Finite Group Databasebeta

← #24915501

Subgroups — #24915501

Order 10624.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14581769C212yesmaximal
2#19127643C212yesmaximal, complemented
3#19127642C212yesmaximal, complemented
4#19127629C212yesmaximal
5#19127626C211yesmaximal, characteristic
6#19127622C212yesmaximal, complemented
7#19127621C212yesmaximal, complemented
8#19127562C211yesmaximal, characteristic
9#397×C83C211yesmaximal, characteristic
10C22×Dic166C2212yes—
11C23×Dic83C2211yescharacteristic
12#14581769—28no—
13C2×C4⋊Dic83C2212yes—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2211yescharacteristic
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C83⋊#95C2212yes—
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C22⋊C4×C83C2212yes—
29C2×C22⋊C4×C83C2211yescharacteristic
30#95×C83C2212yes—
31#95×C83C2211yescharacteristic
32C2×Dic166—24no—
33C2×Dic166—42no—
34C2×Dic166—42no—
35C22×Dic83—41no—
36C22×Dic83—41no—
37C22×Dic83D414yes—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C4⋊Dic83—24no—
43C4⋊Dic83—41no—
44Dic83⋊C4—24no—
45Dic83⋊C4—24no—
46Dic83⋊C4—42no—
47C22⋊Dic83—24no—
48C22⋊Dic83—41no—
49C24×C83C2311yescharacteristic
50C22×C4×C83C2312yes—
51C22×C4×C83C2311yescharacteristic
52C22×C4×C83C2312yes—
53C22×C4×C83C2311yescharacteristic
54C4⋊C4×C83—41no—
55C22⋊C4×C83—24no—
56C22⋊C4×C83—41no—
57C23×C83—21no—
58C23×C83—21no—
59C23×C83—22no—
60C23×C83—22no—
61C23×C83C2×D412yes—
62C23×C83C2411yescharacteristic
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—21no—
66C2×C4×C83—21no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—22no—
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—44no—
85C2×Dic83C2×D414yes—
86Dic166—48no—
87C22×C83—22no—
88C22×C83—24no—
89C22×C83—41no—
90C22×C83—42no—
91C22×C83—41no—
92C22×C83C22×D411yescharacteristic
93C22×C83H(5,𝔽2)11yescharacteristic
94C22×C83H(5,𝔽2)11yescharacteristic
95C22×C832−1+412yes—
96C22×C832−1+412yes—
97C4×C83—42no—
98C4×C83—42no—
99C4×C83—41no—
100C4×C83—41no—
101Dic83—24no—
102Dic83—42no—
103Dic83—42no—
104Dic83—42no—
105C2×C83—42no—
106C2×C83#53411yescharacteristic
107C2×C83#53612yes—
108C2×C83#56311yescharacteristic
109C2×C83#56312yes—
110C2×C83#56311yescharacteristic
111#2817—831nomaximal, Sylow
112C83#281711yescharacteristic, complemented, Sylow
113C2×C22⋊Q8—832no—
114#400—832no—
115#400—832no—
116#397—831no—
117#397—832no—
118#397D8311yescharacteristic
119#395—832no—
120#395—832no—
121#381—831no—
122C22×Q8—832no—
123C23×C4—831no—
124C22⋊Q8—1668no—
125C2×C4⋊C4—832no—
126C2×C4⋊C4—831no—
127C2×C4⋊C4—832no—
128C2×C4⋊C4—832no—
129C2×C4⋊C4—832no—
130C2×C4⋊C4C2×D8311yescharacteristic
131C2×C22⋊C4—831no—
132C2×C22⋊C4—832no—
133C2×C22⋊C4C2×D8312yes—
134C2×C22⋊C4C2×D8311yescharacteristic
135#95—832no—
136#95—832no—
137#95—832no—
138#95—832no—
139#95—832no—
140#95—832no—
141#95—831no—
142#95C2×D8312yes—
143#95C2×D8311yescharacteristic
144C24C22×D8311yescharacteristic
145C2×Q8—3322no—
146C2×Q8—3322no—
147C2×Q8—1664no—
148C22×C4—3321no—
149C22×C4—3321no—
150C22×C4—834no—
151C22×C4—832no—
152C22×C4—832no—
153C22×C4—832no—
154C22×C4—832no—
155C22×C4C22×D8311yescharacteristic
156C22×C4C22×D8311yescharacteristic
157C22×C4C22×D8312yes—
158C22×C4C22×D8312yes—
159C4⋊C4—41no—
160C4⋊C4—3322no—
161C4⋊C4—3321no—
162C4⋊C4—1664no—
163C4⋊C4—1664no—
164C4⋊C4—1664no—
165C22⋊C4—24no—
166C22⋊C4—41no—
167C22⋊C4—3321no—
168C22⋊C4—1664no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—22no—
173C23D4×D8312yes—
174C23C23×D8311yescharacteristic
175Q8—3328no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—3324no—
187C2×C4—834no—
188C2×C4—1662no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4—1662no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1662no—
198C2×C4—1662no—
199C22—22no—
200C22—24no—
201C22—41no—
202C22—42no—
203C22—41no—
204C22C2×D4×D8311yescharacteristic
205C22#1458183711yescharacteristic
206C22#1458183811yescharacteristic
207C22#1458183912yes—
208C22#1458183912yes—
209C4—41no—
210C4—41no—
211C4—42no—
212C4—42no—
213C4—3322no—
214C4—3322no—
215C4—3322no—
216C4—1664no—
217C2—42no—
218C2#1912825911yescharacteristic
219C2#1912827312yes—
220C2#1912845111yescharacteristic
221C2#1912845312yes—
222C2#1912845411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC24×C83C23
hypercenter#397D83
hypocenterC83#2817
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #19127643C2semidirect
3. #19127642C2semidirect
6. #19127622C2semidirect
7. #19127621C2semidirect
112. C83#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#145817692
#191276432
#191276422
#191276292
#191276262
#191276222
#191276212
#191275622
#397×C832
#281783