Finite Group Databasebeta

← #23769540

Subgroups — #23769540

Order 10112.

534 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14579149C211yesmaximal, characteristic, complemented
2#19098485C212yesmaximal, complemented
3#19098483C211yesmaximal, characteristic, complemented
4#19098480C211yesmaximal, characteristic, complemented
5#19098479C212yesmaximal, complemented
6#19098475C211yesmaximal, characteristic, complemented
7#19098472C211yesmaximal, characteristic
8#19098463C212yesmaximal, complemented
9#19098453C212yesmaximal, complemented
10C4×C4⋊Dic79C211yesmaximal, characteristic, complemented
11C2×#126×C79C211yesmaximal, characteristic, complemented
12C22×Dic158C2211yescharacteristic
13C22×D316C2211yescharacteristic
14C22×C22⋊D79C2212yes—
15#14579149—24no—
16C2×C4⋊Dic79C2212yes—
17C2×Dic79⋊C4C2212yes—
18C2×C22⋊Dic79C2212yes—
19C2×(C2×D79)⋊C4C2212yes—
20C2×(C2×D79)⋊C4C2212yes—
21C2×(C2×D79)⋊C4C2212yes—
22C2×(C2×D79)⋊C4C2212yes—
23C2×(C2×D79)⋊C4C2212yes—
24C2×C4×Dic79C2212yes—
25C79⋊#95C2211yescharacteristic
26C79⋊#95C2211yescharacteristic
27C79⋊#95C2212yes—
28C79⋊#95C2212yes—
29#126×C79—24no—
30C2×C4⋊C4×C79C2211yescharacteristic
31C2×C4⋊C4×C79C2211yescharacteristic
32C2×C4⋊C4×C79C2211yescharacteristic
33C2×C22⋊C4×C79C2211yescharacteristic
34C2×C22⋊C4×C79C2211yescharacteristic
35C2×C22⋊C4×C79C2211yescharacteristic
36C2×C42×C79C2211yescharacteristic
37C23×D79C2312yes—
38C2×Dic158—22no—
39C2×Dic158—42no—
40C2×D316—22no—
41C2×D316—42no—
42C2×C22⋊D79—24no—
43C2×C22⋊D79—42no—
44C2×C22⋊D79—42no—
45C22×Dic79C2312yes—
46C22×Dic79C2312yes—
47C22×Dic79C2312yes—
48C4⋊Dic79—24no—
49Dic79⋊C4—42no—
50C22⋊Dic79—42no—
51(C2×D79)⋊C4—24no—
52(C2×D79)⋊C4—24no—
53(C2×D79)⋊C4—24no—
54(C2×D79)⋊C4—42no—
55(C2×D79)⋊C4—42no—
56C4×Dic79—42no—
57C24×C79C2311yescharacteristic
58C22×C4×C79C2311yescharacteristic
59C22×C4×C79C2311yescharacteristic
60C22×C4×C79C2311yescharacteristic
61C22×C4×C79C2311yescharacteristic
62C22×C4×C79C2311yescharacteristic
63C22×C4×C79C2311yescharacteristic
64C4⋊C4×C79—22no—
65C4⋊C4×C79—22no—
66C4⋊C4×C79—22no—
67C22⋊C4×C79—22no—
68C22⋊C4×C79—22no—
69C22⋊C4×C79—22no—
70C42×C79D414yescomplemented
71C23×C79—21no—
72C23×C79—22no—
73C23×C79—22no—
74C23×C79—21no—
75C23×C79—21no—
76C23×C79C2411yescharacteristic
77C22×D79—22no—
78C22×D79—22no—
79C22×D79—22no—
80C22×D79—22no—
81C22×D79—22no—
82C22×D79—22no—
83C22×D79—22no—
84C2×C4×C79—22no—
85C2×C4×C79—21no—
86C2×C4×C79—22no—
87C2×C4×C79—21no—
88C2×C4×C79—21no—
89C2×C4×C79—21no—
90C2×C4×C79—21no—
91C2×C4×C79—22no—
92C2×C4×C79—21no—
93C2×C4×C79—22no—
94C2×C4×C79—21no—
95C2×C4×C79C2×D412yes—
96C2×C4×C79C2×D412yes—
97C2×C4×C79C2×D412yes—
98C22⋊D79—48no—
99C2×Dic79—22no—
100C2×Dic79—22no—
101C2×Dic79—22no—
102C2×Dic79—22no—
103C2×Dic79—22no—
104C2×Dic79—22no—
105C2×Dic79—22no—
106C2×Dic79—22no—
107C2×Dic79#4114yes—
108D316—44no—
109Dic158—44no—
110C22×C79—41no—
111C22×C79—41no—
112C22×C79—41no—
113C22×C79—42no—
114C22×C79—42no—
115C22×C79C22×D411yescharacteristic
116C22×C79C2×#4112yes—
117C22×C79H(5,𝔽2)12yes—
118C22×C79H(5,𝔽2)11yescharacteristic
119C22×C792−1+411yescharacteristic
120C2×D79—42no—
121C2×D79—42no—
122C2×D79—42no—
123C2×D79—42no—
124C2×D79—42no—
125C2×D79—42no—
126C2×D79—42no—
127C4×C79—22no—
128C4×C79—22no—
129C4×C79—22no—
130C4×C79—41no—
131C4×C79—41no—
132C4×C79—41no—
133Dic79—24no—
134Dic79—42no—
135Dic79—42no—
136C2×C79—81no—
