Finite Group Databasebeta

← #24912460

Subgroups — #24912460

Order 10624.

604 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14581790C211yesmaximal, characteristic, complemented
2C2×#14581781C211yesmaximal, characteristic
3C2×#14581780C211yesmaximal, characteristic
4C2×#14581754C211yesmaximal, characteristic
5#19127630C214yesmaximal, complemented
6#19127606C211yesmaximal, characteristic, complemented
7#19127560C214yesmaximal, complemented
8C4×Dic83⋊C4C211yesmaximal, characteristic, complemented
9#391×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2211yescharacteristic
11C22×D332C2211yescharacteristic
12C22×C4×D83C2212yes—
13#14581790—24no—
14#14581781—24no—
15#14581780—24no—
16#14581754—24no—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2214yes—
21C2×(C2×D83)⋊C4C2214yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×C4×Dic83C2211yescharacteristic
24C2×C4×Dic83C2211yescharacteristic
25C83⋊#95C2212yes—
26C83⋊#95C2214yes—
27C22×Q8×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C42×C83C2211yescharacteristic
30#95×C83C2214yes—
31C23×D83C2312yes—
32C2×Dic166—22no—
33C2×Dic166—22no—
34C2×Dic166—22no—
35C2×D332—22no—
36C2×D332—22no—
37C2×D332D414yes—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2311yescharacteristic
41C22×Dic83C2311yescharacteristic
42C2×C4×D83—24no—
43C2×C4×D83—28no—
44Dic83⋊C4—24no—
45Dic83⋊C4—24no—
46Dic83⋊C4—24no—
47(C2×D83)⋊C4—28no—
48(C2×D83)⋊C4—28no—
49(C2×D83)⋊C4—24no—
50C4×Dic83—22no—
51C4×Dic83D414yes—
52C2×Q8×C83—22no—
53C2×Q8×C83—22no—
54C2×Q8×C83—22no—
55C22×C4×C83C2312yes—
56C22×C4×C83C2312yes—
57C22×C4×C83C2312yes—
58C22×C4×C83C2311yescharacteristic
59C4⋊C4×C83—22no—
60C42×C83—22no—
61C23×C83C2411yescharacteristic
62C22×D83—22no—
63C22×D83—24no—
64C22×D83—22no—
65C22×D83—24no—
66C22×D83C2×D414yes—
67Q8×C83—44no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—24no—
75C2×C4×C83—22no—
76C2×C4×C83C2×D412yes—
77C2×C4×C83C2×D412yes—
78C2×C4×C83#4112yes—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83C2×D412yes—
87C2×Dic83C2×D412yes—
88C2×Dic83#4114yes—
89D332—28no—
90C4×D83—48no—
91Dic166—44no—
92C22×C83C22×D411yescharacteristic
93C22×C83C22×D411yescharacteristic
94C22×C83C2×#4112yes—
95C22×C83C2×#4111yescharacteristic
96C22×C832−1+412yes—
97C2×D83—24no—
98C2×D83—28no—
99C2×D83—42no—
100C2×D83—42no—
101C2×D83—44no—
102C4×C83—42no—
103C4×C83—42no—
104C4×C83—42no—
105C4×C83#12114yes—
106Dic83—22no—
107Dic83—42no—
108Dic83—42no—
109Dic83#12414yescomplemented
110C2×C83C2×#12111yescharacteristic
111C2×C83C2×#12411yescharacteristic
112C2×C83#53611yescharacteristic
113C2×C83#54712yes—
114C2×C83#55612yes—
115D83—44no—
116#2547—831nomaximal, Sylow
117C83#254711yescharacteristic, complemented, Sylow
118C2×C4⋊Q8—831no—
119C2×#124—831no—
120C2×#124—831no—
121C2×C4×D4—831no—
122#397—834no—
123#391—831no—
124#391D8311yescharacteristic, complemented
125#380—834no—
126C4×C4⋊C4—831no—
127C22×Q8—831no—
128C22×Q8C2×D8311yescharacteristic
129C22×D4—831no—
130C23×C4—832no—
131C4⋊Q8—1664no—
132#124—1664no—
133#124—1664no—
134C4×D4—1664no—
135C2×C4⋊C4—832no—
136C2×C4⋊C4—832no—
137C2×C4⋊C4—832no—
138C2×C4⋊C4C2×D8311yescharacteristic
139C2×C22⋊C4—832no—
140C2×C22⋊C4—834no—
141C2×C22⋊C4—834no—
142C2×C42—831no—
143C2×C42—831no—
144C2×C42C2×D8311yescharacteristic
145#95—832no—
146#95—834no—
147#95C2×D8314yes—
148C24—832no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—1662no—
153C2×Q8—1662no—
154C2×Q8—1662no—
155C2×D4—834no—
156C2×D4—1662no—
157C2×D4—1662no—
158C22×C4—831no—
159C22×C4—831no—
160C22×C4—832no—
161C22×C4—832no—
162C22×C4—1664no—
163C22×C4—1668no—
164C22×C4C22×D8312yes—
165C22×C4C22×D8312yes—
166C22×C4C22×D8312yes—
167C22×C4C22×D8311yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—1664no—
170C4⋊C4—1664no—
171C4⋊C4—1664no—
172C22⋊C4—1664no—
173C22⋊C4—1668no—
174C22⋊C4—1668no—
175C42—22no—
176C42—834no—
177C42—1662no—
178C23—834no—
179C23—1662no—
180C23—1664no—
181C23—1662no—
182C23—1664no—
183C23C23×D8311yescharacteristic
184Q8—44no—
185Q8—3324no—
186D4—1668no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—3328no—
196C2×C4—832no—
197C2×C4—834no—
198C2×C4—832no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4D4×D8312yes—
207C2×C4D4×D8312yes—
208C2×C4#1305059012yes—
209C22—3324no—
210C22—3322no—
211C22—3322no—
212C22—1668no—
213C22—1664no—
214C22C2×D4×D8311yescharacteristic
215C22C2×D4×D8311yescharacteristic
216C22#41×D8312yes—
217C22C2×#1305059011yescharacteristic
218C22#1458184012yes—
219C4—42no—
220C4—42no—
221C4—42no—
222C4—3322no—
223C4—3322no—
224C4—834no—
225C4—1662no—
226C4#1458176514yes—
227C2—3324no—
228C2C2×#1458176511yescharacteristic
229C2#124×D8311yescharacteristic
230C2#1912827611yescharacteristic
231C2#1912835312yes—
232C2#1912842312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#391×C83C2
prime radicalC22×D332C22
hypercenter#391D83
hypocenterC83#2547
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#14581790C2semidirect
5. #19127630C2semidirect
6. #19127606C2semidirect
7. #19127560C2semidirect
8. C4×Dic83⋊C4C2semidirect
9. #391×C83C2semidirect
109. Dic83#124semidirect
117. C83#2547semidirect
124. #391D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817902
C2×#145817812
C2×#145817802
C2×#145817542
#191276302
#191276062
#191275602
C4×Dic83⋊C42
#391×C832
#254783