Finite Group Databasebeta

← #23766803

Subgroups — #23766803

Order 10112.

610 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 161 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×#14579142C211yesmaximal, characteristic
2C2×#14579117C211yesmaximal, characteristic, complemented
3C22×Dic79⋊C4C211yesmaximal, characteristic, complemented
4#19098494C212yesmaximal, complemented
5#19098486C212yesmaximal, complemented
6#19098459C212yesmaximal, complemented
7#19098451C212yesmaximal, complemented
8#19098443C211yesmaximal, characteristic
9#19098438C211yesmaximal, characteristic, complemented
10#19098426C211yesmaximal, characteristic, complemented
11C2×#123×C79C211yesmaximal, characteristic
12C23×Dic79C2211yescharacteristic
13C23×Dic79C2211yescharacteristic
14#14579142—24no—
15#14579117—24no—
16C2×C4⋊Dic79C2212yes—
17C2×Dic79⋊C4—24no—
18C2×Dic79⋊C4C2212yes—
19C2×Dic79⋊C4C2211yescharacteristic
20C2×Dic79⋊C4C2212yes—
21C2×Dic79⋊C4C2212yes—
22C2×Dic79⋊C4C2211yescharacteristic
23C2×C22⋊Dic79C2211yescharacteristic
24C2×C22⋊Dic79C2211yescharacteristic
25C2×C22⋊Dic79C2212yes—
26C2×C22⋊Dic79C2211yescharacteristic
27C2×C22⋊Dic79C2211yescharacteristic
28C2×C4×Dic79C2212yes—
29C79⋊#95C2212yes—
30C79⋊#95C2212yes—
31C79⋊#95C2212yes—
32C79⋊#95C2212yes—
33C22×D4×C79C2211yescharacteristic
34C23×C4×C79C2211yescharacteristic
35#123×C79—24no—
36C2×C4⋊C4×C79C2212yes—
37C2×C22⋊C4×C79C2212yes—
38C2×C22⋊C4×C79C2211yescharacteristic
39C22×Dic79—24no—
40C22×Dic79—22no—
41C22×Dic79—22no—
42C22×Dic79—22no—
43C22×Dic79D414yes—
44C22×Dic79C2311yescharacteristic
45C22×Dic79C2312yes—
46C22×Dic79C2311yescharacteristic
47C22×Dic79C2311yescharacteristic
48C22×Dic79C2311yescharacteristic
49C22×Dic79C2312yes—
50C4⋊Dic79—24no—
51Dic79⋊C4—24no—
52Dic79⋊C4—24no—
53Dic79⋊C4—22no—
54Dic79⋊C4—22no—
55Dic79⋊C4—24no—
56C22⋊Dic79—22no—
57C22⋊Dic79—22no—
58C22⋊Dic79—24no—
59C22⋊Dic79—22no—
60C22⋊Dic79—22no—
61C4×Dic79—24no—
62C24×C79C2311yescharacteristic
63C24×C79C2311yescharacteristic
64C2×D4×C79—22no—
65C2×D4×C79—22no—
66C2×D4×C79—22no—
67C22×C4×C79—22no—
68C22×C4×C79—22no—
69C22×C4×C79—22no—
70C22×C4×C79C2312yes—
71C22×C4×C79C2311yescharacteristic
72C22×C4×C79C2312yes—
73C4⋊C4×C79—24no—
74C22⋊C4×C79—24no—
75C22⋊C4×C79D414yes—
76C23×C79—22no—
77C23×C79—22no—
78C23×C79—21no—
79C23×C79—21no—
80C23×C79—22no—
81C23×C79—21no—
82C23×C79—22no—
83C23×C79C2×D412yes—
84C23×C79C2×D412yes—
85C23×C79#4112yes—
86C23×C79C2411yescharacteristic
87D4×C79—44no—
88C2×C4×C79—22no—
89C2×C4×C79—22no—
90C2×C4×C79—22no—
91C2×C4×C79—21no—
92C2×C4×C79—22no—
93C2×C4×C79—21no—
94C2×C4×C79—22no—
95C2×C4×C79—21no—
96C2×C4×C79—44no—
97C2×C4×C79C2×D414yes—
98C2×Dic79—28no—
99C2×Dic79—28no—
100C2×Dic79—21no—
101C2×Dic79—22no—
102C2×Dic79—21no—
103C2×Dic79—22no—
104C2×Dic79—21no—
105C2×Dic79—21no—
106C2×Dic79—22no—
107C2×Dic79—21no—
108C2×Dic79—21no—
109C2×Dic79—22no—
110C2×Dic79—22no—
111C2×Dic79—22no—
112C2×Dic79—22no—
113C2×Dic79C2×D412yes—
114C2×Dic79C2×D412yes—
115C2×Dic79#4114yes—
116C22×C79—22no—
117C22×C79—22no—
118C22×C79—24no—
119C22×C79—24no—
120C22×C79—41no—
121C22×C79—41no—
122C22×C79—41no—
123C22×C79—42no—
124C22×C79—42no—
125C22×C79#12114yes—
126C22×C79C22×D411yescharacteristic
127C22×C79C22×D411yescharacteristic
128C22×C79C2×#4112yes—
129C22×C79C2×#4111yescharacteristic
130C22×C792−1+412yes—
131C4×C79—24no—
132C4×C79—42no—
