Finite Group Databasebeta

← #24911090

Subgroups — #24911090

Order 10624.

836 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×#13050591C211yesmaximal, characteristic
2C2×#14581790C211yesmaximal, characteristic, complemented
3C2×#14581780C211yesmaximal, characteristic, complemented
4C2×#14581772C211yesmaximal, characteristic, complemented
5#19127619C214yesmaximal, complemented
6#19127602C211yesmaximal, characteristic, complemented
7#19127575C214yesmaximal, complemented
8C4×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
9#391×C83C211yesmaximal, characteristic, complemented
10C2×#13050591—24no—
11C2×#13050591—24no—
12C2×#13050591—24no—
13C22×Dic166C2211yescharacteristic
14C22×D332C2212yes—
15C22×D332C2211yescharacteristic
16C22×C4×D83C2212yes—
17C22×C4×D83C2211yescharacteristic
18#14581790—24no—
19#14581780—24no—
20#14581772—24no—
21C2×C4⋊Dic83C2212yes—
22C2×Dic83⋊C4C2214yes—
23C2×(C2×D83)⋊C4C2214yes—
24C2×(C2×D83)⋊C4C2214yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2212yes—
29C22×Q8×C83C2211yescharacteristic
30C2×C4⋊C4×C83C2211yescharacteristic
31C2×C42×C83C2211yescharacteristic
32#95×C83C2214yes—
33C23×D83C2312yes—
34C23×D83C2311yescharacteristic
35#13050591—232no—
36C2×Dic166—22no—
37C2×Dic166—22no—
38C2×Dic166—22no—
39C2×D332—24no—
40C2×D332—24no—
41C2×D332—22no—
42C2×D332—24no—
43C2×D332—22no—
44C2×D332D414yes—
45C22×Dic83C2312yes—
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2312yes—
48C2×C4×D83—24no—
49C2×C4×D83—22no—
50C2×C4×D83—24no—
51C2×C4×D83—22no—
52C2×C4×D83—24no—
53C2×C4×D83D414yes—
54C4⋊Dic83—24no—
55Dic83⋊C4—28no—
56(C2×D83)⋊C4—24no—
57(C2×D83)⋊C4—28no—
58(C2×D83)⋊C4—28no—
59(C2×D83)⋊C4—24no—
60C4×Dic83—22no—
61C2×Q8×C83—22no—
62C2×Q8×C83—22no—
63C2×Q8×C83D414yes—
64C22×C4×C83C2312yes—
65C22×C4×C83C2312yes—
66C22×C4×C83C2312yes—
67C22×C4×C83C2311yescharacteristic
68C4⋊C4×C83—22no—
69C42×C83—22no—
70C23×C83C2411yescharacteristic
71C22×D83—22no—
72C22×D83—21no—
73C22×D83—22no—
74C22×D83—21no—
75C22×D83—22no—
76C22×D83—24no—
77C22×D83—22no—
78C22×D83—24no—
79C22×D83C2×D414yes—
80C22×D83C2×D412yes—
81Q8×C83—28no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—24no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83C2×D412yes—
90C2×C4×C83C2×D412yes—
91C2×C4×C83C2×D414yes—
92C2×C4×C83C2×D412yes—
93C2×Dic83—22no—
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83C2×D412yes—
101C2×Dic83#4114yescomplemented
102D332—216no—
103D332—28no—
104C4×D83—216no—
105C4×D83—28no—
106Dic166—44no—
107C22×C83C22×D411yescharacteristic
108C22×C83C22×D411yescharacteristic
109C22×C83C22×D411yescharacteristic
110C22×C83C2×#4112yes—
111C22×C832−1+412yes—
112C2×D83—24no—
113C2×D83—22no—
114C2×D83—22no—
115C2×D83—22no—
116C2×D83—28no—
117C2×D83—42no—
118C2×D83—42no—
119C2×D83—42no—
120C2×D83—42no—
121C2×D83—44no—
122C4×C83—24no—
123C4×C83—42no—
124C4×C83—42no—
125C4×C83#12014yes—
126Dic83—22no—
127Dic83—24no—
128Dic83—42no—
129C2×C83C2×#12011yescharacteristic
130C2×C83#53211yescharacteristic
131C2×C83#53611yescharacteristic
132C2×C83#54712yes—
133C2×C83#54812yes—
134D83—42no—
135D83—44no—
136#2432—831nomaximal, Sylow
137C83#243211yescharacteristic, complemented, Sylow