137C2×C79#53211yescharacteristic
138C2×C79#53511yescharacteristic
139C2×C79#53711yescharacteristic
140C2×C79#53812yes—
141C2×C79#54212yes—
142D79—82no—
143#2662—791nomaximal, Sylow
144C79#266211yescharacteristic, complemented, Sylow
145C2×#126D7911yescharacteristic, complemented
146C2×#124—791no—
147#396—792no—
148#395—791no—
149#394—791no—
150#394—792no—
151#393—791no—
152#392—791no—
153#390—792no—
154#388—792no—
155C4×C4⋊C4—791no—
156C22×Q8—791no—
157C22×D4—792no—
158C22×D4—791no—
159#126—24no—
160#124—1584no—
161C2×C4⋊C4—792no—
162C2×C4⋊C4—792no—
163C2×C4⋊C4C2×D7911yescharacteristic
164C2×C4⋊C4C2×D7911yescharacteristic
165C2×C4⋊C4C2×D7911yescharacteristic
166C2×C22⋊C4—792no—
167C2×C22⋊C4—792no—
168C2×C22⋊C4—792no—
169C2×C22⋊C4—792no—
170C2×C22⋊C4—792no—
171C2×C22⋊C4—792no—
172C2×C22⋊C4C2×D7911yescharacteristic
173C2×C22⋊C4C2×D7911yescharacteristic
174C2×C22⋊C4C2×D7911yescharacteristic
175C2×C42—792no—
176C2×C42C2×D7911yescharacteristic
177#95—791no—
178#95—791no—
179#95—792no—
180#95—792no—
181C24—792no—
182C24C22×D7911yescharacteristic
183C2×Q8—3162no—
184C2×Q8—1582no—
185C2×D4—3162no—
186C2×D4—3162no—
187C2×D4—3162no—
188C2×D4—1584no—
189C2×D4—1582no—
190C22×C4—792no—
191C22×C4—792no—
192C22×C4—792no—
193C22×C4C22×D7911yescharacteristic
194C22×C4C22×D7911yescharacteristic
195C22×C4C22×D7911yescharacteristic
196C22×C4C22×D7911yescharacteristic
197C22×C4C22×D7911yescharacteristic
198C22×C4C22×D7911yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—3162no—
203C4⋊C4—1584no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—3162no—
208C22⋊C4—3162no—
209C22⋊C4—3162no—
210C22⋊C4—1584no—
211C22⋊C4—1584no—
212C22⋊C4—1584no—
213C42—3162no—
214C42C22⋊D7914yescomplemented
215C23—22no—
216C23—22no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—1582no—
221C23—1582no—
222C23—1582no—
223C23—1582no—
224C23—1582no—
225C23—1582no—
226C23—1582no—
227C23C23×D7911yescharacteristic
228Q8—3164no—
229D4—3164no—
230D4—3168no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—794no—
243C2×C4—1582no—
244C2×C4—1582no—
245C2×C4—1582no—
246C2×C4—1582no—
247C2×C4—1582no—
248C2×C4—1582no—
249C2×C4—1582no—
250C2×C4—1582no—
251C2×C4C2×C22⋊D7912yes—
252C2×C4C2×C22⋊D7912yes—
253C2×C4C2×C22⋊D7912yes—
254C22—41no—
255C22—42no—
256C22—42no—
257C22—41no—
258C22—41no—
259C22—3162no—
260C22—3162no—
261C22—3162no—
262C22—3162no—
263C22—3162no—
264C22—3162no—
265C22—3162no—
266C22C22×C22⋊D7911yescharacteristic
267C22#41×D7912yes—
268C22#1457920711yescharacteristic
269C22#1457920712yes—
270C22#1457920811yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—3162no—
278C4—3162no—
279C4—1584no—
280C2—81no—
281C2—6322no—
282C2#1909910911yescharacteristic
283C2#1909912711yescharacteristic
284C2#1909914111yescharacteristic
285C2#1909915212yes—
286C2#1909917912yes—
subgroupG/H
center, FrattiniC23C23×D79
commutator, socle, cosocleC23×C79C24
FittingC2×#126×C79C2
prime radical#19098472C2
hypercenterC2×#126D79
hypocenterC79#2662
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#14579149C2semidirect
2. #19098485C2semidirect
3. #19098483C2semidirect
4. #19098480C2semidirect
5. #19098479C2semidirect
6. #19098475C2semidirect
8. #19098463C2semidirect
9. #19098453C2semidirect
10. C4×C4⋊Dic79C2semidirect
11. C2×#126×C79C2semidirect
70. C42×C79D4semidirect
144. C79#2662semidirect
145. C2×#126D79semidirect
214. C42C22⋊D79semidirect