133C4×C79—41no—
134Dic79—22no—
135Dic79—22no—
136Dic79—24no—
137Dic79—41no—
138Dic79—41no—
139Dic79—42no—
140C2×C79—24no—
141C2×C79—81no—
142C2×C79C2×#12111yescharacteristic
143C2×C79#52511yescharacteristic
144C2×C79#53611yescharacteristic
145C2×C79#54712yes—
146C2×C79#55212yes—
147#2413—791nomaximal, Sylow
148C79#241311yescharacteristic, complemented, Sylow
149C2×#123—791no—
150C2×#123D7911yescharacteristic
151C2×#117—791no—
152C22×C4⋊C4—791no—
153#398—792no—
154#396—792no—
155#389—792no—
156#388—792no—
157#386—791no—
158#385—791no—
159#381—791no—
160C22×D4C2×D7911yescharacteristic
161C23×C4—791no—
162C23×C4—791no—
163C23×C4C2×D7911yescharacteristic
164#123—24no—
165#123—1584no—
166#117—1584no—
167C2×C4⋊C4—792no—
168C2×C4⋊C4—792no—
169C2×C4⋊C4—792no—
170C2×C4⋊C4—791no—
171C2×C4⋊C4—791no—
172C2×C4⋊C4—792no—
173C2×C4⋊C4—1584no—
174C2×C4⋊C4C2×D7912yes—
175C2×C22⋊C4—791no—
176C2×C22⋊C4—791no—
177C2×C22⋊C4—791no—
178C2×C22⋊C4—791no—
179C2×C22⋊C4—792no—
180C2×C22⋊C4C2×D7912yes—
181C2×C22⋊C4C2×D7911yescharacteristic
182C2×C42—792no—
183#95—792no—
184#95—792no—
185#95—792no—
186#95—792no—
187C24C22×D7911yescharacteristic
188C24C22×D7911yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—794no—
196C22×C4—792no—
197C22×C4—791no—
198C22×C4—791no—
199C22×C4—791no—
200C22×C4—792no—
201C22×C4—791no—
202C22×C4—1584no—
203C22×C4—1582no—
204C22×C4—1582no—
205C22×C4—1582no—
206C22×C4C22×D7912yes—
207C22×C4C22×D7911yescharacteristic
208C22×C4C22×D7912yes—
209C4⋊C4—24no—
210C4⋊C4—1584no—
211C4⋊C4—1584no—
212C4⋊C4—1582no—
213C4⋊C4—1582no—
214C4⋊C4—1584no—
215C4⋊C4—1584no—
216C22⋊C4—24no—
217C22⋊C4—1584no—
218C22⋊C4—1582no—
219C22⋊C4—1582no—
220C22⋊C4—1582no—
221C22⋊C4—1582no—
222C22⋊C4C22⋊D7914yes—
223C42—1584no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23—22no—
229C23—21no—
230C23—21no—
231C23D4×D7912yes—
232C23C2×C22⋊D7912yes—
233C23#1192962312yes—
234C23C23×D7911yescharacteristic
235D4—44no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—44no—
245C2×C4—794no—
246C2×C4—792no—
247C2×C4—792no—
248C2×C4—1582no—
249C2×C4—1582no—
250C2×C4—1582no—
251C2×C4—1582no—
252C2×C4—1581no—
253C2×C4—1581no—
254C2×C4—1582no—
255C2×C4—1588no—
256C2×C4—1582no—
257C2×C4—1581no—
258C2×C4—1581no—
259C2×C4—1588no—
260C2×C4—1582no—
261C2×C4—1581no—
262C2×C4—1581no—
263C2×C4C2×C22⋊D7914yes—
264C22—24no—
265C22—24no—
266C22—22no—
267C22—22no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22#1457913214yes—
274C22C2×D4×D7911yescharacteristic
275C22C22×C22⋊D7911yescharacteristic
276C22#41×D7912yes—
277C22C2×#1192962311yescharacteristic
278C22#1457920812yes—
279C4—24no—
280C4—42no—
281C4—41no—
282C4—3162no—
283C4—3161no—
284C4—3161no—
285C4—1582no—
286C4—1582no—
287C4—1584no—
288C2—24no—
289C2—81no—
290C2C2×#1457913211yescharacteristic
291C2#1909907111yescharacteristic
292C2#1909913511yescharacteristic
293C2#1909921212yes—
294C2#1909925012yes—
subgroupG/H
center, FrattiniC23C23×D79
commutator, socle, cosocleC23×C79C24
FittingC2×#123×C79C2
prime radicalC22×D4×C79C22
hypercenterC2×#123D79
hypocenterC79#2413
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#14579117C2semidirect
3. C22×Dic79⋊C4C2semidirect
4. #19098494C2semidirect
5. #19098486C2semidirect
6. #19098459C2semidirect
7. #19098451C2semidirect
9. #19098438C2semidirect
10. #19098426C2semidirect
148. C79#2413semidirect