Finite Group Databasebeta

← #24914429

Subgroups — #24914429

Order 10624.

512 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 133 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, complemented
2#19127643C212yesmaximal, complemented
3#19127631C211yesmaximal, characteristic, complemented
4#19127629C212yesmaximal, complemented
5#19127606C211yesmaximal, characteristic, complemented
6#19127595C212yesmaximal, complemented
7#19127581C212yesmaximal
8#19127576C212yesmaximal, complemented
9#394×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C23×Dic83C2212yes—
12#14581769—28no—
13C2×C4⋊Dic83C2212yes—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C4×Dic83C2211yescharacteristic
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2211yescharacteristic
27C22×D4×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2212yes—
30C2×C22⋊C4×C83C2212yes—
31#95×C83C2211yescharacteristic
32C2×Dic166—41no—
33C2×Dic166—41no—
34C2×Dic166D414yescomplemented
35C22×Dic83—24no—
36C22×Dic83—42no—
37C22×Dic83—42no—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C4⋊Dic83—24no—
43C4⋊Dic83—41no—
44Dic83⋊C4—24no—
45Dic83⋊C4—24no—
46C22⋊Dic83—24no—
47C22⋊Dic83—24no—
48C22⋊Dic83—42no—
49C4×Dic83—41no—
50C24×C83C2312yes—
51C2×D4×C83—24no—
52C2×D4×C83—41no—
53C22×C4×C83C2312yes—
54C22×C4×C83C2311yescharacteristic
55C22×C4×C83C2311yescharacteristic
56C22×C4×C83C2311yescharacteristic
57C4⋊C4×C83—41no—
58C22⋊C4×C83—24no—
59C22⋊C4×C83—24no—
60C23×C83—22no—
61C23×C83—22no—
62C23×C83—22no—
63C23×C83—22no—
64C23×C83—22no—
65C23×C83—22no—
66C23×C83#4114yes—
67C23×C83C2411yescharacteristic
68D4×C83—44no—
69C2×C4×C83—21no—
70C2×C4×C83—21no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—21no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83C2×D412yes—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—48no—
91C2×Dic83C2×D414yes—
92Dic166—44no—
93C22×C83—24no—
94C22×C83—24no—
95C22×C83—24no—
96C22×C83—42no—
97C22×C83—42no—
98C22×C83—42no—
99C22×C83—42no—
100C22×C83C22×D411yescharacteristic
101C22×C83C2×#4112yes—
102C22×C83H(5,𝔽2)11yescharacteristic
103C22×C83H(5,𝔽2)11yescharacteristic
104C22×C832−1+412yes—
105C4×C83—22no—
106C4×C83—42no—
107C4×C83—41no—
108C4×C83—41no—
109Dic83—24no—
110Dic83—42no—
111Dic83—42no—
112Dic83—42no—
113C2×C83—44no—
114C2×C83#53411yescharacteristic
115C2×C83#53912yes—
116C2×C83#54812yes—
117C2×C83#55111yescharacteristic
118C2×C83#56311yescharacteristic
119#2728—831nomaximal, Sylow
120C83#272811yescharacteristic, complemented, Sylow
121C2×C22⋊Q8—832no—
122#400—832no—
123#398—831no—
124#397—832no—
125#394D8311yescharacteristic
126#391—831no—
127#388—832no—
128#386—832no—
129#385—832no—
130C22×Q8—831no—
131C22×D4C2×D8311yescharacteristic
132C23×C4—832no—
133C22⋊Q8—1668no—
134C2×C4⋊C4—831no—
135C2×C4⋊C4—832no—
136C2×C4⋊C4—832no—
137C2×C4⋊C4—832no—
138C2×C4⋊C4C2×D8311yescharacteristic
139C2×C22⋊C4—832no—
140C2×C22⋊C4—832no—
141C2×C22⋊C4—832no—
142C2×C22⋊C4C2×D8312yes—
143C2×C22⋊C4C2×D8312yes—
144C2×C42—831no—
145#95—832no—
146#95—832no—
147#95—832no—
148#95—832no—
149#95—832no—
150#95—831no—
151#95C2×D8311yescharacteristic
152C24C22×D8312yes—
153C2×Q8—3321no—
154C2×Q8—3321no—
155C2×Q8—834no—
156C2×D4—24no—
157C2×D4—41no—
158C22×C4—3322no—
159C22×C4—3322no—
160C22×C4—832no—
161C22×C4—832no—
162C22×C4—832no—
163C22×C4—832no—
164C22×C4—1664no—
165C22×C4C22×D8311yescharacteristic
166C22×C4C22×D8311yescharacteristic
167C22×C4C22×D8312yes—
168C22×C4C22×D8311yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—3321no—
171C4⋊C4—1664no—
172C4⋊C4—1664no—
173C4⋊C4—1664no—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—3322no—
177C22⋊C4—1664no—
178C22⋊C4—1664no—
179C42—3321no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23#1305058914yes—
187C23C23×D8311yescharacteristic
188Q8—3324no—
189D4—44no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—3328no—
200C2×C4—834no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4—1662no—
207C2×C4—1662no—
208C2×C4—1662no—
209C2×C4—1662no—
210C2×C4—1662no—
211C2×C4—1662no—
212C2×C4D4×D8312yes—
213C22—24no—
214C22—24no—
215C22—24no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22C2×D4×D8311yescharacteristic
221C22C2×#1305058912yes—
222C22#1458183711yescharacteristic
223C22#1458183811yescharacteristic
224C22#1458183912yes—
225C4—22no—
226C4—42no—
227C4—41no—
228C4—41no—
229C4—3322no—
230C4—3322no—
231C4—3322no—
232C4—1664no—
233C2—44no—
234C2#1912825911yescharacteristic
235C2#1912829512yes—
236C2#1912835712yes—
237C2#1912838211yescharacteristic
238C2#1912845111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC22×D4×C83C22
hypercenter#394D83
hypocenterC83#2728
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14581769C2semidirect
2. #19127643C2semidirect
3. #19127631C2semidirect
4. #19127629C2semidirect
5. #19127606C2semidirect
6. #19127595C2semidirect
8. #19127576C2semidirect
34. C2×Dic166D4semidirect
120. C83#2728semidirect

Maximal subgroups

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

subgroupindex
C2×#145817692
#191276432
#191276312
#191276292
#191276062
#191275952
#191275812
#191275762
#394×C832
#272883