Finite Group Databasebeta

← #24906397

Subgroups — #24906397

Order 10624.

310 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#19128282C211yesmaximal, characteristic, complemented
2#19127835C212yesmaximal, complemented
3#19127434C212yesmaximal, complemented
4#19127427C211yesmaximal, characteristic, complemented
5#394×C83C211yesmaximal, characteristic, complemented
6C2×#13050590—21no—
7C2×#13050575—22no—
8#14581772—22no—
9#14581767—22no—
10#14581766—21no—
11#14581766C2212yes—
12#14581713—22no—
13C22⋊(C83⋊C8)C2212yes—
14C22×D4×C83C2211yescharacteristic
15C2×C4⋊C4×C83C2211yescharacteristic
16C2×C22⋊C4×C83—21no—
17C2×C22⋊C4×C83—21no—
18#95×C83C2211yescharacteristic
19#13050590—42no—
20#13050590—42no—
21C2×Dic166—21no—
22C2×D332—21no—
23C2×D332—21no—
24C2×D332D412yes—
25C2×C22⋊D83—22no—
26C2×C4×D83—22no—
27#13050575—44no—
28C2×C83⋊C8—22no—
29C4⋊Dic83—21no—
30C4⋊Dic83D412yes—
31(C2×D83)⋊C4—22no—
32(C2×D83)⋊C4—22no—
33C24×C83—21no—
34C2×D4×C83—22no—
35C2×D4×C83—42no—
36C22×C4×C83—21no—
37C22×C4×C83D411yescharacteristic
38C22×C4×C83D411yescharacteristic
39C22×C4×C83C2311yescharacteristic
40C4⋊C4×C83—22no—
41C22⋊C4×C83—42no—
42C22⋊C4×C83—42no—
43C23×C83—22no—
44C23×C83—41no—
45C23×C83—41no—
46C23×C83—41no—
47C23×C83—41no—
48C23×C83—41no—
49C23×C83—41no—
50C23×C83C2×D411yescharacteristic
51C22×D83—22no—
52D4×C83—24no—
53D4×C83—42no—
54C2×C4×C83—21no—
55C2×C4×C83—21no—
56C2×C4×C83—21no—
57C2×C4×C83—41no—
58C2×C4×C83—41no—
59C2×C4×C83—41no—
60C2×C4×C83—41no—
61C2×C4×C83—41no—
62C2×C4×C83D812yescomplemented
63C2×C4×C83C2×D412yes—
64C22⋊D83—42no—
65C22⋊D83—42no—
66C2×Dic83—22no—
67D332—42no—
68D332—42no—
69D332—81no—
70C4×D83—44no—
71Dic166—42no—
72C83⋊C8—42no—
73C22×C83—21no—
74C22×C83—21no—
75C22×C83—42no—
76C22×C83—42no—
77C22×C83—42no—
78C22×C83—81no—
79C22×C83—81no—
80C22×C83—81no—
81C22×C83—81no—
82C22×C83#12011yescharacteristic
83C22×C83C2×D811yescharacteristic
84C22×C83AGL(1,ℤ/8)11yescharacteristic
85C2×D83—42no—
86C2×D83—42no—
87C2×D83—42no—
88C4×C83—22no—
89C4×C83—22no—
90C4×C83—41no—
91C4×C83—81no—
92Dic83—42no—
93C2×C83—21no—
94C2×C83—21no—
95C2×C83—82no—
96C2×C83#44711yescharacteristic
97C2×C83#45411yescharacteristic
98C2×C83B(4,𝔽2)11yescharacteristic
99D83—82no—
100#1598—831nomaximal, Sylow
101C83#159811yescharacteristic, complemented, Sylow
102#537—831no—
103#447—832no—
104#394D8311yescharacteristic, complemented
105#327—832no—
106#324—831no—
107C2×#41—1661no—
108C22×D4C2×D8311yescharacteristic
109C2×D8—1662no—
110C22⋊Q8—1662no—
111#121—832no—
112#121—1662no—
113#121—1661no—
114C2×C4⋊C4C2×D8311yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117#102—1662no—
118C22⋊C8—832no—
119#95C2×D8311yescharacteristic
120C24—21no—
121#41—3322no—
122#41—3322no—
123C2×Q8—1661no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—832no—
127C2×D4—1661no—
128C2×D4—1661no—
129C2×D4—1662no—
130C22×C4—21no—
131C22×C4—1662no—
132C22×C4C22⋊D8311yescharacteristic
133C22×C4C22⋊D8311yescharacteristic
134C22×C4C22×D8311yescharacteristic
135D8—3324no—
136C2×C8—1662no—
137C4⋊C4—22no—
138C4⋊C4—832no—
139C4⋊C4—1661no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—1662no—
143C22⋊C4—1662no—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—1662no—
152C23C2×C22⋊D8311yescharacteristic
153Q8—3322no—
154D4—24no—
155D4—42no—
156D4—3322no—
157D4—3322no—
158D4—3322no—
159D4—3322no—
160D4—6641no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3324no—
170C2×C4—1662no—
171C2×C4#1305057512yescomplemented
172C2×C4D4×D8312yes—
173C8—3322no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—3322no—
184C22—3322no—
185C22—3322no—
186C22#1458176011yescharacteristic
187C22C2×#1305057511yescharacteristic
188C22#1458181711yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—81no—
193C4—3322no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—6642no—
198C2#1912783411yescharacteristic
199C2#1912787311yescharacteristic
200C2#1912788711yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#394×C83C2
socleC22×C83#120
hypercenter#394D83
hypocenterC83#1598
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #19128282C2semidirect
2. #19127835C2semidirect
3. #19127434C2semidirect
4. #19127427C2semidirect
5. #394×C83C2semidirect
62. C2×C4×C83D8semidirect
101. C83#1598semidirect
104. #394D83semidirect
171. C2×C4#13050575semidirect

Maximal subgroups

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

subgroupindex
#191282822
#191278352
#191274342
#191274272
#394×C832
#159883