266C2—81no—
267C2—3162no—
268C2#1909912411yescharacteristic
269C2#1909913311yescharacteristic
270C2#1909914211yescharacteristic
271C2#1909931012yes—
272C2#1909931512yes—
subgroupG/H
center, FrattiniC23C23×D79
commutator, socle, cosocleC23×C79C24
Fitting#397×C79C2
prime radicalC22×C22⋊D79C22
hypercenter#397D79
hypocenterC79#2814
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#14579141C2semidirect
3. #19098495C2semidirect
4. #19098494C2semidirect
5. #19098492C2semidirect
6. #19098486C2semidirect
7. #19098484C2semidirect
9. #19098475C2semidirect
10. #19098422C2semidirect
11. #397×C79C2semidirect
137. C79#2814semidirect
143. #397D79semidirect