Finite Group Databasebeta

← #24910751

Subgroups — #24910751

Order 10624.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×#14581757C211yesmaximal, characteristic, complemented
2C2×D4×Dic83C211yesmaximal, characteristic
3#19127641C211yesmaximal, characteristic, complemented
4#19127640C211yesmaximal, characteristic, complemented
5#19127629C212yesmaximal, complemented
6#19127615C211yesmaximal, characteristic
7#19127613C212yesmaximal, complemented
8C22⋊C4×Dic83C212yesmaximal, complemented
9C22⋊C4×Dic83C212yesmaximal, complemented
10#19127546C211yesmaximal, characteristic, complemented
11#394×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13C23×Dic83C2212yes—
14#14581757—24no—
15D4×Dic83—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2211yescharacteristic
18C2×Dic83⋊C4C2211yescharacteristic
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2212yes—
22C2×C4×Dic83C2211yescharacteristic
23C2×C4×Dic83C2211yescharacteristic
24C2×C4×Dic83C2212yes—
25C2×C4×Dic83C2211yescharacteristic
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2211yescharacteristic
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C22×D4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35C2×C22⋊C4×C83C2212yes—
36#95×C83C2211yescharacteristic
37C2×Dic166—22no—
38C2×Dic166—22no—
39C2×Dic166—22no—
40C22×Dic83—24no—
41C22×Dic83—24no—
42C22×Dic83—24no—
43C22×Dic83C2312yes—
44C22×Dic83C2312yes—
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2311yescharacteristic
49C4⋊Dic83—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52C22⋊Dic83—24no—
53C22⋊Dic83—24no—
54C22⋊Dic83—24no—
55C4×Dic83—22no—
56C4×Dic83—22no—
57C4×Dic83—24no—
58C4×Dic83D414yes—
59C24×C83C2312yes—
60C2×D4×C83—24no—
61C2×D4×C83—22no—
62C22×C4×C83C2312yes—
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C4⋊C4×C83—22no—
67C22⋊C4×C83—24no—
68C22⋊C4×C83—24no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83#4114yes—
76C23×C83C2411yescharacteristic
77D4×C83—28no—
78C2×C4×C83—21no—
79C2×C4×C83—22no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83C2×D412yes—
87C2×C4×C83#4112yes—
88C2×Dic83—21no—
89C2×Dic83—21no—
90C2×Dic83—21no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—48no—
101C2×Dic83C2×D412yes—
102C2×Dic83C2×D412yes—
103C2×Dic83#4112yes—
104C2×Dic83#4114yes—
105Dic166—28no—
106C22×C83—24no—
107C22×C83—24no—
108C22×C83—24no—
109C22×C83—42no—
110C22×C83—42no—
111C22×C83—42no—
112C22×C83—42no—
113C22×C83C22×D411yescharacteristic
114C22×C83C2×#4111yescharacteristic
115C22×C83C2×#4112yes—
116C22×C83C2×#4112yes—
117C22×C83C2×#4111yescharacteristic
118C4×C83—22no—
119C4×C83—22no—
120C4×C83—41no—
121C4×C83—42no—
122Dic83—22no—
123Dic83—22no—
124Dic83—41no—
125Dic83—42no—
126Dic83—42no—
127Dic83#12414yes—
128C2×C83—44no—
129C2×C83#52511yescharacteristic
130C2×C83#52511yescharacteristic
131C2×C83C2×#12411yescharacteristic
132C2×C83#52912yes—
133C2×C83#52912yes—
134#2398—831nomaximal, Sylow
135C83#239811yescharacteristic, complemented, Sylow
136C2×C4×Q8—831no—
137C2×C4×D4—831no—
138#400—831no—
139#400—831no—
140#397—832no—
141#394—831no—
142#394D8311yescharacteristic
143#393—832no—
144C4×C22⋊C4—832no—
145C4×C22⋊C4—832no—
146#376—831no—
147C22×Q8—831no—
148C22×D4C2×D8311yescharacteristic
149C23×C4—832no—
150C4×Q8—1664no—
151C4×D4—1664no—
152C2×C4⋊C4—831no—
153C2×C4⋊C4—831no—
154C2×C4⋊C4—831no—
155C2×C4⋊C4C2×D8311yescharacteristic
156C2×C22⋊C4—832no—
157C2×C22⋊C4—832no—
158C2×C22⋊C4—832no—
159C2×C22⋊C4C2×D8312yes—
160C2×C22⋊C4C2×D8312yes—
161C2×C42—831no—
162C2×C42—831no—
163C2×C42—831no—
164C2×C42—832no—
165#95—832no—
166#95—831no—
167#95—832no—
168#95—832no—
169#95—832no—
170#95—832no—
171#95C2×D8311yescharacteristic
172C24C22×D8312yes—
173C2×Q8—1662no—
174C2×Q8—1662no—
175C2×Q8—1662no—
176C2×D4—24no—
177C2×D4—22no—
178C22×C4—831no—
179C22×C4—831no—
180C22×C4—831no—
181C22×C4—832no—
182C22×C4—832no—
183C22×C4—831no—
184C22×C4—1664no—
185C22×C4—1664no—
186C22×C4—1664no—
187C22×C4C22×D8312yes—
188C22×C4C22×D8311yescharacteristic
189C22×C4C22×D8311yescharacteristic
190C22×C4C22×D8311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—1662no—
193C4⋊C4—1662no—
194C4⋊C4—1662no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—1664no—
198C22⋊C4—1664no—
199C22⋊C4—1664no—
200C42—834no—
201C42—1662no—
202C42—1662no—
203C42—1664no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23#1305058914yes—
211C23C23×D8311yescharacteristic
212Q8—1668no—
213D4—28no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—3328no—
223C2×C4—832no—
224C2×C4—834no—
225C2×C4—832no—
226C2×C4—832no—
227C2×C4—1661no—
228C2×C4—1662no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1661no—
236C2×C4—1661no—
237C2×C4—1661no—
238C2×C4—1661no—
239C2×C4D4×D8312yes—
240C2×C4#1305058912yes—
241C22—24no—
242C22—24no—
243C22—24no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D8311yescharacteristic
249C22#41×D8311yescharacteristic
250C22#41×D8312yes—
251C22C2×#1305058912yes—
252C22C2×#1305058911yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—3321no—
258C4—3322no—
259C4—3322no—
260C4—834no—
261C4—1662no—
262C4—1662no—
263C2—44no—
264C2#1912820711yescharacteristic
265C2#1912821111yescharacteristic
266C2#124×D8311yescharacteristic
267C2#1912822812yes—
268C2#1912823212yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC22×D4×C83C22
hypercenter#394D83
hypocenterC83#2398
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14581757C2semidirect
3. #19127641C2semidirect
4. #19127640C2semidirect
5. #19127629C2semidirect
7. #19127613C2semidirect
8. C22⋊C4×Dic83C2semidirect
9. C22⋊C4×Dic83C2semidirect
10. #19127546C2semidirect
135. C83#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#145817572
C2×D4×Dic832
#191276412
#191276402
#191276292
#191276152
#191276132
C22⋊C4×Dic832
#191275462
#394×C832
#239883