Finite Group Databasebeta

← #24912445

Subgroups — #24912445

Order 10624.

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

normal subgroups only

This group has 576 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×#14581792C211yesmaximal, characteristic, complemented
2C2×#14581779C211yesmaximal, characteristic
3C2×#14581779C211yesmaximal, characteristic
4C2×#14581757C211yesmaximal, characteristic, complemented
5#19127624C214yesmaximal, complemented
6#19127606C211yesmaximal, characteristic, complemented
7#19127594C214yesmaximal, complemented
8C4×Dic83⋊C4C211yesmaximal, characteristic, complemented
9#390×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C22×Dic166C2211yescharacteristic
12#14581792—24no—
13#14581779—24no—
14#14581779—24no—
15#14581757—24no—
16C2×C4⋊Dic83C2214yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2214yes—
21C2×C22⋊Dic83C2212yes—
22C2×C4×Dic83C2211yescharacteristic
23C2×C4×Dic83C2212yes—
24C2×C4×Dic83C2211yescharacteristic
25C83⋊#95C2212yes—
26C83⋊#95C2214yes—
27C22×D4×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2214yes—
30C2×C42×C83C2211yescharacteristic
31C2×Dic166—22no—
32C2×Dic166—22no—
33C2×Dic166—22no—
34C2×Dic166—22no—
35C2×Dic166—22no—
36C2×Dic166D414yes—
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2311yescharacteristic
41C22×Dic83C2311yescharacteristic
42C4⋊Dic83—28no—
43Dic83⋊C4—24no—
44Dic83⋊C4—24no—
45C22⋊Dic83—28no—
46C22⋊Dic83—24no—
47C22⋊Dic83—24no—
48C4×Dic83—22no—
49C4×Dic83—24no—
50C4×Dic83D414yes—
51C24×C83C2312yes—
52C2×D4×C83—22no—
53C2×D4×C83—22no—
54C2×D4×C83—22no—
55C22×C4×C83C2312yes—
56C22×C4×C83C2312yes—
57C22×C4×C83C2311yescharacteristic
58C4⋊C4×C83—22no—
59C22⋊C4×C83—28no—
60C42×C83—22no—
61C23×C83—22no—
62C23×C83—22no—
63C23×C83—24no—
64C23×C83—22no—
65C23×C83—24no—
66C23×C83C2411yescharacteristic
67D4×C83—44no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—24no—
72C2×C4×C83C2×D412yes—
73C2×C4×C83C2×D412yes—
74C2×C4×C83#4114yes—
75C2×C4×C83#4112yes—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—22no—
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×Dic83C2×D414yes—
89Dic166—28no—
90Dic166—44no—
91C22×C83—42no—
92C22×C83—42no—
93C22×C83—42no—
94C22×C83—44no—
95C22×C83—44no—
96C22×C83C22×D411yescharacteristic
97C22×C83C22×D411yescharacteristic
98C22×C83C2×#4112yes—
99C22×C83C2×#4111yescharacteristic
100C22×C832−1+412yes—
101C4×C83—24no—
102C4×C83—42no—
103C4×C83#12114yes—
104Dic83—24no—
105Dic83—22no—
106Dic83—22no—
107Dic83—42no—
108Dic83—42no—
109C2×C83—82no—
110C2×C83C2×#12111yescharacteristic
111C2×C83#53211yescharacteristic
112C2×C83#53611yescharacteristic
113C2×C83#54812yes—
114C2×C83#55612yes—
115#2545—831nomaximal, Sylow
116C83#254511yescharacteristic, complemented, Sylow
117C2×C4⋊Q8—831no—
118C2×#124—831no—
119C2×#124—831no—
120C2×C4×Q8—831no—
121#396—834no—
122#391—831no—
123#390D8311yescharacteristic
124#388—834no—
125C4×C4⋊C4—831no—
126C22×Q8—831no—
127C22×Q8—831no—
128C22×D4C2×D8311yescharacteristic
129C4⋊Q8—1664no—
130#124—1664no—
131#124—1664no—
132C4×Q8—1664no—
133C2×C4⋊C4—832no—
134C2×C4⋊C4—834no—
135C2×C4⋊C4—832no—
136C2×C4⋊C4C2×D8311yescharacteristic
137C2×C22⋊C4—832no—
138C2×C22⋊C4—834no—
139C2×C22⋊C4—832no—
140C2×C22⋊C4C2×D8314yes—
141C2×C42—831no—
142C2×C42—832no—
143C2×C42—831no—
144C2×C42C2×D8311yescharacteristic
145#95—832no—
146#95—834no—
147C24C22×D8312yes—
148C2×Q8—834no—
149C2×Q8—1662no—
150C2×Q8—1662no—
151C2×Q8—1662no—
152C2×Q8—1662no—
153C2×Q8—1662no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C22×C4—831no—
158C22×C4—831no—
159C22×C4—832no—
160C22×C4—832no—
161C22×C4—832no—
162C22×C4C22×D8312yes—
163C22×C4C22×D8312yes—
164C22×C4C22×D8311yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—1664no—
167C4⋊C4—1668no—
168C4⋊C4—1664no—
169C22⋊C4—28no—
170C22⋊C4—1664no—
171C22⋊C4—1664no—
172C22⋊C4—1668no—
173C42—22no—
174C42—834no—
175C42—1664no—
176C42—1662no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—22no—
181C23—24no—
182C23C23×D8311yescharacteristic
183Q8—3324no—
184Q8—1668no—
185D4—44no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—832no—
191C2×C4—832no—
192C2×C4—834no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4D4×D8312yes—
204C2×C4D4×D8312yes—
205C2×C4#1305058914yes—
206C2×C4#1305059012yes—
207C22—44no—
208C22—44no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22C2×D4×D8311yescharacteristic
213C22C2×D4×D8311yescharacteristic
214C22C2×#1305058912yes—
215C22C2×#1305059011yescharacteristic
216C22#1458183912yes—
217C4—24no—
218C4—42no—
219C4—3322no—
220C4—3322no—
221C4—1664no—
222C4—1662no—
223C4—1662no—
224C4#1458176514yes—
225C2—82no—
226C2C2×#1458176511yescharacteristic
227C2#1912824911yescharacteristic
228C2#1912827111yescharacteristic
229C2#1912835712yes—
230C2#1912842112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#390×C83C2
prime radicalC22×D4×C83C22
hypercenter#390D83
hypocenterC83#2545
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#14581792C2semidirect
4. C2×#14581757C2semidirect
5. #19127624C2semidirect
6. #19127606C2semidirect
7. #19127594C2semidirect
8. C4×Dic83⋊C4C2semidirect
116. C83#2545semidirect

Maximal subgroups

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

subgroupindex
C2×#145817922
C2×#145817792
C2×#145817572
#191276242
#191276062
#191275942
C4×Dic83⋊C42
#390×C832
#254583