Finite Group Databasebeta

← #24913328

Subgroups — #24913328

Order 10624.

500 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 131 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×#14581779C211yesmaximal, characteristic
2C2×#14581757C211yesmaximal, characteristic, complemented
3#19127637C212yesmaximal, complemented
4#19127622C211yesmaximal, characteristic, complemented
5#19127621C211yesmaximal, characteristic, complemented
6#19127615C211yesmaximal, characteristic
7#19127613C212yesmaximal, complemented
8#19127595C212yesmaximal, complemented
9#19127594C212yesmaximal, complemented
10#19127568C211yesmaximal, characteristic, complemented
11#394×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13C22×Dic166C2211yescharacteristic
14#14581779—24no—
15#14581757—24no—
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2211yescharacteristic
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2212yes—
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2212yes—
24C2×C4×Dic83C2211yescharacteristic
25C2×C4×Dic83C2211yescharacteristic
26C2×C4×Dic83C2211yescharacteristic
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—
40C2×Dic166—22no—
41C2×Dic166—42no—
42C22×Dic83C2312yes—
43C22×Dic83C2312yes—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C4⋊Dic83—42no—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—42no—
52C22⋊Dic83—24no—
53C22⋊Dic83—24no—
54C22⋊Dic83—24no—
55C22⋊Dic83—42no—
56C4×Dic83—22no—
57C4×Dic83—22no—
58C4×Dic83D414yes—
59C24×C83C2312yes—
60C2×D4×C83—22no—
61C2×D4×C83—42no—
62C22×C4×C83C2312yes—
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C4⋊C4×C83—22no—
67C22⋊C4×C83—24no—
68C22⋊C4×C83—42no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83C2411yescharacteristic
77D4×C83—44no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
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—22no—
93C2×Dic83—21no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83C2×D412yes—
102C2×Dic83C2×D412yes—
103C2×Dic83#4112yes—
104Dic166—28no—
105Dic166—44no—
106C22×C83—42no—
107C22×C83—42no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83—42no—
111C22×C83—42no—
112C22×C83—42no—
113C22×C83C22×D411yescharacteristic
114C22×C83C2×#4111yescharacteristic
115C22×C83C2×#4111yescharacteristic
116C22×C83H(5,𝔽2)12yes—
117C22×C832−1+412yes—
118C4×C83—22no—
119C4×C83—22no—
120C4×C83—41no—
121C4×C83—42no—
122Dic83—22no—
123Dic83—22no—
124Dic83—22no—
125Dic83—42no—
126Dic83—41no—
127Dic83—42no—
128C2×C83—82no—
129C2×C83#52511yescharacteristic
130C2×C83#53511yescharacteristic
131C2×C83#53611yescharacteristic
132C2×C83#54212yes—
133C2×C83#54212yes—
134#2632—831nomaximal, Sylow
135C83#263211yescharacteristic, complemented, Sylow
136C2×#124—831no—
137C2×C4×Q8—831no—
138#399—832no—
139#395—831no—
140#395—831no—
141#394—831no—
142#394D8311yescharacteristic
143#393—832no—
144#388—832no—
145#388—832no—
146#383—831no—
147C22×Q8—831no—
148C22×Q8—831no—
149C22×D4C2×D8311yescharacteristic
150#124—1664no—
151C4×Q8—1664no—
152C2×C4⋊C4—832no—
153C2×C4⋊C4—832no—
154C2×C4⋊C4—831no—
155C2×C4⋊C4—831no—
156C2×C4⋊C4C2×D8311yescharacteristic
157C2×C22⋊C4—832no—
158C2×C22⋊C4—832no—
159C2×C22⋊C4—832no—
160C2×C22⋊C4—832no—
161C2×C22⋊C4C2×D8312yes—
162C2×C22⋊C4C2×D8312yes—
163C2×C42—831no—
164C2×C42—831no—
165C2×C42—831no—
166#95—832no—
167#95—832no—
168#95—831no—
169#95—832no—
170#95—832no—
171#95C2×D8311yescharacteristic
172C24C22×D8312yes—
173C2×Q8—3322no—
174C2×Q8—1662no—
175C2×Q8—1662no—
176C2×Q8—1662no—
177C2×Q8—1662no—
178C2×D4—22no—
179C2×D4—42no—
180C22×C4—831no—
181C22×C4—831no—
182C22×C4—831no—
183C22×C4—832no—
184C22×C4—832no—
185C22×C4—831no—
186C22×C4C22×D8312yes—
187C22×C4C22×D8311yescharacteristic
188C22×C4C22×D8311yescharacteristic
189C22×C4C22×D8311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—3322no—
192C4⋊C4—3322no—
193C4⋊C4—1662no—
194C4⋊C4—1662no—
195C22⋊C4—24no—
196C22⋊C4—42no—
197C22⋊C4—3322no—
198C22⋊C4—1664no—
199C22⋊C4—1664no—
200C22⋊C4—1664no—
201C42—834no—
202C42—1662no—
203C42—1662no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23C23×D8311yescharacteristic
212Q8—3324no—
213Q8—1668no—
214D4—44no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—832no—
224C2×C4—832no—
225C2×C4—832no—
226C2×C4—1661no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1661no—
235C2×C4—1661no—
236C2×C4—1661no—
237C2×C4—1661no—
238C2×C4—1662no—
239C2×C4D4×D8312yes—
240C2×C4#1305058912yes—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D8311yescharacteristic
249C22#41×D8311yescharacteristic
250C22C2×#1305058911yescharacteristic
251C22#1458183712yes—
252C22#1458183912yes—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—3322no—
258C4—3321no—
259C4—3322no—
260C4—1662no—
261C4—1662no—
262C4—1662no—
263C2—82no—
264C2#1912821111yescharacteristic
265C2#1912826611yescharacteristic
266C2#1912827111yescharacteristic
267C2#1912831812yes—
268C2#1912831912yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC22×D4×C83C22
hypercenter#394D83
hypocenterC83#2632
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#14581757C2semidirect
3. #19127637C2semidirect
4. #19127622C2semidirect
5. #19127621C2semidirect
7. #19127613C2semidirect
8. #19127595C2semidirect
9. #19127594C2semidirect
10. #19127568C2semidirect
135. C83#2632semidirect

Maximal subgroups

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

subgroupindex
C2×#145817792
C2×#145817572
#191276372
#191276222
#191276212
#191276152
#191276132
#191275952
#191275942
#191275682
#394×C832
#263283