Finite Group Databasebeta

← #24915799

Subgroups — #24915799

Order 10624.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14581769C211yesmaximal, characteristic
2C22×C4×Dic83C211yesmaximal, characteristic
3#19127605C212yesmaximal, complemented
4#19127596C212yesmaximal, complemented
5#19127587C212yesmaximal, complemented
6#19127583C212yesmaximal, complemented
7#19127574C211yesmaximal, characteristic
8#19127562C211yesmaximal, characteristic
9#19127562C211yesmaximal, characteristic
10#19127559C211yesmaximal, characteristic
11#385×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13C23×Dic83C2211yescharacteristic
14C23×Dic83C2211yescharacteristic
15#14581769—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×C22⋊Dic83C2211yescharacteristic
22C2×C22⋊Dic83C2211yescharacteristic
23C2×C4×Dic83—24no—
24C2×C4×Dic83C2212yes—
25C2×C4×Dic83C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C23×C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2212yes—
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2211yescharacteristic
36#95×C83C2212yes—
37C2×Dic166—22no—
38C2×Dic166—24no—
39C22×Dic83—22no—
40C22×Dic83—22no—
41C22×Dic83—24no—
42C22×Dic83D414yes—
43C22×Dic83Q814yescomplemented
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2312yes—
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2312yes—
49C22×Dic83C2311yescharacteristic
50C4⋊Dic83—22no—
51Dic83⋊C4—24no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54Dic83⋊C4—24no—
55C22⋊Dic83—22no—
56C22⋊Dic83—22no—
57C4×Dic83—24no—
58C4×Dic83—24no—
59C24×C83C2311yescharacteristic
60C22×C4×C83—22no—
61C22×C4×C83—24no—
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2312yes—
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2312yes—
66C4⋊C4×C83—24no—
67C22⋊C4×C83—22no—
68C22⋊C4×C83—22no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83C2×D412yes—
72C23×C83C2×Q812yes—
73C23×C83#4112yes—
74C23×C83C2411yescharacteristic
75C2×C4×C83—28no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83#4114yes—
86C2×Dic83—28no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—28no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83C2×D412yes—
99C2×Dic83C2×D412yes—
100C2×Dic83C2×Q814yes—
101C2×Dic83#4114yes—
102Dic166—44no—
103C22×C83—22no—
104C22×C83—22no—
105C22×C83—24no—
106C22×C83—24no—
107C22×C83C22⋊Q814yes—
108C22×C83C22×D411yescharacteristic
109C22×C83C22×Q811yescharacteristic
110C22×C83C2×#4112yes—
111C22×C83C2×#4112yes—
112C22×C83C2×#4111yescharacteristic
113C4×C83—24no—
114C4×C83—41no—
115C4×C83—41no—
116C4×C83—42no—
117Dic83—24no—
118Dic83—22no—
119Dic83—41no—
120Dic83—41no—
121Dic83—42no—
122Dic83#12314yes—
123C2×C83—24no—
124C2×C83C2×C22⋊Q811yescharacteristic
125C2×C83C2×#12311yescharacteristic
126C2×C83#52511yescharacteristic
127C2×C83#52912yes—
128C2×C83#53312yes—
129#2855—831nomaximal, Sylow
130C83#285511yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—831no—
132C22×C42—831no—
133#391—832no—
134#389—832no—
135#387—832no—
136#387—832no—
137#385—831no—
138#385D8311yescharacteristic
139#381—831no—
140#381—831no—
141#380—831no—
142C22×Q8—831no—
143C23×C4—831no—
144C23×C4—831no—
145C23×C4C2×D8311yescharacteristic
146C22⋊Q8—1664no—
147C2×C4⋊C4—831no—
148C2×C4⋊C4—831no—
149C2×C4⋊C4—831no—
150C2×C4⋊C4—832no—
151C2×C4⋊C4—832no—
152C2×C4⋊C4C2×D8312yes—
153C2×C22⋊C4—831no—
154C2×C22⋊C4—831no—
155C2×C22⋊C4C2×D8311yescharacteristic
156C2×C22⋊C4C2×D8311yescharacteristic
157C2×C42—832no—
158C2×C42—832no—
159C2×C42—1664no—
160#95—832no—
161#95—832no—
162#95—832no—
163#95—832no—
164#95—832no—
165#95—832no—
166#95C2×D8312yes—
167C24C22×D8311yescharacteristic
168C2×Q8—1662no—
169C2×Q8—1664no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—831no—
173C22×C4—834no—
174C22×C4—834no—
175C22×C4—832no—
176C22×C4—831no—
177C22×C4—832no—
178C22×C4—831no—
179C22×C4—831no—
180C22×C4—1664no—
181C22×C4—1662no—
182C22×C4—1662no—
183C22×C4C22×D8312yes—
184C22×C4C22×D8311yescharacteristic
185C22×C4C22×D8311yescharacteristic
186C22×C4C22×D8312yes—
187C4⋊C4—24no—
188C4⋊C4—1662no—
189C4⋊C4—1662no—
190C4⋊C4—1662no—
191C4⋊C4—1664no—
192C4⋊C4—1664no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—1662no—
196C22⋊C4—1662no—
197C42—1664no—
198C42—1664no—
199C23—22no—
200C23—22no—
201C23D4×D8312yes—
202C23Q8×D8312yes—
203C23#1305059012yes—
204C23C23×D8311yescharacteristic
205Q8—3324no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—832no—
217C2×C4—834no—
218C2×C4—832no—
219C2×C4—834no—
220C2×C4—1661no—
221C2×C4—1661no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1662no—
227C2×C4—1668no—
228C2×C4—1668no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4#1305058914yes—
233C22—22no—
234C22—22no—
235C22—24no—
236C22—24no—
237C22#1458177014yes—
238C22C2×D4×D8311yescharacteristic
239C22C2×Q8×D8311yescharacteristic
240C22#41×D8312yes—
241C22C2×#1305058912yes—
242C22C2×#1305059011yescharacteristic
243C4—24no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—3321no—
248C4—3321no—
249C4—3322no—
250C4—834no—
251C4—1664no—
252C4—1662no—
253C2—24no—
254C2C2×#1458177011yescharacteristic
255C2#123×D8311yescharacteristic
256C2#1912820711yescharacteristic
257C2#1912823112yes—
258C2#1912825312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#385×C83C2
prime radicalC24×C83C23
hypercenter#385D83
hypocenterC83#2855
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #19127605C2semidirect
4. #19127596C2semidirect
5. #19127587C2semidirect
6. #19127583C2semidirect
43. C22×Dic83Q8semidirect
130. C83#2855semidirect

Maximal subgroups

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

subgroupindex
C2×#145817692
C22×C4×Dic832
#191276052
#191275962
#191275872
#191275832
#191275742
#191275622
#191275592
#385×C832
#285583