Finite Group Databasebeta

← #24915490

Subgroups — #24915490

Order 10624.

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

normal subgroups only

This group has 528 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×#14581766C211yesmaximal, characteristic
2#19127636C212yesmaximal, complemented
3#19127631C212yesmaximal, complemented
4#19127618C212yesmaximal, complemented
5#19127616C214yesmaximal, complemented
6#19127612C212yesmaximal, complemented
7#19127564C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C83C211yesmaximal, characteristic, complemented
9C22×D332C2211yescharacteristic
10C22×C22⋊D83C2212yes—
11#14581766—24no—
12C2×C4⋊Dic83C2211yescharacteristic
13C2×Dic83⋊C4C2214yes—
14C2×C22⋊Dic83C2212yes—
15C2×(C2×D83)⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2214yes—
17C2×(C2×D83)⋊C4C2214yes—
18C83⋊#95C2212yes—
19C83⋊#95C2212yes—
20C83⋊#95C2214yes—
21C22×Q8×C83C2211yescharacteristic
22C23×C4×C83C2211yescharacteristic
23C22⋊Q8×C83—24no—
24C2×C4⋊C4×C83C2212yes—
25C2×C4⋊C4×C83C2211yescharacteristic
26C2×C22⋊C4×C83C2212yes—
27C23×D83C2312yes—
28C2×D332—22no—
29C2×D332—42no—
30C2×C22⋊D83—24no—
31C2×C22⋊D83—44no—
32C22×Dic83C2312yes—
33C22×Dic83C2314yes—
34C4⋊Dic83—22no—
35Dic83⋊C4—44no—
36C22⋊Dic83—42no—
37(C2×D83)⋊C4—24no—
38(C2×D83)⋊C4—44no—
39(C2×D83)⋊C4—44no—
40C24×C83C2311yescharacteristic
41C2×Q8×C83—22no—
42C2×Q8×C83—42no—
43C22×C4×C83—42no—
44C22×C4×C83D414yes—
45C22×C4×C83C2312yes—
46C22×C4×C83C2312yes—
47C22×C4×C83C2311yescharacteristic
48C22×C4×C83C2311yescharacteristic
49C4⋊C4×C83—22no—
50C4⋊C4×C83—24no—
51C22⋊C4×C83—24no—
52C23×C83—22no—
53C23×C83—22no—
54C23×C83—22no—
55C23×C83C2×D412yes—
56C23×C83C2411yescharacteristic
57C22×D83—22no—
58C22×D83—24no—
59C22×D83—24no—
60C22×D83—24no—
61Q8×C83—44no—
62C2×C4×C83—22no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—24no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—44no—
70C2×C4×C83C2×D412yes—
71C2×C4×C83C2×D412yes—
72C22⋊D83—48no—
73C2×Dic83—22no—
74C2×Dic83—24no—
75C2×Dic83—24no—
76C2×Dic83—24no—
77C2×Dic83—24no—
78D332—44no—
79C22×C83—22no—
80C22×C83—22no—
81C22×C83—22no—
82C22×C83—44no—
83C22×C83C22×D411yescharacteristic
84C22×C83H(5,𝔽2)12yes—
85C22×C83H(5,𝔽2)12yes—
86C22×C832−1+412yes—
87C2×D83—42no—
88C2×D83—42no—
89C2×D83—42no—
90C2×D83—48no—
91C4×C83—22no—
92C4×C83—22no—
93C4×C83—42no—
94C4×C83—42no—
95Dic83—42no—
96Dic83—44no—
97C2×C83—42no—
98C2×C83#53411yescharacteristic
99C2×C83#53511yescharacteristic
100C2×C83#53611yescharacteristic
101C2×C83#56214yes—
102D83—82no—
103#2816—831nomaximal, Sylow
104C83#281611yescharacteristic, complemented, Sylow
105C2×C22⋊Q8D8311yescharacteristic, complemented
106C2×#121—831no—
107#399—832no—
108#398—832no—
109#394—832no—
110#394—834no—
111#393—832no—
112#381—831no—
113C22×Q8C2×D8311yescharacteristic
114C22×D4—832no—
115C22×D4—831no—
116C23×C4C2×D8311yescharacteristic
117C22⋊Q8—24no—
118#121—1664no—
119C2×C4⋊C4—834no—
120C2×C4⋊C4—831no—
121C2×C4⋊C4C2×D8312yes—
122C2×C4⋊C4C2×D8311yescharacteristic
123C2×C22⋊C4—832no—
124C2×C22⋊C4—834no—
125C2×C22⋊C4—834no—
126C2×C22⋊C4—832no—
127C2×C22⋊C4C2×D8312yes—
128#95—832no—
129#95—832no—
130#95—834no—
131C24—832no—
132C24C22×D8311yescharacteristic
133C2×Q8—22no—
134C2×Q8—42no—
135C2×D4—3322no—
136C2×D4—3324no—
137C2×D4—1664no—
138C2×D4—1662no—
139C22×C4—42no—
140C22×C4—834no—
141C22×C4—832no—
142C22×C4C22⋊D8314yes—
143C22×C4C22×D8312yes—
144C22×C4C22×D8312yes—
145C22×C4C22×D8311yescharacteristic
146C22×C4C22×D8311yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—3324no—
150C4⋊C4—1662no—
151C22⋊C4—24no—
152C22⋊C4—3324no—
153C22⋊C4—3324no—
154C22⋊C4—3322no—
155C22⋊C4—1664no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—1662no—
160C23—1664no—
161C23—1664no—
162C23—1664no—
163C23C2×C22⋊D8312yes—
164C23C23×D8311yescharacteristic
165Q8—44no—
166D4—3324no—
167D4—3328no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—1662no—
177C2×C4—1664no—
178C2×C4—1664no—
179C2×C4—1664no—
180C2×C4—1664no—
181C2×C4C2×C22⋊D8312yes—
182C2×C4C2×C22⋊D8312yes—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—44no—
187C22—3328no—
188C22—3322no—
189C22—3322no—
190C22—3322no—
191C22C22×C22⋊D8311yescharacteristic
192C22#1458183712yes—
193C22#1458183812yes—
194C22#1458184012yes—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—3324no—
200C4—3322no—
201C2—42no—
202C2—6642no—
203C2#1912826011yescharacteristic
204C2#1912826511yescharacteristic
205C2#1912827511yescharacteristic
206C2#1912844714yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C22⋊Q8×C83C2
prime radicalC2×#14581766C2
hypercenterC2×C22⋊Q8D83
hypocenterC83#2816
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #19127636C2semidirect
3. #19127631C2semidirect
4. #19127618C2semidirect
5. #19127616C2semidirect
6. #19127612C2semidirect
7. #19127564C2semidirect
8. C2×C22⋊Q8×C83C2semidirect
104. C83#2816semidirect
105. C2×C22⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817662
#191276362
#191276312
#191276182
#191276162
#191276122
#191275642
C2×C22⋊Q8×C832
#281683