Finite Group Databasebeta

← #24913284

Subgroups — #24913284

Order 10624.

488 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 165 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
2#19127640C212yesmaximal, complemented
3#19127637C212yesmaximal
4#19127621C212yesmaximal, complemented
5#19127613C212yesmaximal
6#19127599C212yesmaximal, complemented
7#19127567C211yesmaximal, characteristic, complemented
8#19127561C212yesmaximal
9C2×#117×C83C211yesmaximal, characteristic
10C22×Dic166C2212yes—
11#14581792—24no—
12C2×C4⋊Dic83C2212yes—
13C2×C4⋊Dic83C2212yes—
14C2×C4⋊Dic83C2212yes—
15C2×Dic83⋊C4C2214yes—
16C2×C22⋊Dic83C2214yes—
17C83⋊#95C2214yes—
18C83⋊#95C2212yes—
19C83⋊#95C2212yes—
20C83⋊#95C2214yes—
21C23×C4×C83C2211yescharacteristic
22#117×C83—24no—
23C2×C4⋊C4×C83C2211yescharacteristic
24C2×C4⋊C4×C83C2211yescharacteristic
25C2×C22⋊C4×C83C2211yescharacteristic
26C2×C22⋊C4×C83C2211yescharacteristic
27C2×C42×C83C2211yescharacteristic
28C2×C42×C83C2211yescharacteristic
29C2×Dic166—24no—
30C2×Dic166—44no—
31C22×Dic83C2314yes—
32C22×Dic83C2314yes—
33C4⋊Dic83—24no—
34C4⋊Dic83—24no—
35C4⋊Dic83—24no—
36Dic83⋊C4—44no—
37C22⋊Dic83—44no—
38C24×C83C2311yescharacteristic
39C22×C4×C83—22no—
40C22×C4×C83—22no—
41C22×C4×C83Q814yescomplemented
42C22×C4×C83C2311yescharacteristic
43C22×C4×C83C2311yescharacteristic
44C22×C4×C83C2311yescharacteristic
45C22×C4×C83C2311yescharacteristic
46C22×C4×C83C2311yescharacteristic
47C22×C4×C83C2311yescharacteristic
48C4⋊C4×C83—22no—
49C4⋊C4×C83—22no—
50C22⋊C4×C83—22no—
51C22⋊C4×C83—22no—
52C42×C83—22no—
53C42×C83D414yes—
54C23×C83—21no—
55C23×C83—21no—
56C23×C83—22no—
57C23×C83—22no—
58C23×C83C2×Q812yes—
59C23×C83C2411yescharacteristic
60C2×C4×C83—21no—
61C2×C4×C83—21no—
62C2×C4×C83—22no—
63C2×C4×C83—22no—
64C2×C4×C83—28no—
65C2×C4×C83—21no—
66C2×C4×C83—21no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83C2×D412yes—
70C2×C4×C83C2×D412yes—
71C2×C4×C83C2×D412yes—
72C2×C4×C83C2×Q812yes—
73C2×C4×C83C2×Q812yes—
74C2×C4×C83#4112yes—
75C2×Dic83—24no—
76C2×Dic83—24no—
77C2×Dic83—24no—
78C2×Dic83—24no—
79C2×Dic83—24no—
80C2×Dic83—24no—
81Dic166—48no—
82C22×C83—22no—
83C22×C83—24no—
84C22×C83—41no—
85C22×C83—41no—
86C22×C83—42no—
87C22×C83C22×D411yescharacteristic
88C22×C83C22×Q811yescharacteristic
89C22×C83C2×#4111yescharacteristic
90C22×C83H(5,𝔽2)12yes—
91C22×C832−1+412yes—
92C4×C83—22no—
93C4×C83—22no—
94C4×C83—22no—
95C4×C83—41no—
96C4×C83—41no—
97C4×C83C22⋊Q814yes—
98Dic83—44no—
99Dic83—44no—
100C2×C83—42no—
101C2×C83C2×C22⋊Q811yescharacteristic
102C2×C83#53511yescharacteristic
103C2×C83#53611yescharacteristic
104C2×C83#54212yes—
105C2×C83#54412yes—
106#2628—831nomaximal, Sylow
107C83#262811yescharacteristic, complemented, Sylow
108C2×C4⋊Q8—831no—
109C2×#117D8311yescharacteristic
110#400—832no—
111#399—832no—
112#395—832no—
113#393—832no—
114#389—832no—
115#382—831no—
116#380—832no—
117C22×Q8—832no—
118C23×C4C2×D8311yescharacteristic
119C4⋊Q8—1664no—
120#117—24no—
121C2×C4⋊C4—834no—
122C2×C4⋊C4—832no—
123C2×C4⋊C4—832no—
124C2×C4⋊C4—832no—
125C2×C4⋊C4C2×D8311yescharacteristic
126C2×C4⋊C4C2×D8311yescharacteristic
127C2×C22⋊C4—834no—
128C2×C22⋊C4C2×D8311yescharacteristic
129C2×C22⋊C4C2×D8311yescharacteristic
130C2×C42C2×D8311yescharacteristic
131C2×C42C2×D8311yescharacteristic
132#95—834no—
133#95—832no—
134#95—832no—
135#95—834no—
136C24C22×D8311yescharacteristic
137C2×Q8—3324no—
138C2×Q8—1664no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—834no—
142C22×C4—834no—
143C22×C4Dic16614yescomplemented
144C22×C4C22×D8311yescharacteristic
145C22×C4C22×D8311yescharacteristic
146C22×C4C22×D8311yescharacteristic
147C22×C4C22×D8311yescharacteristic
148C22×C4C22×D8311yescharacteristic
149C22×C4C22×D8311yescharacteristic
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—3324no—
153C4⋊C4—1664no—
154C4⋊C4—1664no—
155C4⋊C4—1664no—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—3324no—
159C42—22no—
160C42C22⋊D8314yes—
161C23—21no—
162C23—21no—
163C23—22no—
164C23—22no—
165C23C2×Dic16612yes—
166C23C23×D8311yescharacteristic
167Q8—3328no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—28no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—1664no—
178C2×C4—1664no—
179C2×C4—1664no—
180C2×C4—1664no—
181C2×C4—1664no—
182C2×C4—1664no—
183C2×C4C2×C22⋊D8312yes—
184C2×C4C2×C22⋊D8312yes—
185C2×C4C2×C22⋊D8312yes—
186C2×C4C2×Dic16612yes—
187C2×C4C2×Dic16612yes—
188C2×C4#1305059012yes—
189C22—24no—
190C22—22no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22C22×C22⋊D8311yescharacteristic
195C22C22×Dic16611yescharacteristic
196C22C2×#1305059011yescharacteristic
197C22#1458183812yes—
198C22#1458183912yes—
199C4—22no—
200C4—22no—
201C4—22no—
202C4—41no—
203C4—41no—
204C4—3324no—
205C4—3324no—
206C4#1458177114yes—
207C2—42no—
208C2C2×#1458177111yescharacteristic
209C2#1912826511yescharacteristic
210C2#1912827211yescharacteristic
211C2#1912832212yes—
212C2#1912832912yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#117×C83C2
prime radicalC24×C83C23
hypercenterC2×#117D83
hypocenterC83#2628
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#14581792C2semidirect
2. #19127640C2semidirect
4. #19127621C2semidirect
6. #19127599C2semidirect
7. #19127567C2semidirect
41. C22×C4×C83Q8semidirect
107. C83#2628semidirect
143. C22×C4Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#145817922
#191276402
#191276372
#191276212
#191276132
#191275992
#191275672
#191275612
C2×#117×C832
#262883