Finite Group Databasebeta

← #24913372

Subgroups — #24913372

Order 10624.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 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×#14581785C211yesmaximal, characteristic
2C2×#14581757C211yesmaximal, characteristic, complemented
3#19127635C212yesmaximal, complemented
4#19127632C211yesmaximal, characteristic, complemented
5#19127624C212yesmaximal
6#19127622C211yesmaximal, characteristic, complemented
7#19127613C211yesmaximal, characteristic
8#19127595C212yesmaximal
9#19127584C212yesmaximal, complemented
10#19127565C211yesmaximal, characteristic, complemented
11#397×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13#14581785—24no—
14#14581757—24no—
15C2×C4⋊Dic83C2212yes—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2211yescharacteristic
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2212yes—
22C2×C22⋊Dic83C2211yescharacteristic
23C2×C22⋊Dic83C2212yes—
24C2×C22⋊Dic83C2211yescharacteristic
25C2×C4×Dic83C2212yes—
26C2×C4×Dic83C2211yescharacteristic
27C83⋊#95C2212yes—
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—
37C2×Dic166—22no—
38C2×Dic166—24no—
39C22×Dic83C2312yes—
40C22×Dic83C2311yescharacteristic
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2312yes—
45C4⋊Dic83—22no—
46C4⋊Dic83—24no—
47Dic83⋊C4—22no—
48Dic83⋊C4—22no—
49Dic83⋊C4—24no—
50Dic83⋊C4—42no—
51Dic83⋊C4—42no—
52C22⋊Dic83—22no—
53C22⋊Dic83—22no—
54C22⋊Dic83—24no—
55C4×Dic83—24no—
56C4×Dic83D414yes—
57C24×C83C2311yescharacteristic
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2312yes—
60C22×C4×C83C2312yes—
61C22×C4×C83C2311yescharacteristic
62C4⋊C4×C83—22no—
63C22⋊C4×C83—22no—
64C22⋊C4×C83—42no—
65C23×C83—21no—
66C23×C83—22no—
67C23×C83—21no—
68C23×C83—22no—
69C23×C83—21no—
70C23×C83C2411yescharacteristic
71C2×C4×C83—22no—
72C2×C4×C83—21no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83C2×D412yes—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—21no—
84C2×Dic83—22no—
85C2×Dic83—21no—
86C2×Dic83—21no—
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83C2×D412yes—
95C2×Dic83C2×D412yes—
96C2×Dic83#4114yes—
97Dic166—28no—
98C22×C83—41no—
99C22×C83—41no—
100C22×C83—41no—
101C22×C83—42no—
102C22×C83—42no—
103C22×C83C22×D411yescharacteristic
104C22×C83C2×#4112yes—
105C22×C83H(5,𝔽2)11yescharacteristic
106C22×C832−1+412yes—
107C22×C832−1+411yescharacteristic
108C4×C83—22no—
109C4×C83—41no—
110C4×C83—42no—
111C4×C83—42no—
112Dic83—24no—
113Dic83—22no—
114Dic83—22no—
115Dic83—41no—
116Dic83—41no—
117Dic83—42no—
118C2×C83—81no—
119C2×C83#52511yescharacteristic
120C2×C83#53611yescharacteristic
121C2×C83#53711yescharacteristic
122C2×C83#54112yes—
123C2×C83#54212yes—
124#2637—831nomaximal, Sylow
125C83#263711yescharacteristic, complemented, Sylow
126C2×#126—831no—
127C2×C4×Q8—831no—
128#398—831no—
129#398—832no—
130#397D8311yescharacteristic
131#396—832no—
132#395—831no—
133#393—831no—
134#388—832no—
135#387—832no—
136#382—831no—
137C22×Q8—831no—
138#126—1664no—
139C4×Q8—1664no—
140C2×C4⋊C4—832no—
141C2×C4⋊C4—832no—
142C2×C4⋊C4—832no—
143C2×C4⋊C4—831no—
144C2×C4⋊C4—831no—
145C2×C4⋊C4—831no—
146C2×C4⋊C4—832no—
147C2×C4⋊C4C2×D8311yescharacteristic
148C2×C22⋊C4—831no—
149C2×C22⋊C4—831no—
150C2×C22⋊C4—832no—
151C2×C22⋊C4C2×D8311yescharacteristic
152C2×C22⋊C4C2×D8312yes—
153C2×C42—831no—
154C2×C42—832no—
155#95—832no—
156#95—831no—
157#95—832no—
158#95—832no—
159#95—832no—
160#95C2×D8311yescharacteristic
161#95C2×D8312yes—
162C24C22×D8311yescharacteristic
163C2×Q8—1664no—
164C2×Q8—1662no—
165C22×C4—831no—
166C22×C4—832no—
167C22×C4—831no—
168C22×C4—832no—
169C22×C4—831no—
170C22×C4—831no—
171C22×C4C22×D8311yescharacteristic
172C22×C4C22×D8312yes—
173C22×C4C22×D8312yes—
174C22×C4C22×D8311yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—3322no—
177C4⋊C4—3322no—
178C4⋊C4—1664no—
179C4⋊C4—1662no—
180C4⋊C4—1662no—
181C4⋊C4—1664no—
182C4⋊C4—1662no—
183C22⋊C4—22no—
184C22⋊C4—42no—
185C22⋊C4—1662no—
186C22⋊C4—1662no—
187C22⋊C4—1664no—
188C42—834no—
189C42—1664no—
190C23—21no—
191C23—22no—
192C23—21no—
193C23—22no—
194C23—21no—
195C23C23×D8311yescharacteristic
196Q8—1668no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—832no—
207C2×C4—832no—
208C2×C4—834no—
209C2×C4—1661no—
210C2×C4—1661no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4—1661no—
214C2×C4—1662no—
215C2×C4—1661no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1661no—
220C2×C4—1662no—
221C2×C4—1661no—
222C2×C4D4×D8312yes—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22C2×D4×D8311yescharacteristic
229C22#41×D8312yes—
230C22#1458183711yescharacteristic
231C22#1458183912yes—
232C22#1458183911yescharacteristic
233C4—22no—
234C4—41no—
235C4—42no—
236C4—42no—
237C4—3321no—
238C4—3321no—
239C4—3322no—
240C4—1664no—
241C4—1662no—
242C4—1662no—
243C2—81no—
244C2#1912820911yescharacteristic
245C2#1912827111yescharacteristic
246C2#1912828011yescharacteristic
247C2#1912830812yes—
248C2#1912831912yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC24×C83C23
hypercenter#397D83
hypocenterC83#2637
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. #19127635C2semidirect
4. #19127632C2semidirect
6. #19127622C2semidirect
9. #19127584C2semidirect
10. #19127565C2semidirect
125. C83#2637semidirect

Maximal subgroups

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

subgroupindex
C2×#145817852
C2×#145817572
#191276352
#191276322
#191276242
#191276222
#191276132
#191275952
#191275842
#191275652
#397×C832
#263783