Finite Group Databasebeta

← #24914366

Subgroups — #24914366

Order 10624.

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

normal subgroups only

This group has 448 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×#14581785C212yesmaximal
2#19127635C212yesmaximal, complemented
3#19127635C212yesmaximal, complemented
4#19127626C211yesmaximal, characteristic
5#19127596C212yesmaximal, complemented
6#19127588C212yesmaximal, complemented
7#19127573C211yesmaximal, characteristic
8#19127559C212yesmaximal
9#397×C83C211yesmaximal, characteristic
10C23×Dic83C2211yescharacteristic
11#14581785—28no—
12C2×C4⋊Dic83C2212yes—
13C2×C4⋊Dic83C2211yescharacteristic
14C2×Dic83⋊C4C2212yes—
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2211yescharacteristic
21C2×C4×Dic83C2212yes—
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
32C22×Dic83—24no—
33C22×Dic83—41no—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C4⋊Dic83—24no—
39C4⋊Dic83—41no—
40Dic83⋊C4—24no—
41Dic83⋊C4—24no—
42Dic83⋊C4—24no—
43Dic83⋊C4—24no—
44Dic83⋊C4—42no—
45C22⋊Dic83—24no—
46C22⋊Dic83D414yes—
47C4×Dic83—24no—
48C24×C83C2311yescharacteristic
49C22×C4×C83C2311yescharacteristic
50C22×C4×C83C2312yes—
51C22×C4×C83C2311yescharacteristic
52C22×C4×C83C2312yes—
53C4⋊C4×C83—41no—
54C22⋊C4×C83—24no—
55C22⋊C4×C83—41no—
56C23×C83—22no—
57C23×C83—21no—
58C23×C83—21no—
59C23×C83—22no—
60C23×C83C2×D412yes—
61C23×C83C2411yescharacteristic
62C2×C4×C83—22no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×Dic83—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—44no—
83C2×Dic83C2×D414yes—
84C2×Dic83#4114yes—
85C22×C83—22no—
86C22×C83—24no—
87C22×C83—42no—
88C22×C83—41no—
89C22×C83—41no—
90C22×C83C22×D411yescharacteristic
91C22×C83C2×#4112yes—
92C22×C83H(5,𝔽2)11yescharacteristic
93C22×C832−1+411yescharacteristic
94C22×C832−1+412yes—
95C4×C83—42no—
96C4×C83—42no—
97C4×C83—41no—
98C4×C83—41no—
99Dic83—24no—
100Dic83—24no—
101Dic83—42no—
102Dic83—42no—
103C2×C83—42no—
104C2×C83#53711yescharacteristic
105C2×C83#54112yes—
106C2×C83#54712yes—
107C2×C83#55111yescharacteristic
108C2×C83#56311yescharacteristic
109#2722—831nomaximal, Sylow
110C83#272211yescharacteristic, complemented, Sylow
111C2×#126—832no—
112#398—832no—
113#398—832no—
114#397—831no—
115#397D8311yescharacteristic
116#389—832no—
117#387—832no—
118#385—831no—
119#380—832no—
120C23×C4—831no—
121#126—1668no—
122C2×C4⋊C4—832no—
123C2×C4⋊C4—831no—
124C2×C4⋊C4—832no—
125C2×C4⋊C4—832no—
126C2×C4⋊C4—832no—
127C2×C4⋊C4—832no—
128C2×C4⋊C4—832no—
129C2×C4⋊C4C2×D8311yescharacteristic
130C2×C22⋊C4—831no—
131C2×C22⋊C4—832no—
132C2×C22⋊C4C2×D8312yes—
133C2×C22⋊C4C2×D8311yescharacteristic
134C2×C42—832no—
135#95—832no—
136#95—832no—
137#95—832no—
138#95—831no—
139#95—832no—
140#95C2×D8312yes—
141#95C2×D8311yescharacteristic
142C24C22×D8311yescharacteristic
143C22×C4—3321no—
144C22×C4—832no—
145C22×C4—832no—
146C22×C4—832no—
147C22×C4—832no—
148C22×C4—1664no—
149C22×C4C22×D8311yescharacteristic
150C22×C4C22×D8311yescharacteristic
151C22×C4C22×D8312yes—
152C22×C4C22×D8312yes—
153C4⋊C4—41no—
154C4⋊C4—3322no—
155C4⋊C4—3321no—
156C4⋊C4—1664no—
157C4⋊C4—1664no—
158C4⋊C4—1664no—
159C4⋊C4—1664no—
160C4⋊C4—1664no—
161C22⋊C4—24no—
162C22⋊C4—41no—
163C22⋊C4—834no—
164C22⋊C4—1664no—
165C42—1664no—
166C23—22no—
167C23—21no—
168C23—22no—
169C23—21no—
170C23D4×D8312yes—
171C23C23×D8311yescharacteristic
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—3324no—
183C2×C4—834no—
184C2×C4—834no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4—1662no—
188C2×C4—1662no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4—1662no—
193C2×C4—1662no—
194C2×C4—1662no—
195C22—24no—
196C22—22no—
197C22—41no—
198C22—41no—
199C22—42no—
200C22C2×D4×D8311yescharacteristic
201C22#41×D8312yes—
202C22#1458183711yescharacteristic
203C22#1458183912yes—
204C22#1458183911yescharacteristic
205C4—42no—
206C4—41no—
207C4—41no—
208C4—42no—
209C4—3322no—
210C4—3322no—
211C4—1664no—
212C4—1664no—
213C2—42no—
214C2#1912827811yescharacteristic
215C2#1912830812yes—
216C2#1912834912yes—
217C2#1912837911yescharacteristic
218C2#1912845111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC24×C83C23
hypercenter#397D83
hypocenterC83#2722
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #19127635C2semidirect
3. #19127635C2semidirect
5. #19127596C2semidirect
6. #19127588C2semidirect
110. C83#2722semidirect

Maximal subgroups

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

subgroupindex
C2×#145817852
#191276352
#191276262
#191275962
#191275882
#191275732
#191275592
#397×C832
#272283