138C22×#41—831no—
139C2×C4⋊Q8—831no—
140C2×#124—831no—
141C2×C22⋊Q8—831no—
142#394—834no—
143#391D8311yescharacteristic, complemented
144#390—831no—
145#385—834no—
146C4×C22⋊C4—831no—
147C2×#41—1664no—
148C2×#41—1664no—
149C2×#41—1664no—
150C22×Q8—831no—
151C22×Q8C2×D8311yescharacteristic
152C22×D4—831no—
153C22×D4—832no—
154C23×C4—831no—
155C23×C4—832no—
156C4⋊Q8—1664no—
157#124—1664no—
158C22⋊Q8—1664no—
159C2×C4⋊C4—832no—
160C2×C4⋊C4—834no—
161C2×C4⋊C4C2×D8311yescharacteristic
162C2×C22⋊C4—832no—
163C2×C22⋊C4—834no—
164C2×C22⋊C4—832no—
165C2×C22⋊C4—834no—
166C2×C42—831no—
167C2×C42C2×D8311yescharacteristic
168#95—832no—
169#95C2×D8314yes—
170C24—831no—
171C24—832no—
172#41—16632no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—1662no—
176C2×Q8—1662no—
177C2×Q8—1662no—
178C2×Q8D33214yes—
179C2×D4—834no—
180C2×D4—1664no—
181C2×D4—1664no—
182C2×D4—1662no—
183C2×D4—1664no—
184C2×D4—1662no—
185C22×C4—831no—
186C22×C4—832no—
187C22×C4—834no—
188C22×C4—832no—
189C22×C4—1664no—
190C22×C4—1664no—
191C22×C4—1662no—
192C22×C4—1664no—
193C22×C4—1662no—
194C22×C4C22×D8312yes—
195C22×C4C22×D8312yes—
196C22×C4C22×D8312yes—
197C22×C4C22×D8311yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—1664no—
200C4⋊C4—1668no—
201C22⋊C4—1664no—
202C22⋊C4—1668no—
203C22⋊C4—1664no—
204C22⋊C4—1668no—
205C42—22no—
206C42—1662no—
207C23—832no—
208C23—834no—
209C23—1661no—
210C23—1662no—
211C23—1662no—
212C23—1661no—
213C23—1662no—
214C23—1664no—
215C23—1662no—
216C23—1664no—
217C23C23×D8311yescharacteristic
218Q8—28no—
219Q8—3324no—
220D4—16616no—
221D4—1668no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—832no—
230C2×C4—834no—
231C2×C4—16616no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1661no—
236C2×C4—1662no—
237C2×C4—1661no—
238C2×C4—1662no—
239C2×C4—1668no—
240C2×C4D4×D8312yes—
241C2×C4D4×D8312yes—
242C2×C4C2×D33214yes—
243C2×C4C2×D33212yes—
244C22—3322no—
245C22—3322no—
246C22—3324no—
247C22—3322no—
248C22—3322no—
249C22—1668no—
250C22—1664no—
251C22—1662no—
252C22—1662no—
253C22—1662no—
254C22C2×D4×D8311yescharacteristic
255C22C2×D4×D8311yescharacteristic
256C22C22×D33211yescharacteristic
257C22#41×D8312yes—
258C22#1458184012yes—
259C4—24no—
260C4—42no—
261C4—42no—
262C4—3322no—
263C4—1664no—
264C4—1662no—
265C4#1458176214yes—
266C2—3322no—
267C2—3324no—
268C2C2×#1458176211yescharacteristic
269C2#1912825011yescharacteristic
270C2#1912827611yescharacteristic
271C2#1912835312yes—
272C2#1912836312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#391×C83C2
prime radicalC22×#13050591C2
hypercenter#391D83
hypocenterC83#2432
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#14581790C2semidirect
3. C2×#14581780C2semidirect
4. C2×#14581772C2semidirect
5. #19127619C2semidirect
6. #19127602C2semidirect
7. #19127575C2semidirect
8. C4×(C2×D83)⋊C4C2semidirect
9. #391×C83C2semidirect
101. C2×Dic83#41semidirect
137. C83#2432semidirect
143. #391D83semidirect

Maximal subgroups

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

subgroupindex
C22×#130505912
C2×#145817902
C2×#145817802
C2×#145817722
#191276192
#191276022
#191275752
C4×(C2×D83)⋊C42
#391×C832
#243283