Finite Group Databasebeta

← #24912947

Subgroups — #24912947

Order 10624.

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

normal subgroups only

This group has 520 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×#14581778C211yesmaximal, characteristic
2C2×#14581757C211yesmaximal, characteristic, complemented
3#19127627C211yesmaximal, characteristic, complemented
4#19127614C211yesmaximal, characteristic, complemented
5#19127613C211yesmaximal, characteristic, complemented
6#19127605C212yesmaximal, complemented
7#19127601C212yesmaximal
8#19127590C212yesmaximal, complemented
9#19127589C212yesmaximal, complemented
10#19127552C211yesmaximal, characteristic, complemented
11#397×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×Dic166C2211yescharacteristic
14C22×C22⋊D83C2211yescharacteristic
15#14581778—24no—
16#14581757—24no—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×C22⋊Dic83C2211yescharacteristic
20C2×C22⋊Dic83C2212yes—
21C2×(C2×D83)⋊C4C2211yescharacteristic
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2212yes—
25C2×C4×Dic83C2212yes—
26C2×C4×Dic83C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2211yescharacteristic
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—24no—
40C2×Dic166—22no—
41C2×Dic166—24no—
42C2×C22⋊D83—22no—
43C2×C22⋊D83—24no—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2312yes—
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2312yes—
48C22×Dic83C2311yescharacteristic
49Dic83⋊C4—24no—
50Dic83⋊C4—24no—
51C22⋊Dic83—22no—
52C22⋊Dic83—24no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—22no—
55(C2×D83)⋊C4—24no—
56(C2×D83)⋊C4—42no—
57C4×Dic83—24no—
58C4×Dic83—42no—
59C4×Dic83D414yes—
60C24×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2312yes—
63C22×C4×C83C2312yes—
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C22⋊C4×C83—22no—
67C22⋊C4×C83—42no—
68C23×C83—21no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—21no—
72C23×C83—21no—
73C23×C83C2411yescharacteristic
74C22×D83—21no—
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—21no—
78C22×D83—21no—
79C2×C4×C83—22no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83C2×D412yes—
89C22⋊D83—44no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83C2×D412yes—
100C2×Dic83C2×D412yes—
101C2×Dic83#4114yes—
102C2×Dic83#4114yes—
103Dic166—28no—
104Dic166—44no—
105C22×C83—41no—
106C22×C83—41no—
107C22×C83—41no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83C22×D411yescharacteristic
111C22×C83C2×#4112yes—
112C22×C83C2×#4112yes—
113C22×C83H(5,𝔽2)11yescharacteristic
114C22×C832−1+411yescharacteristic
115C2×D83—41no—
116C2×D83—41no—
117C2×D83—41no—
118C2×D83—42no—
119C2×D83—42no—
120C4×C83—22no—
121C4×C83—41no—
122C4×C83—42no—
123C4×C83—42no—
124Dic83—24no—
125Dic83—22no—
126Dic83—41no—
127Dic83—42no—
128Dic83#12414yes—
129C2×C83—81no—
130C2×C83#52511yescharacteristic
131C2×C83C2×#12411yescharacteristic
132C2×C83#53711yescharacteristic
133C2×C83#55612yes—
134C2×C83#55912yes—
135D83—81no—
136#2592—831nomaximal, Sylow
137C83#259211yescharacteristic, complemented, Sylow
138C2×#124—831no—
139C2×C4×Q8—831no—
140#397—831no—
141#397D8311yescharacteristic, complemented
142#393—831no—
143#393—831no—
144#391—832no—
145#390—832no—
146#388—832no—
147#388—832no—
148C4×C4⋊C4—831no—
149C22×Q8—831no—
150C22×Q8—831no—
151C22×D4—831no—
152#124—1664no—
153C4×Q8—1664no—
154C2×C4⋊C4—832no—
155C2×C4⋊C4—832no—
156C2×C4⋊C4C2×D8311yescharacteristic
157C2×C22⋊C4—832no—
158C2×C22⋊C4—831no—
159C2×C22⋊C4—832no—
160C2×C22⋊C4—831no—
161C2×C22⋊C4—831no—
162C2×C22⋊C4—832no—
163C2×C22⋊C4C2×D8311yescharacteristic
164C2×C22⋊C4C2×D8312yes—
165C2×C42—832no—
166C2×C42—831no—
167C2×C42—832no—
168#95—831no—
169#95—832no—
170#95—832no—
171#95—832no—
172#95C2×D8311yescharacteristic
173#95C2×D8312yes—
174C24—831no—
175C24C22×D8311yescharacteristic
176C2×Q8—1662no—
177C2×Q8—1664no—
178C2×Q8—1664no—
179C2×Q8—1662no—
180C2×D4—1662no—
181C2×D4—1664no—
182C22×C4—831no—
183C22×C4—832no—
184C22×C4—831no—
185C22×C4—831no—
186C22×C4—832no—
187C22×C4C22×D8311yescharacteristic
188C22×C4C22×D8312yes—
189C22×C4C22×D8312yes—
190C22×C4C22×D8311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—1664no—
193C4⋊C4—1664no—
194C22⋊C4—22no—
195C22⋊C4—42no—
196C22⋊C4—3322no—
197C22⋊C4—1662no—
198C22⋊C4—1664no—
199C22⋊C4—1662no—
200C22⋊C4—1664no—
201C22⋊C4—1662no—
202C42—3322no—
203C42—834no—
204C42—1664no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—1661no—
211C23—1662no—
212C23—1662no—
213C23—1661no—
214C23—1661no—
215C23C23×D8311yescharacteristic
216Q8—3324no—
217Q8—1668no—
218D4—3324no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—832no—
229C2×C4—834no—
230C2×C4—832no—
231C2×C4—834no—
232C2×C4—1661no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1661no—
240C2×C4—1661no—
241C2×C4D4×D8312yes—
242C22—42no—
243C22—42no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—3322no—
248C22—3322no—
249C22—3321no—
250C22—3321no—
251C22—3321no—
252C22C2×D4×D8311yescharacteristic
253C22#41×D8312yes—
254C22#41×D8312yes—
255C22#1458183711yescharacteristic
256C22#1458183911yescharacteristic
257C4—22no—
258C4—42no—
259C4—41no—
260C4—42no—
261C4—3322no—
262C4—3321no—
263C4—834no—
264C4—1664no—
265C4—1662no—
266C2—81no—
267C2—6641no—
268C2#1912820911yescharacteristic
269C2#124×D8311yescharacteristic
270C2#1912828011yescharacteristic
271C2#1912841812yes—
272C2#1912843512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#397D83
hypocenterC83#2592
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#14581757C2semidirect
3. #19127627C2semidirect
4. #19127614C2semidirect
5. #19127613C2semidirect
6. #19127605C2semidirect
8. #19127590C2semidirect
9. #19127589C2semidirect
10. #19127552C2semidirect
11. #397×C83C2semidirect
137. C83#2592semidirect
141. #397D83semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#145817782
C2×#145817572
#191276272
#191276142
#191276132
#191276052
#191276012
#191275902
#191275892
#191275522
#397×C832
#259283